UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Performance of Kalman decision-feedback equalization in HF radio modems Tront, Russell John 1984

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1984_A7 T76.pdf [ 6.18MB ]
Metadata
JSON: 831-1.0096286.json
JSON-LD: 831-1.0096286-ld.json
RDF/XML (Pretty): 831-1.0096286-rdf.xml
RDF/JSON: 831-1.0096286-rdf.json
Turtle: 831-1.0096286-turtle.txt
N-Triples: 831-1.0096286-rdf-ntriples.txt
Original Record: 831-1.0096286-source.json
Full Text
831-1.0096286-fulltext.txt
Citation
831-1.0096286.ris

Full Text

PERFORMANCE OF KALMAN DECISION-FEEDBACK EQUALIZATION IN HF RADIO MODEMS by RUSSELL JOHN TRONT B . S c , The U n i v e r s i t y of V i c t o r i a , 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING  We accept t h i s t h e s i s as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA December 1983 (£) Russell John Tront, 1983  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree a t the  the  University  o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I  further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may  be  department o r by h i s o r her  granted by  the head o f  representatives.  my  It i s  understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be  allowed without my  permission.  Department o f The U n i v e r s i t y o f B r i t i s h 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6  (3/81)  (%AX  !1  }  Columbia  If? / 1  written  i i ABSTRACT  The performance of fast-adapting Kalman algorithms in adjusting the tap weights of d i g i t a l equalizers f o r high-speed, s e r i a l HF radio modems i s investigated.  Experiments on a large r e a l i s t i c set of simulated HF channels,  and on some actual channels, have been conducted using 2400 bps QPSK. Results are presented for l i n e a r and decision-feedback equalizers with symbol and half-symbol ( f r a c t i o n a l l y )  spaced t a p s .  The b i t error rate and the  optimum values of several algorithmic parameters in reference-directed mode, and the r e l a t i v e resistance of the various configurations to "crashing" in d e c i s i o n - d i r e c t e d mode, have been determined as a function of the channel characteristics.  The r e s u l t s show the s u p e r i o r i t y of the Kalman,  f r a c t i o n a l l y - t a p p e d decision-feedback equalizer and point out the uniqueness and r e l a t i v e d i f f i c u l t y of e q u a l i z i n g the HF radio channel.  At slow fade  r a t e s , reference-directed performance within 3-5 dB of optimum i s p o s s i b l e . With fade rates above 0.5 Hz though, the technique becomes f a d e - r a t e l i m i t e d . A t h e o r e t i c a l argument leads to a new d e f i n i t i o n of a "slow fading channel" f o r exponentially-aged Kalman e q u a l i z e r s .  Performance in d e c i s i o n - d i r e c t e d  mode, necessary f o r actual communication, was much poorer than expected.  It  was concluded that the continously-high Kalman Gain necessary on the HF channel r e s u l t s in such adaptation at each i t e r a t i o n that even i s o l a t e d decision errors can s e r i o u s l y disrupt the adaptation process.  In s p i t e of  the d e c i s i o n - d i r e c t e d degradation, the s e r i a l equalizer technique i s shown superior to currently-used p a r a l l e l modems i f the fade rate i s less than 0.5 Hz and the transmissions are peak power l i m i t e d .  iii TABLE OF CONTENTS  ABSTRACT  ii  TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES GLOSSARY OF ABBREVIATIONS ACKNOWLEDGEMENT  iii vi vii i ix x  1.0 INTRODUCTION 1.1  Motivation  1  1.2  Other Work  4  1.3  Description of Project and Thesis  7  2.0 THE HF CHANNEL, ITS DATA COMMUNICATION PROBLEMS, AND SIMULATION 2.1  The HF Radio Channel  11  2.2  The D i f f i c u l t y of HF Data Communication  15  2.3  Discrete Time Simulation of the HF Channel  18  3.0 EQUALIZATION TECHNIQUES AND ALGORITHMS 3.1  E q u a l i z e r Configurations  22  3.2  Equalizer Tap-Weight Adjustment Algorithms  29  3.3  Synchronization and Training  39  iv TABLE OF CONTENTS ( c o n ' t )  4.0 EQUALIZER PERFORMANCE ON SIMULATED CHANNELS 4.1  Basic Symbol-Spaced Equalizer Results and Discussion  43  4.2  Fractional1y-Tapped Equalizer Results and Discussion  57  4.3  Performance as a Function of Other Parameters  63  5.0 CHANNEL ESTIMATION ON SIMULATED AND REAL CHANNELS 5.1  A p p l i c a t i o n of RLS Algorithms to Channel Estimation  70  5.2  Fractionally-Tapped Channel Estimation  74  5.3  Estimation Results on 3 Real Channels  77  5.4  Estimation of Fading Spectra  80  6.0 EQUALIZER PERFORMANCE ON REAL CHANNELS 6.1  Equalizer Performance on Three Real Channels  88  7.0 EQUALIZER PERFORMANCE IN DECISION-DIRECTED MODE 7.1  Methods of Preventing and Recovering from Crashes  91  7.2  Periodic R e t r a i n i n g Performance and Discussion  94  7.3  Constant Modulus Algorithm Performance and Disscussion  106  8.0 CONCLUSION 8.1  Summary and Major Conclusions  109  8.2  Suggestions f o r Future Research  114  V TABLE OF CONTENTS (con't)  APPENDIX A - HF Simulation Software Flow Diagram  117  APPENDIX B - Square-Root-Kalman, Fractionally-Tapped Decision-Feedback Equalizer Algorithm  119  APPENDIX C - D e t a i l s of Real Channel Tests and Data Tapes  123  APPENDIX D - Second-Order Channel Simulator Subroutine  125  APPENDIX E - Fractionally-Tapped Equalizer Subroutine  131  APPENDIX F - Symbol-Spaced Channel Estimator Subroutine  138  REFERENCES  142  vi LIST OF FIGURES  2.1  Ionospheric Sounding  11  2.2  Typical Instantaneous HF Impulse Response  12  2.3  Typical Instantaneous HF Frequency Response  13  2.4  Theoretical Error Rate of QPSK on Various Channels  17  2.5  Possible HF Channel Simulator Configuration  19  3.1  Basic Equalizer Configuration  22  3.2  An LMS Decision Feedback Equalizer  24  3.3  L a t t i c e Stage  25  3.4  A Linear L a t t i c e Equalizer  26  3.5  Convergence Speed of Various Algorithms  31  3.6  CCITT V. 27 Data Scrambler  42  4.1  Three Path Channel (MPGAIN(0:4)  4.2  ( 1 , 0 , 1 , 0 , 1 ) DFE Performance as a Function of q ( . 1 5 , 1.50 Hz)  45  4.3  ( 1 , 0 , 1 , 0 , 1 ) DFE Performance as a Function of q ( . 0 5 , . 5 0 , 5.00 Hz)  46  4.4  Optimum Exponential-Aging Factor  47  4.5  ( 1 , 0 , 1 , 0 , 1 ) DFE Performance as a Function of SNR  48  4.6  ( 1 , 0 , 1 , 0 , 1 ) DFE Performance as a Function of B^  50  4.7  Two-Path Channel (MPGAIN = ( 1 , 0 , 0 , 0 , 1 ) )  51  4.8  ( 1 , 0 , 0 , 0 , 1 ) DFE Performance as a Function of SNR  52  4.9  Unequal-Strength Three-Path Channel (MPGAIN(0:4)  = 1,0,1,0,1)  44  = (.33,0,1,0,-33))  53  4.10 ( . 3 3 , 0 , 1 , 0 . 3 3 ) DFE Performance as a Function of SNR  54  4.11 ( 1 , 0 , 1 , 0 , 1 ) FT DFE Performance as a Function of SNR  58  4.12 T/l and T/2 DFE S e n s i t i v i t y to Sampling Phase  60  4.13 Non-Symbol-Spaced 3-Path Channel (MPDEL=0;JI2;32)  61  vi i LIST OF FIGURES c o n t ' d  4.14 Non-Symbol-Spaced 2-Path Channel (MPDEL=0;28)  61  4.15 Performance on Non-Symbol-Spaced 3-Path Channel  62  4.16 Performance on Non-Symbol-Spaced 2-Path Channel  62  4.17 Simulated Mistuning Results  68  5.1  Channel Estimator  71  5.2  S e n s i t i v i t y of T/l Channel Estimator to Sample Timing  75  5.3  I n s e n s i t i v i t y of FT Channel Estimator to Sample Timing  76  5.4  FT Estimation of Simulated 3-Path Channel  76  5.5  Tape 6 RMS Impulse Response  77  5.6  Tape 9 RMS Impulse Response  78  5.7  Tape 11 RMS Impulse Resonse  78  5.8  Estimated Tape 6 Fading Spectra (Taps 0 and 3)  81  5.9  Estimated Tape 6 Fading Spectra (Taps 7 and 13)  82  5.10 Estimated Tape 9 Fading Spectra (Taps 0 and 3)  83  5.11 Estimated Tape 9 Fading Spectra (Taps 8 and 13)  84  5.12 Estimated Tape 11 Fading Spectra (Taps 0 and 3)  85  5.13 Estimated Tape 11 Fading Spectra (Taps 7 and 11)  86  7.1  P e r i o d i c - R e t r a i n i n g Decision-Directed Results  97  7.2  Decision-Directed Comparison with P a r a l l e l Modems  100  A.l  HF Simulation Flow Diagram  118  vi i i LIST OF TABLES  4.1  Equalizer Performance with Mistuning  69  5.1  Channel Estimator Optimum A as a Function of Bj» f o r SNR=23 dB  74  6.1  Performance on the Taped Real Channels  88  7.1  Decision-Directed Performance with P e r i o d i c R e t r a i n i n g  96  7.2  H a l f - D e c i s i o n - D i r e c t e d Mode Comparison  102  7.3  Performance of Constant Modulus Algorithm  107  C.l  Tape Recording Conditions  123  C.2  S t a r t i n g Sample of M-Sequence on Tapes  124  GLOSSARY OF ABBREVIATIONS  BER  - B i t Error Rate  CMA  -Constant Modulus Algorithm  DFE  -Decision-Feedback Equalizer  E  b  -Energy per b i t  FIR  - F i n i t e Impulse Response  FT  -Fractionally-Tapped  HF  -High Frequency  IIR  - I n f i n i t e Impulse Response  ISI  -Inter-Symbol  LE  - L i n e a r Equalizer  LPF  -Low Pass F i l t e r  MLSE  -Maximum L i k e l i h o o d Sequence Estimation  MUF  -Maximum Usable Frequency  N  -Number of Equalizer Taps  No  -Noise Spectral Density  RLS  -Recursive Least Square  SNR  - S i g n a l - t o - N o i s e Ratio  T  -Symbol Period  TDL  -Tapped Delay Line  Interference  (one-sided)  X ACKNOWLEDGMENT  I would l i k e to thank Dr. Jim Cavers f o r h i s patience, knowledge, and i n f i n i t e enthusiasm in d i r e c t i n g the p r o j e c t , and Dr. M.R. Ito f o r his very wise guidance regarding t h e s i s work i n i n d u s t r y , academic supervision of the work, and his help i n preparation of the t h e s i s and i t s defense.  Glenayre  E l e c t r o n i c s has supported the project in an i n f i n i t e number of ways: f i n a n c i a l , supervisory, access, e t c .  in-house desk space, computer time, word-processing  F i n a l l y , personal f i n a n c i a l support was provided by an  N . S . E . R . C . Postgraduate Scholarship, a U.B.C. Teaching A s s i s t a n t s h i p , Glenayre E l e c t r o n i c s , and i n d i r e c t l y from the B r i t i s h Columbia Science Counci1.  1.  1 -  INTRODUCTION  1.1 Motivation  Shortwave r a d i o , in the high frequency band of 3-30 MHz, o f f e r s a unique, long-range data communication medium.  Audio-bandwidth HF radio i n s t a l l a t i o n s  abound in many f i x e d and mobile operations.  Modems to enable high-speed data  communication on such channels should be well received in such a marketplace. For instance, the a i r l i n e s would l i k e an HF modem compatible in speed with t h e i r VHF 2400 b i t - p e r - s e c o n d ACARS system.  The o i l industry would l i k e to  send secure voice and d i g i t i z e d d r i l l i n g r e s u l t s from r i g s to home base f o r analysis.  Bandwidth-compressed l i n e a r - p r e d i c t i v e coded speech can be sent at  rates as low as 2400 bps, and d i g i t i z e d speech can e a s i l y be encrypted to s a t i s f y the o i l i n d u s t r y ' s need f o r corporate s e c u r i t y .  F i n a l l y , the  m i l i t a r y operates a l l over the world and thus has s i m i l a r long-range data communication a p p l i c a t i o n s .  The HF channel enables o v e r - t h e - h o r i z i o n  communication without the large  d i r e c t i o n a l antennas and c e n t r a l i z e d wide-band nature of s a t e l l i t e communication.  This i s possible because HF r a d i o energy can r e f l e c t  several of the turbulent layers in the ionosphere.  off  It i s characterized as a  fading multipath channel because s e v e r a l , independently-fading r e p l i c a s of the transmitted message are received, each with a s l i g h t l y propagation delay.  different  Present modems f o r the HF channel use low symbol-keying r a t e s .  2 -  This  avoids the intersymbol interference (ISI)  caused by the channel delay spread  through use of very long symbol p e r i o d s .  To a t t a i n the required throughput,  current high-speed modems use m u l t i p l e p a r a l l e l tones to transmit many b i t s per symbol.  It i s also d e s i r a b l e on HF to use the d i v e r s i t y in the d i f f e r e n t  paths f o r reducing the degradation caused by f a d i n g .  Low-speed HF modems can  use bandwidth-expansion techniques to obtain frequency d i v e r s i t y ,  while  high-speed modems have to use redundant tones to obtain d i v e r s i t y . modems have many other drawbacks.  These  They are not very power and bandwidth  e f f i c i e n t , and because of t h e i r analog nature they are expensive and susceptible to component d r i f t .  Recent i n t e r e s t has been shown in high-speed s e r i a l transmission.  In  t h i s method the high keying rate i m p l i c i t l y creates frequency d i v e r s i t y .  The  ISI caused by short duration symbols though, must be removed at the receiver by an adaptive e q u a l i z e r f i l t e r .  This f i l t e r gathers back together the  information smeared out in time by the channel, by deconvolving the transmitted symbol stream out of the received s i g n a l .  The equalizer must of  course be adaptive as the channel d i s t o r t i o n i s c o n t i n u a l l y changing.  It i s conjectured that s e r i a l e q u a l i z e r s , due to t h e i r d i g i t a l nature, might be b u i l t less expensively and be less susceptible to component d r i f t than p a r a l l e l modems.  They would also be more power and bandwidth e f f i c i e n t ,  two important c h a r a c t e r i s t i c s f o r mobile a p p l i c a t i o n s .  S e r i a l equalizer  algorithms when implemented d i g i t a l l y can be very computationally burdensome, but powerful signal processing microprocessors and more e f f i c i e n t algorithms have j u s t become a v a i l a b l e .  -  3 -  Because of the d i f f i c u l t nature of the HF channel, gradient-algorithm l i n e a r equalizers used in o l d e r , high-speed telephone modems are not appropriate.  It has been suggested that a r e c u r s i v e - l e a s t - s q u a r e (RLS)  Kalman algorithm used in a decision-feedback e q u a l i z e r (DFE) might work well on HF.  Kalman algorithms are being applied in telephone modems f o r rapid  i n i t i a l convergence.  These f a s t adapting algorithms would be of continuous  use on the more dynamic HF channel.  S i m i l a r i l y , DFE's have proved superior  to l i n e a r equalizers on other channels.  They are p a r t i c u l a r i l y well suited  to the HF channel i f e r r o r propagation problems can be shown to be not too significant.  This problem i s of serious i n t e r e s t as the fading nature of HF  channels v i r t u a l l y guarantees occasional e r r o r propagation and "crashes" of the e q u a l i z e r .  This project investigates the performance of RLS DFE's under a very wide v a r i e t y of simulated HF operating conditions using 2400 b i t - p e r - s e c o n d quadrature p h a s e - s h i f t keying (QPSK). also checked.  Performance on a few real channels i s  We are p a r t i c u l a r l y interested in determining the adaptation  rate of exponentially-aged RLS algorithms, and in comparing the performance of the decision-feedback and l i n e a r e q u a l i z e r configurations when used on t y p i c a l HF channels.  We also t e s t the resistance to sampling phase of  f r a c t i o n a l l y - t a p p e d (FT) decision-feedback e q u a l i z e r s .  F i n a l l y , we b r i e f l y  i n v e s t i g a t e the performance degradation in d e c i s i o n - d i r e c t e d mode (which must be used f o r true communication) and t e s t several other algorithms and protocols f o r preventing and recovering from crashes.  The reader w i l l f i n d i t helpful to have some knowledge of e i t h e r equalizers or Kalman e s t i m a t i o n .  -  4 -  1.2 Other Work  Previous work on the HF channel i t s e l f , i t s modelling, and i t s data communication problems w i l l be discussed i n Section 2. decision-feedback e q u a l i z e r configuration and of algorithms are well e s t a b l i s h e d . 3.2 r e s p e c t i v e l y .  The advantages of the  recursive-least-square-error  They w i l l be reviewed i n Sections 3.1 and  These techniques are d i r e c t l y applicable to HF  e q u a l i z a t i o n but they have yet to have t h e i r performance q u a n t i f i e d on t h i s d i f f i c u l t channel.  This section w i l l f a m i l i a r i z e the reader with the c u r r e n t l y a v a i l a b l e work on high-speed HF modem performance.  There i s i n f a c t very l i t t l e  r e l i a b l e , openly-published research in the f i e l d . for t h i s .  There are several reasons  F i r s t l y , the channel i s frequency-dependent and continuously  changing, making comparison of new techniques under i d e n t i c a l conditions difficult.  Unlike the a d d i t i v e white gaussian noise channel, there are many,  sometimes d i f f i c u l t - t o - a c q u i r e parameters required to define the channel state.  Secondly, these many parameters are d i f f i c u l t and computationally  burdensome to simulate on a computer, e s p e c i a l l y in r e a l time.  F i n a l l y , much  of the s e r i a l modem research that i s a v a i l a b l e seems purposefully vague, obviously due to proprietary and/or m i l i t a r y considerations.  The most extensive work reported on high-speed HF modem performance was by Watterson [63].  Performance of several commercial and/or m i l i t a r y  p a r a l l e l modems was determined as a function of a few values of multipath spread, fade r a t e , s i g n a l - t o - n o i s e r a t i o (SNR), i n t e r f e r e n c e , and mistuning. Only a few values of each parameter were used to keep t h e i r already voluminous report down to s i z e . They state that because of the formidable number of parameter combinations, the real time required to obtain a complete  -  5 -  set of measurements f o r a t y p i c a l HF modem operating at 2400 bps would be l? about 1 0 y e a r s !  Nonetheless, t h i s work was somewhat of a landmark in  i t s scope and w i l l provide some l i m i t e d data with which to compare our work.  In [48], Pennington reports f o r the UK Admiralty Surface Weapons Establishment on a real and simulated channel comparison between a MIL STD 1886 16 tone DPSK p a r a l l e l modem and an experimental French m i l i t a r y s e r i a l modem.  B i t and b l o c k - e r r o r - r a t e r e s u l t s are presented as a function of SNR  f o r 3 two-path fading channels.  He concludes the s e r i a l modem i s b e t t e r ,  but  unfortunately shows s e r i a l r e s u l t s better than t h e o r e t i c a l l y possible (see Section 2 . 2 ) , and does not reveal the s e r i a l modem c o n f i g u r a t i o n , algorithm, modulation, operating mode, and crash prevention measures.  Perkins and McRae of Harris Corporation [49] report a s i m i l a r p a r a l l e l - v e r s u s - s e r i a l performance comparison on real and simulated channels.  Their s e r i a l modem uses 50% known r e t r a i n i n g symbols at 4800 bps  to achieve a net 2400 bps throughput.  This makes the scheme very robust but  requires a longer e q u a l i z e r to cover the same multipath spread.  Since many  RLS algorithms have a complexity proportional to the square of the e q u a l i z e r length, t h i s may be the reason they required an expensive b i p o l a r 12 b i t processor configured with a separate hardware m u l t i p l i e r c h i p .  They too, do  not reveal t h e i r algorithm except to say i t uses block processing and i t e r a t e s a minimum-mean-square e r r o r (MMSE) algorithm as many times as possible.  This no doubt achieves a decision-feedback e f f e c t s i m i l a r to that  discussed i n Section 3 . 1 .  Their conclusion i s that s e r i a l modems outperform  p a r a l l e l ones, but t h e i r c a l i b r a t e d simulated-channel r e s u l t s are very sketchy making comparison with our r e s u l t s d i f f i c u l t .  -  6 -  Perkins and McRae caution against comparing t h e i r s e r i a l modem r e s u l t s with others, namely Pennington's (which were in some ways worse), as d i f f e r e n t algorithms are probably used.  It i s i r o n i c they don't  volunteer  t h e i r ' s , but t h i s i s obviously an active product area f o r corporate and m i l i t a r y researchers.  As discussed in the next s e c t i o n , t h i s t h e s i s i s  supported in part by Glenayre E l e c t r o n i c s of North Vancouver, B.C. exactly to acquire such algorithmic and performance research.  It w i l l receive only  l i m i t e d d i s t r i b u t i o n f o r the f i r s t year.  The only published work to include the s e r i a l modem algorithm and performance on HF channels i s Hsu [31], and the previous background papers [14,30].  He presents 8PSK r e s u l t s as a function of SNR f o r only one  simulated multipath structure and fading bandwidth (2 equal-mean-square paths separated by one symbol period fading with a 1 Hz half-bandwidth). Unfortunately, both the t h e o r e t i c a l and experimental curves he presents are also better than t h e o r e t i c a l fading-channel l i m i t s , and he presents reference-directed results only. c o r r e c t , was used in t h i s p r o j e c t .  Nonetheless, Hsu's algorithm, which seemed We i n v e s t i g a t e performance on a much  wider v a r i a t i o n of channels including some actual recorded channels.  Hsu's  e q u a l i z e r c o n f i g u r a t i o n was also extended to include f r a c t i o n a l - t a p p i n g , and the performance was characterized by a larger number of performance measures in both reference and d e c i s i o n - d i r e c t e d mode.  F i n a l l y , in [22, 2 3 ] , Falconer et al compare a f r a c t i o n a l l y - t a p p e d RLS DFE with another demodulation method c a l l e d estimation (MLSE).  maximum l i k e l i h o o d sequence  Falconer (and Monson in [44] f o r troposcatter channels)  conclude that MLSE y i e l d s only marginal improvement i n performance at the expense of computational complexity that grows exponentially with multipath spread.  This was of i n t e r e s t as Falconer's work used some of the same real  channel recordings as our p r o j e c t .  7 -  He does not state which part of the tapes  were used though, making comparison with our r e s u l t s q u a l i t a t i v e at best.  We see then, that there i s l i t t l e open or r e l i a b l e previous work a v a i l a b l e in t h i s commercially s t r a t e g i c area.  The work reported herein i s  thus r e l a t i v e l y o r i g i n a l and very t i m e l y .  1.3 Description of Project and Thesis  The main emphasis of the project was to study the performance and other issues surrounding the a p p l i c a t i o n of a decision-feedback e q u a l i z e r in a 2400 bps QPSK s e r i a l modem f o r HF r a d i o .  The research was undertaken in-house at  Glenayre E l e c t r o n i c s Limited i n North Vancouver, B r i t i s h Columbia, to determine the f e a s i b i l i t y of expanding t h e i r product l i n e to include such a modem.  The project grew out of the purchase by Glenayre of another f i r m ' s HF product l i n e which included a l i n e a r ( l a t t i c e ) equalizer research report [ 42 ] .  The report included 3 tapes of d i g i t i z e d , unequalized 2400 bps QPSK  recordings received o f f actual HF channels, as well as the FORTRAN code of a channel simulator and l a t t i c e e q u a l i z e r .  The report presented RLS FT l i n e a r  e q u a l i z e r performance r e s u l t s f o r the 3 tapes of data and a few f i x e d non-fading)  but otherwise H F - l i k e simulated channels.  (ie.  The r e s u l t s presented  were from experiments conducted in reference-directed mode o n l y .  The  unequalized data tapes had several problems and the FORTRAN code was neither documented nor conducive to m o d i f i c a t i o n .  Nevertheless, the project  provided  a well defined s t a r t i n g point f o r a thesis project which would be very applied in nature and provide real world b e n e f i t s .  -  8 -  The research goals of the project were t o :  a)  Determine the b i t - e r r o r - r a t e performance of an RLS DFE as a function of the extensive set of c h a r a c t e r i s t i c s t y p i c a l on the HF channel.  b)  Decide whether RLS Kalman algorithms could adapt q u i c k l y enough to keep up with the moderately-fast HF fade r a t e .  c)  Determine i f an order-of-magnitude improvement in performance could be expected from a DFE as compared to a LE on HF channels.  d)  Confirm the i n s e n s i t i v i t y to sample timing and the improved performance of the f r a c t i o n a l l y - t a p p e d , decision-feedback equalizer (FT DFE) on t y p i c a l HF channels.  e)  Determine the optimum value of various e q u a l i z e r parameters such as forward d e l a y - l i n e length and exponential-aging f a c t o r , as a function of the channel c h a r a c t e r i s t i c s .  f)  Estimate the multipath structure and fading bandwidth of the channels on the tapes.  g)  Try the optimum equalizer on the taped channels and compare with the l a t t i c e report r e s u l t s .  h)  Note the degradation in performance when running in d e c i s i o n directed mode on the simulated and real channels.  This mode i s  necessary f o r actual communication. i)  B r i e f l y i n v e s t i g a t e methods, p r o t o c o l s , and other e q u a l i z e r configurations and algorithms to combat e r r o r propagation and crashing in d e c i s i o n - d i r e c t e d mode.  This f a i r l y extensive set of t a s k s , which required one man-year of e f f o r t , took the f o l l o w i n g form:  -  9 -  a)  F a m i l i a r i z a t i o n with the data communications problems on HF.  b)  Review of the e q u a l i z e r l i t e r a t u r e and the l a t t i c e study.  c)  A decision not to use a l a t t i c e e q u a l i z e r as only a f i x e d order e q u a l i z e r was needed (a l a t t i c e generates a l l o r d e r s ) .  d)  A decision not to use the l a t t i c e study code, but to rewrite i t to be more f l e x i b l e and simulate a more r e a l i s t i c , dynamic channel.  e)  S e l e c t i o n from among the many published RLS algorithms of one or two which could be extended to the complex, f r a c t i o n a l l y - t a p p e d decision-feedback e q u a l i z e r c o n f i g u r a t i o n .  f)  Coding of the channel simulator, e q u a l i z e r ( s ) , and channel estimator.  g)  Gathering of the data through experimental runs on the Glenayre VAX 11/750 computer.  h)  Analysis the data.  The t h e s i s format i s f a i r l y c l e a r from the Table of Contents.  Section 2  covers the HF channel, i t s data communication d i f f i c u l t i e s , and i t s simulation.  Section 3 introduces the various equalizer c o n f i g u r a t i o n s ,  reviews tap-weight adjustment algorithms, and mentions synchronization and training issues.  Section 4 presents the performance of various equalizer configurations on simulated channels.  The r e s u l t s of symbol-spaced e q u a l i z e r s on 2 and 3 path  channels are presented as a function of SNR and fading bandwidth.  The  optimum value f o r the exponential-aging f a c t o r was determined f o r operation i n r e f e r e n c e - d i r e c t e d mode under various channel c o n d i t i o n s .  As the  performance was found to be somewhat f a d e - r a t e l i m i t e d , a new d e f i n i t i o n of "slow f a d i n g " a p p l i c a b l e to exponentially-weighted RLS equalizers i s suggested.  Then, the FT e q u a l i z e r r e s u l t s are presented with special  - 10 emphasis on performance as a function of sampling phase and multipath delay structure.  The FT e q u a l i z e r performance was found to be i n s e n s i t i v e to  sampling phase, and gave much better performance than symbol-spaced equalizers on c e r t a i n multipath channel s t r u c t u r e s .  The l a s t sub-section  discusses performance as a function of forward d e l a y - l i n e length and c a r r i e r frequency mistuning.  At higher channel fade rates the optimum forward length  j u s t spans the multipath spread.  Mistuning was found to be quite degrading  so c a r r i e r frequency a c q u i s i t i o n and tracking w i l l probably be required in an actual modem implementation.  Section 5 discusses channel multipath structure and fading spectra e s t i m a t i o n , and presents estimates of the channels on the 3 t e s t tapes. Section 6 summarizes the real channel tape r e s u l t s and compares them with the l a t t i c e report.  Our l i n e a r e q u a l i z e r r e s u l t s are better than those in the  previous report f o r reasons on which we only speculate.  The f u l l FT DFE  running in r e f e r e n c e - d i r e c t e d mode with a tuning c o r r e c t i o n generally performed very well on the real channels.  Section 7 presents the d e c i s i o n - d i r e c t e d mode r e s u l t s .  Though p e r i o d i c  r e t r a i n i n g proved a robust scheme f o r coping with crashes and somewhat l i m i t i n g the e r r o r propagation, the d e c i s i o n - d i r e c t e d r e s u l t s were s e r i o u s l y degraded from those in reference-directed mode.  S t i l l , the r e s u l t s at slower  fade rates were as good or better than with p a r a l l e l modems.  The mechanism  of the degradation was b r i e f l y investigated and suggested t e s t i n g of an RLS DFE version of T r e i c h l e r ' s Constant Modulus Algorithm (CMA).  F i n a l l y , Section 8 summarizes r e s u l t s and makes suggestions f o r f u r t h e r research.  It a l s o notes that f u r t h e r work i s j u s t s t a r t i n g at Glenayre under  a BCSC Major Research Grant.  - 11 2.  THE HF CHANNEL, ITS DATA COMMUNICATION PROBLEMS, AND SIMULATION  2.1 The HF Radio Channel  The HF channel has been of i n t e r e s t ever since the advent of shortwave radio.  Radio energy in the 3-30 MHz band often has the a b i l i t y to propagate  over the horizon to very d i s t a n t (1000+ km) r e c e i v e r s .  This i s due to  several macroscopic layers i n the ionosphere being able to r e f l e c t the radio energy back to e a r t h .  These layers vary in r e f l e c t i v i t y from second-to-  second due to microscopic ionospheric turbulence.  They also vary i n  p o s i t i o n , a l t i t u d e , and mean-square r e f l e c t i v i t y over minutes or hours.  The  slower v a r i a t i o n i s due to d a i l y and seasonal v a r i a t i o n s , and to s o l a r disturbances.  A sketch showing t y p i c a l ionospheric layers i s presented in Figure 2 . 1 . This graph i l l u s t r a t e s the propagation delay (on a f i x e d l i n k ) off several major layers as a f u n c t i o n of RF frequency.  I _J f  I I f  1  2  I I I  U  MUF  Frequency Figure 2.1  Ionospheric Sounding  - 12 The d e t a i l s of t h i s ionospheric s t r u c t u r e may vary from hour to hour. O c c a s i o n a l l y , at frequencies l i k e f - j , there i s no r e f l e c t i o n f o r hours. At other frequencies such as f , there might be 3 independently and 2  quite r a p i d l y (0.1-10 Hz) Rayleigh fading paths between the transmitter and receiver.  Several authors [13, 62] have confirmed t h i s model.  At frequencies near the maximum usable frequency (MUF), the multipath delay spread i s quite s m a l l .  Radio operators often choose to operate near  the MUF, which varies d i u r n a l l y , because of t h i s reduction i n time dispersion.  However with s e r i a l transmission and an e q u a l i z e r , frequency use  well below the MUF i s p r a c t i c a l and i n f a c t i s improved due the d i v e r s i t y a v a i l a b l e from having several independently-fading paths.  Such frequency use  would also ease the n e c e s s a r i l y global r a t i o n i n g of HF frequencies.  Let h(T,t) be the response of an HF channel at time t+t to an impulse at time t .  At a given time t  Q t  h(t)=h(*,t ) might look l i k e Figure 0  2.2.  h(r )  Propagation Delay  Figure 2.2 Typical Instantaneous Impulse Response  - 13 At another nearby time the 3 d e l t a functions might have d i f f e r e n t amplitudes. They might, in complex baseband representation, have a d i f f e r e n t phase (which could be thought of as r o t a t i o n about the X axis i n Figure 2 . 2 ) .  At a very  d i f f e r e n t time or frequency, they might even have d i f f e r e n t propagation delays T^,  or have d i f f e r e n t root-mean-square amplitudes.  The Fourier transform of h(T,t) with respect to X  is H(f,t).  For HF  radio transmission on an AM SSB audio-bandwidth channel, t h i s frequency response at a given instant might look l i k e Figure 2 . 3 . energy i s passed outside 300-2700 Hz.  Notice that not much  There are also some serious v a r i a t i o n s  i n response, and even n u l l s , within the pass band.  The moderately-rapid  v a r i a t i o n over time and i r r e g u l a r i t y of frequency response on HF i s very u n l i k e most other channels.  In p a r t i c u l a r , i t i s very u n l i k e telephone  channels on which most previous e q u a l i z a t i o n work has been done.  The  frequency response of even an unconditioned telephone l i n e i s generally w i t h i n 5 dB of nominal across most of the band.  10 -\  30OO  Frequency  Figure 2.3  (Hz)  Typical HF Channel Instantaneous Frequency Response  - 14 Let B^r be the 3 dB half-bandwidth of the complex Gaussian multipath fading processes.  The spectrum of the fading i s generally considered [62] to  have a Gaussian b e l l shape.  It therefore r o l l s o f f e x p o n e n t i a l l y , which i s  much f a s t e r and d i f f e r e n t l y than conventional f i l t e r s .  B  f  can also  be thought of as the h a l f - w i d t h of the received spectrum when a pure sine wave i s t r a n s m i t t e d .  It  i s thus sometimes referred to as the doppler spread,  though i t i s due to microscopic motion in the ionosphere rather than gross motion of the r e c e i v e r , t r a n s m i t t e r , or ionospheric l a y e r s .  Values of 0.1-10  Hz are t y p i c a l with the higher bandwidths more common on auroral channels.  We can compute t = l / 2 1 T B d  the fading processes. diversity.  f  as the d e c o r r e l a t i o n time of  This i s a measure of the delay required to get time  Similarily, if X  i s the multipath delay spread (of say  the 3 paths i l l u s t r a t e d i n Figure 2 . 2 ) , then f =l/27TT d  s  i s the  frequency d e c o r r e l a t i o n needed f o r in-channel frequency d i v e r s i t y .  Since  i s 1-5 m i l l i s e c o n d s , fd ranges over 160-30 Hz r e s p e c t i v e l y . P a r a l l e l modems must transmit t h e i r redundant information on sub-channels at least t h i s f a r apart as whole sub-bands of width approximately f^ Hz can fade up and down dependently.  S e r i a l modems must transmit in a bandwidth  greater than t h i s so some of the signal w i l l l i k e l y f a l l outside of n u l l s and pass through the channel.  Such s i g n a l s however, w i l l be badly d i s t o r t e d by  this "frequency-selective fading."  - 15 2.2 The D i f f i c u l t y of HF Data Communication  The major d i f f i c u l t i e s of data communication on HF are: i)  Getting t r a n s m i t t e r and receiver on a frequency where ionospheric reflection is currently possible,  ii) iii) iv) v)  Noise and i n t e r f e r e n c e , Multipath. Fading, C a r r i e r and Timing Recovery.  Some of these problems are common to other communication channels; a l l are more d i f f i c u l t on HF.  Firstly, MHz band.  ionospheric r e f l e c t i o n i s common, but not guaranteed in the 3-30  As i l l u s t r a t e d in the l a s t s e c t i o n , there are o c c a s i o n a l l y gaps in  the band which vary from hour t o hour unpredictably.  If your receiving  s t a t i o n cannot i n i t i a l l y be reached, how do you t e l l the operator what other frequency to try l i s t e n i n g on?  A l s o , HF frequencies are used as m u l t i p l e  access channels so some frequencies may be busy.  High l e v e l protocols such  as the RACE System described in [15] are required f o r automated communication environments.  Secondly, HF has the usual problems of noise and i n t e r f e r e n c e .  To combat  noise a f t e r considerable attenuation of the signal over long transmission d i s t a n c e s , f i x e d s t a t i o n s often use several hundred watts of t r a n s m i t t e r power and d i r e c t i o n a l antennas.  Mobile u n i t s u s u a l l y have neither o p t i o n .  Generally a l l that can be done i s to use a low peak-to-mean power r a t i o modulation ( i e . PSK) to get the most out of the a v a i l a b l e t r a n s m i t t e r power.  - 16 Minor interference i s common on HF as other users can sometimes be using the same frequency on the other side of the world.  This might not p r o h i b i t  the use of the frequency, but i t can complicate c a r r i e r recovery. Interference i s an even more d i f f i c u l t problem on HF because of the fading multipath.  If i t i s narrowband though, we see in the next section that an FT  equalizer i s helpful as i t can function as an adaptive matched f i l t e r and put a notch at the i n t e r f e r i n g frequencies.  T h i r d l y , the multipath on HF can span 5 m i l l i s e c o n d s ( c f . channels less than 1.75 ms).  telephone  Unacceptable intersymbol interference  thus occurs at keying rates as low as 200 symbols/second.  (ISI)  The HF impulse  response u s u a l l y c o n s i s t s of 1 to 4 d i s c r e t e d e l t a f u n c t i o n s .  This r e s u l t s  i n severe n u l l s in the frequency response requiring in-band frequency diversity.  For low data rate modems, bandwidth expansion techniques can be  used as in the Glenayre GLI 102 Pulse Compression Modem [27].  For higher data  r a t e s , m u l t i p l e p a r a l l e l tones with sub-channel d i v e r s i t y are c u r r e n t l y used to transmit many b i t s per symbol p e r i o d .  This technique unfortunately  r e s u l t s in a high peak-to-mean power r a t i o , and less than f u l l bandwidth e f f i c i e n c y due to inter-subchannel guard bands.  A l t e r n a t e l y , i f an adaptive  e q u a l i z e r i s used i n the r e c e i v e r , high key rate s e r i a l transmission can be used to obtain i m p l i c i t frequency d i v e r s i t y .  The e l i m i n a t i o n of the  r e s u l t i n g ISI i s the subject of t h i s t h e s i s .  F o u r t h l y , the HF channel multipaths are subject to independent Rayleigh fading.  As shown i n Figure 2 . 4 , t h i s fading severely l i m i t s the b i t e r r o r  rate (BER) o b t a i n a b l e .  In p a r t i c u l a r , f o r a fading channel with L path  d i v e r s i t y the QPSK e r r o r rate f o r large SNR i s asymptotically proportional to SNR"*-.  The slope of BER versus SNR on a l o g - l o g graph i s thus - L .  one path we see the BER i s l i m i t e d to 1 0 " even at say, 15 dB. 2  For  Compare  - 17 t h i s to the non-fading BER proportional to e r f c ( S N R ^ ) ; 10"  7  values as low as  are possible at 15 dB.  If  10"  \  10  If  HoU-  \  FADIM&  I  ^^^^^  .-5  10  10  10  EL/NO Figure 2.4  30  2D  10  (dB)  E r r o r Rate of QPSK on Various Channels  On the other hand, note the improvements p o s s i b l e with d i v e r s i t y L greater than 1.  The challenge i s to use the multipath f o r d i v e r s i t y rather  than l e t t i n g the r e s u l t i n g ISI degrade the modem performance. job of an adaptive e q u a l i z e r , our main t o p i c .  This i s the  The fade rates on HF channels  require recursive l e a s t square (RLS) algorithms to adapt f a s t enough, but unfortunately, these are very computationally i n t e n s i v e .  A l s o , because the  - 18 multipath delay spread i s so large on HF, long e q u a l i z e r s are r e q u i r e d .  If a  FT DFE i s used, we show in Sections 3.1 and 4.3 that the number of taps (N) required i s about three times the number of symbols spanned by the multipath spread.  Since i t takes 2N i t e r a t i o n s (symbols) f o r RLS algorithms to  converge to w i t h i n 3 dB of optimum e q u a l i z e r output SNR [ 2 8 ] , the length of the equalizer required on HF i s a d e f i n i t e hindrance.  F i n a l l y , fading multipath channels make c a r r i e r a c q u i s i t i o n and t r a c k i n g , receiver sample t i m i n g , and frame synchronization of signals extremely d i f f i c u l t .  suppressed-carrier  As mentioned i n Section 3 . 3 , t h i s project only  i m p l i c i t l y addresses these problems i n s o f a r as f r a c t i o n a l - t a p p i n g makes sample timing phase unimportant.  Regarding the ot h er s, manual techniques  were used so the research e f f o r t could concentrate on equalizer performance.  2.3 Discrete Time Simulation of the HF Channel  One of the project goals was to develop an accurate and f l e x i b l e d i s c r e t e - t i m e HF channel s i m u l a t o r .  This i s a valuable f a c i l i t y as e q u a l i z e r  performance could thus be measured and compared in a repeatable manner as a function of e i t h e r the channel or e q u a l i z e r parameters.  High speed was not  needed as the project experiments could be run as batch computer j o b s .  It was decided to simulate the channel completely in baseband.  This  allowed us to ignore c e r t a i n passband problems but s t i l l simulate in baseband others we were interested i n .  It was also computationally less burdensome,  an important f a c t o r as the HF channel i s an i n t r i c a t e channel to simulate. The chosen modulation was 1200 symbol/second QPSK as the l a t t i c e report channel recordings [42] used t h i s scheme.  QPSK i s an e f f i c i e n t modulation in  - 19 that i t uses only 1 Hz of bandwidth per bit/second throughput and i s constant modulus, thereby having a low peak-to-mean power r a t i o .  A baseband simulator must simulate the transmitted pulse shape, t r a n s mitter b a n d - l i m i t i n g , the fading m u l t i p a t h , noise, and the down-conversion low-pass f i l t e r (LPF).  This i s a p i p e l i n e of l i n e a r f i l t e r s so we can  interchange the order and/or merge some of them.  In f a c t , the t r a n s m i t t e r  b a n d - l i m i t i n g f i l t e r and the receiver LPF both have (baseband) 1200 Hz c u t offs.  Thus f o r a l l p r a c t i c a l purposes only one was needed in the s i m u l a t i o n .  Because radio channels have a f i n i t e length impulse response, our simulator can take the form of a time-varying FIR f i l t e r .  As shown in Figure  2.5 taken from [ 5 2 ] , t h i s can be implemented i n a tapped delay l i n e  (TDL)  configuration.  Additive noise r(/>  Figure 2.5  Possible HF Channel Simulator Configuration  The tap spacing deserves some d i s c u s s i o n .  [52]  Since our complex, baseband  (two-sided) spectrum i s W=2400 Hz wide, the maximum possible r e s o l u t i o n in  - 20 the time domain i s l/2400th of a second ( i e . T/2 where T i s the symbol period).  At the e q u a l i z e r , a path located half way between two T/2 tap  p o s i t i o n s cannot be resolved from two paths, each half as strong, spaced at the adjacent T/2 t a p s .  Thus, a higher r e s o l u t i o n i s simply not needed.  In actual f a c t , a T/8 spacing i s used i n the channel simulator program of Appendix D.  This was done to in order to Nyquist sample the image  frequencies encountered in the down-conversion process, and to e a s i l y s e l e c t the multipath spacing and sample timing to within about t h i s r e s o l u t i o n .  The Gaussian bell-shaped spectra of the path fading must also be simulated.  The path gains are Rayleigh fading processes generated by  f i l t e r i n g complex Gaussian noise generated at 9600 samples/second through a low pass f i l t e r .  (T/8)  This adjustable LPF must approximate the  exponential r o l l - o f f of the fading s p e c t r a .  The r e s u l t i n g bandwidth-reduced,  u n i t - v a r i a n c e processes are then attenuated by the s p e c i f i e d RMS sizes of the several paths.  In Section 4.1 we see the b i t - e r r o r - r a t e of the e q u a l i z e r  r i s e s at a rate greater than ( l o g a r i t h m i c a l l y speaking) 6 dB/octave of fading bandwidth.  Obviously, a one-pole f i l t e r i s not s u i t a b l e as the e q u a l i z e r ' s  s e n s i t i v i t y to fading i s r o l l i n g up f a s t e r than the fading spectrum i s r o l l i n g down.  Hsu [31] used a two-pole Butterworth f i l t e r , while Watterson  [63] used a three-pole f i l t e r .  Our simulator uses a c r i t i c a l l y - d a m p e d  two-pole f i l t e r , which i s more Gaussian bell-shaped than the f l a t - t o p p e d Butterworth.  - 21 A 5 path baseband channel simulator has been coded i n FORTRAN 77, and i s included in Appendix D.  It has the following adjustable f e a t u r e s :  i ) selectable path spacing to w i t h i n 1/8th of a symbol period i i ) v a r i a b l e RMS path amplitudes i i i ) v a r i a b l e - 3 dB fading half-bandwidth iv) v a r i a b l e s i g n a l - t o - n o i s e r a t i o v) simulated RF frequency mistuning v i ) down-conversion LPF c o e f f i c i e n t s read from a f i l e v i i ) s e l e c t a b l e sample timing to w i t h i n 1/8th of a symbol period  The reader should understand three things about the simulation software before proceeding. energy-per-bit decibels.  F i r s t l y , s i g n a l - t o - n o i s e r a t i o s herein are measured as  over the one-sided noise spectral density (E^/NQ) in  This i s a very common measure i n the communications l i t e r a t u r e as  i t compares a l l modulations on a somewhat equal f o o t i n g . Eb/No C/N, the c a r r i e r - t o - n o i s e power r a t i o . =  For QPSK,  Secondly, the multipath  spacing i s s p e c i f i e d in the channel simulator subprogram of Appendix D, and throughout the t h e s i s , by the 5 element integer array MPDEL(0:4).  MPDEL(O)  should .alway be set to zero, while the others specify the r e l a t i v e delay of the remaining 4 paths in units of T/8.  The r e l a t i v e RMS strength of the 5  paths are s p e c i f i e d by the real array MPGAIN(0:4), where MPGAIN(O) should be non-zero so the f i r s t path i s the f i r s t s i g n i f i c a n t one.  MPGAIN(0:4)  is  automatically normalized by the main program f o r u n i t transmitted power; only the r e l a t i v e values are important.  Except in section 4 . 2 , t h i s project used  MPDEL(0:4)=0,8,16,24,32 (incremental symbol spacing) f o r the f i v e paths. Three equal strength paths, each separated by 2 symbol periods, can then be denoted by MPGAIN(0:4)=1,0,1,0,1.  Equivalently though, t h i s could also be  specfied by MPDEL(0:4)=0,16,32,X,X and MPGAIN(0:4)=1,1,1,0,0.  - 22 3.  EQUALIZATION TECHNIQUES AND ALGORITHMS  3.1 Equalizer Configurations  Adaptive e q u a l i z a t i o n f o r data communication i s a r e l a t i v e l y recent t o p i c dating back only to 1965 [12, 4 0 , 5 3 ] .  The channel impulse response spreads  out the information in a s i n g l e symbol over several received sample periods causing intersymbol i n t e r f e r e n c e ( I S I ) . t h i s information back together again.  An equalizer attempts t o "gather" It uses an adaptive f i l t e r to  deconvolve the desired data symbols from the received sample sequence. We w i l l discuss e q u a l i z e r configurations in t h i s sub-section and adaptation algorithms in the n e x t .  One configuration f o r an e q u a l i z e r i s a tapped delay l i n e (TDL), as i l l u s t r a t e d i n Figure 3.1 taken from [ 5 2 ] .  Unequalized  Algorithm for lap gain adjustment  Figure 3.1  Basic Equalizer Configuration [52]  - 23 A l i n e a r e q u a l i z e r (LE) uses a l i n e a r combination of the received sequence to estimate the output symbol.  The estimate i s the vector inner  product of the contents of the TDL with the tap weights. adaptive f i n i t e impulse response (FIR)  filter.  It i s thus an  The tap weights are also  commonly referred to as tap gains or f i l t e r c o e f f i c i e n t s .  The d i f f e r e n c e between the equalized symbol estimate (equalizer output) and the exact symbol that was t r a n s m i t t e d , i s used i n the adaptation algorithm.  Equalizers usually have to be trained on known header symbols in  what i s c a l l e d reference-directed mode.  They are then switched i n t o  d e c i s i o n - d i r e c t e d mode where the receiver does not know what i s being transmitted.  Instead, the equalized symbol estimate quantized (rounded) to  the nearest symbol in the modulation c o n s t e l l a t i o n i s used to c a l c u l a t e the error.  This quantized symbol i s c a l l e d a d e c i s i o n .  It can o c c a s i o n a l l y be  i n c o r r e c t causing some adaption in the wrong d i r e c t i o n .  It  i s conjectured by  most researchers that i f wrong decisions are infrequent then l i t t l e degradation w i l l r e s u l t .  Occasionally though, e r r o r propagation can occur  which sometimes r e s u l t s in the e q u a l i z e r getting l o s t or c r a s h i n g .  A l i n e a r e q u a l i z e r i s l e s s than i d e a l f o r two reasons.  It  theoretically  must be i n f i n i t e l y long, even f o r a f i n i t e - l e n g t h channel response.  Because  intersymbol interference smears information about a symbol among neighbouring received samples, using only a f i n i t e span of samples neglects information which samples outside the span have regarding those i n s i d e the span.  In  a d d i t i o n , there are often zeroes ( n u l l s ) in the HF channel frequency response.  A f i n i t e - l e n g t h FIR f i l t e r i s not as capable as an i n f i n i t e  impulse response (IIR)  f i l t e r in generating the required v a r i a t i o n in inverse  response needed to equalize such a channel.  - 24 An adaptive IIR f i l t e r can be configured by feeding back some weighted, previously-equalized symbols into the summation which estimates the next symbol.  This y i e l d s equalizer poles which can cancel the channel zeroes.  There are two kinds of feedback e q u a l i z e r s : l i n e a r and decision feedback. a linear-feedback equalizer the equalized symbol estimate i s fed back.  In  This  technique i s discussed in [43, 51] and shows l i t t l e improvement over a LE. In a decision-feedback equalizer (DFE), the equalized symbol estimate i s quantized to the nearest symbol and t h i s decision i s fed down the d e c i s i o n feedback part of the TDL.  The technique i s reviewed with an extensive set of  references by B e l f o i r e and Park [ 3 ] .  A DFE equalizer with LMS algorithm  tap-weight adjustment (to be discussed i n Section 3.2) i s i l l u s t r a t e d in Figure 3.2 taken from [52]. Received-Sample Feedforward Part Input  Decision-Feedback Part  (r,)  Figure 3.2  An LMS Decision-Feedback Equalizer [52]  The advantage of a DFE i s i t s a b i l i t y to cancel a l l ISI from previous symbols ( i e . the t a i l s of t h e i r impulse response).  A l s o , a DFE t h e o r e t i c a l l y  requires only a f i n i t e number of decision-feedback t a p s ; j u s t enough to span  - 25 the impulse response d u r a t i o n .  As shown in [12, 5 4 ] , no received-sample  feedback taps are r e q u i r e d .  If there i s no additive noise in the received samples, only one received-sample feedforward tap i s required because no matter how small the f i r s t sample of energy i s from a given transmitted symbol, the ISI can be completely cancelled from i t ]  If there i s noise however, i t behooves the  e q u a l i z e r to weight more forward taps as they too contain useful information. the noise.  Using weight on the forward taps allows p a r t i a l averaging of A DFE r e l i e s heavily on the decision-feedback so i t s performance  should not be severely hindered by a f i n i t e - l e n g t h feedforward TDL.  Wrong decisions can cause the decision-feedback taps to contribute wrong information to the estimation sum.  This would appear to make a DFE more  susceptible to e r r o r propagation and thence crashes.  But because of the  DFE's advantages, i t s chance of making a wrong decision i s much less than that of a l i n e a r e q u a l i z e r .  This t r a d e - o f f i s of serious i n t e r e s t and w i l l  be investigated i n Section 7.  As discussed in Section 1 . 3 , t h i s project started from a l i n e a r l a t t i c e report [42] purchased from another company.  One l a t t i c e stage i s shown in  Figure 3.3 while a l a t t i c e e q u a l i z e r i s shown in Figure 3.4 [both f i g u r e s are taken from 5 2 ] ,  Figure 3.3  L a t t i c e Stage [52]  - 26 -  /(')  *i(0  yU)  2  Stage 2  Stage 1  */v.|C)  * C)  /,(')  * (0 2  A")  Stage N- 1 /,v-|C)  Figure 3.4 A Linear L a t t i c e Equalizer [52]  A l a t t i c e e q u a l i z e r [47,56,57] orthogonalizes the tap adjustment problem so that one tap can be adjusted without d i s t u r b u t i n g the o t h e r s .  This i s  done by computing backward predictors and using these i n the l i n e a r combination forming the equalized symbol e s t i m a t e .  Even l a t t i c e s which use  simple gradient-adaption algorithms (to be discussed in Section 3.2) converge and adapt quite q u i c k l y because of t h i s orthogonalization (see Figure 3 . 6 ) .  L a t t i c e s are i n t e r e s t i n g f o r several other reasons.  They simultaneously  generate outputs f o r a l l e q u a l i z e r lengths (number of t a p s , or order) up to a s p e c i f i e d maximum. [21].  They are configured very s u i t a b l y f o r hardware p i p e l i n i n g  A l s o , they were the f i r s t , r e c u r s i v e - l e a s t - s q u a r e - e r r o r algorithm/  c o n f i g u r a t i o n to be developed with computational complexity proportional to N, the e q u a l i z e r (maximum) l e n g t h .  The o r i g i n a l RLS algorithms had  complexity of order N^.  Unfortunately, e a r l y l a t t i c e work was hindered by d i f f i c u l t and inconsistent n o t a t i o n , no published complex or f r a c t i o n a l l y - t a p p e d formulations of the algorithm, and DFE configurations constrained to use  - 27 the same length forward and DF l i n e s .  Some of these problems have been  addressed in the recent l i t e r a t u r e [ 4 7 , 5 8 ] .  F i n a l l y , l a t t i c e computational  complexity, though proportional to N, i s s t i l l quite intensive (a high constant of  proportionality).  F r a c t i o n a l l y tapped (FT) equalizers [25] form t h e i r equalized symbol estimate from a l i n e a r combination of receiver samples taken at i n t e r v a l s of less than a symbol period (note: only the forward part of an FT DFE i s FT). As discussed i n Section 2 . 3 , with a transmission bandwidth of W the delay structure of the channel should be resolvable to within 1/W.  An e q u a l i z e r  should use t h i s time r e s o l u t i o n in i t s received sample TDL so as to extract a l l the a v a i l a b l e channel and data information from the stream. common i n t e r v a l i s half-symbol (T/2)  The most  spacing as t h i s allows Nyquist rate  sampling of the baseband received audio, although other f r a c t i o n s are also possible [ 6 1 ] .  Note that a T/2 e q u a l i z e r i s s t i l l clocked at the symbol rate  but the received FT samples in the feedforward section are s h i f t e d two p o s i t i o n s each symbol p e r i o d .  The advantages of an FT equalizer a r e : i)  They u s u a l l y provide sampling at the Nyquist rate thus e l i m i n a t i n g the need f o r a matched f i l t e r .  An FT adaptive equalizer i s given  enough information on the received sequence to function as an adaptive d i s c r e t e - t i m e matched f i l t e r . ii)  They e l i m i n a t e the s e n s i t i v i t y to sample timing phase which i s extremely d i f f i c u l t to track on HF.  i i i ) They allow proper r e s o l u t i o n of c l o s e l y or non-symbol-period spaced m u l t i p a t h s , allowing more d i v e r s i t y and thus increased performance.  - 28 The disadvantages of FT e q u a l i z e r s are increased computational burden ( e s p e c i a l l y i f using a complexity N a l g o r i t h m ) , increased memory requirements, and slower adaptation. explored in Section 4 . 3 .  Some of these performance t r a d e o f f s are  F i n a l l y , i f using equalizers with f r a c t i o n a l -  tapping c l o s e r than that given by the Nyquist c r i t e r i o n , the problem of optimum tap weights can be under-determined.  With too many degrees of  freedom, taps can wander o f f toward overflow or underflow numerical values and s t i l l be optimum.  A s o l u t i o n to t h i s problem i s proposed in [26].  Though i t i s not e q u a l i z a t i o n in the symbol-by-symbol sense, maximum l i k e l i h o o d sequence estimation (MLSE) i s another combatting technique which should be mentioned.  intersymbol-interference MLSE minimizes the symbol  e r r o r over an e n t i r e sequence of symbol estimates approximately spanning the channel impulse-response d u r a t i o n .  It i s a t h e o r e t i c a l l y better technique  than symbol-by-symbol e q u a l i z a t i o n , but the V i t e r b i algorithm used in MLSE has a computational complexity of exponential order with increasing response duration.  A l s o , MLSE must run j o i n t l y with a channel estimator i n a decision  d i r e c t e d - l i k e mode.  This in p r a c t i c e may hinder the technique.  Several  authors, Monson [44] f o r the troposcatter channel and Falconer et al [22, 23] f o r HF, have concluded that MLSE performs only s l i g h t l y better than a decision-feedback e q u a l i z e r .  In Section 8.2-we suggest t h i s may be a hasty  conclusion.  In the l i n e a r l a t t i c e report [42], symbol and half-symbol spaced l i n e a r equalizers were t e s t e d .  We on the other hand, are p a r t i c u l a r i l y  interested  i n quantifying the improved performance of DFE's on the HF channel.  In  Section 4 . 1 , symbol-spaced l i n e a r and decision-feedback e q u a l i z e r s are contrasted, while in Section 4.2 we present experimental r e s u l t s comparing T/l and T/2 DFE's.  - 29 3.2  Equalizer Tap-Weight Adjustment Algorithms  For a time varying or unknown channel, the e q u a l i z e r tap weights must adjust f o r the p a r t i c u l a r t r a j e c t o r y of the state of the channel over time.  One of the e a r l i e s t techniques f o r t h i s adaptation was the zero f o r c i n g (ZF) algorithm [40,41,52] which attempts to force the peak d i s t o r t i o n of an i n f i n i t e l i n e a r e q u a l i z e r to zero.  Because the peak d i s t o r t i o n can be shown  a concave f u n c t i o n of the tap weights, a steepest-descent algorithm can be used.  This algorithm forces the c r o s s - c o r r e l a t i o n of the equalizer symbol-  estimation e r r o r (denoted e(k)) with the desired symbol sequence (obtained by d e c i s i o n - d i r e c t i o n ) to zero.  Under some minor r e s t r i c t i v e conditions [52] a  f i n i t e - l e n g t h e q u a l i z e r w i l l converge to the truncated, i n f i n i t e e q u a l i z e r weights.  A l a t e r technique c a l l e d the (stochastic gradient) least mean square (LMS) algorithm [52,64] attempts to minimize E [ e ( k ) ] . 2  The i t e r a t i o n of  the algorithm t r i e s to minimize the c r o s s - c o r r e l a t i o n between e(k) and the received sample stream, not the desired symbol stream as in the ZF algorithm.  If there i s no c o r r e l a t i o n between e(k) and the received samples,  then no f u r t h e r information can be extracted from the samples to help the adaptation mechanism reduce e(k).  - 30 The basic LMS i t e r a t i o n i s :  where  e(k) = d(k) - X ' ( k ) C ( k - l )  (3.1)  C(k) = C ( k - l ) + uX*(k)e(k)  (3.2)  d(k) = the reference or d e c i s i o n at i t e r a t i o n k, Xjk) = the N-vector of concatenated forward and decision-feedback tapped delay l i n e s , C{k)  = the N-vector of tap weights being updated, u = the step s i z e constant,  denotes conjugate, and  denotes transpose.  The algorithm slowly  adjusts £ ( k ) so as remove a l l useful information regarding the desired symbol stream, d ( k ) , from the received sample stream, r(k)  (fed into X.(k)), by  making the e r r o r sequence, e ( k ) , orthogonal to the information i n X(k). Figure 3.2 in the l a s t sub-section i l l u s t r a t e s the LMS tap adjustment mechanism f o r a DFE.  LMS algorithms do not minimize the mean square e r r o r i n an optimum recursive sense. stochastically.  The tap weights only tend i n that d i r e c t i o n If the channel v a r i a t i o n i s very slow and the convergence  time not very c r i t i c a l , they work quite w e l l and are r e l a t i v e l y e f f i c i e n t (requiring only 2N m u l t i p l i c a t i o n s per i t e r a t i o n ) .  As a r e s u l t , LMS  algorithms became quite popular as d i g i t a l hardware came into wide use. v a r i a t i o n s [30,52] have also been i n v e s t i g a t e d .  Unfortunately, LMS algorithms w i l l not converge or adapt as f a s t as required f o r some a p p l i c a t i o n s .  This i s because the technique uses  X*(k)e(k) as an estimate of the gradient (with respect to C{k))  of the  Many  - 31 e r r o r surface [ 5 2 ] , rather than E[X_(k)e(k)]. C[k)  adapt to the proper values.  Only a f t e r many i t e r a t i o n s w i l l  A l s o , LMS convergence rate i s slowed by a  large eigenvalue spread in the channel c o r r e l a t i o n matrix [ 5 7 ] , a r e l a t i v e l y common s i t u a t i o n on the HF channel.  About 20 years ago, recursive least square (RLS) error algorithms were developed Kalman and Bucy [ 3 4 , 3 5 ] .  These algorithms are the optimum  recursive technique f o r minimizing the least square cost f u n c t i o n .  They  maintain i n t e r n a l c o r r e l a t i o n information f o r more properly c a l c u l a t i n g the gradient mentioned above.  For readers u n f a m i l i a r with Kalman e s t i m a t i o n ,  B o z i a c ' s book [5] i s an e x c e l l e n t i n t r o d u c t i o n , while Gelb's [24] i s a more advanced reference.  The increased convergence and adaption speed of RLS algorithms i s demonstrated i n Figure 3.5 taken from [ 5 2 ] .  Godard [28] showed them to  converge, f o r s t a t i c channels with l i t t l e n o i s e , to within 3 dB of the optimum output e r r o r in 2N i t e r a t i o n s .  Channel-correlation matrix Eigenvalue ratio = 21 11-tap equalizer, noise variance = 0.001  0 -3.0  Figure 3.5  100  200  0 300  400  500  600 6'  700  800  900  Number of iterations  Convergence Speed of Various Algorithms [52]  - 32 We now review the various RLS algorthm v a r i a t i o n s .  The context i n which  these algorithms are discussed i s that of the optimum estimation of state variables in a l i n e a r system, given noisy measurements of some 1inear combinations of the s t a t e .  In an e q u a l i z e r , the N optimum tap weights £ variables.  0 P T  (k)  are the state  Kalman estimation i s a technique f o r using noisy measurements to  generate recursive estimates, £ ( k ) , of the state propagating through time v i a the system model:  A(k)C  where  C Pr(k) 0  0 P T  ( k - l ) + W(k)  state vector  (3.3)  (Nxl)  A(k)  state t r a n s i t i o n matrix  W(k)  system d r i v i n g noise  (NxN)  (Nxl)  The measurement model i s :  where  d(k)  X'(k)C  d(k)  measurement  X(k)  measurement vector  V(k)  measurement noise  0 P T  (3.4)  ( k ) + V(k)  (lxl) (Nxl) (lxl)  - 33 If: PACT  CO  =  "  C  a P T  (k))  (C(K)  - C -(k))'*] o n  = the actual state covariance matrix at i t e r a t i o n k (NxN),  Q(k) = E[W(k)W'*(k)] = the expected d r i v i n g noise covariance matrix (NxN),  R(k) = E[V(k)V'*(k)] = the expected measurement noise covariance (1x1),  then the exponentially-aged [59] state estimate update i s :  e(k) = d ( k ) - X ' ( k ) A ( k ) C ( k - l )  (3.5)  C(k) = A ( k ) C ( k - l ) + G(k)e(k)  (3.6)  where: £ ( k ) = the exponentially-aged RLS estimate of £  P(k,k-1) = [ A ( k ) P ( k - l ) A * ( ) / A ] k  + Q(k)  = the a p r i o r i estimate of  0 P T  (k)  (3.7) (k)  G(k) = P ( k , k - l ) X * ( k ) [ R ( k ) + X ' ( k ) P ( k , k - l ) X * ( k ) ] -  1  (3.8)  = the Kalman Gain N-vector  P(k) = [ I - G ( k ) X ' ( k ) ] P ( k , k - l ) = the a p o s t e r i o r i estimate of P  and  X = the exponential-aging constant.  (3.9) A C T  (k)  - 34 The trace of P(k) (the sum of the estimated variances of the elements of the state estimate £ , from C. exponentially-aged,  0PT  ) i s the quantity which i s minimized in the  least-square sense.  In the equalizer a p p l i c a t i o n A(k)  is  usually unknown so X ( K ) C ( k - l ) i s the a p r i o r i symbol estimate, while 1  X'(k)C(k) i s the a p o s t e r i o r i symbol e s t i m a t e .  A valuable c h a r a c t e r i s t i c of  the technique i s that i t provides an estimate, P, of i t s accuracy in estimating C .  o r r  «  Unfortunately, the above equations have computational  complexity proportional  to  A, Q, and the state update equation ( 3 . 3 ) , provide the algorithm with an i n t e r n a l model of system.  If t h i s model information i s not p e r f e c t l y known  (or i f there are numerical p r e c i s i o n d i f f i c u l t i e s in the modelling), then i t i s possible f o r the state estimate C{k) to diverge from C . trace of P(k) would in dicate reasonable.  0PT  ( k ) more than the  The r e l a t i v e l y recent development  of exponential-aging (or weighting) [24,59] combats t h i s problem very w e l l . If A and Q are unknown, we can in f a c t set A=I and Q=0, thus using exponential-aging only as the adaptation mechanism.  For t h i s case the  exponentially-aged, a p o s t e r i o r i symbol-estimation squared e r r o r , T.\  [d(k)-X' ( k ) C ( k ) ] , i s also minimized. 2  With t h i s adaptation  mechanism the performance w i l l not be as good as with f u l l system model information, but the method has been t r i e d with success [31,24] even when the system i s n o n - l i n e a r .  This i s i n f a c t the dynamic-channel e q u a l i z e r case.  The o r i g i n a l a p p l i c a t i o n of RLS algorithms to e q u a l i z a t i o n in t h i s form i s due to Godard [28] who addressed the problem of rapid e q u a l i z e r convergence to a f i x e d , but unknown channel.  He was the f i r s t to show that  the e q u a l i z e r problem could be cast in the form of a Kalman problem without modelling the channel e x p l i c i t l y .  To do t h i s , he reversed the roles of the  measurement, d(k), and the measurement p r e d i c t i o n , X ' ( k ) A ( k ) C ( k - l ) , by  - 35 assuming d(k) was a symbol reference (or decision) while the measurement noise was deemed to occur in the tapped delay l i n e measurement vector,  X(k).  It doesn't r e a l l y matter as long as you regard e ( k ) , the measurement e r r o r which drives the adaptation, to be noisy with variance R(k).  It i s i n t e r e s t i n g to note the s i m i l a r i t y of RLS equations (3.5) and (3.6) when cast in e q u a l i z e r form (normally state t r a n s s i t i o n matrix, A(k), unknown and thus taken to be the i n d e n t i t y  e(k)=d(k) -  is  matrix):  X'(k)C(k-l)  C(k)=C(k-l) + G(k)e(k)  with the LMS equations:  e(k)=d(k) -  X'(k)C(k-l)  C(k)=C(k-l) + uX*(k)e(k)  This i s even more i n t e r e s t i n g i f i t i s noted that equation (3.8) can be rewritten 6(k)=P(k)X*(k)/R(k).  We then see that RLS i s j u s t LMS with the  s c a l a r step s i z e , u, replaced by the r a t i o of the a p o s t e r i o r i covariance, P ( k ) , to measurement noise covariance, R(k).  state  The elements of C_  are thus updated according to the s i z e and complex d i r e c t i o n of the e r r o r e ( k ) , the s i z e and d i r e c t i o n of the elements of Xjk) which through the respective elements of £ might have caused the e r r o r ,  and to the r a t i o of  state covariance to noise covariance (which helps properly specify the gradient of the previously-mentioned error  surface).  - 36 One i n t e r p r e t a t i o n of t h i s , r e a l i z i n g equation (3.9) as the matrix inversion lemma, i s that P i s the inverse of the exponentially-weighted channel c o r r e l a t i o n m a t r i x , P" (k)=E[X*(k)X'(k)]. 1  Some authors  [ 2 0 , 4 7 ] , seemingly separate from the Kalman community, have in f a c t derived s i m i l a r exponentially-aged-only equations d i r e c t l y from the Weiner-Hopf equations and the matrix inversion lemma.  The r e s u l t i n g equations are very  compact because Q=0, A=I, the c a l c u l a t i o n of P ( k , k - 1 ) can be absorbed into the other equations, and R can be taken as 1 .  This w i l l be f u r t h e r discussed  in Section 4 . 3 .  RLS algorithms have evolved in 3 general d i r e c t i o n s .  A l l are equally  f a s t to adapt; i n f a c t they t h e o r e t i c a l l y generate exactly the same r e s u l t s .  F i r s t l y , though Kalman and others have proved the equations converge to the correct s o l u t i o n , system mis-modelling can cause divergence and f i n i t e p r e c i s i o n computations often r e s u l t i n numerical i n s t a b i l i t y .  Mis-modelling  has been attacked by exponential-aging [ 5 9 ] , s e l f - t u n i n g [ 3 2 ] , and l i m i t e d (or s l i d i n g ) memory weighting [ 1 1 , 2 4 ] .  Numerical-precision d i f f i c u l t i e s can  cause P(k) to become n o n - p o s i t i v e d e f i n i t e .  The use o f :  P(k)=[I-G(k)X'(k)]P(k,k-l)[I-G(k)X'(k)]'*+G(k)R(k)G *(k) ,  (3.10)  rather than equation (3.9) preserves p o s i t i v e d e f i n i t e n e s s at the cost of increased computation.  Many more e f f i c i e n t , factored RLS algorithms have been developed which do not store and update P(k) e x p l i c i t l y , but special f a c t o r s of P whose form guarantees p o s i t i v e d e f i n i t e n e s s .  Some examples are the matrix square root  of P and upper triangular/diagonal f a c t o r i z a t i o n s such a s : P(k)=U*(k)D(k)U'(k) These factored algorithms have been extensively treated i n [ 1 , 2 , and 7 ] .  - 37 In [ 3 1 ] , Hsu applies an exponentially-aged v a r i a t i o n of the U*DU' technique, which Carlson [7] c a l l s "factored inverse square-root Kalman," to equalizing the HF channel.  As discussed in Appendix B, t h i s algorithm was  used f o r the e q u a l i z e r in t h i s p r o j e c t .  Interestingly,  a l l the above  factored algorithms both prevent P from l o s i n g p o s i t i v e d e f i n i t e n e s s and increase dynamic range as the square root of most v a r i a b l e s are stored and manipulated.  Hsu does not state his algorithm i s an exponentially-aged one  but i t i s easy to show that his assumption t h a t Q=qAPA'* r e s u l t s in such an algorithm with X= l/(l+q).  See also Section 4 . 3 .  The second d i r e c t i o n RLS equalizer algorithms have taken i s that of RLS lattices.  These f o r m u l a t i o n s , as compared to the gradient l a t t i c e ,  simultaneously generate the RLS s o l u t i o n f o r a l l equalizer lengths up to a selected maximum.  RLS l a t t i c e s were the f i r s t RLS algorithms to be developed  with computational complexity proportional to N (though with a large c o e f f i c i e n t ) rather than N . 2  Some have also developed with square-root  normalization g i v i n g more dynamic range [ 3 7 , 5 0 ] .  The t h i r d d i r e c t i o n RLS work has taken i s toward " f a s t " f i x e d - o r d e r solutions.  These have computational burden proportional to N but with a  smaller c o e f f i c i e n t of p r o p o r t i o n a l i t y than RLS l a t t i c e s . algorithm was termed Fast Kalman [ 3 9 , 2 0 ] .  The f i r s t such  Unfortunately, the Fast Kalman  algorithm proved to be even more numerically unstable than the o r i g i n a l algorithm.  As t h i s project progressed however, even f a s t e r Kalman algorithms  [6, 9] have been developed.  The d i s s e r t a t i o n work by C i o f f i [ 8 ] , and other  forthcoming papers by C i o f f i and K a i l a t h [10,11] point out many of the causes and solutions to the numerical i n s t a b i l i t y .  He also derives the f i r s t  square-root-normalized version of a f a s t , f i x e d - o r d e r Kalman algorithm.  - 38 F i n a l l y , there are two tap-weight adjustment algorithms which do not require any decision d i r e c t i o n .  They converge to the proper tap weights  without presuming the information sequence t r a n s m i t t e d .  These algorithms  should thus never crash in the l i n e a r e q u a l i z e r c o n f i g u r a t i o n .  The f i r s t algorithm, due to Morgan [ 4 6 ] , compares the average received spectrum with the t h e o r e t i c a l spectrum expected.  It then c a l c u l a t e s an e(k)  based on t h i s d i f f e r e n c e only and uses an LMS-type tap update al g or i t h m .  The second algorithm not requiring d e c i s i o n - d i r e c t e d t r a i n i n g was designed f o r use with constant-modulus signal modulations such as QPSK.  The  Constant Modulus Algorithm (CMA) [36, 60] uses only the directed r a d i a l component of e(k)  ( r e c a l l e(k) i s complex in baseband).  Both the above  algorithms depend on the symbol sequence being uncorrelated, or white.  In Section 7, we adapt the CMA algorithm from i t s published LMS l i n e a r equalizer form to an RLS decision-feedback c o n f i g u r a t i o n .  The RLS algorithm  makes i t f a s t e r adapting and the DFE form i s much better on a channel with zeroes.  Of course, the CMA DFE requires decisions f o r the feedback TDL so i t  i s s t i l l somewhat susceptible to crashing i n d e c i s i o n - d i r e c t e d mode.  But  under slower fading c o n d i t i o n s , where the whiteness of the data stream has time to show i t s e l f , the RLS CMA DFE seems more r e s i s t a n t to crashing than a regular RLS DFE.  - 39 3.3 Synchronization and T r a i n i n g  In t h i s sub-section we f i r s t discuss some methods f o r c a r r i e r recovery, receiver sample t i m i n g , and frame s t a r t synchronization.  These t o p i c s were  not of p a r t i c u l a r concern during the c o n t r o l l a b l e or manual conditions which e x i s t e d during our experiments, and though n o n - t r i v i a l , they are not thought t o be insurmountable in p r a c t i c e .  We w i l l b r i e f l y discuss methods used on  other channels and point out some special d i f f i c u l t i e s on HF. s u b - s e c t i o n , some of the desirable conditions and methods f o r  Later in the converging,  t r a c k i n g , and r e t r a i n i n g equalizers are pointed out.  The HF channel i s a very d i f f i c u l t channel to coherently demodulate because of the fading m u l t i p a t h .  Normal s i n g l e - s i d e - b a n d (SSB) radios have  no means f o r c a r r i e r t r a c k i n g so most have a c l a r i f i e r c i r c u i t to manually f i n e tune the f r o n t end of the r e c e i v e r .  Even with the newer synthesized  r a d i o s , the required frequency accuracies demonstrated in Section 4.3 are not possible.  Even i f they were, the doppler s h i f t at a i r c r a f t speeds can e a s i l y  cause a deviation of 15 Hz.  Passband e q u a l i z a t i o n [ 1 8 , 1 9 ] , where e q u a l i z a t i o n takes place before subcarrier down-conversion, has been suggested f o r other multipath channels as the e q u a l i z a t i o n removes the d i s t o r t i o n so standard methods f o r non-fading, non-multipath channels can be used.  This i s not p r a c t i c a l on HF  as a T/8 e q u a l i z e r i s required to sample above the Nyquist rate of the maximum image frequency during down-conversion.  Such an e q u a l i z e r used on a  large-multipath-spread channel such as HF would require over 100 t a p s ; c l e a r l y impractical from an implementational point of view.  - 40 Another scheme proposed in [18] and [ 1 9 ] , and noted herein as f e a s i b l e , would be to use the phase bias in the equalizer output symbol estimates to hold an automatic frequency control (AFC) l o c k . directed (or data-aided) c a r r i e r recovery.  This i s termed d e c i s i o n -  Unfortunately, the scheme  requires the e q u a l i z e r already be converged.  This r a i s e s the question of how  to acquire i n i t i a l c a r r i e r frequency o f f s e t ?  Even i f i t were to work during  convergence, c a r r i e r d e v i a t i o n can make the i d e n t i f i c a t i o n of the correct instant to apply the reference symbols to s t a r t convergence extremely difficult.  One might i n i t i a l l y have to send a p i l o t tone (as p a r a l l e l modems  continuously do) to i n i t i a l l y lock onto.  Due to the frequency s e l e c t i v e  fading on the HF channel, a more diverse set of tones might be required f o r reliable carrier acquisition.  In our experiments c a r r i e r a c q u i s i t i o n was not a problem.  In the  simulated channel experiments our channel simulator and e q u a l i z e r ran i n baseband, though a simulated RF mistuning o f f s e t could be added i f d e s i r e d . During the real channel recordings the t r a n s m i t t e r RF, s u b c a r r i e r , and symbol rate had been cesium locked, as were the r e c e i v e r RF and sampling r a t e .  In  section 5 . 4 , some mistuning was unexplicably noted on these tapes but was corrected manually by the detuning feature mentioned above.  Sampling phase, also known as b i t timing (symbol timing i n the QPSK c a s e ) , requires knowledge of the symbol rate and phase. by RF mistuning or doppler.  It i s not affected  Again, conventional techniques f o r sampling  phase t r a c k i n g perform poorly on HF because of the extreme time d i s p e r s i o n .  As mentioned in the previous s u b - s e c t i o n , f r a c t i o n a l l y - t a p p e d equalizers are a very sensible s o l u t i o n to t h i s problem.  This project v e r i f i e s t h e i r  - 41 performance in the DFE configuration and s p e c i f i c a l l y on HF channels.  The  sample timing was adjusted manually f o r the symbol-spaced equalizers used in the  project.  The f i n a l synchronization problem i s frame sync.  Assuming a known  t r a i n i n g sequence heads each packet, the s t a r t time of the f i r s t energy from t h i s sequence must be located.  Then, the l o c a l l y - g e n e r a t e d , known t r a i n i n g  sequence can be applied to the e q u a l i z e r as a reference during convergence i n reference-directed mode.  An appropriate scenario might be to send a  polyphase Frank sequence, c o r r e l a t e , and note the location of the peaks (which correspond to the m u l t i p a t h s ) .  One can then apply the reference to  t r a i n the equalizer at the time when the f i r s t energy from the f i r s t path i s arriving.  It might be possible to f i n d a sequence with a better ambiguity  function than the Frank sequence, as doppler can be mistaken f o r delay.  In t h i s p r o j e c t , frame sync was not a problem as the channel simulator/ e q u a l i z e r ran in l o c k - s t e p and the real channel tape synchronization was handled manually.  There are many issues that should be pointed out about our l a s t t o p i c i n t h i s s e c t i o n : t r a i n i n g the e q u a l i z e r to converge and f o l l o w the channel. F i r s t l y , the symbol sequence transmitted during convergence and during data transmission should be white, so that the ionospheric condition of the e n t i r e passband i s sounded.  If not, the e q u a l i z e r may adapt (or not adapt) i t s gain  and phase properly across the band.  This could happen when c e r t a i n long  s t r i n g s , say 0° symbols in QPSK, are transmitted.  If the data sequence  suddenly becomes a d i f f e r e n t colour then the e q u a l i z e r may not immediately provide the correct f i l t e r i n g and could even c r a s h .  Polyphase Frank  sequences or Barker codes are very white and would thus make good t r a i n i n g  - 42 sequences.  For the data part of a frame, a scrambler such as s p e c i f i e d i n  CCITT V.27 and i l l u s t r a t e d in Figure 3.6 would be d e s i r a b l e .  The scrambler  s t a r t has to be synchronized but t h i s i s e a s i l y done at the s t a r t of a frame.  In our experiments, we simply used a repeating, 1023 b i t maximal-  length pseudo-random sequence f o r the d a t a .  Figure 3.6  CCITT V.27 Data Scrambler  In the previous s e c t i o n , i t was mentioned that t r a i n i n g can also use the e r r o r i n e i t h e r the received spectrum or symbol modulus.  In the LE  c o n f i g u r a t i o n , no knowledge of the transmitted sequence i s required t o converge nor should such an LE c r a s h .  This w i l l be b r i e f l y investigated in  Section 7.3  P e r i o d i c i n s e r t i o n of r e t r a i n i n g sequences, as discussed in Section 7, i s a robust way to prevent and recover from crashes.  Unfortunately  retraining  can add considerable overhead to the data stream.  I n t e r e s t i n g l y , Hsu [30]  has done some experiments with an ARQ request-training-sequence (RTS) protocol on simulated f u l l - d u p l e x HF channels.  This scheme only requires the  necessary precentage r e t r a i n i n g overhead needed f o r the p a r t i c u l a r state of the channel.  - 43 4.  EQUALIZER PERFORMANCE ON SIMULATED CHANNELS  4.1 Basic Symbol-Spaced Equalizer Results and Discussion  In t h i s S e c t i o n , we i n v e s t i g a t e the b i t e r r o r rate (BER) performance of symbol-spaced equalizers in reference-directed mode on simulated HF channels.  We are p a r t i c u l a r ! ' l y interested in quantizing the performance of  the r e c u r s i v e - l e a s t - s q u a r e - e r r o r  (Kalman) algorithm in a decision-feedback  e q u a l i z e r (DFE), and contrasting i t with the performance obtained using e i t h e r the LMS algorithm or a l i n e a r e q u a l i z e r c o n f i g u r a t i o n .  Results are  presented as a function of s i g n a l - t o - n o i s e r a t i o (SNR), fading bandwidth (Bf)>  symbol-spaced multipath s t r u c t u r e , and exponential-aging  Optimum or nominal values have been used f o r other parameters.  factor. For instance,  we have used optimum sample timing phase, a nominal value f o r the expected Kalman measurement noise R, an equalizer length which j u s t spans the multipath spread, and simulated perfect RF t u n i n g .  In Section 4 . 2 , performance of symbol-spaced and fractional-symbol spaced equalizers are contrasted in r e f e r e n c e - d i r e c t e d mode as a function of sampling phase and non-symbol-spaced multipath delay s t r u c t u r e .  Section 4.3  investigates the e f f e c t of some of the remaining parameters.  Figure 4.1 shows the f i r s t multipath channel structure i n v e s t i g a t e d .  It  i s composed of 3 equi-mean-square fading paths each separated by 2 symbol periods.  A l l channels discussed i n t h i s t h e s i s are Rayleigh fading and path  strengths shown are root-mean-square (RMS) values.  The channel in Figure 4.1  also i l l u s t r a t e s what we term symbol-spaced multipath where the path propagation delays d i f f e r by an i n t e g r a l number of symbol periods.  As  mentioned above, performance on non-symbol-spaced multipath channels i s  - 44 presented in the next s u b - s e c t i o n .  The data transmitted was a r e p e a t i n g ,  1023 b i t maximal-length pseudo-random sequence (M-sequence).  RMS impulse response T T  0  i 2T  3T  4T  D i f f e r e n t i a l Propagation Delay Figure 4.1  Three Path Channel  (MPGAIN(0:4)=1,0,1,0,1))  Only four d e c i s i o n feedback taps are needed f o r the above channel.  On  the other hand, though an i n f i n i t e number are t h e o r e t i c a l l y d e s i r a b l e , f i v e received-sample, feedforward taps are needed to span the m u l t i p a t h .  We show  i n Section 4.3 that t h i s feedforward length i s a good compromise between the t h e o r e t i c a l l y - d e s i r a b l e length, and a short f a s t e r - a d a p t i n g l e n g t h . were 600,000 symbols (500 simulated seconds) long.  A l l runs  A l l SNR f i q u r e s quoted in  t h i s t h e s i s are energy/bit over the noise spectral density  (E /N ). b  0  Because QPSK uses 1 Hz of bandwidth per bit/second throughput, t h i s i s also equivalent to c a r r i e r - t o - n o i s e power r a t i o .  Figures 4.2 and 4.3 present the f i r s t proper c h a r a c t e r i z a t i o n of exponentially-aged Kalman equalizer performance on HF channels.  Raw  p r o b a b i l i t y of b i t e r r o r in reference-directed mode i s shown as a function of Hsu's exponential-aging f a c t o r , q = ( l / X ) - l , f o r s i g n a l - t o - n o i s e r a t i o s 13, 18, 23, and 28 dB.  Results f o r fading bandwidths B  f  0  f  . 1 5 , and 1.5 Hz are  shown i n Figure 4 . 2 , while bandwidths of . 0 5 , . 5 , and 5.0 Hz are i n Figure  - 45 Several things are immediately apparent.  F i r s t , each channel condition  has a q value which gives a minimum p r o b a b i l i t y of e r r o r .  This makes sense  as a small q may give too slow an adaptation r a t e , while a large q causes the e q u a l i z e r to adapt too q u i c k l y thus responding to noise.  Secondly, we see  the optimum value of q i s a function of both SNR and fading bandwidth.  This  too i s l o g i c a l as optimum q must depend in some way on the expected rate of channel v a r i a t i o n versus the noise l e v e l .  I -3  /o  1  -——**"i5 Hz, !8dB  0.15 Hz, I3JB  I.SHz, 23d& /.5Hz,28dB  \ 0.15 Hz,/SdB  0..5Hz,23dB  0J5Hz^2So(B 10  .00  Figure 4.2  JO  .10  .30  ( 1 , 0 , 1 , 0 , 1 ) DFE Performance as a Function of q ( . 1 5 , 1.5 Hz)  - 46 -  10 -  ^5.0 Hz, )3 jB 5.0Hz,  ISOIB  • 0.5 Hz, 13 JB  ^ 0.5Hz, l«dB /6.05Hz,l8a&  ,-3  10  0.5Hz,23dB ^J).5Hz,2&lB 10  10  .00  Figure 4 . 3  JO  .10  .30  ( 1 , 0 , 1 , 0 , 1 ) DFE Performance as a Function of q ( . 0 5 , . 5 , 5 Hz)  Because of the inverse nature of the equalizer problem, i t i s not possible to get a useful a n a l y t i c a l formula f o r the optimum e x p o n e n t i a l aging f a c t o r .  Instead, Figure 4.4 presents a p l o t of optimum aging f a c t o r as  a function of SNR f o r various B . f  we have plotted X o p t  instead  of q as i t i s a more u n i v e r s a l l y understood measure of exponential-aging.  - 47 -  1.00 . 0 5 Mz 0.15 Hz 0.9OPT  0.5 Hz  0.5-  1.5 Uz  - 5 . 0 Hz(FT) 5 . 0 Hz  0.710  30  2D  E Figure 4.4  / No  b  40  (dB)  Optimum Exponential-Aging Factor  When the optimum performance data from Figures 4.2 and 4.3 i s gathered together we can p l o t some very i n t e r e s t i n g performance graphs.  Figure 4.5  shows the b i t - e r r o r - r a t e in r e f e r e n c e - d i r e c t e d mode as a function of SNR f o r various fading bandwidths.  It i s immediately n o t i c a b l e that with high  the performance bottoms out at higher SNR's.  We conclude that  even f a s t - a d a p t i n g RLS algorithms can run into adaptation-rate l i m i t a t i o n s on the HF channel.  This i s a new r e s u l t not a n t i c i p a t e d by the authors who  suggested RLS algorithms f o r e q u a l i z i n g the HF channel. smaller fade rates i s much b e t t e r .  The performance at  The 0.05 Hz l i n e i s within 4-5 dB of the  3 path fading t h e o r e t i c a l l i m i t discussed i n Section 2 . 2 .  In Section 4 . 3 , we  see a longer but i m p r a c t i c a l feedforward length r e s u l t s in performance within about 2 or 3 dB of t h e o r e t i c a l .  Fortunately from an experimental point of  view, the v a r i a t i o n i n fading rate of the HF channel j u s t brackets the  - 48 -  B = 5.0Hz  —--^r^  f  " (Q15Hz LE) \  "—X(0.15Hz LMS)  \  \  \\  >  THEORETICAL  k  \  tO-  \\\ \\\ \\ \  \ \\  -V  10  to  \ \  \ \ \  :  \  \\ \  \ \  \  N.05HZ  \\ \  10  Eb/No Figure 4.5  \v \\ Y  \0.15Hz  30  10  (dB)  \0.05H2  ( 1 , 0 , 1 , 0 , 1 ) DFE Performance as a Function of SNR  t r a c k i n g - r a t e l i m i t a t i o n of the t y p i c a l - l e n g t h RLS equalizer needed on HF. Later in t h i s s u b - s e c t i o n , a f t e r presenting r e s u l t s on a few more channel s t r u c t u r e s , we discuss t h i s l i m i t a t i o n and propose a new d e f i n i t i o n of "slow fading."  - 49 Most of our research takes place in referenee-directed mode and t h u s , as discussed in Section 7, i s somewhat i d e a l i s t i c .  Nonetheless, i t allows easy  i n v e s t i g a t i o n of a v a r i e t y of phenomena without having to deal with crashes, etc.  We note here that the r e s u l t s of Figure 4.5 show the s u p e r i o r i t y of the  s e r i a l modem scheme over that of p a r a l l e l modems [63], when the p a r a l l e l modems are penalized 6 dB by a peak-power l i m i t e d r a d i o .  A more p r a c t i c a l  comparison w i l l be made i n Section 7.2 where we evaluate the s e r i a l modem i n d e c i s i o n - d i r e c t e d mode.  A l i n e a r e q u a l i z e r with the same t o t a l number of taps as the DFE was also t r i e d on the above channel s t r u c t u r e . are also show in Figure 4 . 5 .  These r e s u l t s , which are quite poor,  A longer LE would perform better at low fade  r a t e s , but again, the extra length slows adaptation thus hindering performance on more r a p i d l y - v a r y i n g  channels.  The optimim value of q f o r the  l i n e a r equalizer was found to be less than h a l f that f o r a DFE.  This  suggests the measurement noise of a f i n i t e - l e n g t h LE i s higher than that of a DFE, which i s one i n t e r p r e t a t i o n of why the LE performs so p o o r l y .  Little  f u r t h e r LE reseach was done except to compare r e s u l t s with the l a t t i c e LE report [42] on the three real channel tapes.  The reader w i l l also note two points on the graph which were the best obtained using an LMS (gradient) DFE on a 0.15 Hz fading channel.  These  points confirm that at high s i g n a l - t o - n o i s e r a t i o s , the LMS algorithm cannot perform nearly as well as the f a s t - a d a p t i n g Kalman algorithms on the very dynamic HF channel.  Because the channel was f a d i n g , the eigenvalue r a t i o of  the channel c o r r e l a t i o n matrix, which the LMS algorithm i s s e n s i t i v e t o , l i k e l y took on a v a r i e t y of r e a l i s t i c values.  - 50 In Figure 4.6 we p l o t the same DFE data as i n Figure 4 . 5 , but as a function of fading bandwidth. fading rate. dB/octave.  Note the steep increase in P  g  with  In logarithmic terms, the slope of the SNR=28 dB l i n e i s 7.5 Obviously an equalizer of s u i t a b l e length f o r the HF channel gets  extremely s e n s i t i v e to fade rate around 0.5 Hz.  This steep slope i s not  expected to continue i n d e f i n i t e l y to the l e f t in Figure 4.6 but should bottom out as the channel slows down more.  The p r o b a b i l i t y of e r r o r would then be  -i 10  10  -3  10  10  23d B r -5  10  0.01  1.0  0,1  10.0  (Hz) Figure 4.6  ( 1 , 0 , 1 , 0 , 1 ) DFE Performance as a Function of B  f  - 51 due only to n o i s e , the l e s s - t h a n - i n f i n i t e length forward TDL of the DFE, and the p r o b a b i l i t y that some of the paths would be somewhat faded out.  This graph explains some very poor, e a r l y r e s u l t s we obtained when using a f i l t e r with a - 6 dB/octave r o l l - o f f to bandwidth-limit the fading processes.  Such a one-pole f i l t e r i s a poor approximation to the exponential  r o l l - o f f of the HF channel's Gaussian bell-shaped fading spectrum.  The  e q u a l i z e r ' s s e n s i t i v i t y to fading bandwidth was r o l l i n g up f a s t e r than the bandwidth-limiting f i l t e r was r o l l i n g down the fading spectrum'.  We now present r e s u l t s on two channels with d i f f e r e n t multipath structures.  Figure 4.7 shows the RMS impulse response of a 2 path channel  where the paths are separated by 4 symbol periods (3.3 ms).  Figure 4 . 8  shows the r e s u l t i n g r e f e r e n c e - d i r e c t e d BER performance as a function of SNR. Note the performance i s not as good, due to reduced d i v e r s i t y , as on the previously 3 path channel.  The optimum exponential-aging i s s i m i l a r to the  previously presented r e s u l t s of Figure 4 . 4 .  RMS impulse response  0  IT  2T  3T  4T  D i f f e r e n t i a l Propagation Delay Figure 4.7  Two Path Channel  (MPGAIN=(1,0,0,0,1))  - 52 -  B =5.0Hz f  \ \ \  \  x.  x  \\  ^  \  x-3  VV  \  \ ,  1.5 Hz  \  vv\ \ Vs. N \ * VV\ \ > v\  \  \V  \  >s0.5Hz  THEORETICAL^  V \ \  >  \  \  \  \  \  10  0.15 Hz  \  \ ^  \0.05Hz  \  20  JO  \  30  E /N„ (J&) k  Figure 4.8  ( 1 , 0 , 0 , 0 , 1 ) DFE Performance as a Function of SNR  F i n a l l y , Figure 4.9 shows the RMS impulse response of a 3 path channel with unequal RMS path strengths (MPGAIN(0:4)=.33,0,1,0,.33), symbol periods (3.3 ms).  again spanning 4  Figure 4.10 shows the symbol-spaced DFE performance  on t h i s channel i n reference-directed mode.  The t h e o r e t i c a l curve included  i s f o r 3 equal strength fading paths as the one f o r unequal paths i s rather d i f f i c u l t to c a l c u l a t e .  53  RMS impulse response  t  1  — i —  0  T  3T  21  4T  D i f f e r e n t i a l Propagation Delay Figure 4.9  Unequal-Strength Three Path Channel ( . 3 3 , 0 , 1 , 0 , . 3 3 )  Comparing Figures 4.5 and 4.10, we see the ( . 3 3 , 0 , 1 , 0 , . 3 3 ) r e s u l t s are not as good as the three equal path r e s u l t s .  This i s i n t e r e s t i n g as the  ( . 3 3 , 0 , 1 , 0 , . 3 3 ) channel i s in a sense not as spread. need as long a forward TDL as the ( 1 , 0 , 1 , 0 , 1 ) channel.  It therefore should not We must conclude the  ( 1 , 0 , 1 , 0 , 1 ) channel has more d i v e r s i t y which r e s u l t s in i t s  better  performance.  What can we conclude from the r e s u l t s presented so far?  Obviously,  the  optimum exponential-aging f a c t o r varies s i g n i f i c a n t l y with the channel SNR and B^.  In a d d i t i o n , the b i t error rate can vary quite a b i t with  d e v i a t i o n from optimum weighting.  Is there a way to have the e q u a l i z e r  s e l f - t u n e the exponential-aging factor?  What f i r s t comes to mind i s to  monitor the equalizer output residual e r r o r , E[e ( k ) ] . due to n o i s e , then X could be increased. large due to high B  f  instead?  If i t i s l a r g e ,  But what i f the residual e r r o r i s  We should then use a smaller A .  are unfortunately opposite t r e n d s .  These  If e i t h e r the noise l e v e l or fade rate  were f i x e d , we could s u c c e s s f u l l y s e l f - t u n e the other as in [ 4 , 3 2 , 3 3 ] . Unfortunately,  in foreseeable implementations a nominal value f o r the  exponential-aging f a c t o r may have to be chosen and the ensuing loss of  performance accepted.  Further research, such as examining the higher  s t a t i s t i c s of e(k) [ 3 3 ] , or estimating B from the rate of tap f  v a r i a t i o n , i s d e f i n i n t e l y suggested.  Figure 4.10  ( . 3 3 , 0 , 1 , 0 , . 3 3 ) DFE Performance as a Function of SNR  - 55 We also conclude that the adaptation rate of RLS algorithms i s barely adequate f o r the HF channel.  Auroral channels with high fading bandwidths  w i l l be too r a p i d l y varying f o r the e q u a l i z e r .  As discussed by Monsen [ 4 5 ] ,  HF fade rates are much slower (2-3 orders of magnitude) than the symbol rate.  The HF channel i s thus categorized as "slow f a d i n g . "  Obviously 2  orders of magnitude i s not slow enough f o r the long equalizer required on the HF channel, even when using RLS algorithms.  We now propose a new d e f i n i t i o n f o r slow fading a p p l i c a b l e to exponentially-aged RLS e q u a l i z a t i o n of a dynamic communication channel. Proakis (p. 487 of [52]) suggests that t /T greater than 100 i s a d  good c r i t e r i o n . t , d  A fading bandwidth of 5 Hz has a d e c o r r e l a t i o n t i m e ,  of l/(2/Tx5) seconds.  second.  Thus t /T=38. d  Our symbol p e r i o d , T, was 1/1200 of a  This i s obviously not greater than P r o a k i s '  guide of 100 and suggests why our r e s u l t s on a 5 Hz fading channel bottom out.  Proakis though, does not s p e c i f y how the f i g u r e of 100 was obtained. More importantly, he does not specify whether i t i s a guide f o r the LMS gradient or RLS e q u a l i z e r algorithms; there i s quite a difference in adaptation rate between the two on low noise channels.  A more t h e o r e t i c a l  d e f i n i t i o n i s suggested using Godard's r e s u l t [28] that an RLS algorithm slews the taps weights of an e q u a l i z e r to w i t h i n 3 dB of minimum e q u a l i z e r output e r r o r in 2N i t e r a t i o n s , where N i s the t o t a l number of taps in the equalizer.  Godard's r e s u l t though, i s f o r convergence time on a high SNR,  s t a t i c channel ( i e . A=I and Q=0).  In c o n t r a s t , our e q u a l i z e r does not know  the channel dynamics and has a time delay which i s inherent i n the exponentially-aged t r a c k i n g process.  - 56 A better "slow f a d i n g " c r i t e r i o n f o r dynamic channels tracked by exponential-aged RLS algorithms might therefore be:  t /T » d  2N  or e q u i v a l e n t l y : B « f  where f  f /47TN s  i s the symbol r a t e .  Even t h i s c r i t e r i o n i s only a better guide.  Due to the n o n - l i n e a r  r e l a t i o n s h i p of the e q u a l i z e r taps to the channel fading t a p s , higher order e f f e c t s are p o s s i b l e .  For instance, with 3 channel paths o s c i l l a t i n g at 5 Hz  i t might be possible to f i n d r e l a t i v e phases of the fading where some of the optimum equalizer taps would need to o s c i l l a t e at 15 Hz.  This new concept  strengthens the "much less than" i n e q u a l i t y in the above c r i t e r i o n and points to areas needing f u r t h e r research.  We now see the f a s t e s t fading expected to be properly tracked by a 9 tap e q u a l i z e r i s much less than 1200/(4xft*xN)=10.6 Hz.  This explains why the  e q u a l i z e r was reasonably adept at f o l l o w i n g a . 5 Hz fading bandwidth channel, yet found 5 Hz to be "moderately r a p i d " f a d i n g .  F i n a l l y in t h i s s u b - s e c t i o n , we conclude that the a p p l i c a t i o n of Hsu's algorithm [31] to fading multipath channels r e s u l t s in performance quite d i f f e r e n t from that presented in his paper. Hsu's P  g  As mentioned in Section 1 . 2 ,  versus SNR r e s u l t s have a slope at high SNR which i s  decreasing past - 4 f o r two path d i v e r s i t y . high SNR's of - L f o r L path d i v e r s i t y , and the discussion preceeding i t .  We obtain an asymptotic slope at  in proper agreement with Figure 2.4  - 57 4.2 Fractionally-Tapped Equalizer Performance and Discussion  In t h i s s u b - s e c t i o n , we f i r s t present the r e s u l t s of a (T/2)  fractionally  tapped DFE on one of the same channel structures as used in the l a s t section.  Then we show the v a r i a t i o n in performance of a T/l e q u a l i z e r as a  function of receiver sampling phase and contrast t h i s with the lack of v a r i a t i o n obtained with a T/2 FT DFE.  F i n a l l y , a n o n - i n t e g r a l , symbol-  period-spaced multipath channel i s i n v e s t i g a t e d .  The FT e q u a l i z e r i s show to  o f f e r superior performance on such a channel, even compared to a T/l equalizer operating with optimum sampling phase.  The T/2 spaced FT DFE was exercised on the 3 equi-RMS-strength path channel of Figure 4.1 (MPGAIN(0:4)=1,0,1,0,1).  Rather than having 5  feedforward taps to span the spread, we now need 10 at half the previous spacing.  The optimum X was found to be very s i m i l a r to Figure 4.4 f o r the  symbol-spaced e q u a l i z e r , except the B =5 Hz l i n e was, as shown by the f  dotted l i n e i n Figure 4 . 4 , h i g h e r .  This i n d i c a t e s the FT e q u a l i z e r , being  longer, cannot be pushed as hard to track a r a p i d l y changing channel.  It  seems there may be two time constants in an e q u a l i z e r : the exponential one controls the rate at which P can adjust to new channel s t a t i s t i c s , while the 2N slewing constant of Godard in conjunction with the exponential one c o n t r o l s how f a s t the taps can be r e c u r s i v e l y slewed, given white data and a good estimate of P, to new v a l u e s .  This seems to be a new hypothesis, at  least in the e q u a l i z e r communications l i t e r a t u r e .  Figure 4.11 shows the optimum FT DFE performance in reference-directed mode on the three path channel as a function of SNR f o r various fading bandwidths.  - 58 -  I3  5.0Hz  ^k^L5Hz  \  \\  \  \ \\\ \  \\  \  \ \ \^ \  K v A UOM& DFE--V  t?1  0.05 H* \ THEORETICAL^  10  \0.I5 Hz  i  \  10  Figure 4.11  V  \o.05Hz^  10  30  ( 1 , 0 , 1 , 0 , 1 ) FT DFE Performance as a Function of SNR  The performance of the f r a c t i o n a l l y - t a p p e d , decision-feedback equalizer i s s l i g h t l y d i f f e r e n t from the symbol-spaced equalizer r e s u l t s of Figure 4.5.  With high s i g n a l - t o - n o i s e r a t i o s the FT e r r o r rate f o r 1.5 and 5 Hz  fading i s about 3 times that of the symbol-spaced e q u a l i z e r .  Tests showed  t h i s could be a t t r i b u t e d to the FT DFE having about 50% more taps and thus slower adaptation a b i l i t y .  In c o n t r a s t , at low SNR's and fade rates the FT  - 59 e r r o r rates are lower than the T/l DFE.  This i s l i k e l y due to the FT  equalizer being able to pick up some of the information smeared between symbols centers by the receiver low-pass f i l t e r .  Another way to consider  t h i s i s that the adaptive m a t c h e d - f i l t e r c a p a b i l i t y of the FT equalizer gives improved performance when i t i s not adaptation-rate l i m i t e d by i t s  larger  number of t a p s .  Also shown in Figure 4.11 i s the mean and (logarithmic)  standard  deviation of nine 600,000 symbol simulations of a longer FT DFE on a channel with B =0.05 Hz and 18 dB. f  The FT DFE had a feedforward TDL length  which spanned almost twice the multipath spread.  The nine runs, each with a  d i f f e r e n t i n i t i a l random number seed, were done f o r two reasons.  Firstly,  it  gave us an idea of the s t a t i s t i c a l deviation at very low e r r o r rates and with very slow f a d i n g .  The deviation should be less f o r e r r o r rates greater than  10*^ and fading bandwidths f a s t e r than 0.15 Hz.  Because the square-root  Kalman algorithm use herein was order t r i n computational burden, longer runs of t h i s long FT DFE would have taken over 10 hours of VAX 11/750 computer time.  Secondly, we got a accurate estimate of the performance with  a longer, but somewhat i m p r a c t i c a l length FT DFE at very slow fade r a t e s .  We  see that performance w i t h i n 2-3 dB of t h e o r e t i c a l i s p o s s i b l e .  We now move on to compare the performance of T/l and FT equalizers as a function of r e c e i v e r sample t i m i n g .  This has been investigated by a number  of authors f o r l i n e a r e q u a l i z e r s , but seemingly only by G i t l i n and Weinstein [25] f o r DFE's.  We would l i k e to confirm t h e i r r e s u l t s on the HF channel.  The f o l l o w i n g r e s u l t s are f o r the 3 path channel of Figure 4.1 (MPGAIN(0:4)=1,0,1,0,1) with SNR=18 dB and B =0.5 Hz. f  Figure 4.12  compares the performance of the equalizers as a function of received sample  - 60 timing phase.  The abscissa i s l a b e l l e d TIMING, a program input which  s p e c i f i e s the timing phase in units of T/8.  With TIMING=0, sampling occurs  at the beginning of a symbol period f o r a path with propagation delay MPDEL(I)=0 (or 8 , 1 6 , 2 4 , 3 2 ) .  TIMING=3 or 4 i s symbol center and approximately  optimum phase, while TIMING=7 i s j u s t g e t t i n g the t a i l of the low-passf i l t e r e d symbol pulse-shape.  The f i g u r e shows a wide v a r i a t i o n of b i t error  rate with sampling phase f o r the symbol-spaced (T/l)  equalizer.  The T/2-  spaced FT DFE, which only needed to be tested over a h a l f - s y m b o l - p e r i o d v a r i a t i o n , was e s s e n t i a l l y immune to sampling phase.  Note however, that the  T/l r e s u l t s at best sampling phase are l i t t l e worse than the T/2 r e s u l t s f o r t h i s special symbol-spaced multipath channel.  As noted i n Sections 2.1 and  3 . 3 , acquiring and t r a c k i n g optimum sample timing phase i s very d i f f i c u l t on the HF channel, and symbol-spaced channels are an unusual case anyway. s i m i l a r r e s u l t s were obtained on a symbol-spaced 2 path channel.  FT  /0  0  /  2  3  ^  TIMING Figure 4.12  5  6  7  (ftKsec.)  T/l and T/2 DFE S e n s i t i v i t y to Sampling Phase  Very  - 61 We now address multipath channels which, as discussed i n Section 2, may not have a delay structure which i s symbol-period spaced.  The concept of  symbol-spaced multipath structure i s best understood by comparing the channel responses of Figures 4.1 and 4 . 1 3 .  The few authors who address HF  e q u a l i z a t i o n have not i n v e s t i g a t i e d performance under these quite t y p i c a l channel c o n d i t i o n s .  We investigated the non-symbol-spaced multipath channels  shown i n Figure 4.13 (MPDEL(0:4)=0,8,J_2,24,32 and MPGAIN(0:4)= 1 , 0 , 1 , 0 , 1 ) , and Figure 4.14 (MPDEL(0:4)=0,8,16,24,28 and MPGAIN(0:4)= 1 , 0 , 0 , 0 , 1 ) . r e s u l t s are shown i n Figures 4.15 and 4.16.  1  RMS Impulse Response  —r0  T  2T  3T  4T  D i f f e r e n t i a l Propagation Delay Figure 4.13  Non-Symbol-Spaced 3 Path Channel (MPDEL=0 ;1_2;32)  RMS Impulse Response  —r~ 0  T  2T  •  3T  -r~ 4T  D i f f e r e n t i a l Propagation Delay Figure 4.14  Non-Symbol-Spaced 2 Path Channel (MPDEL=0;28)  The  -  x-3  -V 10  /  2  3  TIMING  ^  5  6  7  (ftKsEC.)  Figure 4.15 Performance on Non-Symbol-Spaced 3 Path Channel  —  *  —  t  —  •  FT  -V 10  TIMINGFigure 4.16  (JthsEC.)  Performance on Non-Symbol-Spaced 2 Path Channel  - 63 The T/l DFE again has a d e f i n i t e sampling-phase s e n s i t i v i t y .  We found  that with unusual multipath structures i t was hard even to guess what the optimum phase would be f o r symbol-spaced e q u a l i z e r .  Also note the FT DFE  has roughly 1/4 the e r r o r rate of the T/l e q u a l i z e r at optimum sampling phase.  This contrasts the symbol-spaced multipath r e s u l t s of Figure 4.12  where the T/l DFE only performed poorer at non-optimum sampling phase.  The  T/l DFE s u f f e r s with the non-symbol-spaced multipath because no matter which timing phase i s used, i t w i l l never be correct f o r a l l signal p a t h s l  The FT  e q u a l i z e r does not have t h i s problem.  In summary, FT e q u a l i z a t i o n r e s u l t s in a s l i g h t l y slower adaptation rate and requires more computation per symbol.  But because Nyquist rate sampling  allows an adaptive m a t c h e d - f i l t e r e f f e c t and an a b i l i t y to resolve nonsymbol-spaced m u l t i p a t h , i t r e s u l t s in better performance on many t y p i c a l channels and eliminates the necessity f o r r e c e i v e r sampling synchronization.  4.3 Performance as a Function of Other Parameters  In t h i s section we present b i t e r r o r rate performance as a function of the expected measurement noise covariance, of e q u a l i z e r length, and frequency mistuning.  It has been found by several authors [28,31] that the value of the expected measurement noise variance, R, i n the exponentially-aged RLS algorithm i s not very important.  For an SNR of 20 dB i t should be about . 0 1 ,  but i t s only e f f e c t i s to s l i g h t l y change the i n i t i a l performance during convergence.  Ironically,  Hsu says his R (  in his notation)  is  ineffective  over several orders of magnitude but then suggests his algorithm i s better  - 64 because of the f l e x i b l i t y of having an extra v a r i a b l e to adjust [ 3 1 ] .  Other  authors who use exponentially-aged RLS equalizers derived d i r e c t l y from the Weiner Hopf equations [eg. 20], also have a second v a r i a b l e , the inverse of the i n i t i a l value of the diagonal elements of P, which a f f e c t s convergence similarily.  As mentioned i n Section 3 . 2 , the Kalman gain can also be w r i t t e n  Gjk)=P(k)X_ (k)/R(k). equivalent.  For k=l the two techniques are obviously  In a moment we see f u r t h e r evidence why they are l i k e l y  equivalent f o r a l l k.  Once converged, we found R to be i n e f f e c t i v e over 15 orders of magnitude.  If R was extremely large ( r e l a t i v e to 1) the AGC nature of the  e q u a l i z e r was very slow during convergence. QPSK demodulation. This was c u r i o u s .  This of course does not a f f e c t  Once converged, t h i s sluggish AGC behavior disappeared. Examination of the denominator of Equation 3.8 (for the  Kalman Gain) showed that no matter what value was used f o r R, the r e l a t i v e sizes of the two terms were eventually determined only by the exponentialaging f a c t o r .  This leads us to a new i n t e r p r e t a t i o n of the e x p o n e n t i a l l y -  aged Kalman algorithm.  If the algorithm uses exponential-aging as i t s only  adaption mechanism ( i e . Q=0), then only one parameter i s needed a f t e r covergence to s p e c i f y , in a general way, the weight ( i e . the Kalman Gain) given to each new measurement. factor,  This parameter, the optimum exponential-aging  X=l/(l+q), determines how f a s t the algorithm's adaptation to channel  v a r i a t i o n (and to noise) w i l l be.  We can associate the exponential-weighting  f a c t o r with the r e l a t i v e sizes of the true system's rate of v a r i a t i o n , Q, and measurement n o i s e , R. the r a t i o Q/R.  The s i z e of the Kalman gain w i l l i n general depend on  I n t e r e s t i n g l y , exponential-only RLS algorithms derived  d i r e c t l y from the Weiner-Hopf equation make no mention of Q or R.  - 65 As discussed i n Section 4 . 1 , we were not able to determine the dependence of the optimum exponential-aging on channel fading bandwidth and s i g n a l - t o noise r a t i o .  Nonetheless, the o r i g i n of t h i s i n s e n s i t i v i t y to R, and the  equivalence of the i n i t i a l value of R to other authors variables has not been previously p o s t u l a t e d .  In l i g h t of the above d i s c u s s i o n , i t i s reasonable to surmise the " s i z e " of the P matrix may not p e r f e c t l y r e f l e c t the covariance of the state in an exponential-aged Kalman algorithm.  This i s discussed in [6elb74] where  exponential-weighting i s addressed in the context of keeping P large so as to avoid system mismodelling-induced divergence.  Our next t o p i c i s e q u a l i z e r performance as a function of the number of feed-forward t a p s .  We know the decision-feedback tapped delay l i n e should be  long enough to span the multipath spread, but how much of a compromise can be made in the supposedly, i n f i n i t e l y - l o n g feedforward TDL?  L i t t l e information  has been published on t h i s t o p i c .  In l i g h t of previous r e s u l t s we suspect a short forward TDL would reduce adaptation t i m e .  Conversely, a longer one would allow better c a n c e l l a t i o n of  ISI from future symbols.  This c a n c e l l a t i o n i s not extremely important to a  DFE though, unless there i s considerable noise in the s i g n a l .  Experimental  computer simulations were done on the two d i f f e r e n t 3 path channels i l l u s t r a t e d i n Figures 4.1 (MP6AIN(0:4)= 1 , 0 , 1 , 0 , 1 ) and 4.10 (MP6AIN(0:4)= .33,0.1.,33). were t e s t e d .  An SNR of 18 dB was selected and fade rates of 0.5 and 5.0 Hz  - 66 On the 5.0 Hz fading channel the fade rate was so l i m i t i n g that the received sample feedforward TDL need j u s t span the multipath spread.  Any  longer or shorter length was less than optimum.  For the 0.5 Hz fading channel, the optimum feed-forward equalizer length spans about 1.5 times the multipath delay spread. channel (MPGAIN(0:4)=1,0,1,0,1)  For the " f u l l y spread"  or when using a symbol-spaced e q u a l i z e r , a  s l i g h t l y longer span was b e t t e r .  On the l e s s spread channel (MPGAIN(0:4)=  . 3 3 , 0 , 1 , 0 , . 3 3 ) or with an FT e q u a l i z e r , a s l i g h t l y shorter span was more desirable.  These trends seem quite reasonable as on a f u l l y spread channel  the information from a given symbol i s smeared more strongly (though not farther).  An FT e q u a l i z e r i s generally slower adapting because i t  requires  more taps f o r a given multipath spread; a shorter one seems to track f a s t e r and give s i m i l a r or better performance even though i t i s not as t h e o r e t i c a l l y optimal.  Interestingly,  on very slow fading channels a FT DFE with a  feed-forward TDL length of twice the multipath spread comes withing 2 or 3 dB of t h e o r e t i c a l fading channel performance (see Figure 4 . 1 1 ) , but i s l i k e l y i m p r a c t i c a l from the point of view of computational burden and use on moderately-rapid fading channels.  A DFE feedforward span of about the  multipath spread i s thus probably a p r a c t i c a l compromise f o r the HF channel. We note that the conditions under which t h i s compromise would cause the most penalty i s very slow f a d i n g , noisy radio l i n k s .  The HF channel worst-case multipath spread i s about 6 ms.  Data on the  p r o b a b i l i t y d i s t r i b u t i o n of spread i s not a v a i l a b l e though, so the best choice of e q u a l i z e r feedforward span w i l l have to be guess; 3-4 milliseconds seems reasonable. Interestingly,  More data i n t h i s regard would be of u s e f u l .  p a r a l l e l modems degrade quite r a p i d l y when the spead exceeds  2-3 milliseconds [ 6 3 ] .  - 67 To end t h i s sub-section we discuss frequency mistuning.  This can be  caused by e i t h e r the doppler s h i f t of mobile units or d i f f e r e n t l o c a l o s c i l l a t o r frequencies in the receiver and t r a n s m i t t e r .  Frequency mistuning  i s also discussed in Section 6 regarding our real channels tapes.  The e f f e c t of mistuning i s to require the e q u a l i z e r taps to r o t a t e the complex plane) at a frequency equivalent to the mistuning.  (in  The real and  imaginary parts of the taps must therefore o s c i l l a t e at t h i s frequency.  The  e f f e c t on the components of the various taps i s thus quite s i m i l a r to random 2D f a d i n g ; a degradation s i m i l a r to increasing fading bandwidth i s expected.  The mistuning can be simulated by i n c r e a s i n g l y t w i s t i n g each successive complex baseband value of the received samples fed to the e q u a l i z e r .  For  1200 symbols/second, the increment i s 360°/1200= 0.3° per Hz of m i s tuning.  A three-path channel structure as i n Figure 4.1 was used in the  experiments with an SNR=23 dB and B =o.5 Hz. f  Several runs of 60,000  symbols with varying amounts of mis-tuning were done.  Figure 4.17 i s a repeat of Figure 4.5 but with the r e s u l t s of 0 . 5 , 1.5, and 5.0 Hz mis-tuning added to the 0.5 Hz f a d i n g .  The r e s u l t i s an e r r o r  rate roughly half that a s i m i l a r amount of fading bandwidth would cause. With 5 Hz mistuning and 0.5 Hz fading bandwidth, one would surmise a l l of the t r a c k i n g required of the equalizer tap weights i s in the range 4 . 5 - 5 . 5 Hz, while 5 Hz fading with no mistuning d i s t r i b u t e s a s i m i l a r amount of fading energy over - 5 to 5 Hz.  Most of t h i s fading energy i s below 4.5 Hz in  absolute value, leading one to suspect mis-tuning would be harder to follow than f a d i n g .  Why do our r e s u l t s c o n t r a d i c t t h i s ?  Maybe our hypothesis in  Section 4.1 regarding the possible need f o r the e q u a l i z e r taps to adapt at 15  - 68 Hz on a 3 path channel with 5 Hz fading was c o r r e c t .  If a l l the taps are  r o t a t i n g i n unison, no higher-order e f f e c t s may be p o s s i b l e .  Several other i n t e r e s t i n g points were noticed during the mis-tuning experiments.  As shown i n Table 4 . 1 , the optimum exponential-aging  changes with applied m i s - t u n i n g .  factor  This seems reasonable as the c a r r i e r phase  i s changing f a s t e r r e s u l t i n g i n the e q u a l i z e r taps needing to adjust f a s t e r .  10  ^  6-5.0Hz f  X  10  «0.5Hz 4-  M15TUNIN&»5.DHz  ~ B =0.5 Hz + MISTWNINS-* 1.5 Hz F  V>CB =0.5Hz + MISTUNING-=0.5'Hz F  -V B = 0.5Hz. F  10  10  10  30  Et/No Figure 4.17  Simulated Mistuning Results  - 69 Table 4.1 Mistuning (Hz) 0.0 0.5 1.5 5.0  Optimum  .92 .89 .86 .78  Equalizer Performance with Mistuning RMS Symbol Error  RMS Phase Error  Mean Phase Error  Bit Error Rate  .207 .216 .256 .425  8.8° 9.4 11.6 20.9  .1° 1.8 4.8 10.9  .00027 .00037 .00097 .017  The second i n t e r e s t i n g point i s that a bias was noted in the mean equalized symbol-phase e r r o r .  This b i a s , which manifests i t s e l f from a trend  i n the symbol-phase e r r o r , i s what drives the r o t a t i o n of the equalizer taps.  It i s never e n t i r e l y removed as the e q u a l i z e r functions as a  f i r s t - o r d e r phased-locked loop and thus cannot track a frequency o f f s e t with zero e r r o r .  This can also be seen i n Table 4 . 1 .  It was postulated that t h i s phase bias could be the lag from an exponentially-weighted control device t r y i n g to track a phase ramp. Unfortunately, t h i s elementary control systems theory could not be made to f i t the data very c l o s e l y , possible because the e q u a l i z e r has two time constants as discussed in Section 4 . 2 .  Nonetheless, i t seems that a  second-order, phase-locked-loop automatic frequency control might be made t o track t h i s bias as the e f f e c t i s monotonic and the mistuning may not change very r a p i d l y .  Further research in t h i s area i s already underway, though as  discussed in Section 3 . 3 , i n i t i a l c a r r i e r frequency a c q u i s i t i o n may be a problem because the technique depends on the equalizer already being converged.  - 70 5.  CHANNEL ESTIMATION ON SIMULATED AND REAL CHANNELS  5.1 A p p l i c a t i o n of RLS Algorithms to Channel Estimation  In Section 6 the equalizers are tested on three actual HF channel transmissions recorded on tape. i n Appendix C.  The d e t a i l s of the recordings are contained  In order to evaluate e q u a l i z e r performance on the real  channel tapes i t i s necessary to estimate the s e v e r i t y of the multipath d i s t o r t i o n present during the r e c o r d i n g .  Though i t i s d i f f i c u l t to estimate,  or even define the amount of additive noise on fading multipath channels,  it  i s possible to estimate t h e i r r e l a t i v e l y short term impulse response and thence fading s p e c t r a .  From t h i s information i t i s possible to determine  d i v e r s i t y , fading bandwidth, and mistuning.  As shown i n the previous  s e c t i o n , these parameters a f f e c t the error rate performance j u s t as s e r i o u s l y as s i g n a l - t o - n o i s e r a t i o .  This sub-section introduces the a p p l i c a t i o n of  recursive-least-square-error  algorithms to channel e s t i m a t i o n , while Section  5.2 t e s t s a f r a c t i o n a l l y - t a p p e d channel estimator on a simulated channel. Section 5.3 presents the FT estimation of the 3 real channel tapes while Section 5.4 presents the method and r e s u l t s of the spectral analysis of the fading.  Channel estimation i s the inverse of the e q u a l i z a t i o n problem. e q u a l i z e r , a channel estimator can use a simple tapped-delay-1ine  Like an structure.  As shown in Figure 5.1 though, the received samples are used as the reference response while the known transmitted symbols are fed down the l i n e .  The received samples are not from a d i s c r e t e c o n s t e l l a t i o n but represent an analog complex baseband waveform.  The channel estimator has no decision  device; i t always runs in reference-directed mode.  Note the resemblance of  - 71 the estimator to the channel model of Figure 2 . 5 ; Kalman algorithms always incorporate a model of the system within them.  TRANSMITTED  *i  SYMBOLS  RECEIVED SYMBOLS ' k MLMtt  TAP-WEIGHT  ADJUSTMENT AL&ORITHM  EXPECTED Noxse. LEVEL LzKPECTED FAt>E KATE.  * -*  Figure 5.1 Channel Estimator  The channel estimator problem i s also l i n e a r .  One i s t r y i n g to estimate  the Rayleigh-fading path gains generated by a l i n e a r process, namely, complex gaussian noise f i l t e r e d by a second-order low-pass f i l t e r .  This contrasts  the e q u a l i z e r case where the tap weights being estimated are generated by a n o n - l i n e a r process; i n f a c t , they are the inverse of the f i l t e r formed by the channel path gains.  When the channel model parameters are known from e i t h e r the channel simulator or a n a l y s i s of past i n p u t s , i t i s p o s s i b l e to use a more i n t e l l i g e n t estimator adaptation method than simple e x p o n e n t i a l - a g i n g .  What  i s needed i s the system noise covariance matrix Q, the measurement noise covariance R, and the state t r a n s i t i o n matrix A.  With these q u a n t i t i e s , we  can use the f u l l Kalman update equations ( 3 . 3 - 3 . 9 ) .  - 72 Because a one-pole model f o r the path fading process was very simple t o c o n s t r u c t , some experimentation was done with Q, R, and A in a one-pole (internal  system model) estimator using the f u l l Kalman update.  a two-pole simulated channel was used.  Necessarily,  The one-pole estimator was given the  p o s i t i o n and RMS-size information regarding the 3 paths i n the channel.  The best r e s u l t s were obtained with some one-pole modelling and some exponential-aging.  This seems reasonable as a one-pole estimator should not  perform optimally on a two-pole channel.  On the other hand, e x p o n e n t i a l -  aging i s a u n i v e r s a l l y applicable but not very i n t e l l i g e n t adaptation mechanism.  It does not use any a p r i o r i s t a t i s t i c a l knowledge regarding the  s i z e and p o s i t i o n of the paths nor how f a s t they may change.  A combination  of both schemes allows the estimator use of some s t a t i s t i c a l knowledge of the path behavior and takes care of any "apparent divergence" [24] caused by the mis-modelling.  These r e s u l t s in dic at e that i f a better A and/or Q could be  suggested f o r the d i f f i c u l t e q u a l i z e r problem, i t s performance might be improved too.  Quicker exponential-aging causes l e s s e r r o r i n the estimation of the s i g n i f i c a n t paths i n the channel.  However, quicker aging also r e s u l t s i n  more e r r o r i n the estimation of the less s i g n i f i c a n t paths.  The experiments  also raised some question as to how best to measure o v e r a l l performance. Should one use the RMS sample estimation e r r o r or the RMS tap errors summed over the taps?  The minima of these two measures of performance as a function  of X do not even c o i n c i d e i f using mixed one-pole modelling and exponentialaging.  For an unknown channel the tap e r r o r cannot be c a l c u l a t e d , so RMS  sample estimation e r r o r i s probably the best general-purpose performance measure and w i l l be used i n Section 5 . 3 .  - 73 Several other i n t e r e s t i n g things were noted when using mixed one-pole modelling and exponential-aging.  The accuracy of the tap estimates were only  weakly affected by the SNR, i n d i c a t i n g the estimator was doing a good job of t r a c k i n g the channel in s p i t e of the noise.  Secondly, the input value of the  expected measurement n o i s e , R, was only e f f e c t i v e i f Q was not the zero matrix ( i e . not exponential-aging o n l y ) .  This v e r i f i e s the hypothesis of  Section 4.3 that i t i s exponential-aging as the sole method of adaptation that causes the extreme i n s e n s i t i v i t y to R. were noted when using MPGAIN(0:4)=1,0,1,0,1.  F i n a l l y , computer underflows These disappeared when  MPGAIN(0:4)=1,.001,1,.001,1 was used f o r the simulated channel RMS path amplitudes.  It i s suspected that with time, estimates of path amplitudes  f i x e d at zero can be estimated so accurately as to underflow the computer's op real number range (10 ).  Assuming the estimator cannot determine Q, R, and A in some s e l f - t u n i n g way [ 4 ] , exponential-aging i s the only adaptation mechanism f o r an unknown channel.  Performance under t h i s mechanism was investigated next.  It was  found that minimum RMS sample-estimation e r r o r and minimum RMS t a p - e s t i m a t i o n error better c o i n c i d e when using exponential-aging only than when using a mixed adaptation mechanism.  Optimum X i s only weakly dependent on SNR but  quite dependent on fading bandwidth. optimum A data of Table 5.1 was noted.  For a 3 p a t h , 23 dB channel the The optimum exponential-aging i s  quite s h o r t , e s p e c i a l l y r e l a t i v e to the amount required when a mixed adaptation mechanism i s used.  - 74 Table 5.1  Channel Estimator Optimum X as a Function of B f o r SNR=23 dB f  Fading Bandwidth  Optimum X  .05 Hz  .96  .15 Hz  .92  .50 Hz  .84  1.50 Hz  .72  5.00 Hz  .60  In summary, we have shown how a channel estimator can be constructed and how the RLS algorithm i s used within i t .  We b r i e f l y tested the estimator on  a simulated channel with and without the estimator having some knowledge of the channel.  Knowledge of the channel c e r t a i n l y helps the e s t i m a t i o n , but  without f u r t h e r research into s e l f - t u n i n g mechanisms, exponential-aging w i l l have to be used as the sole adaptation method.  5.2 Fractionally-Tapped Channel Estimation  The simple channel estimator described in the previous sub-section has a sample timing s e n s i t i v i t y s i m i l a r to symbol-spaced e q u a l i z e r s .  Because our  transmitted signal has a complex (2 sided) baseband bandwidth of 2400 Hz, we must estimate the channel impulse response with a time r e s o l u t i o n of 1/2400 second; otherwise c l o s e l y or unusually spaced components w i l l not be resolvable.  An FT channel estimator can be constructed somewhat d i f f e r e n t l y than a FT equalizer.  While an FT e q u a l i z e r uses received data at twice the symbol rate  - 75 and produces an equalized symbol stream at the symbol r a t e , an FT estimator must estimate the FT received data stream at twice the symbol r a t e .  This  might be done by r e p l i c a t i n g every transmitted symbol fed down the TDL and c l o c k i n g the estimator twice as f a s t .  It can also be accomplished by using  two symbol-spaced estimators running o f f the same TDL.  One estimates the odd  received symbols while the other estimates the even ones. response i s the interleaved r e s u l t of the two.  The impulse  Interestingly, this i s  equivalent to running a symbol-spaced channel estimator with two d i f f e r e n t sample timings separated by a h a l f symbol p e r i o d , on the same data.  Figure 5.2 i l l u s t r a t e s the e f f e c t of changing the timing of a symbolspaced estimator by a h a l f symbol-period.  Figure 5.3 shows r e s u l t s when t h i s  change occurs while using an FT e s t i m a t o r .  Both f i g u r e s are from some short  (6000 symbol) estimation runs on Tape 11.  Figure 5.2  S e n s i t i v i t y of T/l Channel  Estimator to Sample Timing  Figure 5.3  I n s e n s i t i v i t y of FT Channel  Estimator to Sample Timing  Figure 5.4 i l l u s t r a t e s the performance of the FT estimator on a simulated 3 equi-rms-strength path channel.  Note the paths are not p a r t i c u l a r ! ' l y  narrow as the have been smeared by the low pass f i l t e r .  RMS Impulse Response  o  Figure 5.4  x  if  6 8 10 FT Tap Number  IZ  /y  FT Estimation of Simulated 3 Path Channel  - 77 5.3 Estimation of Three Real Channels  Using the demodulation software from the e q u a l i z e r , the channel estimator was fed with FT-spaced samples of 310,000 symbols (approx. 4.3 minutes) s t a r t i n g from the beginning of the M-sequence on each of Tapes 6, 9, and 11. Because the channels were unknown, only exponential-aging was used as an adaption mechanism ( i t would be i n t e r e s t i n g to rerun the estimator someday giving i t more a p r i o r i  information from our i n i t i a l e s t i m a t i o n ) .  Figures  5 . 5 , 5 . 6 , and 5.7 show the RMS magnitude of the equivalent complex baseband impulse responses of Tapes 6, 9, and 11, r e s p e c t f u l l y .  The p l o t s are against  FT tap number; each approximately 1/2400 of a second apart.  RMS Impulse Response  o  3.  y  6  8  to  jz  FT Tap Number  Figure 5.5  Tape 6 RMS Impulse Response  TAPE <? RMS Impulse Response  -2.5T•5.0T o  i  V  6  -I %  /0  FT Tap Number Figure 5.6 Tape 9 RMS Impulse Response  X  f 6 8 FT Tap Number  10  12  Figure 5.7 Tape 11 RMS Impulse Response  - 79 A s i m i l a r estimation of Tape 11 was reported by Falconer et al [22,23] f o r a run of approximately 30,000 symbols.  Which 30,000 symbols of the over  300,000 on the tape was not s t a t e d , but our path p o s i t i o n s and RMS path amplitudes are approximately the same s i z e . path was down 6dB from the f i r s t .  For instance Falconer's second  Our r e s u l t s f o r the f i r s t 60,000 symbols  gave - 4 . 7 d B , while Figure 5.7 shows 310,000 gave - 8 . 3 d B .  These r e s u l t s at  least bracket his average amplitudes.  L u c k i l y , each of the three channels has a d i f f e r e n t number of paths providing us with a v a r i e t y of d i v e r s i t y .  Also note that Tape 9's impulse  response has a non-symbol-spaced multipath s t r u c t u r e .  Unfortunately,  l i t t l e record of the ambient s i g n a l - t o - n o i s e r a t i o was  kept during the the tape recordings.  As RMS symbol estimation error i s not a  good i n d i c a t i o n of the noise l e v e l on dynamic channels, we t r i e d a novel analysis on a simulated version of Tape 11's channel.  Using the r e l a t i v e RMS  amplitudes and p o s i t i o n s from Figure 5.7 and the de-tuning and fading bandwidth determined in the next s u b - s e c t i o n , we t r i e d simultations of Tape l l ' s c h a r a c t e r i s t i c s with d i f f e r e n t noise l e v e l s u n t i l we found one that resulted in the same error rate as the actual channel.  The simulated  conditions used were MP6AIN(0:4)=1,0,0.58,0,0.18 and MPDEL(0:4)=0,8,16,24,32 (from Figure 5 . 7 ) , Figure 5 . 1 0 ) .  and detuning=0.7 Hz and fading bandwidth=0.2 Hz (from  Under these c o n d i t i o n s , a simulated SNR of about 14 dB gave  the same e r r o r rate as Tape 11.  - 80 5.4 Estimation of Fading Spectra  The fading spectra of the path gains was determined using a complex f a s t Fourier transform.  The estimated values of the 14 f r a c t i o n a l l y - s p a c e d tap  gains ( i . e . the estimated, instantaneous complex-baseband impulse response) were output by the channel estimator to a f i l e once every 60 symbols (20 times per simulated second).  The r e s u l t i n g data f i l e was composed of 14 time  h i s t o r i e s , each ten records of 512 tap gain estimates.  For each h i s t o r y ,  ten  512-sample long FFT's were performed and the ten r e s u l t i n g spectra averaged. This gave a frequency r e s o l u t i o n of 0.04 Hz over ( - 1 0 , 10Hz).  A three-point  running-average f i l t e r was used to smooth the remaining Gibbs phenomena o s c i l l a t i o n s from the spectra.  Referring to the tap numbers of Figure 5 . 5 ,  the Tape 6 spectra f o r taps 0, 3, 7, 13 are presented in Figures 5.8 and 5.9.  A l l spectra are p l o t t e d with a u t o - s c a l i n g so the reader should be  c a r e f u l to note the scale of the ordinate in each p l o t .  For Tape 9, spectra  of taps 0 , 3, 8, and 13 are presented i n Figures 5.10 and 5 . 1 1 .  Finally,  Tape 11 spectra of taps 0, 3, 7, 11 are shown in Figures 5.12 and 5 . 1 3 .  Several features are n o t a b l e .  The l i n e spectra of the largest taps have  a fading half-bandwidth of roughly 0.2 Hz centered at 0 . 7 , 1.6, and 0.75 Hz f o r Tapes 6, 9, and 11, r e s p e c t i v e l y . unknown.  The o r i g i n of the displacement i s  The t r a n s m i t t i n g and receiving stations were f i x e d and used cesium  clocks to lock the d a t a , s u b - c a r r i e r , and c a r r i e r frequencies.  There should  have thus been no doppler s h i f t due to s t a t i o n motion, or mistuning. postulated that the center a l t i t u d e of the r e f l e c t i n g ionospheric might be moving several 10's of meters per second.  It was  layers  On each tape though, the  frequency displacements of the d i f f e r e n t paths was the same.  It seems  u n l i k e l y that several independent layers would be moving at the same r a t e .  S P E C T R A L  TAPE 6 TAP 0  D E N S I T Y  .5  10  FREQUENCY  15  2.0  1.5  3.0  (Hz)  31.7  5  P E C T R A L  TAPE b TAP 3  0  E N S I T Y  .5  1.0  FREQUENCY  Figure 5.8  (Hz)  Estimated Tape 6 Fading Spectra (Taps 0 and 3)  - 82 -  Figure 5.9  Estimated Tape 6 Fading Spectra (Taps 7 and 13)  .77  FREQUENCY  (Hz)  35.0  FREQUENCY  Figure 5.10  (Hz)  Estimated Tape 9 Fading Spectra (Taps 0 and 3)  11.5  ure 5.11  Estimated Tape 9 Fading Spectra (Taps 8 and 13)  85 -  -.*5  O  ^5  FREQUENCY  Figure 5.12  1  (Hz)  7i  2.0  Estimated Tape 11 Fading Spectra (Taps 0 and 3)  7."?  i  1  -.5  O  1  .5  •  •  1.0  FREQUENCY  •  1.5  2J0  (Hz)  2.7  _r•5  + O  , .5  , 1.0  FREQUENCY  , 1.5  .  , 2J0  (Hz)  ure 5 . 1 3 , Estimated Tape 11 Fading Spectra (Taps 7 and 11)  - 87 The f a c t that the fading energy in the main paths was i n narrow l i n e s (0.2 Hz half-bandwidth) suggests a tuning correction would be d e s i r a b l e to reduce the " e f f e c t i v e " fading bandwidth the e q u a l i z e r must t r a c k . with t h i s c o r r e c t i o n are discussed in the next s e c t i o n .  Results  It i s also  i n t e r e s t i n g that some of the spectra of the l e s s e r - s t r e n g t h paths are plateau-like.  Are the weaker paths, which on a l l three tapes are l a t e r  (longer), from higher more-rapidly fading (turbulent) layers of the ionosphere?  Could they be the r e s u l t of several i n t e r n a l r e f l e c t i o n s within  the ionosphere, each in turn broadening the spectrum?  We are u n c e r t a i n .  S i m i l a r i l y , i t i s unclear why the spectra of n o n - s i g n i f i c a n t taps ( i e . 0 or 13) have t h i s same p l a t e a u - l i k e c h a r a c t e r i s t i c . be white (uniform in spectral density) the paths.  One might suspect they would  i f they were f a r enough removed from  - 88 6.  EQUALIZER PERFORMANCE ON REAL CHANNELS  6.1 Equalizer Performance on Three Real Channels  As discussed i n Sections 1.3 and 5, and d e t a i l e d i n Appendix C, three tapes of d i g i t i z e d HF audio output were a v a i l a b l e from the MDA/CRC l a t t i c e study [ 4 2 ] .  The received s i g n a l , 1201.739 symbol/s QPSK on a 1505.107 Hz  s u b - c a r r i e r , had been d i g i t i z e d at 9600.00 Hz.  The e q u a l i z a t i o n experiments  concentrated on the f i r s t 300,000 symbols (4.2 minutes) of M-sequence data. Table 6.1 presents our reference-directed r e s u l t s along with those of the MDA/CRC study.  These are some of the few FT DFE r e s u l t s ever reported on  actual HF r a d i o channels.  Table 6.1  Performance on the Taped Real Channels  Research Sponsors  Equalization  MDA/CRC  Unequalized LE FT LE  Gl enayre/ UBC  LE DFE DFE FT DFE  ReTuning no no no no no yes yes  Tape 6 BER  Tape 9 BER  Tape 11 BER  .027 .015 .007  .139 .130 .082  .080 .016 .009  .0077 .0077 .0058 .0052  .098 .079 .068 .047  .0078 .0049 .0047 .0030  Note 1: MDA/CRC t e s t s were 310,000 symbols long, while Glenayre/UBC's were 300,000 symbols long. Note 2: MDA/CRC t e s t s did not c a l c u l a t e b i t BER, only symbol e r r o r r a t e . The MDA/CRC BER r e s u l t s shown are one h a l f t h e i r reported symbol e r r o r r a t e , and are therefore s l i g h t l y o p t i m i s t i c .  - 89 In l i g h t of our knowledge of the taped channels from Section 5, several i n t e r e s t i n g trends are notable:  i)  Tape 6, which has only 1 s i g n i f i c a n t path, shows r e l a t i v e l y  improvement with e q u a l i z a t i o n .  little  The DFE gives no improvement over the LE.  Most gains which show up in the table are l i k e l y due to correction of mistuning, and f r a c t i o n a l - t a p p i n g picking up information smeared by the bandlimiting f i l t e r .  ii)  Tape 9, a two path channel contaminated by impulse noise (possibly  Russian o v e r - t h e - h o r i z o n radar [ 2 3 , 4 2 ] ) , has a very high error r a t e .  This  masks much of the r e l a t i v e improvement possible with the various techniques. The DFE performs only s l i g h t l y better than the LE, p a r t l y because the second path i s weak and adds l i t t l e d i v e r s i t y . non-symbol-spaced multipath s t r u c t u r e .  Tape 9 i s the only one to have a It thus showed the largest absolute,  though not r e l a t i v e improvement in BER when switching to FT e q u a l i z a t i o n . Re-tuning also y i e l d s a s i g n i f i c a n t improvement on Tape 9 as i t had the greatest mistuning.  i i i ) Tape 11, being a 3 path channel with s i g n i f i c a n t I S I , r e l a t i v e improvement of the DFE over the LE.  shows the largest  It also shows the most o v e r a l l  improvement.  iv)  The MDA/CRC l i n e a r e q u a l i z e r r e s u l t s should be comparable to those of  the Glenayre/UBC research done h e r e i n . difference f o r tapes 6 and 11.  Unfortunately, there i s a f a c t o r of 2  This i s c u r i o u s .  It may be a t t r i b u t a b l e to  our c a r e f u l optimization of the exponential-weighting f a c t o r , or p o s s i b l y less n u m e r i c a l - p r e c i s i o n e r r o r i n our square-root-normalized f i x e d - o r d e r  - 90 Kalman implementation.  Re-running the MDA/CRC a n a l y s i s was not deemed  worthwhile to resolve t h i s .  v)  Note that our re-tuned FT DFE r e s u l t s on 300,000 symbols of Tape 11 give  a 0.003 b i t - e r r o r - r a t e .  This i s i d e n t i c a l to that found in [22,23] f o r  30,000 symbols in an unspecified part of Tape 11.  These r e s u l t s of course, represent only a small sample of p o s s i b l e HF channels.  They are not p a r t i c u l a r i l y well spread in the sense t h a t , on a l l  three tapes, the secondary paths were more than 7 dB down i n RMS strength from the main paths.  Since there i s a serious lack of information regarding  the p r o b a b i l i t y of r e l a t i v e RMS path strengths and fading bandwidths on HF channels, i t i s hard to judge how t y p i c a l these recorded channels a r e . F i n a l l y , l i t t l e idea of the s i g n a l - t o - n o i s e r a t i o of the taped channels was a v a i l a b l e except as discussed in Section 5.3 and Appendix C.  Future  experiments should consider t h i s problem, though true SNR data i s hard to measure, or even define f o r fading multipath channels.  relatively  91 7.  EQUALIZER PERFORMANCE IN DECISION-DIRECTED MODE  7.1 Methods of Preventing and Recovering from Crashes  Data communication always requires quantization of received s i g n a l s to the nearest symbol in the modulation c o n s t e l l a t i o n .  We have seen that  d i g i t a l equalizers often use these decisions in t h e i r adaptation or estimation mechanism.  If too many decision errors are made, e r r o r  propagation can occur and p o s s i b l y lead to an e q u a l i z e r c r a s h .  We have  avoided t h i s problem u n t i l now by operating in reference-directed mode. Unfortunately, t h i s mode i s not r e a l l y "communication" but i s of course simpler and more t h e o r e t i c a l l y t r a c t a b l e [16].  Reference-directed mode i s  used by most authors f o r t h e i r analysis because on most channels at reasonable e r r o r rates i t i s usually only s l i g h t l y o p t i m i s t i c .  We s h a l l see that Kalman e q u a l i z a t i o n of the HF channel does not allow t h i s assumption.  Being a randomly fading channel, o c c a s i o n a l l y a l l  diversity  paths w i l l fade simultaneously v i r t u a l l y guaranteeing an e q u a l i z e r c r a s h .  In  c o n t r a s t , the response of even an unconditioned telephone channel i s normally w i t h i n 5 dB of nominal across the audio band. moderately r a p i d l y in time.  In a d d i t i o n , HF channels vary  As we saw in Section 4 t h i s rapid fading  challenges the adaptation c a p a b i l i t i e s of even RLS algorithms.  F i n a l l y , the  l o c a l i z e d nature of the HF impulse response may contribute to the increased e r r o r rate with a DFE.  In the next sub-section we therefore propose that a  low reference-directed p r o b a b i l i t y - o f - e r r o r  should not be the sole c r i t e r i o n  f o r judging the v a l i d i t y of assumptions about d e c i s i o n - d i r e c t e d experiments.  92 We f i r s t operated in d e c i s i o n - d i r e c t e d mode by using an automatic crash detection and recovery scheme.  The e q u a l i z e r monitors an exponentially  weighted measure of i t s RMS output symbol estimation e r r o r . (decision-directed)  When t h i s  output SNR exceeds a c e r t a i n t r i g g e r l e v e l , i n d i c a t i n g a  c r a s h , the e q u a l i z e r switches to r e f e r e n c e - d i r e c t e d mode.  When the output  SNR decreases back to (say) 60% of the t r i g g e r l e v e l , the equalizer switches back into d e c i s i o n - d i r e c t e d mode.  This i s cheating in a sense, as we assume  a reference symbol stream i s always a v a i l a b l e .  On the other hand, i t gave us  a f i r s t look at d e c i s i o n - d i r e c t i o n and some rough estimates of the s u p r i s i n g l y high rate of crashes.  It i s a l s o very s i m i l a r in nature to Hsu's  ARQ request-training-sequence method [30] f o r f u l l duplex channels.  Since f u l l duplex l i n k s are not common in HF communication, we next consider i n s e r t i n g p e r i o d i c r e t r a i n i n g sequences i n the data stream.  This i s  a commonly suggested and very robust method of crash prevention and recovery.  The e q u a l i z e r knows that (say) the f i r s t 20% of every data frame  i s a r e t r a i n i n g sequence.  This requires considerable overhead but allows the  e q u a l i z e r to operate f o r a part of each frame in r e f e r e n c e - d i r e c t e d mode.  If  a crash has occurred, or i s close to o c c u r r i n g , periodic r e t r a i n i n g allows complete recovery f o r the next data frame.  During r e t r a i n i n g i t may be advantageous to reset c e r t a i n i n t e r n a l variables. regardless.  This i s not absolutely necessary as the e q u a l i z e r w i l l  reconverge  If the e q u a l i z e r has crashed though, i t may reconverge f a s t e r ,  more completely, or with a shorter r e t r a i n i n g sequence i f the tap weight vector C_(k) i s reset to zero. auto-crash detection mechanism.  This i s an ideal a p p l i c a t i o n f o r the  The covariance matrix P can also be r e s e t , u s u a l l y to the matrix.  identity  This slows reconvergence and i s not t h e o r e t i c a l l y necessary as P i s  unaffected by wrong d e c i s i o n s .  As discussed i n Section 3.2 though, r e s e t t i n g  P cleans out any round-off errors that might be b u i l d i n g up.  Hsu's square-  root Kalman algorithm [31] used herein was always completely stable f o r 600,000 symbols using 32 b i t f l o a t i n g - p o i n t a r i t h m e t i c .  Some of the Fast  Kalman algorithms discussed i n Section 3.2 are not so s t a b l e .  Two new LMS-based algorithms f o r the l i n e a r e q u a l i z e r configuration have recently appeared in the l i t e r a t u r e . f o r d r i v i n g the adaptation. spectrum.  Both use rather novel measures of e r r o r  Morgan [46] uses the e r r o r in the expected power  The o th er , to be discussed i n Section 7 . 3 , i s f o r constant modulus  modulations; only the e r r o r in the symbol modulus i s used in the adaptation mechanism.  Neither of these algorithms feed on decisions so when used in a  l i n e a r equalizer they do not require the knowledge of the transmitted sequence to converge, nor should they ever crash!.  If a l i n e a r e q u a l i z e r could be used on an HF channel these two techniques would eliminate many problems.  We concluded i n Section 4 though, that a  l i n e a r e q u a l i z e r performs very poorly on an HF channel.  Fortunately i t  is  possible to adapt the constant modulus technique to use both the f a s t adapting RLS algorithm and the DFE c o n f i g u r a t i o n , but with the loss of the above-mentioned advantages.  Nonetheless, since the symbol-estimation modulus  e r r o r i s never wrong, a DFE using t h i s e r r o r measure might not crash as often.  We i n v e s t i g a t e t h i s p o s s i b i l i t y i n Section 7 . 3 .  94 7.2 Periodic Retraining Performance and Discussion  In t h i s s e c t i o n , we t e s t the performance of various equalizers in d e c i s i o n - d i r e c t e d mode with p e r i o d i c r e t r a i n i n g .  Our main i n t e r e s t w i l l be  to note the degradation in performance of an FT DFE operating in d e c i s i o n directed mode as compared to r e f e r e n c e - d i r e c t e d mode.  We w i l l only  t e s t a LE or experiment on one of the real channel tapes.  briefly  An explanation i s  suggested f o r the s u r p r i s i n g l y substantial degradation in d e c i s i o n - d i r e c t e d mode, the degraded performance i s compared with p a r a l l e l modems, and an algorithm we c a l l a p o s t e r i o r i Kalman i s t r i e d .  Since the FT DFE we use has 14 t a p s , we should choose a r e t r a i n i n g block length (in symbols) of about twice t h i s so as to give the e q u a l i z e r time to re-converge [28].  A Frank sequence was used f o r r e t r a i n i n g , f o r as discussed  i n Section 3 . 3 , i t i s very white and properly sounds the ionosphere across the passband.  Since the 4-phase Frank sequence i s 16 symbols long, a  r e t r a i n i n g sequence length of 32 symbols allows e x a c t l y two f u l l Frank sequences in the r e t r a i n i n g block.  A l l p e r i o d i c r e t r a i n i n g experiments were  performed with r e s e t t i n g of i n t e r n a l v a r i a b l e s ( £ and P) so as to be able to compare with other authors r e s u l t s [22,23].  The 3 path channel structure of Figure 4.1 (MPGAIN(0:4)=1,0,1,0,1) investigated under 4 channel c o n d i t i o n s : B^=0.15 and 1.5 Hz.  was  E /N =13 and 23 dB with b  Q  Considering the r e t r a i n i n g sequence to be  overhead, we tabulated the data e r r o r rate only, not i n c l u d i n g r e t r a i n i n g . We were also interested in the mean-time-to-crash.  Both these performance  f i g u r e s were measured as a function of r e t r a i n i n g p e r i o d i c i t y because p e r i o d i c r e t r a i n i n g can, in addition to recovering from crashes, also ease d e c i s i o n - d i r e c t e d degradation, thus changing the e r r o r r a t e .  Similar  95 experimental r e s u l t s , e s p e c i a l l y as a function of r e t r a i n i n g p e r i o d i c i t y , have never previously been presented.  We comment that the net throughput  in  our p e r i o d i c - r e t r a i n i n g experiments i s not 2400 bps, but somewhat less due to the r e t r a i n i n g overhead.  Table 7.1 presents our d e c i s i o n - d i r e c t e d r e s u l t s as a function of packet frame composition f o r 60,000 symbol simulation runs. aging from r e f e r e n c e - d i r e c t e d mode was used.  Optimum e x p o n e n t i a l -  A 32/64 designation indicates a  96 symbol frame i n which the f i r s t 32 symbols (1 block) i s the r e t r a i n i n g sequence, and the next 64 symbols (2 blocks) i s the M-sequence data.  The  performance on channels with fading bandwidths of 1.5 Hz was very poor. Figure 7 . 1 , which p l o t s some of the r e s u l t s in comparison to the referencedirected and t h e o r e t i c a l performance, shows the e r r o r rate to be 5 to 10 times higher than in r e f e r e n c e - d i r e c t e d mode.  We surmise that when the  equalizer adaptation rate i s taxed, even a few wrong decisions can cause i t to f a l l behind enough to c r a s h .  This i s evident in the blocks-uncrashed  column of Table 7.1 which i s computed assuming that a crashed data block i s one with more than 16 of 64 b i t s wrong.  A few b r i e f computations show that  the b i t error rate shown in Table 7.1 i s almost e n t i r e l y due to crashes near the end of frames, not to any s i g n i f i c a n t number of random errors during uncrashed portions of the frame.  The only exception i s the case of  B =0.15 Hz, 32/64 r e t r a i n i n g , and SNR=13 dB. f  Under the l a t t e r  c o n d i t i o n s , the chance of crashing i s low and the noise i s h i g h , r e s u l t i n g i n about half the errors being random rather than end of frame b u r s t s .  This  suggests t h a t any data transmitted on our proposed modem should be r e l a t i v e l y t o l e r a n t to burst e r r o r s .  Generally, the r e s u l t s on 0.15 Hz fading channels are not too bad in terms of absolute e r r o r r a t e , but they also show a 5-10 times increase i n  Equalizer  E  b/No  Blks UnCrashed  Data BER  Blks UnCrashed  32/1024  32/256  32/64  32/32  Data BER  Blks UnCrashed  Data BER  32/4096  Blks UnCrashed  Data BER  Blks UnCrashed  Data BER  122.6/128  .020  97.0/128  .123  DFE  0.15Hz  23 dB  1.994/2  .0016  7.96/8  .002  31.52/32  .0068  DFE  0.15  13  1.93 /2  .032  7.50/8  .038  28.78/32  .056  DFE  1.5  23  1.81 /2  .048  5.49/8  .154  8.3 /32  .368  LE  1.5  23  1.69 /2  • 128  DFE  1.5  13  1.61 /2  .108  3.44/8  .277  .910/1  TABLE 7.1  .069  Decision Directed Performance with P e r i o d i c Retraining  -  97 -  Eb/No W&) Figure 7.1  P e r i o d i c - R e t r a i n i n g Decision-Directed Results  e r r o r rate over s i m i l a r reference-directed r e s u l t s .  This i s c u r i o u s , but at  low error rates i t probably only takes a small degrading influence to change the error rate s u b s t a n t i a l l y .  Another way to think of t h i s i s that the  increased e r r o r rate due to d e c i s i o n - d i r e c t e d degradation was, in a d d i t i v e terms, much less than f o r the higher 1.5 Hz f a d i n g .  98 It i s i n t e r e s t i n g to consider the trends as a function of the r e t r a i n i n g p e r i o d i c i t y in Table 7.1 and Figure 7 . 1 .  Notice that the 0.15 Hz r e s u l t s  show r e l a t i v e l y l i t t l e degradation with decreased r e t r a i n i n g p e r i o d i c i t y . The 0.15 Hz, 32/4096 r e s u l t s show the mean-time-to-crash i s approximately 100 blocks.  In c o n t r a s t , the 1.5 Hz r e s u l t s degrade s i g n i f i c a n t l y with decreased  r e t r a i n i n g and the mean-time-to-crash i s only 10 blocks.  This i n d i c a t e s the  equalizer w i l l only run 1/IOth as long at the higher fade rate before crashing.  Periodic r e t r a i n i n g was also t r i e d on Tape 11 to compare with the r e s u l t s of Falconer et al [ 2 2 , 2 3 ] .  Using an FT DFE (with tuning c o r r e c t i o n ) on the  f i r s t 300,000 M-sequence symbols, we achieved a b i t e r r o r rate of 0.010; t r i p l e the r e f e r e n c e - d i r e c t e d r a t e .  We were forced by our software system to  use 32/320 r e t r a i n i n g , while Falconer used 30/330 frames.  S t i l l , this bit  e r r o r rate i s e x a c t l y the same as he acheived on 30,000 symbols i n an unspecified area of Tape 11.  We have previously seen that a l i n e a r e q u a l i z e r generally performs poorly.  A l i n e a r e q u a l i z e r however, i s not susceptible to e r r o r propagation  from decision-feedback.  Since our DFE r e s u l t s at high fade rates were so  d i s a p p o i n t i n g , we postulated a LE might do as well or better under such conditions.  A LE was t r i e d on the 3 p a t h , 23 dB, 1.5 Hz fading channel using  32/64 r e t r a i n i n g .  As shown i n Table 7 . 1 , the e r r o r rate of the LE only  doubled from that under the equivalent reference-directed conditions but s t i l l the absolute b i t - e r r o r - r a t e remained worse than f o r a DFE.  In order t o put the d e c i s i o n - d i r e c t e d r e s u l t s in perspective, we simulated some of the channel conditions used in Watterson and M i n i s t e r ' s p a r a l l e l modem study [63].  They used a two-path fading channel with e q u a l -  RMS-strength paths separated by 1 m i l l i s e c o n d delay spread. s l i g h t l y worse channel with 1.6 m i l l i s e c o n d s spread.  99 -  We used a  Their s p e c i f i c a t i o n of  fading rate was based on twice the standard deviation of the Gaussian b e l l shaped spectra, r e q u i r i n g conversion to our - 3 dB half-bandwidth.  The ACQ-6  and USC-10 16-tone QPSK p a r a l l e l modems were tested by them at 0.12 and 0.6 Hz fading half-bandwidths and gave i d e n t i c a l r e s u l t s .  The MX-190 25-tone  QPSK p a r a l l e l modem, which has i n t e r n a l (25,16) e r r o r c o r r e c t i o n coding, was tested at 0.6 and 3.0 Hz.  The comparison of the p a r a l l e l tone modems with our 32/128 p e r i o d i c a l l y r etra i ne d FT DFE i s shown i n Figure 7 . 2 .  Because most transmitters are peak  power l i m i t e d and the p a r a l l e l modems have a p r a c t i c a l peak-to-mean power r a t i o of about 4, the p a r a l l e l modems i n Figure 7.2 have been penalized by 6 dB and the abscissa labeled SNR rather than E./N . b o  We see the DFE  does very well against even the MX-190 modem when i t i s so p e n a l i z e d , i f the fade rate i s less than about 0.5 Hz. t h i s comparison.  Several comments should be made about  F i r s t l y , the MX-190 i s an extremely complex and expensive  modem due to i t s use of 25 tones and coding.  Secondly, i t i s extremely  s e n s i t i v e to multipath spread above 1 m i l l i s e c o n d , i t s e r r o r rate increasing by a f a c t o r of 10 at 2 m i l l i s e c o n d s (see Figure 64 of [ 6 3 ] ) .  The DFE r e s u l t s  presented herein used an equalizer which spanned 3.2 m i l l i s e c o n d s ; i t s performance would not l i k e l y degrade by more than a f a c t o r of 2-4 u n t i l the spread exceeds 3.2 ms.  On the other hand, with 32/128 r e t r a i n i n g , the net  throughput of the s e r i a l modem i s only 0.8x2400 bps.  100  3.0 Hz DFE 10 -  NX. VS.  \>v\  10--  \  \  \  \  \  1  > 0.6 Hz DFE  \  x-3  \  10-  3.0 Hz MX-190 •* 0.6 Hz ACQ-6 * 0.12Hz ACQ- 6  \ \  \  V  N  \  ^•0.6 Hz MX-190  2 PATH \ THEORETICAL'. \  \  \ \  \ \  to  \ \  to  20  SNR  1  • 0.12 Hz DFE  30  HO  (JB)  Figure 7.2 Decision-Directed Comparison with P a r a l l e l Modems  - 101 Since the net data rate of the s e r i a l modem could be raised without too serious a penalty, i t seems the technique may have quite a f u t u r e .  If the  nature of the d e c i s i o n - d i r e c t e d degradation could be ascertained, maybe f u r t h e r performance gains could be made. r e s u l t s so disappointing?  Why were the d e c i s i o n - d i r e c t e d mode  There are several unique features about the HF  channel which could cause more degradation than say, telephone channels.  Is  i t the f a d i n g , the fade r a t e , or the l o c a l i z e d nature of the HF channel' impulse response?  Regarding the l a t t e r , we suggest that such an impulse  response causes only one or two of the decision-feedback tap-weights to be significant.  This tap-weight structure could s i g n i f i c a n t l y disturb the  e q u a l i z a t i o n of l a t e r symbols i f wrong decisions are fed down the d e c i s i o n feedback tapped delay l i n e .  Unfortunately i t i s d i f f i c u l t to t e s t t h i s  hypothesis i n a s t r a i g h t forward manner.  Comparing performance on a more  uniform RMS impulse response channel would be meaningless since there would be a s i g n i f i c a n t change in d i v e r s i t y .  The d e c i s i o n - d i r e c t e d degradation of a DFE can be examined in a more fundamental way by comparing the performance in two new forms of " h a l f d e c i s i o n - d i r e c t e d " mode.  In one mode, which we w i l l c a l l "perfect-feedback"  mode, the symbols s h i f t e d i n t o the decision-feedback tapped-delay-1ine are perfect reference symbols, while the e r r o r , e ( k ) , used by the Kalman algorithm i s s t i l l formed using d e c i s i o n s . e r r o r , " i s j u s t the opposite.  The other mode, c a l l e d "perfect  It uses e r r o r information c a l c u l a t e d from a  perfect reference, but feeds decisions down the decision-feedback part of the tapped-delay-1ine.  By comparing the r e s u l t s of perfect-feedback, p e r f e c t -  e r r o r , and f u l l d e c i s i o n - d i r e c t e d modes, i t i s possible f o r the f i r s t time to i d e n t i f y the major cause of crashing on a moderately-rapid fading HF channel.  - 102 Table 7.2 presents the r e s u l t s of t h i s comparison on a 3 p a t h , 20 dB, 1.5 Hz fading channel with 32/256 r e t r a i n i n g . better.  The p e r f e c t - e r r o r mode i s f a r  This i n d i c a t e s the accuracy of e(k) i s c r i t i c a l to the performance  of the equalizer at higher fade rates while occasional wrong decisions fed back into the estimation sum are not p a r t i c u l a r i l y  disruptive.  I n t e r e s t i n g l y , t h i s shows that the l o c a l i z e d nature of the HF impulse response is not as d i s r u p t i v e as some of the other degrading e f f e c t s of the HF channel.  Since a l i n e a r e q u a l i z e r i s j u s t as dependent on e(k) as a DFE,  t h i s explains why i t has s i g n i f i c a n t d e c i s i o n - d i r e c t e d degradation t o o .  Half  d e c i s i o n - d i r e c t i o n d e f i n i t e l y needs f u r t h e r research, e s p e c i a l l y under a wider v a r i e t y of channel c o n d i t i o n s .  Table 7.2  H a l f - D e c i s i o n - D i r e c t e d Mode Comparison  B i t Error Rate  Mode  Mean # Blocks Uncrashed  Perfect Feedback  .138  5.6 / 8  Perfect Error  .042  7.7 / 8  F u l l Dec. D i r e c t i o n  .154  5.5 / 8  In l i g h t of the h a l f - d e c i s i o n - d i r e c t e d r e s u l t s , why i s e(k) such a serious c o n t r i b u t e r to d e c i s i o n - d i r e c t e d degradation?  Compared to some  Kalman a p p l i c a t i o n s , the a d d i t i v e noise in the e q u a l i z e r measurement equation i s quite s m a l l , -10 to - 2 0 dB, and the rate of v a r i a t i o n s i g n i f i c a n t .  For  t h i s reason the optimum exponential-aging i s quite short, o r , the Kalman gain i s generally quite l a r g e .  This gives each new measurement considerable  weight compared to the past state and even one wrong decision a f f e c t i n g e(k) may cause s i g n i f i c a n t adaptation in the wrong d i r e c t i o n and s t a r t an error propagation b u r s t .  This probably explains why even the 0.15 Hz r e s u l t s ,  where the e q u a l i z e r should not be that f a d e - r a t e taxed, also show s i g n i f i c a n t  - 103 degradation when using the optimum reference-directed  exponential-aging  f a c t o r , but in d e c i s i o n - d i r e c t e d mode.  A very b r i e f i n v e s t i g a t i o n showed the optimum d e c i s i o n - d i r e c t e d aging to be approximately h a l f as f a s t as that in reference-directed mode.  Further  research to determine the optimum exponential-aging f a c t o r in d e c i s i o n directed mode as a function of SNR, fading bandwidth, and r e t r a i n i n g periodicity is indicated.  Because the r e s u l t s in Section 4.1 show  considerable s e n s i t i v i t y to slowing the aging though, v a s t l y improved performance i s not expected.  Much of the previous DFE research in d e c i s i o n - d i r e c t e d mode has used the LMS algorithm which must, f o r s t a b i l i t y , use a very small step s i z e s c a l a r gain) to even be s t a b l e .  (or  Researchers who have used the Kalman  algorithm have used i t mainly f o r very f a s t convergence, which i s done in reference-directed mode.  Depending on the i n i t i a l values of P and R, the  Kalman gain may be very large during convergence.  Unless Q or the  exponential-aging f a c t o r are very s i g n i f i c a n t though, the gain w i l l automatically decrease to smaller values, p o s s i b l y of the same order as the t y p i c a l LMS s c a l a r step s i z e , f o r the post-convergence operation.  decision-directed  This ofcourse i s what i s d e s i r a b l e on almost s t a t i c channels, but  i t also explains why previous researchers neither noted nor predicted such s i g n i f i c a n t d e c i s i o n - d i r e c t e d degradation.  To q u a l i t a t i v e l y v e r i f y our high Kalmam Gain hypothesis, the equalizer was modified to run as a f i l t e r rather than a p r e d i c t o r .  Normally, the a  p r i o r i symbol estimate i s the inner product X_' ( k ) £ ( k - l ) because the state t r a n s i t i o n matrix i s given by A(k)=I.  In t h i s mode, there i s s t i l l a s i z a b l e  RMS symbol e r r o r a f t e r e q u a l i z a t i o n (see Table 4.1 f o r example), f a r larger  - 104 than the SNR would suggest.  When the new quantity X' (k)C_(k) i s c a l c u l a t e d ,  t h i s a p o s t e r i o r i RMS symbol e r r o r i s s i g n i f i c a n t l y smaller because with the large gain c a p a b i l i t y of the Kalman algorithm, £ ( k ) has been adapted to take out much of the e r r o r .  The a p o s t e r i o r i RMS symbol e r r o r also seems to be of  a more correct magnitude considering the channel SNR.  In reference-directed mode the a p o s t e r i o r i Kalman e q u a l i z e r gives the i l l u s i o n of i n c r e d i b l e performance, i f the Kalman gain i s kept large, because the i t e r a t i o n of the symbol estimate can p u l l a wrong estimate across a decision boundary.  In d e c i s i o n - d i r e c t e d mode however, the i t e r a t i o n only  p u l l s bad symbol estimates toward wrong d e c i s i o n s .  Consequently, the e r r o r  rate i s exactly the same as the normal e q u a l i z e r and in f a c t , the tap weight vector C{k) takes the same t r a j e c t o r y i n time.  In d e c i s i o n - d i r e c t e d simulations, the a p o s t e r i o r i RMS error remains very small u n t i l a d e c i s i o n e r r o r i s made.  Then, the e r r o r increases greatly and  the e q u a l i z e r often crashes.  Several conclusions can be drawn from these a p o s t e r i o r i  experiments.  The low SNR's t y p i c a l to the e q u a l i z e r problem and the r e l a t i v e l y f a s t dynamics of the HF channel cause the optimum exponential-aging f a c t o r to be very s h o r t , r e s u l t i n g i n s i g n i f i c a n t adaptation at each i t e r a t i o n .  Secondly,  the reason f o r the larger RMS symbol-error encountered with the a p r i o r i algorithm i s that the one step delay in updating the taps i s s i g n i f i c a n t at the moderately-high HF fade rates which the Kalman algorithm i s capable of tracking.  In one i t e r a t i o n the optimum C{k) changes so much that i t  somewhat obsolete to use C_(k-1) in the symbol e s t i m a t i o n . another d e f i n i t i o n of slow f a d i n g .  is  This could lead to  It also reinforces the suggestion i n  Section 5 that a better value than the i d e n t i t y f o r the  - 105 equalizer state t r a n s i t i o n matrix would be of i n t e r e s t .  F i n a l l y , an  automatic crash detection mechanism would be more s e n s i t i v e i f i t monitored the a p o s t e r i o r i RMS e r r o r .  In summary, we t r i e d the e q u a l i z e r i n d e c i s i o n - d i r e c t e d mode using periodic r e t r a i n i n g .  The r e s u l t s were marginal f o r high fading bandwidth  channels, though they did agree with F a l c o n e r ' s r e s u l t s f o r Tape 11 and do out perform even the best p a r a l l e l modems at low fade rates on radios that are peak-power l i m i t e d .  In order that f u r t h e r performance gains might be  uncovered, several reasons f o r the s i g n i f i c a n t degradation in d e c i s i o n directed mode were hypothesized and t e s t e d .  At higher fade rates the  dependence on an accurate e(k) was noted as a s i g n i f i c a n t cause, because RLS algorthms can adapt f a s t enough to f o l l o w channels where s i g n i f i c a n t change in the tap weights i s required with each new symbol.  Wrong decisions may  therefore cause s i g n i f i c a n t adaptation in the wrong d i r e c t i o n and lead to e r r o r propagation.  The optimum exponential-aging i s l i k e l y s l i g h t l y  d i f f e r e n t in d e c i s i o n - d i r e c t e d mode than in reference-directed mode.  In  other a p p l i c a t i o n s such as control systems, the f a s t - a d a p t i n g Kalman algorithms are not used in a d e c i s i o n - d i r e c t e d manner.  Previous e q u a l i z a t i o n  work has used e i t h e r the slowly-adapting LMS algorithm o r , because the channels investigated were almost s t a t i c , the Kalman gain has been allowed to (somewhat automatically) decrease f o r the d e c i s i o n - d i r e c t e d operation.  Since  we may be the f i r s t to use a large Kalman Gain in d e c i s i o n - d i r e c t e d mode, t h i s explains why other researchers wrongly assume that low p r o b a b i l i t y - o f error i s a good enough c r i t e r i o n to assume l i t t l e difference in performance between r e f e r e n c e - d i r e c t e d and d e c i s i o n - d i r e c t e d mode.  - 106 7.3 Constant Modulus Algorithm Performance and Discussion  In t h i s section we b r i e f l y investigate the Constant Modulus Algorithm (CMA).  It was developed from some general concepts of Godard [29] by  Treichler,  Larimore and Agee [36,60].  It i s novel in t h a t , in the LE  c o n f i g u r a t i o n , i t needs no reference to converge and should never c r a s h . T r e i c h l e r suggested i t could be modified to the DFE configuration and use an RLS a l g o r i t h m .  It i s believed that we present the f i r s t such RLS r e s u l t s .  Our purpose i s to determine i f a one-dimensional error ( i e . modulus only) which i s always p e r f e c t l y c o r r e c t , r e s u l t s in better performance i n d e c i s i o n directed mode than a two-dimensional error i n which one dimension, the phase, can be corrupted by decision e r r o r s .  This i s p a r t i c u l a r i l y relevant i n l i g h t  of the conclusions of the l a s t sub-section that much of the e r r o r propagation tendancy o r i g i n a t e s from decision e r r o r s corrupting  e(k).  Because the HF channel i s r a p i d l y f a d i n g , we desire a f a s t adapting RLS version of the CMA.  In [ 3 6 ] , the CMA convergence performance i s examined f o r  the f a m i l y of modulus cost f u n c t i o n s : J =E[(|d(k)| -|x'(k)C(k-l)| ) ] a  ab  a  b  We see the least squares cost function i s  r e s u l t i n g in the state  update: e  CMA< ) k  C(k)  =  Cl-IJL'  (k)c.(k-i) | D*CX' (k)C.(k-l )/| x.' (k)c.(k-l) | D  = C(k-l) + 6(k)e  C M A  (k)  The second equation i s the same as (3.6) but with a d i f f e r e n t e r r o r : a d i r e c t e d modulus e r r o r . modulus e r r o r .  The f i r s t f a c t o r in Q ^ ( k ) i s the s c a l a r e  The second i s a unit vector in the complex plane pointing i n  the d i r e c t i o n of X . ' ( k ) C ( k - l ) .  We see e » ( k ) r M  i s exactly that  - 107 complex number which when added t o X ' ( k ) C ( k - l ) , the symbol estimate, lengthens i t to the correct modulus.  This RLS-modified algorithm worked in the l i n e a r e q u a l i z e r configuration so we immediately t r i e d i t with the DFE.  The CMA algorithm does not  b a s i c a l l y change the functioning of any of the various equalizer configurations introduced i n Section 3 . 1 , i t j u s t uses only one part of  e(k).  In order to compare the modified CMA and our previous algorithms we ran a RLS CMA DFE in p e r i o d i c r e t r a i n i n g mode.  During r e t r a i n i n g i t was switched  to the normal, r e f e r e n c e - d i r e c t e d algorithm so as to receive the exact same retraining.  Table 7.3 presents some preliminary r e s u l t s from a few short  simulations of a 3 path channel using a T/l DFE with 32/256 r e t r a i n i n g .  Table 7.3  Channel Condition 1.50 Hz  0.15 Hz  23 dB  13 dB  Performance of Constant Modulus Algorithm  A .83  .03  Equalizer  Data BER  Mean # Blocks Uncrashed  DFE  .130  5.8 / 8  CMA DFE  .273  3.6 / 8  DFE  .054  7.3 / 8  CMA DFE  .046  7.7 / 8  The CMA e q u a l i z e r has a better e r r o r rate and mean-time-to-crash at slow fade rates while the regular algorithm i s superior at 1.5 Hz.  The CMA, because i t  receives only r a d i a l e r r o r information, depends heavily on the data stream being uncorrelated and i t needs time to use t h i s c h a r a c t e r i s t i c f o r gathering the phase i n f o r m a t i o n .  Its adaptation rate i s therefore slower than the  regular DFE which, i n s p i t e of d e c i s i o n - d i r e c t e d contamination of e ( k ) , does better at high fade r a t e s .  - 108 In summary, the CMA seems to o f f e r improved performance at slower fade rates.  Further i n v e s t i g a t i o n i s needed to determine where the advantage ends  and whether t h i s endpoint i s dependent on various channel or equalizer factors.  Perhaps a superior hybrid technique could be devised that uses "one  and a h a l f dimensions" of e r r o r information: the modulus error and an f r a c t i o n of the phase e r r o r .  A l s o , i f a DFE detects i t has crashed, i t might  t r y to recover as a CMA LE without having to wait f o r the next r e t r a i n i n g sequence.  In l i g h t of our CMA r e s u l t s , some i n v e s t i g a t i o n of the spectrum  technique of [Morgan78] may be warranted i f i t generates a two-dimensional error and i s not too computationally burdensome.  - 109 8.  CONCLUSION  8.1 Summary And Major Conclusions  The HF r a d i o channel i s a long range, but uniquely d i f f i c u l t channel to work w i t h .  It has a long, but l o c a l i z e d , fading impulse response which  causes wide, r a p i d l y moving n u l l s in the frequency domain.  Modems designed  f o r use on HF frequencies must therefore be considerably d i f f e r e n t from those used on other channels.  Recently, d i g i t a l l y - i m p l e m e n t e d s e r i a l modems used on telephone or troposcatter channels have been implemented using r e c u r s i v e - l e a s t - s q u a r e (Kalman) algorithms to r e a l i z e very f a s t converging e q u a l i z a t i o n .  These  algorithms have been suggested f o r HF modem equalizers because they adapt very q u i c k l y .  L i t t l e previous work s p e c i f i c to the a p p l i c a t i o n of RLS algorithms to HF equalization e x i s t s .  Only small amount of corporate and m i l i t a r y research  has been published or was a v a i l a b l e .  Unfortunately much of i t used e i t h e r a  s t a t i c or very narrow v a r i e t y of channel c o n d i t i o n s , was purposefully u n d e t a i l e d , presented unbelievable r e s u l t s , or never investigated e q u a l i z e r performance in d e c i s i o n - d i r e c t e d mode.  This project set out to determine how well RLS algorithms could perform in the HF modem a p p l i c a t i o n using a 2400 bps QPSK s e r i a l modulation.  It was  also hoped to i n v e s t i g a t e some of the accompanying synchronization, p r o t o c o l , software and hardware c o n s i d e r a t i o n s , but i t q u i c k l y became obvious that even c h a r a c t e r i z i n g the performance as a function of the very large v a r i e t y of p o s s i b l e of HF channel parameters was a formidable t a s k .  A l s o , the  -  performance of several e q u a l i z e r configurations was of p r i o r i t y  no  -  interest.  This c h a r a c t e r i z a t i o n as a function of both the channel and configuration i s necessary to determine the implementation d e t a i l s of a s e r i a l HF modem.  An extensive c h a r a c t e r i z a t i o n of e q u a l i z e r performance was performed as a function of both e q u a l i z e r and HF channel parameters.  The e q u a l i z e r  parameters investigated were configuration ( l i n e a r versus decision-feedback e q u a l i z a t i o n , f r a c t i o n a l versus symbol-spaced t a p s ) , mode (reference versus d e c i s i o n - d i r e c t e d ) , sample t i m i n g , and l e n g t h .  The channel parameters varied  were s i g n a l - t o - n o i s e r a t i o , mistuning (or doppler s h i f t ) , number of multipaths and t h e i r RMS strengths, fading bandwidth, and delay spacing (including less than symbol-period r e s o l u t i o n spacing).  The major conclusions of the project were:  a)  The f r a c t i o n a l l y - t a p p e d decision-feedback e q u a l i z e r (FT DFE) was d e f i n i t e l y the superior c o n f i g u r a t i o n .  Performance within 2-3 dB of  t h e o r e t i c a l was found p o s s i b l e on very slow fading channels i n reference-directed mode.  The DFE was able to provide 1-2 orders of  magnitude reduction i n e r r o r rate compared to a l i n e a r e q u a l i z e r on a v a r i e t y of HF channels.  In a d d i t i o n , the f r a c t i o n a l l y - t a p p e d DFE proved  to be i n s e n s i t i v e to sampling phase and provided much better performance on channels with non-symbol-period-spaced m u l t i p a t h , channels often overlooked by other authors.  b)  The decision-feedback equalizers were f a d e - r a t e l i m i t e d to approximately 0.5 Hz.  Even though the RLS (Kalman) algorithm i s f a s t e r adapting than  the LMS algorithmm, and in some cases produced an e r r o r rate 2 orders of magnitude s m a l l e r , i t i s not f a s t enough to track the e n t i r e range of  -  fade rates t y p i c a l of the HF channel.  -  A new d e f i n i t i o n of slow fading  which depends on the e q u a l i z e r length was proposed f o r RLS e q u a l i z e r s .  in  exponentially-aged  This was q u a l i t a t i v e l y confirmed by the  bit-error-rate  r e s u l t s , and by the f a c t that the best performance on channels fading f a s t e r was obtained with a feedforward t a p p e d - d e l a y - l i n e length which j u s t spanned the multipath spread.  The received-sample feedforward  line  should t h e o r e t i c a l l y be i n f i n i t e l y long.  Operation of the equalizers in d e c i s i o n - d i r e c t e d mode caused s i g n i f i c a n t degradation.  Other authors have always assumed that with a low b i t -  e r r o r - r a t e , the d i f f e r e n c e between reference and d e c i s i o n - d i r e c t e d mode should be s m a l l .  This does not seem to be the case i f operating in  d e c i s i o n - d i r e c t e d mode with a large Kalman Gain, as, i s required in the HF equalizer a p p l i c a t i o n .  S i g n i f i c a n t adaptation takes place at each  i t e r a t i o n and thus even i s o l a t e d wrong decisions can cause a serious disturbance in the adaptation, r e s u l t i n g in f u r t h e r e r r o r s and posssibly a crash.  This suggests wrong decisions fed back into a DFE symbol  estimation are not a serious f a c t o r i n the HF a p p l i c a t i o n .  A l s o , the  e r r o r rate of a s e r i a l modem w i l l l i k e l y be dominated by the crashes and cause errors to occur in b u r s t s .  In s p i t e of the d e c i s i o n - d i r e c t e d degradation, an FT DFE-based s e r i a l modem could in many cases perform better than a p a r a l l e l tone modem i f , as i s necessary on peak-power l i m i t e d t r a n s m i t t e r s , the p a r a l l e l modems are properly penalized f o r t h e i r high peak-to-mean power r a t i o .  - 112 e)  I d e n t i f i e d the equivalence of the exponential-aging formulation derived from the Weiner-Hopf equations [20], with that of the Kalman community i f Q=0 and R=l [ 2 4 , 5 9 ] .  Showed that i n Hsu [31] the assumption of  Q=qAPA'* was equivalent to exponential-aging.  Noted that i f using  exponential-aging and Q=0, then R i s i n e f f e c t u a l a f t e r convergence.  This  i s l i k e l y due to the RLS algorithm only r e a l l y needing one parameter, the exponential-aging time constant, to specify in a general sense the desired amount of adaptation at each i t e r a t i o n .  f)  Determined the optimum r e f e r e n c e - d i r e c t e d value f o r the exponential-aging constant.  It i s independent of number of paths, t h e i r strengths and  t h e i r spacing, but generally dependent on SNR, fade r a t e , number of equalizer t a p s , configuration (LE vs DFE), mode (decision vs r e f e r e n c e - d i r e c t e d ) , and r e t r a i n i n g p e r i o d i c i t y .  g)  Noted that the Constant Modulus Algorithm, when adapted to an RLS DFE, may provide superior performance in d e c i s i o n - d i r e c t e d mode.  The o v e r a l l research project has been a success, e s p e c i a l l y in showing the improved performance of the RLS FT DFE over e i t h e r LMS, symbol-spaced, or l i n e a r equalizers on the HF channel.  A very good channel simulator was  constructed so that the d i f f i c u l t HF channel could be tamed and experiments w i t h , f o r example, fading channels with c o n t i n o u s l y - v a r y i n g , channelc o r r e l a t i o n - m a t r i x eigenvalue r a t i o s could be conducted. new r e s u l t s in d e c i s i o n - d i r e c t e d mode were uncovered.  F i n a l l y , important  - 113 Though the project concentrated on the problems and advantages of various equalizer configurations and adaptation algorithms, we have been assured of the p r a c t i c a l i t y of the s e r i a l technique by our near-optimal slow-fading reference-directed r e s u l t s and by d e c i s i o n - d i r e c t e d r e s u l t s often better than obtainable with current p a r a l l e l tone modems.  The d e c i s i o n - d i r e c t e d absolute e r r o r rates a v a i l a b l e on e i t h e r type of modem are quite s i g n i f i c a n t , i n d i c a t i n g that some sort of ARQ scheme w i l l be required when t r a n s m i t t i n g data other than d i g i t i z e d speech (which i s quite t o l e r a n t to e r r o r s ) .  Since the s e r i a l scheme r e s u l t s in burst e r r o r s ,  the same b i t - e r r o r - r a t e p a r a l l e l modems.  for  i t w i l l require less retransmissions that the  A l s o , the equalizers were tested on the severest channels:  those with several equal-RMS-strength multipaths fading at s i g n i f i c a n t rates.  Maybe equal strength paths are f a i r l y a t y p i c a l and our r e s u l t s o v e r l y  pessimistic.  S i m i l a r l y , maybe fade rates over 0.15 Hz, say, are uncommon and  the t r a c k i n g rate l i m i t s of the Kalman DFE w i l l not be a problem.  S e r i a l modems should be able to challenge the current p a r a l l e l modems in performance, e f f i c i e n c y , and because of t h e i r d i g i t a l nature,  reliability.  The challenge now i s to both gather more channel tapes and to work out the p r a c t i c a l i t i e s of s e r i a l modem implementation, p a r t i c u l a r i l y  carrier  frequency a c q u i s i t i o n and t r a c k i n g , and handling the computational burden in a compact economical way.  - 114 8.2 Suggestions f o r Further Research  Many areas have been i n v e s t i g a t e d , but some only b r i e f l y .  Other  complementary areas, such as maximum l i k e l i h o o d sequence e s t i m a t i o n , should c e r t a i n l y be examined.  A c r i t i c a l eye i s needed as many f a c t o r s are  dependent on many o t h e r s , yet i t i s not p r a c t i c a l to extensively check every dependency.  The following i s a l i s t of areas considered e i t h e r p r a c t i c a l , important, or loose ends needing a t t e n t i o n :  a)  Further q u a n t i f i c a t i o n of e q u a l i z e r performance i n d e c i s i o n - d i r e c t e d mode is required.  In order to obtain the optimum performance and aging  f a c t o r , more complete r e s u l t s as a function of fading bandwidth, SNR, r e t r a i n i n g p e r i o d i c i t y , and aging f a c t o r are needed.  b)  To reduce d e c i s i o n - d i r e c t e d degradation a thorough i n v e s t i g a t i o n of Constant Modulus Algorithm performance as a function of fade r a t e , SNR, r e t r a i n i n g p e r i o d i c i t y , and aging f a c t o r i s i n d i c a t e d .  This w i l l  determine the conditions under which the CMA performs better than the f u l l - e r r o r algorithm, and i f the aging f a c t o r i s d i f f e r e n t f o r t h i s algorithm.  A "one and a h a l f dimension of e r r o r " hybrid CMA algorithm  which uses the modulus e r r o r and a f r a c t i o n (perhaps h a l f ) of the phase e r r o r might be better suited than e i t h e r technique.  c)  Which Kalman algorithm should be used in an actual modem?  For  computational e f f i c i e n c y , a " f a s t " Kalman algorithm whose complexity i s proportional to the number of taps i s probably d e s i r a b l e .  Such  algorithms are often numerically unstable but there are several new, even  - 115 f a s t e r Kalman algorithms a v a i l a b l e , some with i n s t a b i l i t y detection and recovery mechanisms.  What w i l l be the e f f e c t of 16 b i t arithmetic on the  choice of a l g o r i t h m ; w i l l square-root normalization be necessary f o r increased dynamic range?  Can the various algorithms be decomposed to run e f f i c i e n t l y on a p i p e l i n e of several TMS320 processor chips?  Can more than one l o g i c a l stage be  put on a sin gle TMS320 pipe computational element?  If not, the modem may  have to wait about one year f o r the next generation TMS320 processor.  Several researchers have shown that maximum l i k e l i h o o d sequence estimation (MLSE) o f f e r s only marginally better performance than a DFE. Our d e c i s i o n - d i r e c t e d Kalman DFE r e s u l t s though, show the degradation from i s o l a t e d decision e r r o r s to be a s i g n i f i c a n t problem due to high Kalman Gain.  Since t h i s appears to be unique to our a p p l i c a t i o n perhaps  MLSE, which makes sequence rather than symbol-by-symbol d e c i s i o n s , should not be ruled out.  The problem of c a r r i e r frequency a c q u i s i t i o n and t r a c k i n g , and frame s t a r t l o c a t i o n (for use i n s t a r t i n g the r e f e r e n c e - d i r e c t e d  convergence)  needs i n v e s t i g a t i o n under fading multipath c o n d i t i o n s .  More data from HF channels i s needed to estimate whether multipath channels with equal-RMS-strength paths are common.  The p r o b a b i l i t y  d i s t r i b u t i o n of multipath delay spread and fading bandwidth i s also of interest.  We w i l l then be able to determing the length of e q u a l i z e r  required and whether the l i m i t e d t r a c k i n g rates noted i n the project w i l l be a commonly occuring hindrance.  - 116 h)  Future work might also investigate the p o s s i b i l i t y of s e l f - t u n i n g the system model in the Kalman algorithm.  If an estimate of A, Q, or the  optimum exponential-aging was a v a i l a b l e , better e q u a l i z e r performance, or channel estimator performance f o r MLSE, might be obtainable.  As t h i s t h e s i s i s being w r i t t e n , Glenayre Electronics i s s t a r t i n g work on a B r i t i s h Columbia Science Council Major Research Grant to develop some research-oriented prototype hardware and software.  The project w i l l be based  on the TMS 320 signal processing microprocessor and the r e s u l t s of t h i s thesis.  Some of the d i f f i c u l t i e s not yet resolved w i l l be addressed, more  real channel data i s expected to be gathered, and another graduate student w i l l be involved in the work.  - 117 APPENDIX A - HF Simulation Flow Diagram  The diagram in the next page i l l u s t r a t e s the timing of the flow of complex symbols and received samples in the main program (not l i s t e d i n the t h e s i s ) , and i s included f o r i n t e r e s t  only.  -"CHDLY"  7  (TAPE^.  REFERENCE  Z  -"CHDLY"  "EQDLY"  "GETSAM" and  ref.  "ENRX"  "GETSYM"  "IROD"  "DUNTX"  -"EQDLY"  "DENTX"  dir.  dec. d i r . « •  SIMULATED  SEQUENCE  BASEBAND  DIFFERENTIAL  GENERATOR  "UNTX"  ENCODING  "ENTX"  "REFSEQ"  CHANNEL  "DENTX"  DECISION DEVICE ERROR  "ENRX"  EQUALIZER sim. c h .  "FUNHSU"  "LBBSCH"  "ENEQ"  STATISTICS ANALYSIS  "ERRMAG" "PHERR"  "NUMERR"  "ERSTAT" "ENEQ"  NOTES:  1) Values i n quotation marks are a c t u a l symbolic names, e i t h e r v a r i a b l e s or s u b r o u t i n e s , used i n the code. 2) Z "  1  DIFFERENTIAL  "DEEQ"  DECODING  represents a one symbol d e l a y . DIFFERENTIAL  "EQDLY"  DECODING  z  Figure A . l  -"EqOLY"  Simulation Flow Diagram  "DDERX"  "DENRX"  00 I  - 119 APPENDIX B  Square-Root-Kalman, Fractionally-Tapped  Decision-Feedback  Equalizer Algorithm  A l l RLS (or Kalman) algorithms t h e o r e t i c a l l y generate the same recursive update of the state vector.  Some though, are more computationally e f f i c i e n t ,  more numerically s t a b l e , or use less numerical dynamic range than others.  The algorithm used i n t h i s project i s taken from Hsu [ 3 1 ] .  It i s an  exponentially-aged RLS algorithm based on the diagonal and upper t r i a n g u l a r matrix f a c t o r i z a t i o n Carlson [7] c a l l s factored inverse square-root Kalman:  P(k) = U*(k)D(k)U'(k) where P(k) i s the estimate of the state covariance m a t r i x , D(k) i s a diagonal matrix f a c t o r of P, U(k) i s an upper t r i a n g u l a r matrix double f a c t o r of P, (with i m p l i c i t unit d i a g o n a l ) , * 1  indicates complex conjugate, and indicates matrix transpose.  If P becomes non-positive d e f i n i t e due to round-off e r r o r the algorithm becomes unstable.  The s t o r i n g and updating of D and U instead of P, prevents  U*DU' from l o s i n g i t s p o s i t i v e d e f i n i t e n e s s without increasing computation s i g n i f i c a n t l y .  - 120 Even with 32 b i t f l o a t i n g - p o i n t a r i t h m e t i c , the o r i g i n a l Kalman algorithm implemented with Equation 3.9 f a i l s due to the loss of p o s i t i v e d e f i n i t e n e s s of P a f t e r roughly 200 symbols.  Using Equation 3.10 to update P s t a b i l i z e s  the algorithm but takes several times as much computational time.  The  factored algorithm i s stable using 32 b i t f l o a t i n g point f o r more than 600,000 symbols, i s computationally more e f f i c i e n t than Equations 3.9 or 3.10, and uses less memory.  A l s o , the use of D and U reduces the problems of  numerical dynamic range as i t i s s i m i l a r to s t o r i n g and updating the matrix square-root of P [ 1 , 2 ] .  Hsu does not discuss his algorithm as an exponentially-aged one. Instead, he takes Q(k) = qA(k)P(k-l)A* (k) and adjusts q.  As mentioned  i n Section 3.2 though, i t i s r e l a t i v e l y easy to show t h i s equivalent to exponential-aging with  X=l/(l+q).  See also Section 4 . 3 .  We extended Hsu's algorithm to handle the l i n e a r and decision-feedback configuration w i t h , or without f r a c t i o n a l - t a p p i n g of the input s e c t i o n .  This  i s a f a i r l y s t r a i g h t forward process f o r algorithms with computational burden proportional to the number of taps squared (as opposed to f a s t Kalman's).  It  simply requires c a r e f u l manipulation of the values i n the tapped delay l i n e X.  If using a DFE, X i s two concatenated TDL's: the f i r s t part i s received  samples, the l a t t e r part i s d e c i s i o n s .  If f r a c t i o n a l - t a p p e d with spacing  T/2, the f i r s t part i s twice as long as normal and the samples are s h i f t e d two elements down the l i n e each symbol i t e r a t i o n . of Hsu's algorithm i s presented in Appendix E.  The FORTRAN implementation  - 121 Hsu's algorithm, as extracted d i r e c t l y from [ 3 1 ] , i s included at the end of t h i s appendix.  The reader i s encouraged not to examine i t too c l o s e l y  unless he reads [31] as i t i s optimized by Hsu f o r storage and computational speed.  It i s included here only f o r completeness.  The reader i s advised  t h a t Hsu also suggests a revised version of his algorithm in [31] which i s more s u i t a b l e f o r f i x e d - p o i n t arithmetic implementation.  Notes on Hsu's a l g o r i t h m :  1)  U i s a s t r i c t l y upper t r i a n g u l a r matrix and i s i n i t i a l i z e d to 0 (though i t i m p l i c i t l y has a u n i t d i a g o n a l ) .  2)  D i s a diagonal matrix i n i t i a l i z e d to the i d e n t i t y matrix and can ofcourse be stored as a v e c t o r .  3)  £ i s the e q u a l i z e r tap-weight vector (state v e c t o r ) .  4)  r  5)  I|< i s the reference symbol or d e c i s i o n .  6)  q = (1/X)-1 i s Hsu's exponential-aging f a c t o r .  7)  $ = R/(l+q) = RX , where R i s the measurement noise covariance.  i s the tapped-delay-1ine vector (measurement m a t r i x ) .  VI.  COMPUTATIONAL  PROCEDURES  Significant computer storage can be saved for numerical computation of the square root Kalman algorithms. The computational procedures described here are such that the lower triangular portion and the diagonal elements of U are not used (the diagonal elements of U are unity). The vector V is replaced by vector G for storage reduction. The computational procedures of the square root (U-D factorization) formulation defined in Section I V are given by N (6.1) 7=1  (6.2)  2  fj<=  PI,J * X  */*»  +  j = 1,2,-••,JV,  g = d (k-\)f , J  j  y = 2,3,-••,7V,  J  (6.3) (6.4) (6.5)  j =: 2 , 3 , - - - , J V ,  a = « , _ , + $,/}•, y  (6.6) (6.7)  _ J_  (6.8)  d (k) = d {k - l)h„ly, t  (6-9)  x  (6.10)  \ = -fa,  (6.11)  1  (6.12) (6.13)  dj(k) = dj(k - l)A,j8y,  (6.14)  Pi = ft*\,.  (6-15)  ft = ft + g / l * -  (6.16)  M = £| + V  A l l constants, except {, a j= 1,2,---,/Y, y, h , q, d j = l,,---,N, and /3 defined i n (6.1) to (6.16), are complex quantities. The quantities defined i n (6.1) to (6.9) are computed first. Equations (6.10) to (6.16) are evaluated recursively for /' = 2,3,- • -,N, and (6.14) to (6.16) are computed i n order for i = 1,2,- • • J — 1. The equalizer tap weights C = ( C , , C , - • -,C ) are given by Jt  2  q  N  (6.17)  e = *(k)y, Cj=Cj + gje,  Jt  y = 1,2, •••,#,  (6.18)  where the computation of the final Kalman gain, gj = j = 1,2,- • - ,N, is avoided by adding an intermediate step defined in (6.17). Equations (6.1) to (6.18) are evaluated for every new input signal X defined i n (3.1).  u's Square-Root Kalman Equalizer Algorithm [31]  - 123 APPENDIX C - D e t a i l s of Real Channel Tests and Data Tapes  As discussed in Sections 1 . 3 , 5, and 6, three tapes of d i g i t i z e d HF audio were a v a i l a b l e from the l a t t i c e study [ 4 2 ] .  The transmissions were of  1201.739 symbol/second d i f f e r e n t i a l l y - e n c o d e d QPSK on a 1505.107 Hz subcarrier.  A l l the data rates and frequencies, including the c a r r i e r  o s c i l l a t o r s , were locked to a cesium clock and should have been good to 1 part in 1 0 . 1 1  The transmitted data used a 45° t w i s t on the QPSK  c o n s t e l l a t i o n r e s u l t i n g i n 45°, 135°, 225°, and 315° symbols.  This was  o r i g i n a l l y done so t h a t , when d i f f e r e n t i a l l y - e n c o d e d , the c o n s t e l l a t i o n would rotate at l/8th the symbol rate so the symbol rate might be locked onto. This mechanism was not used in our  project.  The transmissions took place between Cobden, Ontario and the CRC labs at S h i r l e y Bay, Ontario, a distance of about 60 k i l o m e t e r s .  The transmitter  power was 16 watts, no noticable ground wave was received, and the Tape 6 SNR was about 18 dB (measured by dropping the transmitter power t i l the signal spectrum f e l l into the n o i s e ) .  Table C . l Tape 6 9 11  Frequency  Tape Recording Conditions  Local Time  Comments  11:40  Steady, strong c l e a r s i g n a l .  MHz  15:00  Strong interference at 10 pulses/second.  5.2675 MHz  17:40  Strong signal with deep fading at  7.5555 MHz 11.623  0.5 Hz.  The t e s t data tapes started with a preamble of noise and several unusual synchronizing sequences which were not of p a r t i c u l a r i n t e r e s t .  The main data  sequence was a repeating 1023 b i t maximal-length pseudo-random sequence (M-sequence) generated using a 10 b i t s h i f t r e g i s t e r with taps on b i t s 7 and  - 124 10.  The s t a r t i n g phase was considered to be 10 s h i f t s past a l l V s .  Note  that when t h i s b i t - s t r e a m i s encoded into 2 - b i t QPSK symbols, the r e s u l t i s a 1023 symbol long repeating pseudo-random symbol stream.  The received signal was d i g i t i z e d at 9600.00 Hz onto standard 1/2 inch 9 track computer t a p e s .  The records used a f i x e d 512 byte length.  was a tape header of 512 bytes.  Record 0  The f o l l o w i n g records were composed of 256  words of 16 b i t s , each word s t o r i n g the unsigned 12 b i t output from an a n a l o g u e - t o - d i g i t a l converter.  There was a s l i g h t bias in the conversion  discovered in the l a t t i c e study, so 2040 should be taken as zero.  The proper sampling phase was obtained manually by down-converting, low-pass f i l t e r i n g , and looking f o r the maximum envelope of the 1505.107 Hz subcarrier.  With no multipath t h i s would give symbol center, and in f a c t on  Tape 11 which has 3 paths, t h i s proved a d i f f i c u l t chore.  Once approximate  symbol center was found, the s t a r t of the M-sequence (needed to converge the e q u a l i z e r i n r e f e r e n c e - d i r e c t e d mode) was located by looking at every eighth sample.  Once the e q u a l i z e r could be converged we varied the s t a r t i n g sample  s l i g h t l y to f i n d the optimum values shown i n Table C.2.  Table C.2  S t a r t i n g Sample of M-Sequence on Tapes  Tape 6  Record 122, Sample 232  Tape 9  Record 107, Sample 59  Tape 11  Record 100, Sample 4  Unfortunately, because 9600.00/1201.739 i s not exactly 8, a sample s l i p p i n g scheme was necessary to hold correct sampling phase.  Once every  86.38147 symbols the tape demodulation program had to generate a FT sample only 3, rather than 4 samples a f t e r the previous one.  - 125 APPENDIX D - Second-Order Channel Simulator Subroutine  The operation of the subroutine i s r e l a t i v e l y simple. runs at the (simulated) receiver sampling rate of 9600 Hz.  Internally,  it  The input symbols  are r e p l i c a t e d 8 times (to get from 1200 to 9600 Hz and preserve the rectangular pulse shape).  These are then clocked into the TDL named DELAYD  which i s long enough to span 6 symbols of multipath spread.  TDL DELAYD,  which f o r e f f i c i e n c y i s operated as a c i r c u l a r b u f f e r , thus has 49 taps. Only 5, whose p o s i t i o n s are specifed by the elements of the path spacing vector MPDEL(0:4), are considered to have non-zero path g a i n .  The 5 path  signals tapped out are weighted by the respective products of the 5 independently-fading u n i t - v a r i a n c e processes G(0:4) and the 5 RMS path amplitudes MPGAIN(0:4), and then summed.  Complex additive white Gaussian  noise i s added to give the i n f i n i t e - b a n d w i d t h , noisy fading-multipath s i g n a l , CHOUT.  This signal i s fed into another ( c i r c u l a r buffer) TDL named LPFLIN where i t i s delayed and then convolved with the b a n d - l i m i t i n g low-pass f i l t e r coefficients.  The delay i s arranged so t h a t , including the inherent LPF  d e l a y , the t o t a l delay of the f i r s t path (assumed to have MPDEL(0)=0 and MPGAIN(O) not 0) i s a known i n t e g r a l number (DELAY) of symbol periods.  This  band-limited sample stream i s then sub-sampled at 2400 Hz, with the s p e c i f i e d sample t i m i n g , r e s u l t i n g i n a T/2 FT sample stream ready to be equalized.  If  TIMING=0, samples are taken near the beginning of a symbol received on the f i r s t path. symbol.  TIMING=7 i s near the end, and 3 or 4 i s near the middle of a  The sample timing of the other paths depend on both TIMING and MPDEL  of course.  Frequency mistuning i s simulated in the main program by r o t a t i n g  the input samples fed to the e q u a l i z e r a few degrees more (in the complex plane) each successive symbol.  - 126 -  *  SUBROUTINE CDTPCH  (SYMBIN,SYMOUT,MPGAIN,MPDEL,SNR,FCNORM, TIMING,FT,GTAFS)  c c ADAFTED 21/2/84 BY RUSS TRONT c -FROM LBBSCH TO USE A CRITICALLY-DAMPED, RATHER c THAN BUTTERWORTH, TWO-POLE FILTER TO BANDWIDTH RESTRICT THE FADING PROCESSES. A CRITICALLY-DAMPED FILTER BETTER c APPROXIMATES THE GAUSSIAN-BELL SHAPED SPECTRUM OF HF FADING c AS COMPARED TO A BUTTERWORTH WHICH IS TOO FLAT TOPPED. c c CRITICALLY-DAMPED TWO-POLE CHANNEL: c THIS ROUTINE PROVIDES A SAMPLE-SPACED (T/8 OF SYMBOL) c COMPLEX BASEBAND HF CHANNEL MODEL (FINITE BANDWIDTH, 5 COMPLEX c c FADING PATHS WITH INDIVIDUAL RELATIVE PROPAGATION DELAYS c AND MEAN SQUARE GAINS, AND ADDITIVE WHITE GAUSSIAN NOISE. c THE FINITE BANDWIDTH IS PROVIDED BY A LOW-PASS FILTER WHOSE c (ODD LENGTH) IMPULSE RESPONSE IS READ IN FROM UNIT 7. c THE 5 COMPLEX RAYLEIGH FADING MULTIPATHS ARE PLACABLE TO WITHIN c 1/8TH OF A SYMBOL PERIOD RELATIVE PROPAGATION DELAY WITH MPDEL(0:4), c AND EACH HAVE INDIVIDUALLY SELECTABLE RMS AMPLITUDES, MPGAIN(0:4). c THE RMS PATH GAINS ARE ASSUMED TO BE NORMALIZED BEFORE BEING c PASSED TO THIS SUBROUTINE (IE. SUM OF MPGAIN(I)**2 SHOULD BE 1.) c FCNORM IS THE -3DB CUT-OFF FREQUENCY, NORMALIZED BY THE SYMBOL RATE, c OF THE CRITICALLY-DAMPED TWO-POLE FILTERS CONTROLLING THE FADING c BANDWIDTH OF THE INDIVIDUAL PATH GAINS. c THE PROPAGATION DELAY OF THIS CHANNEL IS CHDLY (ASSUMING MPGAIN(O).NE.O c AND MPDEL(0)=0.). c cc SPECIAL SPECIFICATIONS: c c c c c c c c c c c c c c c c  IMPLICIT NONE  SAVE  REAL NGAUSS  REAL*8 DEXP INTEGER NPATH,NECHO PARAMETER (NPATH=5) PARAMETER (NECHO=NPATH-1) INTEGER MAXDEL,BUFSIZ PARAMETER (MAXDEL=60) PARAMETER (BUFSIZ=MAXDEL+1>  -VAX EXTENTION TO FORTRAN-77 TO FORCE DECLARATION OF ALL VARIABLES. -FORCES RETENTION OF LOCAL VARIABLES BETWEEN CALLS LIKE FORTRAN IV! -REAL FUNCTION WHICH RETURNS A NORMALIZED GAUSSIAN RANDOM NUMBER (MEAN=0,VARIANCE=1). -DOUBLE PRECISION EXP FUNCTION.  -CONFIGURE FOR NUMBER OF MULTIPATHS.  -CONFIGURE FOR MAX NUMBER OF SAMPLES DELAY SPREAD IN CHANNEL (60=.005/(1/12000)).  INTEGER MAXSIZ,MAXSM1,MAXSP8,MAXSP7 PARAMETER (MAXSIZ=49) FARAMETER (MAXSM1=MAXSIZ-1) PARAMETER (MAXSP8=MAXSIZ+8) PARAMETER (MAXSP7=MAXSP8-1) -MAX SIZE OF LPF, AND LPFLIN. LPFLIN IS A CIRCULAR DELAY LINE OF SAMPLE SPACED CHANNEL OUTPUTS, WHICH IS AT LEAST THE LENGTH OF THE LPF ROUNDED UP TO THE NEAREST 8 SAMPLES I  c c c c c c c- ARGUEMENT DECLARATIONS: c c COMPLEX SYMBIN c c COMPLEX SYMOUT(2)  -THE COMPLEX INPUT SYMBOL -THE COMPLEX OUTPUTS OF THE CHANNEL:  - 127 -  c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c  IF TIMING=3, SYMOUTd > IS THE FRACTIONALLY-TAPPED OUTPUT SAMPLE ON THE FIRST FATH AT ROUGHLY SYMBOL CENTER, WHILE SYMOUTf 2) IS THE SAMPLE AT INTERSYMBOL TIMES. REAL MPGAIN(0:NECHO)  INTEGER MPDEL(0:NECHO)  REAL SNR  REAL FCNORM  INTEGER TIMING  LOGICAL FT  COMPLEX GTAPS(0:NECHO)  -THE INDIVIDUAL MULTIPATH RMS AMPLITUDES. SHOULD BE NORMALIZED BY THE MAIN PROGRAM SO THE SUM OF MPGAIN(I)**2 FOR ALL I IS 1.0 BEFORE BEING PASSED TO THIS SUBPROGRAM. -THE FIVE DELAYS (IN UNITS OF 1/8TH SYMBOL) SELECTING THE POSITIONS OF THE FIVE MULTIPATHS "MPGAIN". TO SIMULATE OLD SYMBOL-SPACED CHANNEL CHOOSE MPDEL(0:4)=(0,8,16,24,32). MPDEL(0) SHOULD ALWAYS BE 0 SO THE SHORTEST PATH HAS ZERO RELATIVE DELAY. -THE DESIRED CARRIER TO NOISE RATIO IN DB, OF THE GAUSSIAN ADDITIVE NOISE. (FOR QPSK, SNR=Eb/No SINCE 1 BPS/HZ OF BANDWIDTH). -THE 3DB FADING BANDWIDTH OF THE PATH GAIN SPECTRUMS (FCUT), NORMALIZED BY THE SYMBOL RATE. -A PARAMETER TO VARY TIMING PHASE. MUST BE IN RANGE (0,7). THREE IS NORMAL PUTTING SYMOUTd) ABOUT SYMBOL CENTER ON PATHS WITH MPDEL( )=0. IF FRACTIONALLY TAPPED, MUST BE IN RANGE (0,3). -.TRUE. IF SECOND ELEMENT OF SYMOUT IS TO BE GENERATED FOR USE BY AN FT EQUALIZER. .FALSE. EXECUTES FASTER.  c c c c c c c c c OTHER DECLARATIONS: c  -THE CURRENT VALUE OF THE 5 PATH GAINS G(I)*MPGAIN(I>. NOTE GTAPS IS PASSED OUT "CHDLY" SYMBOLS EARLIER THAN THE SYMBOL IT GENERATES DUE TO THE LPF DELAY. ONLY USED BY CHANNEL ESTIMATOR SOFTWARE.  C C C C C C C  -THE DELAY LINE OF INPUT SYMBOLS (REPLICATED 8 TIMES AS THE SAMPLE RATE IS 8 TIMES THE SYMBOL RATE). DELAYD IS USED IN THE CONVOLUTION OF THE INPUT SYMBOLS AND THE MULTIPATH IMPULSE RESPONSE.  C C C C C  COMPLEX DELAYD(0:MAXDEL) DATA DELAYD/BUFSIZ*(0.,0.)/  COMPLEX NOISE(0:NECHO)  COMPLEX G(0:NECHO) DATA G/NPATH*(1.,0. )/ INTEGER POINTR DATA POINTR/0/  -THE UNIT-VARIANCE COMPLEX (2D) NOISE WHICH DRIVES THE RAYLEIGH FADING PROCESSES.  -THE UNIT-VARIANCE RAYLEIGH FADING PROCESSES.  -POINTER TO PRESENT LOCATION IN CIRCULAR  - 128 -  C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C  C C  c  C  BUFFER "DELAYD". COMPLEX CHOUT  COMPLEX LFFLIN(0:MAXSP7) DATA LPFLIN/MAXSP8*(0.,0.)/  REAL LPF(0:MAXSM1)  INTEGER DELAY  INTEGER LPFSIZ,LPFSM1  INTEGER LPFLSZ,LPFLM1  INTEGER LPFPTR DATA LPFPTR/0/  COMPLEX LPFOUT COMPLEX E(0:NECHO) DATA E/NPATH*(0. , 0.) /  REAL A0,B1PRIM,B2 REAL AT  -THE CHANNEL OUTFUT BEFORE LPF, AT ANY SAMPLE TIME.  LFFLIN IS A CIRCULAR DELAY LINE OF SAMPLE SPACED CHANNEL OUTPUTS (BEFORE LPF) USED IK THE LPF CONVOLUTION. IT IS AT LEAST THE LENGTH OF THE LPF ROUNDED UP TO THE NEAREST 8 SAMPLES (NEAREST INTEGER NUMBER OF SYMBOLS)I -THE LPF IMPULSE RESPONSE OF LENGTH LPFSIZ (ODD1), READ IN FROM UNIT 7 USING F0RMAT(I15/(E15.7)). THE 115 IS THE LPFSIZ, THE RESPONSE IN E15.7 FOLLOWS ONE PER LINE. -THE LPF DELAY ROUNDED UP TO THE NEAREST NUMBER OF SYMBOLS -THE SIZE (AND SIZE-1) OF THE LPF READ IN FROM UNIT 7. -THE SIZE AND SIZE-1 OF THE PART OF LPFLIN ACTUALLY USED.  -THE POINTER TO THE PRESENT LOCATION IN CIRCULAR DELAY LINE "LPFLIN". -THE RESULT OF THE LPF CONVOLUTION.  -THE DIFFERENCE BETWEEN THE LAST AND SECOND TO LAST FADING FILTER OUTPUTS. THESE ARE USED IN THE FILTERS RATHER THAN THE SECOND-TO-LAST FILTER OUTPUTS FOR BETTER NUMERICAL PRECISION AT VERY LOW Kl/1000) FCNORM. -THE CRITICALLY-DAMPED TWO-POLE DISCRETE TIME FILTER COEFFICIENTS. -THE CRITICALLY-DAMPED FILTER "A" TIMES THE SYMBOL PERIOD, T, WHERE "A" IS FROM THE FILTER TRANSFER FUNCTION: H(S)=A**2/((S+A)A*2>.  REAL STUFF REAL RATIO INTEGER I,J,SCRTCH REAL R00T2 REAL PI DATA PI/3.1415927/ LOGICAL FCALL DATA FCALL/.TRUE./  IF (FCALL) THEN FCALL-.FALSE.  -THEN THIS IS THE FIRST CALL.  - 129 -  c 9 C 10  C C C  c c c c c c c c c c c c c c  -CHECK TIMING <4 IF FT. IF (FT.AND.(TIMING.GT.3)) THEN WRITE(6,5) FORMAT('0**** ERROR **** - IF FRACTIONALLY TAPPED, THEN ', 'TIMING MUST BE LESS THAN 4') STOP ENDIF -READ IN LPFSIZ, LPF COEFFICIENTS. READ(7,9)LPFSIZ,(LPF(I),1=0,LPFSIZ-1) F0RMAT(I15/(E15.7)) LFFSM1=LPFSIZ-1 IF(LPFSIZ.EQ.(LPFSIZ/2)A2)THEN WRITE(6,10) FORMAK'OAAA* ERROR A A A * - LOW PASS FILTER SIZE LFPSIZ'/ MUST BE ODD!') STOP ENDIF  DELAY=(7+LPFSM1/2)/8  LPFLSZ=8ADELAY+1+LFFSM1/2 LFFLM1=LPFLSZ-1 :  -LPF DELAY ROUNDED UP TO NEAREST INTEGER NUMBER OF SYMBOLS. -SET UP LPF CIRCULAR DELAY LINE "LPFLIN" TO GIVE DELAY OF AN INTEGER NUMBER OF SYMBOLS, EVEN THOUGH LPF DELAY IS "LPFSMl"/2 SAMPLES (LPFSM1/16 SYMBOLS).  -CALCULATE THE RATIO OF SIGNAL VARIANCE (1. ) TO THE ADDITIVE NOISE VARIANCE TO BE ADDED BEFORE THE LPF, KNOWING THAT THE LPF WILL REDUCE THE NOISE VARIANCE BY A FACTOR EQUAL TO THE SQUARE ROOT OF THE SUM OF THE LPF IMPULSE RESPONSE SQUARED.  STUFF=0. DO 20 I=0,LFFSM1 STUFF=STUFF+LPF(I)AA2 CONTINUE RATI0=(10.*A(SNR/20.))ASQRT(STUFF) C C C C  c  C C  c  C C C  c c c  C  c c c  C C  R0OT2=SQRT(2.)  -SET UP CRITICALLY-DAMPED 2 POLE FADING-PROCESS FILTER COEFFICIENTS. LAPLACE TRANSFORM OF FILTER IS H(S)=AAA2/((S+A)AA2). YOU WILL NOTE IT HAS DC GAIN OF 1, -3DB CUT-OFF FREQUENCY FCUT=AASQRT(ROOT2-l.WTWOFI, AND EQUIVALENT NOISE BANDWIDTH OF 1.22AFCUT. AT=2.APIAFCNORM/SQRT(ROOT2-l.) -AO IS THE FILTER COEF OF THE INPUT. AO=ATAATAEXP(-1.*AT) -CHANGE THE GAIN OF THE FILTER SO THAT UNIT-VARIANCE IN RESULTS IN UNIT-VARIANCE OUT, IRRESPECTIVE OF FCUT. NOTE THE UNITVARIANCE NOISE IN IS WHITE TO FSYMB/2. THAT'S WHERE THE 2 COMES FROM! -BE CAREFUL NOT TO DIVIDE BY ZERO IF FCNORM=0. IF (FCNORM.NE.0.) THEN A0=A0/SQRT< 1.22A2. AFCNORM) ELSE A0 = 0. ENDIF -USE DOUBLE PRECISION FOR DIFFERENCE OF TWO NUMBERS VERY NEAR 1. SEE FILTER ITERATION BELOW FOR EXPLAINATION FOR USE OF B1PRIM RATHER THAN BI. B1PRIM=-1.D0A(1.D0-DEXP(-1.D0*AT)>*A2 B2=-1.AEXP(-2.AAT) ENDIF -ITERATE THE "NPATH" FADING PROCESSES.  -  C C C C C  DO 100 I=0,NECHO -NGAUSS IS A ZERO MEAN, UNIT VARIANCE GAUSSIAN RANDOM NUMBER GENERATOR. NORMALIZE COMPLEX NOISE TO MEAN SQUARE MAGNITUDE OF UNITY. NOISE(I)=CMPLX(NGAUSS(),NGAUSS())/R00T2  C C C C C C C C  -THIS IS THE CRITICALLY-DAMPED 2 POLE FILTER ITERATION SPECIALLY ADAPTED FOR NUMERICAL PRECISION AT VERY LOW FCNORM. AS EXPLAINED IN J.K.CAVERS' MEMO OF 12/12/83, THE TECHNIQUE INVOLVES ITERATING THE CHANGE, E ( I ) , IN THE LAST OUTPUT RATHER THAN THE OUTPUT ITSELF. E( I)=AO*NOISE(I)+B1FRIM*G(I)-B2*E(I) G(I)=G(I)+E(I)  C C C C 100 C C C C  C C C  150 C C C  GTAPS(I)=G(I)*MPGAIN(I) CONTINUE  -CREATE 8 SAMPLES FOR EACH SYMBOL. RUN THE DELAY LINE AHEAD 8 SYMBOLS WITH THE SAME INPUT.  DO 200 1=0,7 POINTR=POINTR-l IF(POINTR.LT.0)POINTR=MAXDEL DELAYD(POINTR)=SYMBIN  -DO THE CONVOLUTION OF THE CHANNEL RESPONSE WITH THE SAMPLE SEQUENCE.  CHOUT=(0.,0.) DO 150 J=0,NECHO CHOUT=CHOUT+MPGAIN(J)*G(J)* DELAYD(MOD(POINTR+MPDELCJ),BUFSIZ)) CONTINUE  *  -ADD NOISE WITH THE PROPER SNR TO EACH OUTPUT CHOUT=CHOUT+CMPLX(NGAUSS(),NGAUSS())/(R00T2*RATI0)  C C C  -UPDATE THE LPF DELAY LINE OF CHOUT'S.  LFFPTR=LFFPTR-1 IF (LPFFTR.LT.O) LPFPTR=LPFLM1 LPFLIN(LPFPTR)=CHOUT  C C C C C C  -IF BIT TIMING IS RIGHT, THEN DO THE LPF CONVOLUTION. THE FOLLOWING CODE DOES NOT PRODUCE A CORRECT SYM0UT(2) FOR TIMING)=4  1 2  170 C  C 200  -COMPUTE PATH GAINS FOR USE WHEN CALLED BY CHANNEL ESTIMATOR SOFTWARE SYSTEM.  A  IF((I.EQ.TIMING) .OR. (FT.AND.(I.EQ.MOD(TIMING+4,8))) ) THEN LPFOUT=(0.,0.) SCRTCH=LPFPTR+LPFLSZ-LPFSIZ DO 170 J=0,LPFSM1 LPFOUT=LPFOUT+LPF(J)* LPFLIN(MOD(SCRTCH+J,LPFLSZ)) CONTINUE IF (I.EQ.TIMING) THEN SYMOUTd) =LPF0UT ELSE SYMOUT(2)=LPFOUT ENDIF ENDIF  CONTINUE RETURN END  130 -  - 131 APPENDIX E - Fractionally-Tapped Decision-Feedback Equalizer Subroutine  See Appendix B f o r discussion of the algorithm.  SUBROUTINE FUNHSU 1 2 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C  (YIN,IREF,IEST,IDEC,EPSIL, NIFFOR,NIFBCK,NDFBCK, DECDIR,TWIST,Q,R,FT)  WRITTEN BY RUSS TRONT L A S T MODIFICATION 1/11/81 TO ADD ENTRY FUNSET T H I S ROUTINE IMPLEMENTS A FRACTIONALLY ( T / 2 ) TAPPED VERSION OF T H E "UNREVISED" SQUARE ROOT KALMAN EQUALIZER ALGORITHM DESCRIBED BY F.M. HSU I N I E E E TRANS. ON INFO. THEORY, VOL. I T - 2 5 ( 5 ) , S E P T . 8 2 , P. 753-763 (FOR R E V I S E D HSU, S E E SUBROUTINE 'HSU'). MORE S P E C I F I C A L L Y , I T IMPLEMENTS HSU'S "REVISED" EQUATIONS 6.1-6.7, 6.19-6.21, 6.11, 6.22, AND 6.13-6.18, BUT UNREVISES THEM BY S E T T I N G HT=0l I HAVE USED E S S E N T I A L L Y T H E SAME NOTATION AS HSU, E X C E P T I U S E R FOR X I ' . T H I S I S A SHORT BUT VERY COMPLICATED ALGORITHM, OPTIMIZED BY HSU FOR SPEED, STORAGE, AND A*ASNEAKY*AA E F F I C I E N T CALCULATION OF UPPER TRIANGULAR AND DIAGONAL MATRICES. T H E EQUALIZER T A K E S A NEW FRACTIONALLY T A P P E D VECTOR OF INPUTS 'YIN', AND AN OLD FEEDBACK REFERENCE OR D E C I S I O N 'IROD' (DEPENDING ON WHETHER I T I S I N REFERENCE OR D E C I S I O N D I R E C T E D MODE), AND S H I F T S THEM INTO T H E TAPPED DELAY L I N E 'X' (WHICH I S R E A L L Y TWO SEPARATE TDL'S CONCATENATED, ONE OF INPUT DATA 'YIN' , AND ONE OF F E D BACK REFERENCE OR D E C I S I O N SYMBOLS 'IROD'). I T THEN CALCULATES T H E NEW OUTPUT E S T I M A T E ' I E S T ' AS T H E INNER PRODUCT O F T H E OLD TAP WEIGHT VECTOR ' C AND TAPS 'X', AND QUANTIZES I T TO T H E NEAREST SYMBOL I N ' I D E C . I T THEN CALCULATES A NEW V A L U E FOR 'IROD' DEPENDING ON T H E MODE, AND THENCE T H E ERROR ' E P S I L ' I N T H E E S T I M A T E USING E I T H E R T H E REFERENCE OR T H E D E C I S I O N . T H E NEW KALMAN GAIN 'G' I S THEN CALCULATED U S I N G T H E ABOVE REFERENCED ALGORITHM. F I N A L L Y , T H E ERROR AND T H E NEW KALMAN GAIN ARE USED TO UPDATE T H E TAP WEIGHT VECTOR ' C .  c C SPECIAL SPECIFICATIONS: I M P L I C I T NONE C -VAX EXTENTION TO FORTRAN-77 TO C FORCE DECLARATION OF A L L V A R I A B L E S . C SAVE C -FORCES RETENTION OF SUBROUTINE C LOCAL VARIABLES BETWEEN C A L L S , C L I K E GOOD OLD FORTRAN I V I C C ENTRY FUNSET C -RESETS INTERNAL VARIABLES U,D,C, C AND IROD. C INTEGER NTAPMX,NTAPM2 PARAMETER (NTAPMX=26) PARAMETER (NTAPM2=NTAPMX*NTAPMX) C -PARAMETERS TO S E T ARRAY S I Z E S TO C CORRESPOND TO MAXIMUM E X P E C T E D C NUMBER OF T A P S . C C ARGUEMENT DECLARATIONS: C COMPLEX Y I N ( 2 ) C - T H E FRACTIONALLY TAPPED INPUT SYMBOLS. C I F TIMING I N CHANNEL I S S E T FOR MIDC SYMBOL, THEN C Y I N ( l ) I S AT OPTIMAL SAMPLE TIME. C Y I N ( 2 ) I S A T SUB-OPTIMAL SAMPLE T I M E . COMPLEX I R E F C -THE INPUT REFERENCE SYMBOL. C I T I S USED I N REFERENCE DIRECTED C MODE ONLY, TO C A L C U L A T E T H E C T H E ERROR. AND I N T H E FOLLOWING C A L L C AS A FEEDBACK SYMBOL FOR T H E T D L .  -  c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c  COMPLEX IEST COMPLEX IDEC  COMPLEX EPSIL  INTEGER NIFFOR  INTEGER NIFBCK  INTEGER NDFBCK  LOGICAL DECDIR  LOGICAL TWIST  REAL Q  REAL R  -THE EQUALIZED OUTPUT ESTIMATE. -THE OUTPUT DECISION OF THE EQUALIZER, 'IEST' QUANTIZED TO THE NEAREST SYMBOL IN THE SIGNAL CONSTELLATION. -EPSILON, THE OUTPUT SYMBOL ERROR. IN DECISION DIRECTED MODE, IT IS 'IDEC'-'IEST'. IN REF. DIRECTED MODE, IT IS 'IREF'-'IEST'. -THE NO. OF INPUT FEED-FORWARD TAPS, INCLUDING THE ONE "CENTER" OR FEEDNEUTRAL TAP. NIFFOR SHOULD NOT BE CHANGED ON SUCCESSIVE CALLS. -THE NO. OF INPUT (NOT DECISION) FEED BACK TAPS. SHOULD NOT BE CHANGED ON SUCCESSIVE CALLS. -THE NO. OF DECISION FEED BACK TAPS. DON'T CHANGE ON LATER CALLS. -MODE FLAG THAT IS .TRUE. IF IN DECISION DIRECTED MODE. IF .FALSE. THEN IN REFERENCE DIRECTED MODE. -IN DD MODE, 'IDEC IS USED FOR BOTH THE CALCULATION OF 'EPSIL', AND AS A FEED BACK SYMBOL. -IN REF. DIR. MODE, 'IREF' IS USED TO CALCULATE 'EPSIL', AND AS A FEEDBACK SYMBOL. -.TRUE. IF 45 DEGREE TWIST ON INPUT SIGNAL CONSTELLATION WAS IN EFFECT WHEN THE SYMBOL ^CURRENTLY** BEING EQUALIZED WAS ENCODED. IF TWIST CHANGED IN MAIN, MAY HAVE TO WAIT FOR PROPAGATION DELAYS BEFORE FEEDING CHANGED VALUE TO THIS SUBPROG. -A SMALL PARAMETER DESCRIBED IN HSU AS SET PROPORTIONAL TO THE FADE RATEI -THE MEASUREMENT NOISE COVARIANCE. NOTE: XI=R/(1+Q) IS A SMALL PARAMETER DESCRIBED IN HSU AS APPROXIMATELY THE EXPECTATION OF THE MEAN SQUARE ERROR (SAY .01 FOR 20DB SNR)  LOGICAL FT  c c cc OTHER DECLARATIONS: c c c c c  REAL XI  INTEGER NTAPS INTEGER NN,NN1,NN2  -XI=R/(1+Q) -TOTAL NO. OF TAPS. NTAPS=NIFFOR+NIFBCK+NDFBCK.  133  -  -  C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C  134  -NIFFOR+NIFBCK, NIFFOR+NIFBCK+1, NIFFOR+NIFBCK+2. LOGICAL TOGGLE DATA TOGGLE/.FALSE./  COMPLEX IROD DATA IROD/(0.,0.)/  COMPLEX X(NTAPMX) DATA X/NTAPMX*(0.,0.)/  COMPLEX C(NTAPMX) DATA C/NTAPMX*(0.,0.)/  COMPLEX G(NTAPMX)  COMPLEX U(NTAPMX,NTAPMX) DATA U/NTAPM2*(0.,0.)/  REAL D(NTAPMX) DATA D/NTAPMX*(1. ,0.)/  COMPLEX F(NTAPMX)  COMPLEX LAMBDA(NTAPMX)  COMPLEX BETA1  REAL BETA REAL ALPHA(NTAPMX)  -FLIPS BACK AND FORTH FROM CALL TO CALL IF TWIST=.TRUE. CONTROLS WHICH SIGNAL CONSTELLATION THE QUANTIZER QUANTIZES TO!  -LOCAL STORAGE FOR THE REFERENCE OR DECISION USED IN CALCULATING THE ERROR, AND FED-BACK INTO THE TDL ON THE NEXT CALL. IN REF. DIRECTED MODE, IT IS 'IREF'. IN DECISION DIRECTED MODE, IT IS 'IDEC.  -THE TAPPED DELAY LINE COMPOSED OF (NIFFOR) FED-FORWARD INPUT SYMBOLS, FOLLOWED BY (NIFBCK) INPUT SYMBOLS, FOLLOWED BY (NDFBCK) FEED-BACK DECISIONS (OR REFERENCE) SYMBOLS, FOLLOWED BY (NTAPMX-NTAPS) UNUSED VALUES. IN THE KALMAN FORMULATION OF AN EQUALIZER, X ALSO PLAYS THE ROLE OF THE MEASUREMENT MATRIX, COMMONLY REFERRED TO IN THE LITERATURE AS H(K).  -THE TAP WEIGHT VECTOR. IN THE KALMAN FORMULATION OF AN EQUALIZER, C IS THE STATE VECTOR BEING ESTIMATED. -THE KALMAN GAIN MATRIX (IN THE EQUALIZER CASE, A VECTOR). INITIALLY, G IS USED FOR SCRATCH STUFF, THEN MODIFIED TO BE KALMAN GAIN/LAST GAMMA. MULTIPLICATION BY GAMMA OCCURS IN TAP UPDATE.  -AN UPPER TRIANGULAR MATRIX THAT IS ONE OF THE FACTORS OF P=CONJG(U)*D*TRANSPOSE(U) DECOMPOSITION. THE DIAGONAL AND LOWER TRIANGLE ARE NOT USED. (THE THEORY IMPLICITLY ASSUME THE DIAGONAL IS ALL (l.,0.))  -A REAL DIAGONAL MATRIX (ONE OF THE FACTORS OF P), STORED AS A VECTOR. -A VECTOR DEFINED BY HSU AS TRANSPOSE(U)*CONJG(X>, USED FOR INTERMEDIATE CALCULATIONS. -A VECTOR DEFINED BY HSU AS F*GAMMA, AND USED FOR INTERMEDIATE CALCULATIONS. -USED BY HSU TO REDUCE ARRAY REFERENCES AND STORAGE. -DEFINED IN EQN. 6.21  -  -  c c c c c c c c c c c c c  -USED BY HSU IN INTERMEDIATE CALCULATIONS. ALPHA(NTAPS) CORRESPONDS TO THE ALPHA DEFINED AFTER EQN. 4.4 REAL HT  REAL HQ REAL GAMMA COMPLEX E  -DEFINED AFTER EQN. 6.22, AND SET TO ZERO IN THE UNREVISED HSU ALGORITHM. -HQ=1+Q -DEFINED IN 6.19 AND 6.22 -DEFINED IN EQN. 6.17  INTEGER I,J  CAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA  C C C  C C C  COMPLEX Z45 DATA Z45/(.7071068,.7071068)/ -USED IN QUANTIZATION TO "UNTWISTED" CONSTELLATION. COMPLEX QUANT,ZDUMMY QUANT(ZDUMMY)=CMPLX(SIGN(.7071068,REAL(ZDUMMY)), 1 SIGN(.7071068,AIMAG(ZDUMMY))) -STATEMENT FUNCTION TO QUANTIZE ZDUMMY TO THE NEAREST SYMBOL IN THE "TWISTED" CONSTELLATION.  C**************************************************************************  c  C C C C  C  XI=R/(1+Q) NTAPS=NIFFOR+NIFBCK+NDFBCK NN=NIFFOR+NIFBCK NN1=NN+1 NN2=NN+2  -CONVERT MEASUREMENT NOISE COVARIANCE INTO XI FOR USE IN UNHSU ALGORITHM. -CALC AND CHECK TOTAL NO. OF TAPS.  IF (NTAPS.GT.NTAPMX) THEN WRITE(6,99999) FORMAT(' ***** ERROR ***** - TOO MANY EQUALIZER TAPS'/ 99999 - INCREASE SIZE OF PARAMETER', "NTAPMX"'/ IN EQUALIZER SUBROUTINE HSU. STOP ENDIF C C C -UPDATE THE TAPPED DELAY LINE. C FOR NON-FRACTIONALLY TAPPED, C SHIFT THE WHOLE LINE TO THE C RIGHT, ADD THE NEW INPUT TO C THE BEGINNING, AND OVERWRITE THE c APPROPRIATE ELEMENT WITH THE C FEED-BACK SYMBOL 'IROD', WHICH IS C EITHER 'IREF' OR 'IDEC' DEPENDING C ON MODE. IF NO FEEDBACK TAPS, C BE CAREFULI IF FRACTIONALLY TAPPED c BE EVEN MORE CAREFUL! C IF (.NOT.FT) THEN DO 100 I=NTAPS,2,-1 X(I)=X(I-1) 100 CONTINUE C X(1)=YIN(1) IF (NDFBCK.GT.O) X(NN1)=IR0D  135 -  -  ELSE  110  120  C C C C  500 C C C C C C C C C C C  c  c c c c c c c c c c c c c c c c c c c  190  DO 110 I=NN,3,-1 X(I)=X(I-2) CONTINUE X(2)=YIN(1) X(1)=YIN(2) IF (NDFBCK.GT.O) THEN DO 120 I=NTAPS,NN2,-1 X(I)=X(I-1) CONTINUE X(NN1)=IR0D ENDIF ENDIF  IEST=C(1)*X(1) DO 500 J=2,NTAPS IEST=IEST+C(J)*X(J) CONTINUE  -CALCULATE THE NEW OUTPUT ESTIMATE AS THE INNER PRODUCT OF THE TAP WEIGHT VECTOR AND THE TAPS.  -FORM DECISION 'IDEC BASED ON A QUANTIZATION OF 'IEST' TO NEAREST SYMBOL IN THE SIGNAL CONSTELLATION. 'QUANT' IS A STATEMENT FUNCTION WHICH QUANTIZES TO THE "TWISTED" SIGNAL CONSTELLATION. IF TWISTED CONSTELLATION WAS USED DURING ENCODING, THEN THE ONE TO QUANTIZE TO TOGGLES BACK AND FORTH BETWEEN TWISTED AND UNTWISTED ONES. THINK ABOUT ITJ IF (TWIST) TOGGLE= .NOT. TOGGLE IF (TOGGLE) THEN IDEC=QUANT(IEST) ELSE IDEC=QUANT(Z45*IEST > *CONJG(Z45 > ENDIF -DEPENDING ON WHETHER IN REFERENCE OR DECISION DIRECTED MODE, SET 'IROD' TO EITHER 'IREF' OR 'IQUANT'. IF (DECDIR) THEN IROD=IDEC ELSE IROD=IREF ENDIF -CALCULATE THE ERROR IN IEST (IF IN DECISION DIRECTED MODE, CALCULATE THE SUSPECTED ERROR).  EPSIL=IROD-IEST  -NOW CALCULATE THE NEW KALMAN GAIN USING HSU'S ALGORITHM. -THE KALMAN GAIN MATRIX G, FOR A WHILE IS USED FOR TEMPORARY STORAGE OF AN INTERMEDIATE VECTOR HSU CALLS V. -NOTE THERE IS A LOT OF REAL/COMPLEX MIXED MODE ARITHMETIC  F(l)=CONJG(X(D) G(1)=D(1)*F(1) ALPHA(1> =XI+G(1)*CONJG(F(1))  DO 200 J=2,NTAPS F(J)=C0NJG(X(J)) DO 190 1=1,J-l F(J)=F(J)+U(I,J)*CONJG(X(I)) CONTINUE G(J)=D(J)*F(J) ALPHA(J)=ALPHA(J-1)+G(J)*C0NJG(F(J))  136  -  200 C  CONTINUE  HQ=1+Q C C C CAAAAAHT=ALPHA(NTAPS)AQ HT=0. CAAAAA  C  290 300 C C C  400 C  C C  -THIS SMALL MODIFICATION IS A QUICK AND DIRTY CONVERSION OF THE REVISED HSU ALGORITHM, TO THE UNREVISED ONE  GAMMA=1./(ALPHA(1)+HT) D(1)=D( 1)*HQA((XI+HT)AGAMMA) DO 300 J=2,NTAPS BETA=ALPHA (J-D+HT LAMBDA(J)=-1.AF(J)AGAMMA GAMMA=1./(ALPHA<J)+HT) D(J)=D(J)AHQA(BETAAGAMMA) DO 290 1=1,J-l BETA1=U(I,J) U(I,J)=BETA1+C0NJG(G(I))ALAMBDA(J) G(I)=G(I)+G(J)AC0NJG(BETA1) CONTINUE CONTINUE  E=EPSILAGAMMA DO 400 J=l,NTAPS C(J)=C(J)+G(J)AE CONTINUE  -NOW UPDATE THE TAP WEIGHTS WITH THE KALMAN GAIN A ERROR A GAMMA.  RETURN  C CAAA*AAAAAA**AAAAAAA*AAAAAA**A*AA*AAAA*A*AAAA*AAAAAAA*AAA*A****AA*AAA C  C C C  ENTRY FUNSET THIS ENTRY RESETS MATRIX P, VECTOR C, AND SCALAR IROD. NOTE P IS REALLY CONJGfU)ADATRANSPOSE(U), SO RESETTING IT TO AN IDENTITY MATRIX INVOLVES CLEARING U, AND SETTING D TO ALL ONES.  CAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA  C C C  5000 5001 C C C  DU)=1. C(1)=(0.,0.) DO 5001 J=2,NTAPS D(J)=1. C(J)=(0.,0.) DO 5000 1=1,J-l U(I,J) = (0. ,0.) CONTINUE CONTINUE IROD'=(0. ,0.) RETURN END  -ONLY RESET THE USED ENTRIES.  -NOW RESET IROD  138 APPENDIX F - Symbol-Spaced Channel Estimator Subroutine  See also Section 5.1 f o r discussion of the channel estimator.  -  *  SUBROUTINE CHEST(REF,REC, EST,DECAY,LAMBDA, MFGSQ,R,NTAPS,C)  c c . WRITTEN 20/6/83 BY RUSS TRONT c IMPLEMENTS A KALMAN CHANNEL ESTIMATOR. USES AN INTERNAL c MODEL OF AN HF RADIO CHANNEL BASED ON PATH GAINS DERIVED c FROM COMFLEX GAUSSIAN NOISE FILTERED BY A 1ST ORDER c AUTO-REGRESSIVE PROCESS, AND SCALED BY AN RMS AMPLITUDE c MULTIPATH GAIN. EXPONENTIAL-AGING IS ALSO AVAILABLE. c THE ALGORITHM IS IMPLEMENTED AS SIMPLE STRAIGHT FORWARD c BRUTE FORCE (IE. ORDER NTAPS**2) KALMAN (AS OPPOSED TO c FAST KALMAN OR SQUARE-ROOT NORMALIZED KALMAN). SEE FOR c INSTANCE: A.GELB, "APPLIED OPTIMAL ESTIMATION", MIT PRESS, c 1974, EQUATIONS 8.1-16 AND 8.4-4. GELB'S NOTATION IS USED, c EXCEPT HIS S=l/LAMBDA. c c cc SFECIAL DECLARATIONS: c c c c  IMPLICIT NONE  SAVE INTEGER NTAPMX PARAMETER (NTAPMX=20)  -VAX EXTENSION TO FORTRAN-77 TO FORCE DECLARATION OF ALL VARIABLES. -SAVES ALL VARIABLE BETWEEN CALLS, LIKE GOOD OLD FORTRAN IVI  c c c c ARGUEMENT DECLARATIONS:  -MAX NUMBER OF TAPS IN THE TAPPED DELAY LINE CHANNEL MODEL.  c c  -THE CURRENT SYMBOL BEING.TRANSMITTED.  c c c c c c c c c c c c c c c c c c c c c c c c c c c  COMFLEX REF  COMPLEX REC  COMPLEX EST REAL DECAY  REAL LAMBDA  INTEGER NTAPS REAL MPGSQ(O-.NTAPS-l)  REAL R  COMPLEX C(NTAPS)  c c cc OTHER DECLARATIONS: COMPLEX ERROR  -THE CURRENT SYMBOL BEING RECEIVED, CONTAMINATED BY INTERSYMBOL INTERFERENCE FROM PAST SYMBOLSI -THE CURRENT ESTIMATE OF THE RECEIVED SYMBOL. -THE DECAY CONSTANT OF THE FIRST ORDER RAYLEIGH FADING PATH MODEL: C(K,T)=DECAY*C(K,T-1)+ SQRT(1-DECAY**2)*W(K,T) WHERE W IS WHITE AND 2D GAUSSIAN. IF DECAY=1, THEN GELB'S Q (THE SYSTEM, NOT MEASUREMENT, NOISE COVARIANCE) =0, AND ONLY EXPONENTIAL AGING IS USED FOR ADAPTION. -EXPONENTIAL AGING FACTOR, EQUIVALENT TO THE RECIPROCAL OF GELB'S VARIABLE S. -THE NUMBER OF TAPS IN THE ESTIMATOR. -THE EXPECTED MEAN SQUARE AMPLITUDES OF THE TAPS, COMPUTED BY DSTRMP FROM MPGAIN,MPDEL, AND FT. -THE VARIANCE OF THE EXPECTED MEASUREMENT NOISE. -THE CURRENT MULTIPATH TAP WEIGHTS.  -THE ESTIMATION ERROR IN THE CURRENT SYMBOL.  139  -  -  c c c c c c c c c c c c c c c  c c  COMPLEX P(NTAPMX,NTAPMX) -THE ESTIMATED TAP COVARIANCE MATRIX. COMPLEX X(NTAPMX)  COMPLEX ALPHA  COMPLEX G(NTAPMX)  -THE MEASUREMENT MATRIX. IN THE ESTIMATOR CASE, THE TAPPED DELAY LINE (VECTOR) OF TRANSMITTED SYMBOLS. -THE DENOMINATOR OF THE KALMAN GAIN (SHOULD BE REAL BUT CONSTRUCTED FROM COMPLEX EXPRESSION), -KALMAN GAIN VECTOR.  COMPLEX TEMP(NTAPMX,NTAPMX) - CI-G*TRANSP(X)3 COMPLEX Tl(NTAPMX,NTAPMX) - TEMP*P LOGICAL INIT DATA INIT/.FALSE./ REAL DSQ,OMDSQ  -IS INITIALIZATION DONE ON FIRST CALL YET? - DECAY**2 AND 1-DECAY**2  INTEGER I,J,K C THE CODE: C IF (.NOT.INIT) THEN IF (NTAPS.GT.NTAPMX) THEN WRTTE(6,9999)NTAPS FORMAT(' ***** ERROR ***** IN SUBROUTINE CHEST:'/ 9999 * ' NTAPS=',G14.7,' TOO LARGE!', * ' CHANGE PARAMETER NTAPMX.') STOP ENDIF DSQ=DECAY**2 OMDSC=l.-DSQ  10 C C C 100  140 -  DO 10 1=1,NTAPS P(I,I)=(1.,0) CONTINUE INIT=.TRUE. ENDIF -SHIFT THE TRANSMITTED SYMBOLS DOWN THE TDL. DO 100 I=NTAPS,2,-1 X(I)=X(I-1) CONTINUE X(1)=REF -FORM THE A PRIORI ERROR COVARIANCE P(K|K-1)=PHI*P(K-1)*CONJGTRANSP(PHI) +Q(K) =P(K-1)*DECAY**2 +(1-DECAY**2)*DIAGMTX(MPGSQ)  DO 1000 1=1,NTAPS DO 900 J=l,NTAPS C***********P(I,J)=DSQ*P(I,J) C -THIS MOD GIVESTHE EXPONENTIAL AGING. P(I,J>=DS0*P<I,J>/LAMBDA IF (I.EQ.J) P(I,J)=P(I,J)+0MDSQ*MPGSQ(I-1) 900 CONTINUE 1000 CONTINUE C C -FORM EST=TRANSP(X)*PHI*C AND C ALPHA=DENOMINATOR OF G C =R+TRANSP(X)*P*CONJG(X) AND C NUMERATOR OF G = P*CONJG(X). EST=(0.,0.)  -  .  1100 1200 C C C C C  1250 1300 C C C  1400 1500 1600 C C C  1800 1900 2000 C C  141  ALPHA=R DO 1200 1=1,NTAPS EST=EST+X(I)*DECAYAC(I) G(I)=(0.,0.) DO 1100 J=l,NTAPS G(I)=G(I>+P(I,J)*CONJG<X(J)) CONTINUE ALPHA=ALPHA+X(I)*G(I) CONTINUE -FORM ERROR=THE ESTIMATION ERROR, THE KALMAN GAIN=NUMERATOR/REAL(ALPHA), UPDATE THE STATE VECTOR (IE. PATH TAP WEIGHTS), AND CALCULATE TEMP=CI-G*TRANSP(X>3.  ERROR=REC-EST DO 1300 1=1,NTAPS G(I)=G(I)/REAL(ALPHA) C(I)=DECAY*C(I)+G(I)*ERROR DO 1250 J=l,NTAPS IF (I.EQ.J) THEN TEMP(I,I)=(1.,0.) ELSE TEMP(I,J)=(0.,0.) ENDIF TEMP(I,J)=TEMP(I,J)-G(I)*X(J) CONTINUE CONTINUE  -CALCULATE T1=TEMP*P =CI-G*TRANSP(X)3*P  DO 1600 1=1,NTAPS DO 1500 K=l,NTAPS T1(I,K)=(0.,0.) DO 1400 J=l,NTAPS T1(I,K)=T1(I,K)+TEMP(I,J)*P(J,K) CONTINUE CONTINUE CONTINUE  -FINALLY FORM THE A POSTIORI P= Tl*CONJGTRANSP(TEMP>+G*R*CONJGTRANSP(G).  DO 2000 1=1,NTAPS DO 1900 K=l,NTAPS P(I,K)=(0.,0.) DO 1800 J=l,NTAPS P<I,K)=P(I,K)+T1(I,J)*C0NJG(TEMP(K,J)> CONTINUE P(I,K)=P(I,K)+G(I)*R*CONJG(G(K)) CONTINUE CONTINUE RETURN END  -  - 142 REFERENCES  [1]  G . J . Beirman, "Measurement Updating Using the U-D F a c t o r i z a t i o n , " IEEE Conf. on Decision and C o n t r o l , Houston, TX, December, 1975.  [2]  G . J . Beirman, F a c t o r i z a t i o n Methods f o r Discrete Sequential Estimation, Academic P r e s s , New York, 1977.  [3]  C.A. B e l f i o r e and J . H . Park J r . ,  "Decision Feedback E q u a l i z a t i o n , " Proc.  of the IEEE, V. 67(8), Aug. 1979, pp. 1143-1156.  [4]  S . A . B i l l i n g s and C . J . H a r r i s , Self-Tuning and Adaptive C o n t r o l : Theory and A p p l i c a t i o n s , I . E . E . E . , New York, 1981.  [5]  S.M. Boziac, D i g i t a l and Kalman F i l t e r i n g , Edward Arnold L t d . , London, 1979.  [6]  G. Carayannis, D. Manolakis, N. K a l o u p t s i d i s , "A Fast Sequential Algorithm f o r Least-Squares F i l t e r i n g and P r e d i c t i o n , " IEEE Trans, on ASSP, V. ASSP-31(6), December 1983, pp. 1394-1402..  [7]  N.A. Carlson and A . F . Culmore, " E f f i c i e n t Algorithms f o r On-Board Array P r o c e s s i n g , " I n t l . Conf. on Comm., Boston, MA, June 1979.  [8]  J . M . C i o f f i , " F a s t , Fixed-Order, Least-Squares Algorithms f o r Communications A p p l i c a t i o n s , " Ph.D. D i s s . , Stanford U., 1983.  - 143 [9]  J . M . C i o f f i and T. K a i l a t h , "Fast, Fixed-Order,  Least-Squares Algorithms  f o r Adaptive F i l t e r i n g , " IEEE I n t l . Conf. on ASSP, Boston, MA, A p r i l 1983.  [10] J.M. C i o f f i and T. K a i l a t h , " F a s t , Recursive-Least-Squares,  Transversal  F i l t e r s f o r Adpative F i l t e r i n g , " in r e v i s i o n , IEEE Trans, on ASSP.  [11] J . M . C i o f f i and T. K a i l a t h , "Windowing Methods and t h e i r  Efficient  Transversal F i l t e r Implementation f o r the RLS A d a p t i v e - F i l t e r i n g C r i t e r i o n , " submitted to IEEE Trans, on ASSP.  [12] D.C. C o l l , D.A. George, "The Reception of Time Dispersed P u l s e s , " Conf. Record 1965 IEEE Ann. Commun. Conv., pp. 749-752.  [13] F. David, A.G. Franco, H. Sherman, L.B. Shucavage,  "Correlation  Measurements on an HF Transmission L i n k , " IEEE Trans, on Comm. T e c h . , V. COM-17(2), A p r i l 1969.  [14] H. de Pedro, F. Hsu, A. Giordano, J . P r o a k i s , "Signal Design f o r High Speed S e r i a l Transmission on Fading Dispersive Channels," National Telecom. C o n f . , Birmingham, AL, December 1978.  [15] E.W. Derbyshire, "Radiotelephone Automatically Rates HF Channels to Improve Message Quality to Remote S u b s c r i b e r s , " Communications Systems, June/July 1982.  [16] D.L. D u t t w e i l l e r , J . E . Mazo, "An Upper Bound on the Error P r o b a b i l i t y in Decision-Feedback E q u a l i z a t i o n " , IEEE Trans, on Info. Theory, V. I T - 2 0 ( 4 ) , J u l y 1974, pp. 490-497.  - 144 [17] D.D. Falconer, G . J . F o s c h i n i , "Theory of Minimum Mean-Square-Error QAM Systems Employing Decision Feedback E q u a l i z a t i o n , " BSTJ, V. 52(10), December 1973, pp. 1821-1849.  [18] D.D. Falconer, " J o i n t l y Adaptive Equalization and C a r r i e r Recovery in Two-Dimensional D i g i t a l Communications Systems," BSTJ, V. 55(3), March 1976, pp. 317-334.  [19] D.D. Falconer, "Analysis of a Gradient Algorithm f o r Simultaneous Passband Equalization and C a r r i e r Phase Recovery," BSTJ, V. 55(4), A p r i l 1976, pp. 409-428.  [20] D.D. Falconer, L. Ljung, " A p p l i c a t i o n of Fast Kalman Estimation to Adaptive E q u a l i z a t i o n , " IEEE Trans, on Comm., V. C0M-26(10), October 1978, pp. 1439-1446.  [21] D.D. Falconer, V . B . Lawrence, S . K . Tewksbury,  "Processor-Hardware  Considerations f o r Adaptive D i g i t a l F i l t e r Algorithms," I n t l . Conf. on Comm., S e a t t l e , WA, June 1980.  [22] D.D. Falconer et a l , "Comparison of DFE and MLSE Receiver Performance on HF Channels," submitted to Globecom ' 8 3 .  [23] D.D. Falconer, H.M. Hafeg, A.U.H. Shiekh, E. E l e f t h e r i o u , M. Tobis,  "A  Comparative Study of Maximum Likelihood and Adaptive Equalization f o r High Rate, High Frequency Data Communications," Dept. of Systems and Computer Engineering, Carlton U n i v e r s i t y ,  1983 (Limited  Distribution).  [24] A. Gelb ( E d . ) , Applied Optimal E s t i m a t i o n , MIT Press, Cambridge, 1974.  - 145 [25] R.D. G i t l i n , S . B . Weinstein, "Fractionally-Spaced E q u a l i z a t i o n : An Improved D i g i t a l Transversal E q u a l i z e r , " BSTJ, V. 60(2), February 1981, pp. 275-296.  [26] R.D. G i t l i n , H.C. Meadors, and S . B . Weinstein, "The Tap-Leakage Algorithms: An Algorithm f o r the Stable Operation of a D i g i t a l l y Implemented, F r a c t i o n a l l y Spaced Adaptive E q u a l i z e r , " BSTJ, V. 61(8), October 1982, pp. 1817-1839.  [27] Glenayre E l e c t r o n i c s L t d . , GL1102 Pulse Compression Modem Engineering Manual, Glenayre Doc. GL3000-030, Issue 2, October 1983.  [28] D. Godard, "Channel E q u a l i z a t i o n Using a Kalman F i l t e r f o r Fast Data Transmission," IBM J . of Res. Develop., May 1974, pp. 267-273.  [29] D. Godard, "Self-Recovering Equalization and C a r r i e r Tracking in Two-Dimensional Data Communications Systems," IEEE Trans, on Comm., V. C0M-28(11), November 1980.  [30] F.M. Hsu et a l , "Adaptive Equalization Techniques f o r High-Speed Transmission on Fading Dispersive HF Channels," National Telecom. Conf., Houston, TX, 1980.  [31] F.M. Hsu, "Square Root Kalman F i l t e r i n g f o r High-Speed Data Received over Fading Dispersive HF Channels," IEEE Trans, on I n f o . Theory, V. I T - 2 8 ( 5 ) , September 1982, pp. 753-763.  [32] A.H. J a z w i n s k i , "Adaptive F i l t e r i n g " , Automatica, V. 5, 1969, pp. 475-485.  - 146 [33] A.H. Oazwinski, S t o c a s t i c Processes and F i l t e r i n g Theory, Academic Press, New York, 1970.  [34] R.E. Kalman, "A New Approach to Linear F i l t e r i n g and P r e d i c t i o n Problems," Trans. ASME, Series D, Journal of Basic Engineering, V. 8 2 , 1960, pp. 35-45.  [35] R.E. Kalman, R.S. Bucy, "New Results in Linear F i l t e r i n g and P r e d i c t i o n Theory," Trans. ASME, Series D, Journal of Basic Engineering, V. 8 3 , 1961, pp. 95-108.  [36] M.G. Larimore, J . R . T r e i c h l e r ,  "Convergence Behaviour of the Constant  Modulus A l g o r i t h m , " I n t l . Conf. on ASSP, Boston, MA, A p r i l 1983.  [37] D.T.L. Lee, M. Morf, B. F r i e d l a n d e r , "Recursive Least Squares Ladder Estimation Algorithms," IEEE Trans, on ASSP, V. ASSP-29, June 1981, pp. 627-641.  [38] F. L i n g , J . G . P r o a k i s , "Generalized Least-Squares L a t t i c e Algorithms and t h e i r A p p l i c a t i o n to Decision Feedback E q u a l i z a t i o n , " I n t l . Conf. on ASSP, P a r i s , May 1982.  [39] L. Ljung, M. Morf, D. Falconer, "Fast C a l c u l a t i o n of Gain Matrices f o r Recursive Estimation Schemes," I n t . J . C o n t r o l , V. 27(1), 1978, pp. 1-19.  [40] R.W. Lucky, "Automatic Equalization f o r D i g i t a l Communication," BSTJ, V. 4 4 ( 4 ) , Part 1, A p r i l 1965, pp. 547-588.  - 147 [41] R.W. Lucky, H.R. Rudin, "An Automatic Equalizer f o r General-Purpose Communication Channels," BSTJ, V. 46, November 1967, pp. 2179-2208.  [42] MacDonald, D e t t w i l e r , and Assoc. L t d . , "High Speed HF Modem using Adaptive E q u a l i z a t i o n , " F i r s t to Fourth Milestone Reports, Prepared f o r DOC Communications Research Center, Contract #0ST81-00057, 1981.  [43] F.R. Magee J r . ,  " F i l t e r i n g Techniques f o r Time Dispersive Channels,"  Ph.D. Thesis, Northeastern U n i v e r s i t y , 1972.  [44] P. Monsen, "Adaptive E q u a l i z a t i o n of the Slow Fading Channel," IEEE Trans, on Comm., V. C0M-22(8), August 1974, pp. 1063-1075.  [45] P. Monsen, "Fading Channel Communications," IEEE Comm. Magazine, Jan.  1980, pp. 16-25.  [46] D.R. Morgan, "Adaptive Multipath Cancellation f o r D i g i t a l Data Communication," IEEE Trans, on Comm., V. C0M-26(9), September 1978, pp. 1380-1390.  [47] M.S. M u e l l e r , "Least-Squares Algorithms f o r Adaptive E q u a l i z e r s , " BSTJ, V. 60(8), October 1981, pp. 1905-1925.  [48] J . Pennington, "Comparative Measurements of P a r a l l e l and S e r i a l 2.4 kbps Modems," Second Conf. on HF Com. Sys. and Techniques, IEE, London, February 1982.  [49] F.A. P e r k i n s , D.D. McRae, "A High-Performance HF Modem," I n t l . E l e c . Expo, Hanover, May 1982.  Defense  - 148 [50] B. Porat, B. F r i e d l a n d e r , M. Morf, "Square Root Covariance Ladder Algorithms," IEEE Trans, on Auto. C o n t r o l , V. AC-27(4), August 1982, pp. 813-829.  [51] J . G . P r o a k i s , "Advances in E q u a l i z a t i o n f o r Intersymbol  Interference"  Advances In Communications Systems: Theory and A p p l i c a t i o n s  in  (A.J.  V e r t e r b i , E d . ) , Academic Press, New York, 1975.  [52] J . G . P r o a k i s , D i g i t a l Communications, M c G r a w - H i l l , New York, 1983.  [53] H.R. Rudin, "Automatic Equalization Using Transversal F i l t e r s , " IEEE Spectrum, V. 4, January 1967, pp. 5 3 - 5 9 .  [54] J . S a l z , "Optimum Mean-Square Decision Feedback E q u a l i z a t i o n , " BSTJ, V. 52(8), October 1973, pp. 1341-1373.  [55] E.H. S a t o r i u s , S.T. Alexander, "Channel Equalization Using Adaptive L a t t i c e A l g o r i t h m s , " IEEE Trans, on Comm., V. C0M-27(6), June 1979, pp. 899-905.  [56] E.H. S a t o r i u s , M.J. Shensa, "On The A p p l i c a t i o n of Recursive Least Squares Methods to Adaptive P r o c e s s i n g , " I n t l . Workshop on Appls. of Adaptive C o n t r o l , Yale U., New Haven, CT, August 1979.  [57] E.H. S a t o r i u s , J . D . Pack, " A p p l . of Least Squares L a t t i c e Algorithms to Adaptive E q u a l i z a t i o n , " IEEE Trans, on Comm., V. COM-29, pp. 136-142, February 1981.  - 149 [58] M.J. Shensa, "A Least-Squares L a t t i c e Decision-Feedback E q u a l i z e r , " I n t l . Conf. on Comm., S e a t t l e , WA, June 1980.  [59] H.W. Sorensen and J . E . Sacks, "Recursive Fading Memory F i l t e r i n g , " Information Sciences, V. 3 , 1971, pp. 101-119.  [60] J . R . T r i e c h l e r ,  B.G. Agee, "A New Approach to Multipath Correction of  Constant Modulus S i g n a l s , " IEEE Trans, on ASSP, V. ASSP-31(2), A p r i l 1983, pp. 459-471.  [61] G. Ungerboeck,  " F r a c t i o n a l Tap-Spacing Equalizer and Consequences f o r  Clock Recovery i n Data Modems," IEEE Trans, on Comm., V. C0M-24(8), August 1976, pp. 856-864.  [62] C.C. Watterson, J . R . Juroshek, W.D. Bensema, "Experimental Confirmation of an HF Channel Model," IEEE Trans, on Comm. Tech., V. C0M-18(6), December 1970, pp. 792-803.  [63] C.C. Watterson and C M . M i n i s t e r , "HF Channel-Simulator Measurements and Performance Analyses on the USC-10, ACQ-6 and MX-190 PSK Modems," U.S. Dept. of Commerce, OT Report 75-56 (NTIS COM-75-11206), 1975.  [64] B. Widrow, J . McCool, M. B a l l , "The Complex LMS A l g o r i t h m , " Proc. of the IEEE, V. 63, A p r i l 1975.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0096286/manifest

Comment

Related Items