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

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PERFORMANCE OF KALMAN DECISION-FEEDBACK EQUALIZATION IN HF RADIO MODEMS by RUSSELL JOHN TRONT B . S c , The University 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 th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH December 1983 (£) Russell John Tront, COLUMBIA 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission f o r extensive copying of t h i s thesis for s c h o l a r l y purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date (%AX !1} If?1/ DE-6 (3/81) i i ABSTRACT The performance of fast -adapting Kalman algorithms in adjusting the tap weights of d i g i t a l equalizers for high-speed, s e r i a l HF radio modems i s invest igated. 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 inear and decision-feedback equalizers with symbol and half-symbol ( f r a c t i o n a l l y ) spaced taps . 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 decis ion-d i rected mode, have been determined as a function of the channel c h a r a c t e r i s t i c s . The resul ts show the super ior i ty of the Kalman, f rac t iona l l y - tapped decision-feedback equal izer 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 equal iz ing the HF radio channel. At slow fade rates, reference-directed performance within 3-5 dB of optimum is poss ib le . With fade rates above 0.5 Hz though, the technique becomes fade-rate l i m i t e d . A theoret ical argument leads to a new d e f i n i t i o n of a "slow fading channel" for exponentially-aged Kalman equa l i ze rs . Performance in dec is ion -d i rected mode, necessary for actual communication, was much poorer than expected. It was concluded that the continously -high Kalman Gain necessary on the HF channel resul ts in such adaptation at each i t e r a t i o n that even iso lated decision errors can ser ious ly disrupt the adaptation process. In spi te of the dec is ion -d i rected degradation, the s e r i a l equal izer technique is 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 . i i i TABLE OF CONTENTS ABSTRACT TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES GLOSSARY OF ABBREVIATIONS ACKNOWLEDGEMENT 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 Equal izer Configurations 22 3.2 Equalizer Tap-Weight Adjustment Algorithms 29 3.3 Synchronization and Training 39 i i i i i vi v i i i ix x iv TABLE OF CONTENTS (con'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 Appl icat ion 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 Equal izer 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 Retraining 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 fo 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 - Detai ls of Real Channel Tests and Data Tapes 123 APPENDIX D - Second-Order Channel Simulator Subroutine 125 APPENDIX E - Fractional ly -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 Equal izer 24 3.3 Lat t ice Stage 25 3.4 A Linear Lat t i ce Equal izer 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) = 1 ,0 ,1 ,0 ,1 ) 44 4.2 (1 ,0 ,1 ,0 ,1 ) DFE Performance as a Function of q ( .15, 1.50 Hz) 45 4.3 (1 ,0 ,1 ,0 ,1 ) DFE Performance as a Function of q ( .05 , .50, 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) = ( . 3 3 , 0 , 1 , 0 , - 3 3 ) ) 53 4.10 ( .33 ,0 ,1 ,0 .33) 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 cont '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 Insens 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 Per iod ic -Ret ra in ing Decision-Directed Results 97 7.2 Decis ion-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 Equal izer 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 Per iodic Retraining 96 7.2 Half -Decis ion-Directed Mode Comparison 102 7.3 Performance of Constant Modulus Algorithm 107 C. l Tape Recording Conditions 123 C.2 Star t ing Sample of M-Sequence on Tapes 124 GLOSSARY OF ABBREVIATIONS BER - B i t Error Rate CMA -Constant Modulus Algorithm DFE -Decision-Feedback Equal izer E b -Energy per b i t FIR - F i n i t e Impulse Response FT -Fract ional ly -Tapped HF -High Frequency IIR - I n f i n i t e Impulse Response ISI - Inter-Symbol Interference LE -L inear Equal izer LPF -Low Pass F i l t e r MLSE -Maximum Likel ihood Sequence Estimation MUF -Maximum Usable Frequency N -Number of Equal izer Taps No -Noise Spectral Density (one-sided) 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 X ACKNOWLEDGMENT I would l i k e to thank Dr. Jim Cavers f o r h is patience, knowledge, and i n f i n i t e enthusiasm in d i rec t ing the pro ject , and Dr. M.R. Ito fo r his very wise guidance regarding thes is work in industry , academic supervision of the work, and his help in preparation of the thes i s and i t s defense. Glenayre E lectronics 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, in-house desk space, computer t ime, word-processing access, e tc . 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 Ass is tantsh ip , Glenayre E lec t ron ics , 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 rad io , in the high frequency band of 3-30 MHz, of fers 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 ixed 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 -per -second 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 resu l ts from r igs to home base for ana lys i s . 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 industry 's need fo 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 app l i ca t ions . The HF channel enables over - the -hor iz ion communication without the large d i rec t iona l antennas and centra l i zed wide-band nature of s a t e l l i t e communication. This is possible because HF radio energy can r e f l e c t o f f several of the turbulent layers in the ionosphere. It i s characterized as a fading multipath channel because severa l , independently-fading rep l i cas of the transmitted message are received, each with a s l i g h t l y d i f fe rent propagation delay. - 2 -Present modems fo r the HF channel use low symbol-keying r a t e s . This avoids the intersymbol interference (ISI) caused by the channel delay spread through use of very long symbol per iods. To at ta in the required throughput, current high-speed modems use mult ip le p a r a l l e l tones to transmit many b i t s per symbol. It i s also desirable on HF to use the d i v e r s i t y in the d i f fe rent paths f o r reducing the degradation caused by fad ing . 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 . These modems have many other drawbacks. 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 interest has been shown in high-speed s e r i a l t ransmission. 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 equal izer 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 equal izer must of course be adaptive as the channel d i s to r t ion is cont inual ly changing. It i s conjectured that s e r i a l equal i zers , 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 fo r mobile app l icat ions . Ser ia l equal izer 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 just 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 -algor i thm l inear equalizers used in o lder , high-speed telephone modems are not appropriate. It has been suggested that a recurs ive - least -square (RLS) Kalman algorithm used in a decision-feedback equal izer (DFE) might work well on HF. Kalman algorithms are being applied in telephone modems fo r rapid i n i t i a l convergence. These fas 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 inear 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 error propagation problems can be shown to be not too s i g n i f i c a n t . This problem is of serious interest as the fading nature of HF channels v i r t u a l l y guarantees occasional er ror propagation and "crashes" of the equal izer . This project investigates the performance of RLS DFE's under a very wide var iety of simulated HF operating condit ions using 2400 bi t -per -second quadrature phase-shi f t keying (QPSK). Performance on a few real channels i s also checked. 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 inear equal izer configurations when used on t yp ica l HF channels. We also test the resistance to sampling phase of f rac t iona l l y - tapped (FT) decision-feedback equa l i ze rs . F i n a l l y , we b r i e f l y invest igate the performance degradation in dec is ion -d i rected mode (which must be used fo r true communication) and test several other algorithms and protocols f o r preventing and recovering from crashes. The reader w i l l f ind i t helpful to have some knowledge of e i ther equal izers or Kalman est imat ion. - 4 -1.2 Other Work Previous work on the HF channel i t s e l f , i t s modell ing, and i t s data communication problems w i l l be discussed in Section 2. The advantages of the decision-feedback equal izer configurat ion and of recurs ive - least -square -er ror algorithms are well es tab l i shed . They w i l l be reviewed i n Sections 3.1 and 3.2 respect ive ly . These techniques are d i r e c t l y appl icable to HF equal izat ion but they have yet to have t h e i r performance quant i f ied 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 current ly ava i lab le work on high-speed HF modem performance. There i s in fac 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 . There are several reasons fo r t h i s . F i r s t l y , the channel i s frequency-dependent and continuously changing, making comparison of new techniques under ident i ca l condit ions d i f f i c u l t . Unlike the addit ive 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 s t a t e . Secondly, these many parameters are d i f f i c u l t and computationally burdensome to simulate on a computer, espec ia l l y in real t ime. F i n a l l y , much of the s e r i a l modem research that i s ava i lab le seems purposefully vague, obviously due to proprietary and/or m i l i t a r y considerat ions. 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 funct ion of a few values of multipath spread, fade ra te , s igna l - to -no i se r a t i o (SNR), interference, 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 for 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, th i s work was somewhat of a landmark in i t s scope and w i l l provide some l imi ted data with which to compare our work. In [48], Pennington reports fo 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 lock -e r ro r - ra te resu l ts are presented as a funct ion of SNR for 3 two-path fading channels. He concludes the s e r i a l modem i s bet ter , but unfortunately shows s e r i a l resul ts 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 conf igurat ion, algorithm, modulation, operating mode, and crash prevention measures. Perkins and McRae of Harris Corporation [49] report a s imi la 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 ret ra in ing symbols at 4800 bps to achieve a net 2400 bps throughput. This makes the scheme very robust but requires a longer equal izer to cover the same multipath spread. Since many RLS algorithms have a complexity proportional to the square of the equal izer length, t h i s may be the reason they required an expensive b ipolar 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 te rates a minimum-mean-square error (MMSE) algorithm as many times as poss ib le . This no doubt achieves a decision-feedback e f fec t s imi la 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 ca l ib rated simulated-channel resu l ts are very sketchy making comparison with our resul ts 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 resul ts with others, namely Pennington's (which were in some ways worse), as d i f fe ren t algorithms are probably used. It i s i ron ic they don't volunteer t h e i r ' s , but t h i s i s obviously an act ive product area fo r corporate and m i l i t a r y researchers. As discussed in the next sec t ion , t h i s thes is i s supported in part by Glenayre E lectronics of North Vancouver, B.C. exactly to acquire such algorithmic and performance research. It w i l l receive only l imited d i s t r i b u t i o n fo 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 resu l ts as a funct ion of SNR for 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 theoret ical and experimental curves he presents are also better than theoret ica l fading-channel l i m i t s , and he presents reference-directed resu l ts only . Nonetheless, Hsu's algor i thm, which seemed cor rect , was used in t h i s project . We invest igate performance on a much wider var ia t ion of channels including some actual recorded channels. Hsu's equal izer conf igurat ion 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 dec is ion -d i rected mode. F i n a l l y , in [22, 23] , Falconer et al compare a f rac t iona l l y - tapped RLS DFE with another demodulation method ca l led maximum l ike l ihood sequence estimation (MLSE). Falconer (and Monson in [44] for troposcatter channels) conclude that MLSE y i e l d s only marginal improvement in performance at the expense of computational complexity that grows exponential ly with multipath spread. This was of in terest as Falconer's work used some of the same real - 7 -channel recordings as our pro ject . He does not state which part of the tapes were used though, making comparison with our resu l ts qua l i ta t i ve 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 ava i lab le in th i s commercially s t rateg ic area. The work reported herein is thus r e l a t i v e l y o r i g i n a l and very t ime ly . 1.3 Description of Project and Thesis The main emphasis of the project was to study the performance and other issues surrounding the appl icat ion of a decision-feedback equal izer in a 2400 bps QPSK s e r i a l modem for HF rad io . The research was undertaken in-house at Glenayre E lectronics Limited in 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 ne 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 ) equal izer research report [42] . The report included 3 tapes of d i g i t i z e d , unequalized 2400 bps QPSK recordings received of f actual HF channels, as wel l as the FORTRAN code of a channel simulator and l a t t i c e equal i zer . The report presented RLS FT l inear equal izer performance resu l ts fo r the 3 tapes of data and a few f i xed ( i e . non-fading) but otherwise HF - l ike simulated channels. The resu l ts presented were from experiments conducted in reference-directed mode only . The unequalized data tapes had several problems and the FORTRAN code was neither documented nor conducive to modi f i ca t ion . Nevertheless, the project provided a well defined s ta r t ing point for a thesis project which would be very applied in nature and provide real world benef i ts . - 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 charac te r i s t i cs t yp ica l on the HF channel. b) Decide whether RLS Kalman algorithms could adapt quickly 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 equal izer (FT DFE) on t yp ica l HF channels. e) Determine the optimum value of various equal izer 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 equal izer 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 fo r actual communication. i ) B r i e f l y invest igate methods, protocols , and other equal izer configurations and algorithms to combat error propagation and crashing in dec is ion -d i rected mode. This f a i r l y extensive set of tasks , which required one man-year of e f f o r t , took the fo l lowing form: - 9 -a) Fami l ia r i za t ion with the data communications problems on HF. b) Review of the equal izer 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 equal izer as only a f i xed order equal izer was needed (a l a t t i c e generates a l l o rders ) . 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) Select ion from among the many published RLS algorithms of one or two which could be extended to the complex, f ract iona l l y - tapped decision-feedback equal izer conf igurat ion . f ) Coding of the channel s imulator , equa l i ze 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 thesis format is f a i r l y c lear 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 s imulat ion. Section 3 introduces the various equal izer conf igurat ions, reviews tap-weight adjustment algorithms, and mentions synchronization and t r a i n i n g issues. Section 4 presents the performance of various equal izer configurations on simulated channels. The resu l ts of symbol-spaced equal izers 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 fac tor was determined fo r operation i n reference-directed mode under various channel condi t ions . As the performance was found to be somewhat fade- rate l i m i t e d , a new d e f i n i t i o n of "slow fading" appl icable to exponentially-weighted RLS equalizers is suggested. Then, the FT equal izer resu l ts are presented with special - 10 -emphasis on performance as a funct ion of sampling phase and multipath delay st ructure . The FT equal izer performance was found to be insens i t ive to sampling phase, and gave much better performance than symbol-spaced equalizers on cer ta in multipath channel s t ructures . The last 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 jus t spans the multipath spread. Mistuning was found to be quite degrading so ca r r i e r frequency acquis i t ion and tracking w i l l probably be required in an actual modem implementation. Section 5 discusses channel multipath structure and fading spectra est imat ion, and presents estimates of the channels on the 3 test tapes. Section 6 summarizes the real channel tape resu l ts and compares them with the l a t t i c e report . Our l i near equal izer resu l ts are better than those in the previous report fo r reasons on which we only speculate. The f u l l FT DFE running in reference-directed mode with a tuning correct ion general ly performed very well on the real channels. Section 7 presents the dec is ion -d i rected mode r e s u l t s . Though per iodic ret ra in ing proved a robust scheme fo r coping with crashes and somewhat l i m i t i n g the error propagation, the dec is ion -d i rected resu l ts were ser ious ly degraded from those in reference-directed mode. S t i l l , the resu l ts 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 test ing 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 resu l ts and makes suggestions for further research. It a lso notes that further work i s just s ta r t ing 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 interest ever since the advent of shortwave rad io . 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 distant (1000+ km) receivers . This i s due to several macroscopic layers in the ionosphere being able to re f lec t the radio energy back to earth . 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 in 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 var ia t ion i s due to da i l y and seasonal v a r i a t i o n s , and to so lar disturbances. A sketch showing t yp ica 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 xed l i nk ) off several major layers as a funct ion of RF frequency. I I I I I _J I U f 1 f 2 MUF Frequency Figure 2.1 Ionospheric Sounding - 12 -The de ta i l s of t h i s ionospheric structure may vary from hour to hour. Occasional ly , at frequencies l i k e f - j , there i s no r e f l e c t i o n fo r hours. At other frequencies such as f 2 , there might be 3 independently and quite rap id ly (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 is quite smal l . Radio operators often choose to operate near the MUF, which var ies d i u r n a l l y , because of t h i s reduction in time d ispers ion . However with s e r i a l transmission and an equal i zer , frequency use well below the MUF i s p rac t i ca l and in fac t i s improved due the d i v e r s i t y ava i lab le from having several independently-fading paths. Such frequency use would also ease the necessar i l y global ra t ion ing 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 0) might look l i k e Figure 2 . 2 . h(r ) Propagation Delay Figure 2.2 Typical Instantaneous Impulse Response - 13 -At another nearby time the 3 de l ta functions might have d i f fe rent amplitudes. They might, in complex baseband representat ion, have a d i f ferent phase (which could be thought of as rotat ion about the X axis i n Figure 2 . 2 ) . At a very d i f ferent time or frequency, they might even have d i f fe rent propagation delays T^, or have d i f fe rent root-mean-square amplitudes. The Fourier transform of h(T,t) with respect to X i s 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 . Notice that not much energy i s passed outside 300-2700 Hz. There are also some serious var ia t ions i n response, and even n u l l s , within the pass band. The moderately-rapid var ia t ion over time and i r r e g u l a r i t y of frequency response on HF i s very unl ike most other channels. In p a r t i c u l a r , i t i s very unl ike telephone channels on which most previous equal i zat ion work has been done. The frequency response of even an unconditioned telephone l i n e i s general ly with in 5 dB of nominal across most of the band. 10 -\ 30OO Frequency (Hz) Figure 2.3 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 general ly considered [62] to have a Gaussian be l l shape. I t therefore r o l l s of f exponent ia l ly , which i s much faster 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 ha l f -w idth of the received spectrum when a pure sine wave i s t ransmitted. 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 receiver , t ransmit ter , or ionospheric layers . Values of 0.1-10 Hz are t yp ica l with the higher bandwidths more common on auroral channels. We can compute t d =l/21TB f as the decorrelat ion time of the fading processes. This i s a measure of the delay required to get time d i v e r s i t y . S i m i l a r i l y , i f X i s the multipath delay spread (of say the 3 paths i l l u s t r a t e d in Figure 2 .2 ) , then f d=l/27TT s i s the frequency decorrelat ion needed f o r in-channel frequency d i v e r s i t y . Since is 1-5 mi l l i seconds , fd ranges over 160-30 Hz respect ive ly . P a r a l l e l modems must transmit t h e i r redundant information on sub-channels at least th is f a r apart as whole sub-bands of width approximately f^ Hz can fade up and down dependently. Ser ia 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 nu l l s and pass through the channel. Such s ignals however, w i l l be badly d is tor ted by t h i s " f requency-select ive f a d i n g . " - 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 ransmitter and receiver on a frequency where ionospheric re f lec t ion i s current ly poss ib le , i i ) Noise and inter ference, i i i ) Mult ipath. iv ) Fading, v) Car r ie 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. F i r s t l y , ionospheric r e f l e c t i o n i s common, but not guaranteed in the 3-30 MHz band. As i l l u s t r a t e d in the las t sec t ion , there are occasional ly gaps in the band which vary from hour to hour unpredictably. I f your receiv ing s tat ion 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 lso , HF frequencies are used as mult ip le access channels so some frequencies may be busy. High level protocols such as the RACE System described in [15] are required for automated communication environments. Secondly, HF has the usual problems of noise and interference. To combat noise af ter considerable attenuation of the signal over long transmission distances, f i xed stat ions often use several hundred watts of t ransmitter power and d i r e c t i o n a l antennas. Mobile units usual ly have neither opt ion. Generally a l l that can be done is to use a low peak-to-mean power r a t i o modulation ( i e . PSK) to get the most out of the avai lable t ransmitter power. - 16 -Minor interference is common on HF as other users can sometimes be using the same frequency on the other side of the world. This might not prohibi t the use of the frequency, but i t can complicate c a r r i e r recovery. Interference is an even more d i f f i c u l t problem on HF because of the fading mul t ipath . 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 in te r fe r ing frequencies. Th i rd l y , the multipath on HF can span 5 mi l l iseconds (c f . telephone channels less than 1.75 ms). Unacceptable intersymbol interference (ISI) thus occurs at keying rates as low as 200 symbols/second. The HF impulse response usual ly consists of 1 to 4 d iscrete de l ta funct ions . This resu l ts i n severe nu l l s in the frequency response requir ing in-band frequency d i v e r s i t y . For low data rate modems, bandwidth expansion techniques can be used as in the Glenayre GLI 102 Pulse Compression Modem [27]. For higher data ra tes , mult ip le p a r a l l e l tones with sub-channel d i v e r s i t y are current ly used to transmit many b i t s per symbol per iod . This technique unfortunately resu l ts 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 te rnate l y , i f an adaptive equal izer i s used in the rece iver , 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 el iminat ion 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 . Fourthly , the HF channel multipaths are subject to independent Rayleigh f a d i n g . 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 ror rate (BER) obtainable. In p a r t i c u l a r , for a fading channel with L path d i v e r s i t y the QPSK error rate for large SNR i s asymptotical ly proportional to SNR"*-. The slope of BER versus SNR on a log - log graph i s thus - L . For one path we see the BER i s l imi ted to 1 0 " 2 even at say, 15 dB. Compare - 17 -t h i s to the non-fading BER proportional to e r f c ( S N R ^ ) ; values as low as 10" 7 are possible at 15 dB. If 10" 10 If 10 .-5 10 ^^^^^ \ HoU- \ FADIM& I 10 2D 30 E L / N O (dB) 10 Figure 2.4 Error Rate of QPSK on Various Channels On the other hand, note the improvements possible with d i v e r s i t y L greater than 1. The challenge i s to use the multipath fo 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. This i s the job of an adaptive equal i zer , our main t o p i c . The fade rates on HF channels require recursive least square (RLS) algorithms to adapt fas t enough, but unfortunately , these are very computationally in tens ive . A lso , because the - 18 -multipath delay spread is so large on HF, long equal izers are required. 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 te ra t ions (symbols) for RLS algorithms to converge to wi th in 3 dB of optimum equal izer output SNR [28] , the length of the equal izer required on HF i s a de f in i te hindrance. F i n a l l y , fading multipath channels make c a r r i e r acqu is i t ion and t rack ing , receiver sample t iming , and frame synchronization of suppressed-carrier signals extremely d i f f i c u l t . As mentioned in Section 3 . 3 , t h i s project only i m p l i c i t l y addresses these problems insofar as f rac t iona l - tapp ing makes sample t iming phase unimportant. Regarding the others, manual techniques were used so the research e f f o r t could concentrate on equal izer 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 iscrete - t ime HF channel s imulator . This i s a valuable f a c i l i t y as equal izer performance could thus be measured and compared in a repeatable manner as a function of e i ther the channel or equal izer parameters. High speed was not needed as the project experiments could be run as batch computer jobs . It was decided to simulate the channel completely in baseband. This allowed us to ignore cer ta in 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 factor 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 band - l im i t ing , the fading mult ipath, noise, and the down-conversion low-pass f i l t e r (LPF). This i s a p ipe l ine of l inear 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 ransmitter band- l imi t ing f i l t e r and the receiver LPF both have (baseband) 1200 Hz cu t -o f f s . Thus fo r a l l p r a c t i c a l purposes only one was needed in the s imulat ion . Because radio channels have a f i n i t e length impulse response, our simulator can take the form of a t ime-varying FIR f i l t e r . As shown in Figure 2.5 taken from [52] , t h i s can be implemented in a tapped delay l i n e (TDL) conf igurat ion. Additive noise r(/> Figure 2.5 Possible HF Channel Simulator Configuration [52] The tap spacing deserves some d iscuss ion . Since our complex, baseband (two-sided) spectrum is W=2400 Hz wide, the maximum possible resolut ion in - 20 -the time domain i s l/2400th of a second ( i e . T/2 where T i s the symbol per iod) . At the equal izer , a path located half way between two T/2 tap posit ions cannot be resolved from two paths, each half as strong, spaced at the adjacent T/2 taps . Thus, a higher resolut ion is simply not needed. In actual f a c t , a T/8 spacing i s used in 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 select the multipath spacing and sample timing to within about t h i s reso lu t ion . The Gaussian bel l -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 (T/8) through a low pass f i l t e r . This adjustable LPF must approximate the exponential r o l l - o f f of the fading spectra . The resu l t ing bandwidth-reduced, unit -var iance processes are then attenuated by the spec i f ied 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 equal izer r i ses at a rate greater than ( logar i thmica l l y speaking) 6 dB/octave of fading bandwidth. Obviously, a one-pole f i l t e r i s not su i table as the equa l i ze 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 fas te r than the fading spectrum is 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 cr i t ica l ly -damped two-pole f i l t e r , which is more Gaussian bel l -shaped than the f la t - topped Butterworth. - 21 -A 5 path baseband channel simulator has been coded i n FORTRAN 77, and is included in Appendix D. It has the fol lowing adjustable features : i ) selectable path spacing to with in 1/8th of a symbol period i i ) var iable RMS path amplitudes i i i ) var iable - 3 dB fading half-bandwidth iv) var iable 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 ) selectable sample timing to within 1/8th of a symbol period The reader should understand three things about the simulation software before proceeding. F i r s t l y , s i g n a l - t o - n o i s e ra t ios herein are measured as energy-per -bi t over the one-sided noise spectral density (E^/NQ) in d e c i b e l s . This i s a very common measure in 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 . For QPSK, 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 . Secondly, the multipath spacing i s speci f ied 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 re la t i ve delay of the remaining 4 paths in units of T/8. The re la t i ve RMS strength of the 5 paths are speci f ied 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) i s automatical ly normalized by the main program for unit transmitted power; only the re la t i ve 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) fo 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 equal izat ion fo r data communication is a r e l a t i v e l y recent topic dating back only to 1965 [12, 40, 53] . The channel impulse response spreads out the information in a s ingle symbol over several received sample periods causing intersymbol interference ( IS I ) . An equal izer attempts to "gather" th is information back together again. 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 equal izer configurat ions in t h i s sub-section and adaptation algorithms in the next . One configuration fo r an equal izer i s a tapped delay l i n e (TDL), as i l l u s t r a t e d in Figure 3.1 taken from [52] . Unequalized Algorithm for lap gain adjustment Figure 3.1 Basic Equalizer Configuration [52] - 23 -A l inear equal izer (LE) uses a l inear 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. It i s thus an adaptive f i n i t e impulse response (FIR) f i l t e r . 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 di f ference between the equalized symbol estimate (equalizer output) and the exact symbol that was t ransmitted, i s used in the adaptation algorithm. Equalizers usual ly have to be trained on known header symbols in what is ca l led reference-directed mode. They are then switched into dec is ion -d i rected mode where the receiver does not know what is being t ransmit ted . Instead, the equalized symbol estimate quantized (rounded) to the nearest symbol in the modulation cons te l la t ion i s used to ca lcu late the e r r o r . This quantized symbol i s ca l led a dec i s ion . It can occasional ly be incorrect causing some adaption in the wrong d i r e c t i o n . It is 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, error propagation can occur which sometimes resu l ts in the equal izer gett ing lost or crashing. A l inear equal izer i s less than ideal fo r two reasons. It t h e o r e t i c a l l y must be i n f i n i t e l y long, even for 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 ins ide the span. In add i t ion , there are often zeroes (nul ls ) 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 var iat ion 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-equal ized symbols into the summation which estimates the next symbol. This y ie lds equal izer poles which can cancel the channel zeroes. There are two kinds of feedback equal i zers : l i near and decision feedback. In a l inear-feedback equal izer the equalized symbol estimate i s fed back. This technique is discussed in [43, 51] and shows l i t t l e improvement over a LE. In a decision-feedback equal izer (DFE), the equalized symbol estimate i s quantized to the nearest symbol and t h i s decision i s fed down the dec is ion -feedback part of the TDL. The technique i s reviewed with an extensive set of references by Be l fo i re and Park [3 ] . A DFE equal izer with LMS algorithm tap-weight adjustment (to be discussed in Section 3.2) i s i l l u s t r a t e d in Figure 3.2 taken from [52]. Input (r,) Received-Sample Feedforward Part Decision-Feedback Part Figure 3.2 An LMS Decision-Feedback Equalizer [52] The advantage of a DFE is 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 so , 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 taps ; just enough to span - 25 -the impulse response durat ion . As shown in [12, 54] , no received-sample feedback taps are required. If there i s no addit ive 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 is noise however, i t behooves the equal izer to weight more forward taps as they too contain useful information. Using weight on the forward taps allows p a r t i a l averaging of the noise. A DFE r e l i e s heavi ly 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 error 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 inear equa l i ze r . This t rade -of f i s of serious interest 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 inear l a t t i c e report [42] purchased from another company. One l a t t i c e stage is shown in Figure 3.3 while a l a t t i c e equal izer i s shown in Figure 3.4 [both f igures are taken from 52] , Figure 3.3 Lat t ice Stage [52] - 26 -yU) /(') *i(0 Stage 1 /,(') Stage 2 * 2 C) * 2 (0 A") Stage N - 1 */v.|C) /,v-|C) Figure 3.4 A Linear Lat t i ce Equalizer [52] A l a t t i c e equal izer [47,56,57] orthogonalizes the tap adjustment problem so that one tap can be adjusted without d isturbut ing the others . This i s done by computing backward predictors and using these in the l inear combination forming the equalized symbol est imate. 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 quick ly because of t h i s orthogonal izat ion (see Figure 3 .6 ) . Lat t i ces are in terest ing fo r several other reasons. They simultaneously generate outputs fo r a l l equal izer lengths (number of taps, or order) up to a speci f ied maximum. They are configured very su i tab ly fo r hardware p ipe l in ing [ 2 1 ] . A l so , they were the f i r s t , recurs ive - least - square -er ror algorithm/ conf igurat ion to be developed with computational complexity proportional to N, the equal izer (maximum) length . The o r i g i n a l RLS algorithms had complexity of order N^. Unfortunately, ear ly l a t t i c e work was hindered by d i f f i c u l t and inconsistent notat ion , no published complex or f ract iona l l y - tapped 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 [47,58] . 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 p ropor t iona l i t y ) . F rac t iona l l y tapped (FT) equal izers [25] form t h e i r equalized symbol estimate from a l i near combination of receiver samples taken at in terva ls of less than a symbol period (note: only the forward part of an FT DFE i s FT). As discussed in Section 2 . 3 , with a transmission bandwidth of W the delay structure of the channel should be resolvable to within 1/W. An equal izer should use t h i s time resolut ion in i t s received sample TDL so as to extract a l l the avai lable channel and data information from the stream. The most common interva l i s half-symbol (T/2) spacing as t h i s allows Nyquist rate sampling of the baseband received audio, although other f ract ions are also possible [61] . Note that a T/2 equal izer i s s t i l l clocked at the symbol rate but the received FT samples in the feedforward section are sh i f ted two posi t ions each symbol per iod. The advantages of an FT equal izer are: i ) They usual ly provide sampling at the Nyquist rate thus e l iminat ing the need for a matched f i l t e r . An FT adaptive equal izer i s given enough information on the received sequence to funct ion as an adaptive d iscrete - t ime matched f i l t e r . i i ) They el iminate 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 resolut ion of c lose l y or non-symbol-period spaced mult ipaths, allowing more d i v e r s i t y and thus increased performance. - 28 -The disadvantages of FT equal izers are increased computational burden (espec ia l l y i f using a complexity N algor i thm), increased memory requirements, and slower adaptation. Some of these performance tradeoffs are explored in Section 4 . 3 . F i n a l l y , i f using equalizers with f r a c t i o n a l -tapping c loser 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 of f toward overflow or underflow numerical values and s t i l l be optimum. A solut ion to t h i s problem i s proposed in [26]. Though i t i s not equal izat ion in the symbol-by-symbol sense, maximum  l ike l ihood sequence estimation (MLSE) i s another intersymbol- interference combatting technique which should be mentioned. MLSE minimizes the symbol error over an ent i re sequence of symbol estimates approximately spanning the channel impulse-response durat ion . It i s a t h e o r e t i c a l l y better technique than symbol-by-symbol equa l i za t ion , but the V i terb i algorithm used in MLSE has a computational complexity of exponential order with increasing response durat ion. A l so , MLSE must run j o i n t l y with a channel estimator in a decision d i r e c t e d - l i k e mode. This in pract ice may hinder the technique. Several authors, Monson [44] fo 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 equa l i zer . In Section 8.2-we suggest th i s may be a hasty conclus ion. In the l i near l a t t i c e report [42], symbol and half-symbol spaced l inear equal izers were tes ted . 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 inear and decision-feedback equal izers are contrasted, while in Section 4.2 we present experimental resu l ts comparing T/l and T/2 DFE's. - 29 -3.2 Equal izer Tap-Weight Adjustment Algorithms For a time varying or unknown channel, the equal izer tap weights must adjust for the pa r t i cu la r t ra jec to ry of the state of the channel over t ime. One of the e a r l i e s t techniques f o r t h i s adaptation was the zero forc ing (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 inear equal izer to zero. Because the peak d i s t o r t i o n can be shown a concave funct ion of the tap weights, a steepest-descent algorithm can be used. This algorithm forces the c ross -cor re la t ion of the equal izer symbol-estimation error (denoted e(k)) with the desired symbol sequence (obtained by dec is ion -d i rec t ion ) 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 equal izer w i l l converge to the truncated, i n f i n i t e equal izer weights. A la te r technique ca l led the (stochast ic gradient) least mean square  (LMS) algorithm [52,64] attempts to minimize E [ e 2 ( k ) ] . The i te ra t ion of the algorithm t r i e s to minimize the c ross -co r re la t ion between e(k) and the received sample stream, not the desired symbol stream as in the ZF algor i thm. If there is no cor re lat ion between e(k) and the received samples, then no fur ther 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 : e(k) = d(k) - X ' (k )C(k - l ) (3.1) C(k) = C (k - l ) + uX*(k)e(k) (3.2) where d(k) = the reference or decis ion at i t e ra t ion 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 ze 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 error sequence, e(k) , orthogonal to the information in X(k). Figure 3.2 in the las t sub-section i l l u s t r a t e s the LMS tap adjustment mechanism for a DFE. LMS algorithms do not minimize the mean square error in an optimum recursive sense. The tap weights only tend in that d i rec t ion s t o c h a s t i c a l l y . I f the channel var ia t ion i s very slow and the convergence time not very c r i t i c a l , they work quite wel l and are r e l a t i v e l y e f f i c i e n t (requir ing only 2N m u l t ip l i ca t ions 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. Many var iat ions [30,52] have also been invest igated. Unfortunately, LMS algorithms w i l l not converge or adapt as f a s t as required for some app l i ca t ions . This i s because the technique uses X*(k)e(k) as an estimate of the gradient (with respect to C{k)) of the - 31 -error surface [52] , rather than E[X_(k)e(k)]. Only af ter many i te ra t ions w i l l C[k) adapt to the proper values. A lso , LMS convergence rate i s slowed by a large eigenvalue spread in the channel co r re la t ion matrix [57] , a r e l a t i v e l y common s i tuat ion on the HF channel. About 20 years ago, recursive least square (RLS) error algorithms were developed Kalman and Bucy [34 ,35] . These algorithms are the optimum  recursive technique f o r minimizing the least square cost funct ion . They maintain internal co r re la t ion information f o r more properly ca l cu la t ing the gradient mentioned above. For readers unfamil iar with Kalman est imat ion, Boziac 's book [5] i s an excel lent in t roduct ion , while Gelb's [24] i s a more advanced reference. The increased convergence and adaption speed of RLS algorithms is demonstrated in Figure 3.5 taken from [ 5 2 ] . Godard [28] showed them to converge, fo r s t a t i c channels with l i t t l e noise, to within 3 dB of the optimum output error 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 0 100 200 300 400 500 6' Number of iterations 00 700 800 900 -3.0 Figure 3.5 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 in which these algorithms are discussed is that of the optimum estimation of state var iables in a l inear system, given noisy measurements of some 1inear combinations of the s ta te . In an equal izer , the N optimum tap weights £ 0 P T ( k ) are the state va r iab les . Kalman estimation i s a technique fo r using noisy measurements to generate recursive estimates, £ ( k ) , of the state propagating through time v ia the system model: A ( k ) C 0 P T ( k - l ) + W(k) (3.3) where C 0Pr(k) A(k) state vector (Nxl) state t r a n s i t i o n matrix (NxN) W(k) system dr i v ing noise (Nxl) The measurement model i s : d(k) X ' ( k ) C 0 P T ( k ) + V(k) (3.4) where d(k) measurement ( l x l ) X(k) measurement vector (Nxl) V(k) measurement noise ( l x l ) - 33 I f : PACT CO = " C a P T ( k ) ) ( C ( K ) - C o n - ( k ) ) ' * ] = the actual state covariance matrix at i t e ra t ion k (NxN), Q(k) = E[W(k)W'*(k)] = the expected dr iv ing noise covariance matrix (NxN), R(k) = E[V(k)V'*(k)] = the expected measurement noise covariance (1x1), then the exponential ly-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 £ 0 P T ( k ) P(k,k-1) = [A (k )P (k - l )A* ( k ) /A ] + Q(k) (3.7) = the a p r i o r i estimate of (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 ) (3.9) = the a pos te r io r i estimate of P A C T ( k ) and X = the exponential -aging constant. - 34 -The trace of P(k) (the sum of the estimated variances of the elements of the state estimate £ , from C. 0 P T ) i s the quantity which i s minimized in the exponential ly -aged, least -square sense. In the equal izer appl icat ion A(k) i s usually unknown so X 1 (K )C(k - l ) is the a p r i o r i symbol estimate, while X'(k)C(k) i s the a pos te r io r i symbol est imate. 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 internal model of system. If t h i s model information is not per fect l y known (or i f there are numerical prec is ion d i f f i c u l t i e s in the model l ing) , then i t i s possible for the state estimate C{k) to diverge from C. 0 P T (k) more than the trace of P(k) would indicate reasonable. 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 exponential ly -aged, a poster io r i symbol-estimation squared er ror , T.\ [d(k)-X' (k )C (k ) ] 2 , i s also minimized. 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 in fac t the dynamic-channel equal izer case. The o r ig ina l appl icat ion of RLS algorithms to equal izat ion in t h i s form i s due to Godard [28] who addressed the problem of rapid equal izer convergence to a f i x e d , but unknown channel. He was the f i r s t to show that the equal izer 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 decis ion) while the measurement noise was deemed to occur in the tapped delay l i ne measurement vector, X(k). It doesn't r e a l l y matter as long as you regard e(k) , the measurement error which drives the adaptation, to be noisy with variance R(k). It i s in terest ing to note the s i m i l a r i t y of RLS equations (3.5) and (3.6) when cast in equal izer form (normally state t ranss i t i on matr ix , A(k) , i s unknown and thus taken to be the indent i ty matr i x ) : e(k)=d(k) - 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 interest ing 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 is just LMS with the scalar step s i z e , u, replaced by the r a t i o of the a pos te r io r i state covariance, P ( k ) , to measurement noise covariance, R(k) . The elements of C_ are thus updated according to the s ize and complex d i rec t ion of the error e ( k ) , the s i ze and d i rec t ion of the elements of Xjk) which through the respective elements of £ might have caused the e r ro 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 interpretat ion 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 cor re lat ion matr ix , P" 1(k)=E[X*(k)X'(k)]. Some authors [20,47] , seemingly separate from the Kalman community, have in fact derived s i m i l a r exponential ly -aged-only equations d i r e c t l y from the Weiner-Hopf equations and the matrix inversion lemma. The resu l t ing equations are very compact because Q=0, A=I, the ca lcu la t ion 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 further discussed in Section 4 . 3 . RLS algorithms have evolved in 3 general d i rec t ions . A l l are equally fas t to adapt; in fac t they t h e o r e t i c a l l y generate exact ly 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-modell ing can cause divergence and f i n i t e precis ion computations often resu l t in numerical i n s t a b i l i t y . Mis-modell ing has been attacked by exponential -aging [59] , s e l f - t u n i n g [32] , and l imi ted (or s l id ing ) memory weighting [11, 24] . Numerical -precision d i f f i c u l t i e s can cause P(k) to become non-posi t ive 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 pos i t i ve def in i teness 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 factors of P whose form guarantees pos i t i ve def in i teness . Some examples are the matrix square root of P and upper tr iangular/diagonal fac to r i za t ions such as : P(k)=U*(k)D(k)U'(k) These factored algorithms have been extensively treated in [1 ,2 , and 7 ] . - 37 -In [31 ] , Hsu applies an exponentially-aged var ia t ion of the U*DU' technique, which Carlson [7] c a l l s "factored inverse square-root Kalman," to equaliz ing the HF channel. As discussed in Appendix B, t h i s algorithm was used fo r the equal izer in t h i s pro ject . In terest ing ly , a l l the above factored algorithms both prevent P from losing pos i t i ve def ini teness and increase dynamic range as the square root of most var iables are stored and manipulated. Hsu does not state his algorithm is an exponentially-aged one but i t i s easy to show that his assumption that Q=qAPA'* resu l ts in such an algorithm with X= l/(l+q). See also Section 4 . 3 . The second d i rec t ion RLS equal izer algorithms have taken i s that of RLS l a t t i c e s . These formulat ions, as compared to the gradient l a t t i c e , simultaneously generate the RLS solut ion f o r a l l equal izer 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 iv ing more dynamic range [37,50] . The th i rd d i rec t ion RLS work has taken is toward " fas t " f ixed-order s o l u t i o n s . These have computational burden proportional to N but with a smaller c o e f f i c i e n t of propor t ional i ty than RLS l a t t i c e s . The f i r s t such algorithm was termed Fast Kalman [39, 20] . Unfortunately, the Fast Kalman algorithm proved to be even more numerically unstable than the o r ig ina l a lgor i thm. As t h i s project progressed however, even f a s t e r Kalman algorithms [6, 9] have been developed. The d isser ta t ion work by C i o f f i [8 ] , and other forthcoming papers by C i o f f i and Kai lath [10,11] point out many of the causes and solut ions 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 xed -order Kalman algori thm. - 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 ransmit ted . These algorithms should thus never crash in the l inear equal izer conf igurat ion. The f i r s t algor i thm, due to Morgan [46] , compares the average received spectrum with the theoret ica l spectrum expected. It then calculates an e(k) based on t h i s d i f ference only and uses an LMS-type tap update algor i thm. The second algorithm not requir ing dec is ion -d i rected t ra in ing was designed fo r use with constant-modulus signal modulations such as QPSK. The Constant Modulus Algorithm (CMA) [36, 60] uses only the directed rad ia l component of e(k) ( reca 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 equal izer form to an RLS decision-feedback conf igurat ion . The RLS algorithm makes i t fas ter 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 in dec is ion -d i rected mode. But under slower fading condit ions, where the whiteness of the data stream has time to show i t s e l f , the RLS CMA DFE seems more res is tant to crashing than a regular RLS DFE. - 39 -3.3 Synchronization and Training In t h i s sub-section we f i r s t discuss some methods fo r c a r r i e r recovery, receiver sample t iming , and frame star t synchronizat ion. These topics were not of par t i cu la r concern during the cont ro l lab le or manual conditions which existed during our experiments, and though n o n - t r i v i a l , they are not thought to be insurmountable in p rac t i ce . 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. Later in the sub-sect ion, some of the desirable conditions and methods f o r converging, t rack ing , and re t ra in ing equal izers 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 mul t ipath . Normal s ing le -s ide -band (SSB) radios have no means fo r c a r r i e r t racking 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 ront end of the rece iver . Even with the newer synthesized radios , the required frequency accuracies demonstrated in Section 4.3 are not poss ib le . 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 equal izat ion [18,19] , where equal izat ion takes place before subcarr ier down-conversion, has been suggested f o r other multipath channels as the equal izat ion 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 is not p rac t i ca l on HF as a T/8 equal izer i s required to sample above the Nyquist rate of the maximum image frequency during down-conversion. Such an equal izer used on a large-mult ipath-spread channel such as HF would require over 100 taps ; c l e a r l y impract ical from an implementational point of view. - 40 -Another scheme proposed in [18] and [19] , and noted herein as f e a s i b l e , would be to use the phase bias in the equal izer output symbol estimates to hold an automatic frequency control (AFC) l o c k . This i s termed d e c i s i o n -directed (or data-aided) ca r r i e r recovery. Unfortunately, the scheme requires the equal izer already be converged. This ra ises the question of how to acquire i n i t i a l c a r r i e r frequency of fset? Even i f i t were to work during convergence, c a r r i e r deviat ion 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 tar t convergence extremely d i f f i c u l t . 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 se lect ive fading on the HF channel, a more diverse set of tones might be required fo r r e l i a b l e c a r r i e r a c q u i s i t i o n . In our experiments c a r r i e r acqu is i t ion was not a problem. In the simulated channel experiments our channel simulator and equal izer ran in baseband, though a simulated RF mistuning of fset could be added i f des i red . During the real channel recordings the t ransmitter RF, subcarr ier , and symbol rate had been cesium locked, as were the receiver RF and sampling rate . In sect ion 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 t iming (symbol t iming in the QPSK case) , requires knowledge of the symbol rate and phase. It i s not affected by RF mistuning or doppler. Again, conventional techniques for sampling phase tracking perform poorly on HF because of the extreme time d ispers ion . As mentioned in the previous sub-sect ion , f ract iona l l y - tapped equal izers are a very sensible solut ion 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 fo 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 ta r t time of the f i r s t energy from t h i s sequence must be located. Then, the loca l ly -generated , known t ra in ing sequence can be applied to the equal izer as a reference during convergence in reference-directed mode. An appropriate scenario might be to send a polyphase Frank sequence, co r re la te , and note the locat ion of the peaks (which correspond to the mult ipaths) . One can then apply the reference to t r a i n the equal izer at the time when the f i r s t energy from the f i r s t path i s a r r i v i n g . It might be possible to f ind a sequence with a better ambiguity funct ion than the Frank sequence, as doppler can be mistaken f o r delay. In t h i s p ro ject , frame sync was not a problem as the channel simulator/ equal izer ran in lock-step and the real channel tape synchronization was handled manually. There are many issues that should be pointed out about our las t topic in t h i s s e c t i o n : t r a i n i n g the equal izer to converge and fo l low 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 condit ion of the ent i re passband is sounded. If not, the equal izer may adapt (or not adapt) i t s gain and phase properly across the band. This could happen when cer ta in long s t r ings , 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 equal izer may not immediately provide the correct f i l t e r i n g and could even crash. 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 spec i f ied in CCITT V.27 and i l l u s t r a t e d in Figure 3.6 would be des i rab le . The scrambler s ta r t has to be synchronized but t h i s i s e a s i l y done at the s tar t of a frame. In our experiments, we simply used a repeating, 1023 b i t maximal-length pseudo-random sequence fo r the data . Figure 3.6 CCITT V.27 Data Scrambler In the previous sec t ion , i t was mentioned that t ra in ing can also use the error in e i ther the received spectrum or symbol modulus. In the LE conf igurat ion , no knowledge of the transmitted sequence i s required to converge nor should such an LE crash. This w i l l be b r i e f l y investigated in Section 7.3 Per iodic inser t ion of re t ra in ing sequences, as discussed in Section 7, i s a robust way to prevent and recover from crashes. Unfortunately re t ra in ing can add considerable overhead to the data stream. In terest ing ly , 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 re t ra in ing overhead needed f o r the par t i cu la 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 Sect ion, we invest igate the b i t error rate (BER) performance of symbol-spaced equal izers in reference-directed mode on simulated HF channels. We are par t i cu la r ! ' l y interested in quantizing the performance of the recurs ive - least - square -e r ro r (Kalman) algorithm in a decision-feedback equal izer (DFE), and contrast ing i t with the performance obtained using e i ther the LMS algorithm or a l inear equal izer conf igurat ion. Results are presented as a funct ion 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 st ructure , and exponential-aging f a c t o r . Optimum or nominal values have been used f o r other parameters. For instance, we have used optimum sample timing phase, a nominal value for the expected Kalman measurement noise R, an equal izer length which just spans the multipath spread, and simulated perfect RF tuning. In Section 4 . 2 , performance of symbol-spaced and fract ional -symbol spaced equalizers are contrasted in reference-directed mode as a function of sampling phase and non-symbol-spaced multipath delay s t ructure . Section 4.3 investigates the e f fect of some of the remaining parameters. Figure 4.1 shows the f i r s t multipath channel structure invest igated . It i s composed of 3 equi-mean-square fading paths each separated by 2 symbol per iods. A l l channels discussed in t h i s thes is 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 integral number of symbol periods. As mentioned above, performance on non-symbol-spaced multipath channels i s - 44 -presented in the next sub -sect ion . The data transmitted was a repeat ing, 1023 b i t maximal-length pseudo-random sequence (M-sequence). RMS impulse response 0 T T i 3T 2T 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 decis ion 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 des i rab le , f i v e received-sample, feedforward taps are needed to span the mult ipath. We show in 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 faster -adapt ing length . A l l runs were 600,000 symbols (500 simulated seconds) long. A l l SNR f iqures quoted in t h i s thes is are energy/bit over the noise spectral density (E b /N 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 character izat ion of exponentially-aged Kalman equal izer performance on HF channels. Raw probab i l i t y of b i t error in reference-directed mode i s shown as a function of Hsu's exponential -aging fac to r , q=( l/X) - l , for s i g n a l - t o - n o i s e ra t ios 13, 18, 23, and 28 dB. Results f o r fading bandwidths B f 0 f .15, and 1.5 Hz are shown in Figure 4 . 2 , while bandwidths of . 0 5 , . 5 , and 5.0 Hz are in Figure - 45 -Several things are immediately apparent. F i r s t , each channel condit ion has a q value which gives a minimum probab i l i t y of e r ro r . This makes sense as a small q may give too slow an adaptation ra te , while a large q causes the equal izer to adapt too quickly thus responding to noise. Secondly, we see the optimum value of q i s a funct ion 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 var ia t ion versus the noise l e v e l . I -3 /o 1 10 0.15 Hz, I3JB \ -——**"i5 Hz, !8dB I.SHz, 23d& /.5Hz,28dB 0.15 Hz,/SdB 0..5Hz,23dB 0J5Hz^ 2So(B .00 JO .10 .30 Figure 4.2 ( 1 , 0 , 1 , 0 , 1 ) DFE Performance as a Function of q ( .15 , 1.5 Hz) - 46 -10 -10 ,-3 10 10 ^5.0 Hz, )3 jB 5.0Hz, ISOIB • 0.5 Hz, 13 JB /6.05Hz,l8a& ^ 0.5Hz, l«dB 0.5Hz,23dB ^J) .5Hz,2&lB .00 JO .10 .30 Figure 4.3 ( 1 , 0 , 1 , 0 , 1 ) DFE Performance as a Function of q ( .05 , . 5 , 5 Hz) Because of the inverse nature of the equal izer problem, i t i s not possible to get a useful ana ly t i ca l formula f o r the optimum exponent ial -aging fac to r . Instead, Figure 4.4 presents a p lot of optimum aging factor as a function of SNR for various B f . we have plotted X o p t instead of q as i t i s a more un iversa l l y understood measure of exponential -aging. - 47 -1.0-0.9-OPT 0.5 -0.7-0 . 0 5 Mz 0.15 Hz 0.5 Hz 1.5 Uz - 5 . 0 Hz(FT) 5.0 Hz 10 2 D 30 4 0 E b / No (dB) Figure 4.4 Optimum Exponential-Aging Factor When the optimum performance data from Figures 4.2 and 4.3 i s gathered together we can p lo t some very in terest ing performance graphs. Figure 4.5 shows the b i t - e r r o r - r a t e in reference-directed mode as a function of SNR for various fading bandwidths. It i s immediately not icable that with high the performance bottoms out at higher SNR's. We conclude that even fast -adapt ing 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 resu l t not ant ic ipated by the authors who suggested RLS algorithms fo r equal iz ing the HF channel. The performance at smaller fade rates i s much bet ter . The 0.05 Hz l i n e i s within 4-5 dB of the 3 path fading theore t i ca l l i m i t discussed i n Section 2 . 2 . In Section 4 . 3 , we see a longer but impract ical feedforward length resu l ts 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 var ia t ion in fading rate of the HF channel just brackets the - 48 -tO-: 10 -V to — - - ^ r ^ B f = 5.0Hz " (Q15Hz LE) \ \ \ > THEORETICAL \ \ \ k \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "—X(0.15Hz LMS) \ \ v \ \\ \ Y \ \ \ \ \ N . 0 5 H Z \ \\ \ \0.15Hz  \0.05H2 10 10 Eb/No (dB) 30 Figure 4.5 (1 ,0 ,1 ,0 ,1 ) DFE Performance as a Function of SNR t rack ing - ra te 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 equal izer needed on HF. Later in t h i s sub -sect ion , a f ter presenting resu l ts on a few more channel s t ructures , 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 f a d i n g . " - 49 -Most of our research takes place in referenee-directed mode and thus, as discussed in Section 7, i s somewhat i d e a l i s t i c . Nonetheless, i t allows easy invest igat ion of a var iety of phenomena without having to deal with crashes, e t c . We note here that the resu l ts of Figure 4.5 show the super io 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 imi ted radio . A more p rac t i ca l comparison w i l l be made in Section 7.2 where we evaluate the s e r i a l modem in dec is ion -d i rected mode. A l inear equal izer with the same to ta l number of taps as the DFE was also t r i e d on the above channel s t ruc tu re . These r e s u l t s , which are quite poor, are also show in Figure 4 . 5 . A longer LE would perform better at low fade ra tes , but again, the extra length slows adaptation thus hindering performance on more rapid ly -vary ing channels. The optimim value of q f o r the l inear equal izer was found to be less than hal f that fo 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 interpretat ion of why the LE performs so poor ly . L i t t l e fur ther LE reseach was done except to compare resu l ts 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 wel l as the fast -adapt ing Kalman algorithms on the very dynamic HF channel. Because the channel was fad ing , the eigenvalue r a t i o of the channel cor re la t ion matr ix , which the LMS algorithm i s sens i t i ve t o , l i k e l y took on a var iety of r e a l i s t i c values. - 50 -In Figure 4.6 we plot the same DFE data as in Figure 4 . 5 , but as a function of fading bandwidth. Note the steep increase in P g with fading r a t e . In logarithmic terms, the slope of the SNR=28 dB l i n e i s 7.5 dB/octave. Obviously an equal izer of sui table length for the HF channel gets extremely sens i t i ve 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 robab i l i t y of error would then be 10 -i 10 23d B r 10 -3 10 -5 10 0.01 0,1 1.0 10.0 (Hz) Figure 4.6 ( 1 , 0 , 1 , 0 , 1 ) DFE Performance as a Function of B f - 51 -due only to noise, 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 probab i l i t y that some of the paths would be somewhat faded out. This graph explains some very poor, ear l y resu l ts 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- l imit 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 equa l i ze 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 fas te r than the bandwidth- l imit ing f i l t e r was r o l l i n g down the fading spectrum'. We now present resu l ts on two channels with d i f fe rent multipath s t ructures . 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 resu l t ing reference-directed BER performance as a function of SNR. Note the performance is 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 imi la r to the previously presented resu l ts 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 -x-3 10 \ \ B f =5.0Hz \ x x. ^ \ \ \ \ V V \ \ v v \ \ Vs. N \ VV > v\ \ , 1.5 Hz * \ \ \ \ \ THEORETICAL^ V >s0.5Hz V \ 0.15 Hz \ > \ \ \ \ ^ \ \0.05Hz \ \ \ \ JO 20 30 Ek/N„ (J&) 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), again spanning 4 symbol periods (3.3 ms). Figure 4.10 shows the symbol-spaced DFE performance on t h i s channel in reference-directed mode. The theoret i ca l curve included i s f o r 3 equal strength fading paths as the one fo 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 0 — i — 3T 1 4T T 21 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 ) resu l ts are not as good as the three equal path r e s u l t s . This i s in terest ing as the ( . 3 3 , 0 , 1 , 0 , . 3 3 ) channel i s in a sense not as spread. It therefore should not need as long a forward TDL as the (1 ,0 ,1 ,0 ,1 ) channel. We must conclude the (1 ,0 ,1 ,0 ,1) channel has more d i v e r s i t y which resu l ts in i t s better performance. What can we conclude from the resu l ts presented so far? Obviously, the optimum exponential -aging factor varies s i g n i f i c a n t l y with the channel SNR and B^. In add i t ion , the b i t error rate can vary quite a b i t with deviat ion from optimum weighting. Is there a way to have the equal izer se l f - tune the exponential-aging factor? What f i r s t comes to mind is to monitor the equal izer output residual e r r o r , E[e (k ) ] . If i t is large , due to noise , then X could be increased. But what i f the residual er ror i s large due to high B f instead? We should then use a smaller A . These are unfortunately opposite t rends . If e i ther the noise level or fade rate were f i x e d , we could successfu l ly se l f - tune the other as in [4 ,32,33] . Unfortunately, in foreseeable implementations a nominal value fo r the exponential-aging fac tor 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) [33] , or estimating B f from the rate of tap 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 for the HF channel. Auroral channels with high fading bandwidths w i l l be too rapid ly varying for the equa l i ze r . As discussed by Monsen [45] , HF fade rates are much slower (2-3 orders of magnitude) than the symbol r a t e . The HF channel i s thus categorized as "slow f a d i n g . " Obviously 2 orders of magnitude is not slow enough fo r the long equal izer 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 appl icable to exponentially-aged RLS equal izat ion of a dynamic communication channel. Proakis (p. 487 of [52]) suggests that t d /T greater than 100 i s a good c r i t e r i o n . A fading bandwidth of 5 Hz has a decorrelat ion t ime, t d , of l/(2/Tx5) seconds. Our symbol per iod , T, was 1/1200 of a second. Thus t d/T=38. This i s obviously not greater than Proakis ' guide of 100 and suggests why our resu l ts on a 5 Hz fading channel bottom out. Proakis though, does not speci fy how the f igure of 100 was obtained. More importantly , he does not specify whether i t i s a guide for the LMS gradient or RLS equal izer algorithms; there i s quite a dif ference in adaptation rate between the two on low noise channels. A more theoret ica l d e f i n i t i o n i s suggested using Godard's resu l t [28] that an RLS algorithm slews the taps weights of an equal izer to wi th in 3 dB of minimum equal izer output error in 2N i t e r a t i o n s , where N i s the tota l number of taps in the equa l i ze r . Godard's resu 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 cont rast , our equal izer does not know the channel dynamics and has a time delay which is inherent in the exponentially-aged tracking process. - 56 -A better "slow fad ing" c r i t e r i o n fo r dynamic channels tracked by exponential-aged RLS algorithms might therefore be: t d /T » 2N or equiva lent ly : B f « f s/47TN where f i s the symbol ra te . Even t h i s c r i t e r i o n is only a better guide. Due to the non- l inear re lat ionship of the equal izer taps to the channel fading taps , higher order e f fec ts are poss ib le . 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 ind r e l a t i v e phases of the fading where some of the optimum equal izer taps would need to o s c i l l a t e at 15 Hz. This new concept strengthens the "much less than" inequal i ty in the above c r i t e r i o n and points to areas needing fur ther research. We now see the fas tes t fading expected to be properly tracked by a 9 tap equal izer i s much less than 1200/(4xft*xN)=10.6 Hz. This explains why the equal izer was reasonably adept at fo l lowing a .5 Hz fading bandwidth channel, yet found 5 Hz to be "moderately rap id" fad ing . F i n a l l y in t h i s sub-sect ion , we conclude that the appl icat ion of Hsu's algorithm [31] to fading multipath channels resul ts in performance quite d i f fe rent from that presented in his paper. As mentioned in Section 1 .2 , Hsu's P g versus SNR resu l ts 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 . We obtain an asymptotic slope at high SNR's of - L for L path d i v e r s i t y , in proper agreement with Figure 2.4 and the discussion preceeding i t . - 57 -4.2 Fractionally-Tapped Equal izer Performance and Discussion In th i s sub-sect ion, we f i r s t present the resul ts of a (T/2) f r a c t i o n a l l y tapped DFE on one of the same channel structures as used in the las t sect ion . Then we show the var ia t ion in performance of a T/l equal izer as a function of receiver sampling phase and contrast t h i s with the lack of var ia t ion obtained with a T/2 FT DFE. F i n a l l y , a non - in tegra l , symbol-period-spaced multipath channel i s invest igated . The FT equal izer i s show to of fer superior performance on such a channel, even compared to a T/l equal izer 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 fo r the symbol-spaced equal i zer , except the Bf=5 Hz l i n e was, as shown by the dotted l i n e i n Figure 4 . 4 , h igher . This indicates the FT equal i zer , being longer, cannot be pushed as hard to track a rap id ly changing channel. It  seems there may be two time constants in an equal i zer : 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 controls how f a s t the taps can be recurs ive ly slewed, given white data and a good estimate of P, to new values . This seems to be a new hypothesis, at least in the equal izer 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 fo r various fading bandwidths. - 58 -I 3 t?1 10 5.0Hz \ \ \ \ \ \ \ \ \ \ \ \ ^ k ^ L 5 H z \ \^ \ \ K v UOM& DFE--V 0.05 H* \ THEORETICAL^ A i \ V \0.I5 Hz \o .05Hz^ 10 10 30 Figure 4.11 (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 equal izer i s s l i g h t l y d i f fe rent from the symbol-spaced equal izer resu l ts of Figure 4 . 5 . With high s i g n a l - t o - n o i s e ra t ios the FT error rate for 1.5 and 5 Hz fading i s about 3 times that of the symbol-spaced equa l i ze r . Tests showed t h i s could be att r ibuted to the FT DFE having about 50% more taps and thus slower adaptation a b i l i t y . In contrast , at low SNR's and fade rates the FT - 59 -error rates are lower than the T/l DFE. This is l i k e l y due to the FT equal izer 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 matched- f i l te r c a p a b i l i t y of the FT equal izer gives improved performance when i t i s not adaptation-rate l imi ted by i t s larger number of taps. Also shown in Figure 4.11 i s the mean and ( logarithmic) standard deviat ion of nine 600,000 symbol simulations of a longer FT DFE on a channel with B f=0.05 Hz and 18 dB. The FT DFE had a feedforward TDL length which spanned almost twice the multipath spread. The nine runs, each with a d i f fe rent i n i t i a l random number seed, were done for two reasons. F i r s t l y , i t gave us an idea of the s t a t i s t i c a l deviat ion at very low error rates and with very slow fad ing . The deviat ion should be less f o r error rates greater than 10*^ and fading bandwidths fas te r than 0.15 Hz. Because the square-root Kalman algorithm use herein was order t r in computational burden, longer runs of t h i s long FT DFE would have taken over 10 hours of VAX 11/750 computer t ime. Secondly, we got a accurate estimate of the performance with a longer, but somewhat impract ical length FT DFE at very slow fade ra tes . We see that performance with in 2-3 dB of theoret ica l i s poss ib le . We now move on to compare the performance of T/l and FT equal izers as a function of receiver sample t iming . This has been investigated by a number of authors for l i n e a r equal i zers , but seemingly only by G i t l i n and Weinstein [25] for DFE's. We would l i k e to confirm t h e i r resu l ts on the HF channel. The fo l lowing resu l t s are fo r the 3 path channel of Figure 4.1 (MPGAIN(0:4)=1,0,1,0,1) with SNR=18 dB and B f=0.5 Hz. Figure 4.12 compares the performance of the equal izers as a function of received sample - 60 -t iming phase. The abscissa i s labe l led TIMING, a program input which spec i f ies the timing phase in units of T/8. With TIMING=0, sampling occurs at the beginning of a symbol period for a path with propagation delay MPDEL(I)=0 (or 8 ,16,24,32) . TIMING=3 or 4 i s symbol center and approximately optimum phase, while TIMING=7 is jus t gett ing the t a i l of the low-pass-f i l t e r e d symbol pulse-shape. The f igure shows a wide var ia t ion of b i t error rate with sampling phase f o r the symbol-spaced (T/l) equa l i ze r . The T/2-spaced FT DFE, which only needed to be tested over a half -symbol -per iod 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 resu l ts at best sampling phase are l i t t l e worse than the T/2 resu l ts f o r t h i s special symbol-spaced multipath channel. As noted in Sections 2.1 and 3 . 3 , acquiring and t racking 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. Very s i m i l a r resu l ts were obtained on a symbol-spaced 2 path channel. /0 FT 0 / 2 3 ^ 5 6 7 T I M I N G (ftKsec.) Figure 4.12 T/l and T/2 DFE S e n s i t i v i t y to Sampling Phase - 61 -We now address multipath channels which, as discussed in Section 2, may not have a delay structure which is symbol-period spaced. The concept of symbol-spaced multipath structure is best understood by comparing the channel responses of Figures 4.1 and 4 .13 . The few authors who address HF equal izat ion have not invest igat ied performance under these quite t yp ica l channel condit ions. We investigated the non-symbol-spaced multipath channels shown in 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 ) . The resu l ts are shown i n Figures 4.15 and 4.16. RMS Impulse Response 1 2T — r -3T 0 T   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~ 2T • 3T - r ~ 4T 0 T 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) x-3 -V 10 --/ 2 3 ^ 5 6 7 T I M I N G (ftKsEC.) Figure 4.15 Performance on Non-Symbol-Spaced 3 Path Channel 10 -V — * — t — • FT TIMING- (JthsEC.) Figure 4.16 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 for symbol-spaced equa l i ze r . Also note the FT DFE has roughly 1/4 the error rate of the T/l equal izer at optimum sampling phase. This contrasts the symbol-spaced multipath resu l ts of Figure 4.12 where the T/l DFE only performed poorer at non-optimum sampling phase. The T/l DFE suffers with the non-symbol-spaced multipath because no matter which timing phase i s used, i t w i l l never be correct for a l l signal pathsl The FT equal izer does not have t h i s problem. In summary, FT equal izat ion resu 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 matched - f i l te r e f fec t and an a b i l i t y to resolve non-symbol-spaced mul t ipath , i t resu l ts in better performance on many typ ica l channels and el iminates the necessity for receiver sampling synchronization. 4.3 Performance as a Function of Other Parameters In t h i s section we present b i t er ror rate performance as a function of the expected measurement noise covariance, of equal izer length, and frequency mistuning. I t has been found by several authors [28,31] that the value of the expected measurement noise variance, R, in 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 fec t i s to s l i g h t l y change the i n i t i a l performance during convergence. I r o n i c a l l y , Hsu says his R ( in his notation) i s inef fect i ve 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 var iable to adjust [31] . 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 va r iab le , the inverse of the i n i t i a l value of the diagonal elements of P, which af fects convergence s i m i l a r i l y . As mentioned in Section 3 . 2 , the Kalman gain can also be wr i t ten Gjk)=P(k)X_ (k)/R(k). For k=l the two techniques are obviously equivalent . In a moment we see further evidence why they are l i k e l y equivalent fo r a l l k. Once converged, we found R to be ine f fec t i ve over 15 orders of  magnitude. If R was extremely large ( re la t i ve to 1) the AGC nature of the equal izer was very slow during convergence. This of course does not af fect QPSK demodulation. Once converged, t h i s sluggish AGC behavior disappeared. This was cur ious. 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 exponential -aging f a c t o r . This leads us to a new interpretat ion of the exponent ia l ly -aged Kalman algori thm. If the algorithm uses exponential-aging as i t s only adaption mechanism ( i e . Q=0), then only one parameter i s needed af ter covergence to spec i f y , in a general way, the weight ( i e . the Kalman Gain) given to each new measurement. This parameter, the optimum exponential-aging f a c t o r , X=l/(l+q), determines how fas t the algorithm's adaptation to channel va r ia t ion (and to noise) w i l l be. We can associate the exponential-weighting fac tor with the r e l a t i v e s izes of the true system's rate of v a r i a t i o n , Q, and measurement noise, R. The s i ze of the Kalman gain w i l l in general depend on the ra t io Q/R. In te res t ing ly , 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 in 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 ig in 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 postulated. In l i gh t of the above d iscuss ion , i t i s reasonable to surmise the " s i z e " of the P matrix may not per fect l y r e f l e c t the covariance of the state in an exponential-aged Kalman algorithm. This is 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 topic i s equal izer performance as a function of the number of feed-forward taps. 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 resu l ts we suspect a short forward TDL would reduce adaptation t ime. Conversely, a longer one would allow better cancel lat ion of ISI from future symbols. This cance l la t ion 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 fe rent 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)= . 3 3 , 0 . 1 . , 3 3 ) . An SNR of 18 dB was selected and fade rates of 0.5 and 5.0 Hz were tested . - 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 just span the multipath spread. Any longer or shorter length was less than optimum. For the 0.5 Hz fading channel, the optimum feed-forward equal izer length spans about 1.5 times the multipath delay spread. For the " f u l l y spread" channel (MPGAIN(0:4)=1,0,1,0,1) or when using a symbol-spaced equal i zer , a s l i g h t l y longer span was bet ter . On the less spread channel (MPGAIN(0:4)= . 3 3 , 0 , 1 , 0 , . 3 3 ) or with an FT equal i zer , a s l i g h t l y shorter span was more d e s i r a b l e . 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 f a r t h e r ) . An FT equal izer i s general ly slower adapting because i t requires more taps f o r a given multipath spread; a shorter one seems to track fas te 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 opt imal . In te res t ing ly , 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 theoret ica l fading channel performance (see Figure 4 .11) , but i s l i k e l y impract ical 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 rac t i ca l compromise f o r the HF channel. We note that the condit ions under which t h i s compromise would cause the most penalty i s very slow fad ing , noisy radio l i n k s . The HF channel worst-case multipath spread is about 6 ms. Data on the probabi l i t y d i s t r i b u t i o n of spread i s not avai lab le though, so the best choice of equal izer feedforward span w i l l have to be guess; 3-4 mil l iseconds seems reasonable. More data in t h i s regard would be of u s e f u l . In terest ing ly , p a r a l l e l modems degrade quite rapid ly when the spead exceeds 2-3 mil l iseconds [63] . - 67 -To end t h i s sub-sect ion we discuss frequency mistuning. This can be caused by e i ther the doppler s h i f t of mobile units or d i f fe rent local o s c i l l a t o r frequencies in the receiver and t ransmit te r . Frequency mistuning i s also discussed in Section 6 regarding our real channels tapes. The e f fect of mistuning is to require the equal izer taps to rotate (in the complex plane) at a frequency equivalent to the mistuning. 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 ef fect on the components of the various taps is thus quite s i m i l a r to random 2D fad ing ; a degradation s i m i l a r to increasing fading bandwidth i s expected. The mistuning can be simulated by increasingly twis t ing each successive complex baseband value of the received samples fed to the equa l i ze r . For 1200 symbols/second, the increment is 360°/1200= 0.3° per Hz of mis -tun ing . A three-path channel structure as i n Figure 4.1 was used in the experiments with an SNR=23 dB and B f=o.5 Hz. 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 resu l ts of 0 . 5 , 1.5, and 5.0 Hz mis-tuning added to the 0.5 Hz fad ing . The resul t i s an error 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 tracking required of the equal izer tap weights i s in the range 4 . 5 - 5 . 5 Hz, while 5 Hz fading with no mistuning d i s t r ibu tes 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 fol low than fad ing . Why do our resu l ts contradict th is? Maybe our hypothesis in Section 4.1 regarding the possible need f o r the equal izer taps to adapt at 15 - 68 -Hz on a 3 path channel with 5 Hz fading was cor rec t . If a l l the taps are rotat ing in unison, no higher-order e f fects may be poss ib le . Several other in te res t ing points were noticed during the mis-tuning experiments. As shown in Table 4 . 1 , the optimum exponential-aging fac tor changes with applied mis - tun ing . This seems reasonable as the c a r r i e r phase i s changing fas te r resu l t ing in the equal izer taps needing to adjust f a s t e r . 10 10 -V 10 ^ 6f-5.0Hz X «0.5Hz 4-~ BF=0.5 Hz + V>CBF=0.5Hz + BF = 0.5Hz. M15TUNIN&»5.DHz MISTWNINS-* 1.5 Hz MISTUNING-=0.5'Hz 10 10 Et/No 30 Figure 4.17 Simulated Mistuning Results - 69 -Table 4.1 Equalizer Performance with Mistuning Mis - Optimum RMS RMS Mean B i t tuning Symbol Phase Phase Error (Hz) Error Error Error Rate 0.0 .92 .207 8.8° .1° .00027 0.5 .89 .216 9.4 1.8 .00037 1.5 .86 .256 11.6 4.8 .00097 5.0 .78 .425 20.9 10.9 .017 The second interest ing point i s that a bias was noted in the mean equalized symbol-phase e r r o r . This b ias , which manifests i t s e l f from a trend i n the symbol-phase er ror , i s what drives the rotat ion of the equal izer taps . It i s never e n t i r e l y removed as the equal izer functions as a f i r s t - o r d e r phased-locked loop and thus cannot track a frequency o f fse t with zero e r r o r . This can also be seen i n Table 4 . 1 . It was postulated that th i s phase bias could be the lag from an exponentially-weighted control device t r y ing 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 equal izer 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 to track t h i s bias as the e f fect is 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 acquis i t ion may be a problem because the technique depends on the equal izer already being converged. - 70 -5. CHANNEL ESTIMATION ON SIMULATED AND REAL CHANNELS 5.1 Appl icat ion of RLS Algorithms to Channel Estimation In Section 6 the equal izers are tested on three actual HF channel transmissions recorded on tape. The de ta i l s of the recordings are contained in Appendix C. In order to evaluate equal izer performance on the real channel tapes i t i s necessary to estimate the sever i ty of the multipath d i s t o r t i o n present during the recording. Though i t i s d i f f i c u l t to estimate, or even define the amount of addit ive noise on fading multipath channels, i t 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 spectra . From t h i s information i t is possible to determine d i v e r s i t y , fading bandwidth, and mistuning. As shown in the previous sect ion , these parameters af fect the error rate performance just as ser ious ly as s igna l - to -no i se r a t i o . This sub-section introduces the appl icat ion of recurs ive - least - square -er ror algorithms to channel est imation, while Section 5.2 tests a f rac t iona l l y - tapped 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 resu l ts of the spectral analysis of the fad ing . Channel estimation is the inverse of the equal izat ion problem. Like an equa l i zer , a channel estimator can use a simple tapped-delay-1ine st ructure . 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 discrete conste l la t ion 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 SYMBOLS *i 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 is t ry ing 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 equal izer case where the tap weights being estimated are generated by a non- l inear process; in 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 ei ther the channel simulator or analys is of past inputs, i t i s possible to use a more i n t e l l i g e n t estimator adaptation method than simple exponential -aging. 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 quant i t ies , we can use the f u l l Kalman update equations ( 3 . 3 - 3 . 9 ) . - 72 -Because a one-pole model fo r the path fading process was very simple to construct , 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. Necessar i ly , a two-pole simulated channel was used. The one-pole estimator was given the pos i t ion and RMS-size information regarding the 3 paths in the channel. The best resu l ts were obtained with some one-pole modelling and some exponent ial -aging. This seems reasonable as a one-pole estimator should not perform optimally on a two-pole channel. On the other hand, exponent ial -aging i s a un iversa l l y appl icable 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 ize and posi t ion 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-model l ing. These resu l ts indicate that i f a better A and/or Q could be suggested fo r the d i f f i c u l t equal izer problem, i t s performance might be improved too. Quicker exponential -aging causes less error in the estimation of the s i g n i f i c a n t paths in the channel . However, quicker aging also resu l ts in more error in 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 overa l l performance. Should one use the RMS sample estimation error 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 coincide i f using mixed one-pole modelling and exponential -aging. For an unknown channel the tap error cannot be ca lcu lated , so RMS sample estimation error i s probably the best general-purpose performance measure and w i l l be used in Section 5 .3 . - 73 -Several other in terest ing 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, ind icat ing the estimator was doing a good job of t racking the channel in spi te of the noise. Secondly, the input value of the expected measurement noise, 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 on ly ) . 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. F i n a l l y , computer underflows were noted when using MPGAIN(0:4)=1,0,1,0,1. These disappeared when MPGAIN(0:4)=1,.001,1,.001,1 was used for the simulated channel RMS path amplitudes. It i s suspected that with t ime, estimates of path amplitudes f ixed 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 for an unknown channel. Performance under t h i s mechanism was investigated next. It was found that minimum RMS sample-estimation error and minimum RMS tap-est imation error better coincide 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. For a 3 path, 23 dB channel the optimum A data of Table 5.1 was noted. The optimum exponential -aging i s quite short , espec ia l l y r e l a t i v e to the amount required when a mixed adaptation mechanism is used. - 74 -Table 5.1 Channel Estimator Optimum X as a Function of B f for SNR=23 dB 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 est imation, but without further 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 Fractional ly -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 equa l i ze rs . Because our transmitted signal has a complex (2 sided) baseband bandwidth of 2400 Hz, we must estimate the channel impulse response with a time resolut ion of 1/2400 second; otherwise c lose l y or unusually spaced components w i l l not be reso lvable . An FT channel estimator can be constructed somewhat d i f f e r e n t l y than a FT equa l i ze r . While an FT equal izer uses received data at twice the symbol rate - 75 -and produces an equalized symbol stream at the symbol ra te , an FT estimator must estimate the FT received data stream at twice the symbol rate . 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 locking the estimator twice as f a s t . It can also be accomplished by using two symbol-spaced estimators running of f the same TDL. One estimates the odd received symbols while the other estimates the even ones. The impulse response i s the interleaved resu l t of the two. In terest ing ly , t h i s i s equivalent to running a symbol-spaced channel estimator with two d i f fe rent sample timings separated by a half symbol per iod , on the same data. Figure 5.2 i l l u s t r a t e s the e f fec t of changing the timing of a symbol-spaced estimator by a half symbol-period. Figure 5.3 shows resu l ts when t h i s change occurs while using an FT est imator . Both f igures 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 par t icu lar ! ' l y narrow as the have been smeared by the low pass f i l t e r . RMS Impulse Response o x if 6 8 10 IZ /y FT Tap Number Figure 5.4 FT Estimation of Simulated 3 Path Channel - 77 -5.3 Estimation of Three Real Channels Using the demodulation software from the equal i zer , the channel estimator was fed with FT-spaced samples of 310,000 symbols (approx. 4.3 minutes) s ta r t ing 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 in terest ing to rerun the estimator someday giv ing i t more a p r i o r i information from our i n i t i a l est imat ion) . 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 plots 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 -I /0 o i V 6 % FT Tap Number Figure 5.6 Tape 9 RMS Impulse Response X f 6 8 10 12 FT Tap Number 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] for a run of approximately 30,000 symbols. Which 30,000 symbols of the over 300,000 on the tape was not s tated , but our path posi t ions and RMS path amplitudes are approximately the same s i z e . For instance Falconer 's second path was down 6dB from the f i r s t . Our resu l t s fo r the f i r s t 60,000 symbols gave -4 .7dB, while Figure 5.7 shows 310,000 gave -8 .3dB . These resu l ts at least bracket his average amplitudes. Luck i l y , each of the three channels has a d i f fe rent number of paths providing us with a var iety 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 ructure . 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 ind icat ion of the noise level on dynamic channels, we t r i e d a novel analysis on a simulated version of Tape 11's channel. Using the re la t i ve RMS amplitudes and pos i t ions from Figure 5.7 and the de-tuning and fading bandwidth determined in the next sub-sect ion, we t r i e d simultations of Tape l l ' s charac te r i s t i cs with d i f fe rent noise leve ls u n t i l we found one that resulted in the same error rate as the actual channel. The simulated condit ions 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 ) , and detuning=0.7 Hz and fading bandwidth=0.2 Hz (from Figure 5.10) . Under these condit ions, a simulated SNR of about 14 dB gave the same error rate as Tape 11. - 80 -5.4 Estimation of Fading Spectra The fading spectra of the path gains was determined using a complex fast Fourier transform. The estimated values of the 14 f rac t iona l l y - spaced 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 resu l t ing 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 est imates. For each h i s to ry , ten 512-sample long FFT's were performed and the ten resu l t ing spectra averaged. This gave a frequency resolut ion of 0.04 Hz over ( -10, 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 fo r taps 0, 3, 7, 13 are presented in Figures 5.8 and 5 . 9 . A l l spectra are p lotted with auto -scal ing so the reader should be careful 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 in Figures 5.10 and 5.11. F i n a l l y , Tape 11 spectra of taps 0, 3, 7, 11 are shown in Figures 5.12 and 5.13. Several features are notable . 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 for Tapes 6, 9, and 11, respect i ve ly . The o r i g i n of the displacement is unknown. The t ransmitt ing and receiving stat ions were f ixed and used cesium clocks to lock the data, sub -ca r r ie 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 tat ion motion, or mistuning. It was postulated that the center a l t i tude of the r e f l e c t i n g ionospheric layers might be moving several 10's of meters per second. On each tape though, the frequency displacements of the d i f fe rent paths was the same. It seems un l ike l y that several independent layers would be moving at the same rate . S P E C T R A L D E N S I T Y TAPE 6 TAP 0 .5 10 FREQUENCY (Hz) 15 2.0 31.7 5 P E C T R A L 0 E N S I T Y TAPE b TAP 3 .5 1.0 1.5 FREQUENCY (Hz) 3.0 Figure 5.8 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 (Hz) Figure 5.10 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^ 7i 2.0 FREQUENCY1 (Hz) Figure 5.12 Estimated Tape 11 Fading Spectra (Taps 0 and 3) 7."? i 1 1 • • • - .5 O .5 1.0 1.5 2J0 FREQUENCY (Hz) 2.7 _r- + , , , . , •5 O .5 1.0 1.5 2J0 FREQUENCY (Hz) ure 5 . 1 3 , Estimated Tape 11 Fading Spectra (Taps 7 and 11) - 87 -The fact that the fading energy in the main paths was in narrow l ines (0.2 Hz half-bandwidth) suggests a tuning correct ion would be desirable to reduce the "e f fec t i ve " fading bandwidth the equal izer must t rack . Results with t h i s correct ion are discussed in the next sect ion . I t i s also in terest ing that some of the spectra of the lesser -s t rength paths are p l a t e a u - l i k e . Are the weaker paths, which on a l l three tapes are la te r ( longer) , from higher more-rapidly fading (turbulent) layers of the ionosphere? Could they be the resul t of several internal re f lec t ions within the ionosphere, each in turn broadening the spectrum? We are uncerta in . S i m i l a r i l y , i t i s unclear why the spectra of non-s ign i f icant 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 . One might suspect they would be white (uniform in spectral density) i f they were fa r enough removed from the paths. - 88 -6. EQUALIZER PERFORMANCE ON REAL CHANNELS 6.1 Equalizer Performance on Three Real Channels As discussed in Sections 1.3 and 5, and detai led i n Appendix C, three tapes of d i g i t i z e d HF audio output were avai lable from the MDA/CRC l a t t i c e study [42] . The received s i g n a l , 1201.739 symbol/s QPSK on a 1505.107 Hz sub -car r ie r , had been d i g i t i z e d at 9600.00 Hz. The equal izat ion 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 resu l ts along with those of the MDA/CRC study. These are some of the few FT DFE resul ts ever reported on actual HF radio channels. Table 6.1 Performance on the Taped Real Channels Research Equal izat ion Re- Tape 6 Tape 9 Tape 11 Sponsors Tuning BER BER BER MDA/CRC Unequalized no .027 .139 .080 LE no .015 .130 .016 FT LE no .007 .082 .009 Gl enayre/ LE no .0077 .098 .0078 UBC DFE no .0077 .079 .0049 DFE yes .0058 .068 .0047 FT DFE yes .0052 .047 .0030 Note 1: MDA/CRC tests were 310,000 symbols long, while Glenayre/UBC's were 300,000 symbols long. Note 2: MDA/CRC tests did not ca lcu late b i t BER, only symbol error rate . The MDA/CRC BER resu l ts shown are one hal f t h e i r reported symbol error ra te , 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 in terest ing 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 l i t t l e improvement with e q u a l i z a t i o n . 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 correct ion of mistuning, and f rac t iona l - tapp ing picking up information smeared by the bandlimiting f i l t e r . i i ) Tape 9, a two path channel contaminated by impulse noise (possibly Russian over - the-hor izon radar [23, 42] ) , 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, par 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 . Tape 9 i s the only one to have a non-symbol-spaced multipath s t ruc ture . It thus showed the largest absolute, though not re la t i ve improvement in BER when switching to FT equal i zat ion . 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 ISI , shows the largest r e l a t i v e improvement of the DFE over the LE. It also shows the most overa l l improvement. iv ) The MDA/CRC l i n e a r equal izer resu l ts should be comparable to those of the Glenayre/UBC research done here in . Unfortunately, there i s a fac tor of 2 dif ference fo r tapes 6 and 11. This i s cur ious . It may be at t r ibutable to our carefu l optimization of the exponential -weighting f a c t o r , or poss ib ly less numerical -precis ion error in our square-root-normalized f ixed-order - 90 -Kalman implementation. Re-running the MDA/CRC analys is was not deemed worthwhile to resolve t h i s . v) Note that our re-tuned FT DFE resu 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 ident ica l to that found in [22,23] f o r 30,000 symbols in an unspecif ied part of Tape 11. These resu l ts of course, represent only a small sample of poss ib le HF channels. They are not p a r t i c u l a r i l y well spread in the sense that , 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 robab 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 typ ica l these recorded channels are . 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 avai lable 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 r e l a t i v e l y hard to measure, or even define for fading multipath channels. 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 ignals 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 equal izers often use these decisions in t h e i r adaptation or estimation mechanism. If too many decision errors are made, error propagation can occur and possibly lead to an equal izer crash. We have avoided t h i s problem unt 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 ractable [16]. Reference-directed mode i s used by most authors fo r t h e i r analysis because on most channels at reasonable error rates i t is usual ly only s l i g h t l y o p t i m i s t i c . We shal l see that Kalman equal izat ion of the HF channel does not allow th is assumption. Being a randomly fading channel, occasional ly a l l d i v e r s i t y paths w i l l fade simultaneously v i r t u a l l y guaranteeing an equal izer c rash . In contrast , the response of even an unconditioned telephone channel i s normally within 5 dB of nominal across the audio band. In a d d i t i o n , HF channels vary moderately rap id ly in t ime. 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 loca l i zed nature of the HF impulse response may contribute to the increased error rate with a DFE. In the next sub-sect ion 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 fo r judging the v a l i d i t y of assumptions about dec is ion -d i rected experiments. 92 -We f i r s t operated in dec is ion -d i rected mode by using an automatic crash  detection and recovery scheme. The equal izer monitors an exponential ly weighted measure of i t s RMS output symbol estimation e r ro r . When t h i s (decis ion-directed) output SNR exceeds a certa in t r igger l e v e l , ind icat ing a crash , the equal izer switches to reference-directed mode. When the output SNR decreases back to (say) 60% of the t r igger l e v e l , the equal izer switches back into dec is ion -d i rected 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 dec i s ion -d i rec t ion and some rough estimates of the supr is ingly high rate of crashes. It i s a lso 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 inks are not common in HF communication, we next consider inser t ing per iodic ret ra in ing sequences in the data stream. This i s a commonly suggested and very robust method of crash prevention and recovery. The equal izer knows that (say) the f i r s t 20% of every data frame i s a ret ra in ing sequence. This requires considerable overhead but allows the equal izer to operate for a part of each frame in reference-directed mode. If a crash has occurred, or i s c lose to occurr ing, per iodic re t ra in ing allows complete recovery fo r the next data frame. During re t ra in ing i t may be advantageous to reset cer ta in internal v a r i a b l e s . This i s not absolutely necessary as the equal izer w i l l reconverge regardless. If the equal izer has crashed though, i t may reconverge f a s t e r , more completely, or with a shorter re t ra in ing sequence i f the tap weight vector C_(k) i s reset to zero. This i s an ideal appl icat ion f o r the auto-crash detection mechanism. The covariance matrix P can also be reset , usual ly to the ident i t y matrix . This slows reconvergence and is not t h e o r e t i c a l l y necessary as P i s  unaffected by wrong dec is ions . As discussed in Section 3.2 though, reset t ing P cleans out any round-off errors that might be bui ld ing up. Hsu's square-root Kalman algorithm [31] used herein was always completely stable for 600,000 symbols using 32 b i t f l oa t ing -po in t ar i thmet ic . Some of the Fast Kalman algorithms discussed in Section 3.2 are not so s tab le . Two new LMS-based algorithms for the l inear equal izer configurat ion have recently appeared in the l i t e r a t u r e . Both use rather novel measures of error for d r i v ing the adaptation. Morgan [46] uses the error in the expected power spectrum. The other , to be discussed in Section 7 .3 , i s f o r constant modulus modulations; only the error in the symbol modulus i s used in the adaptation mechanism. Neither of these algorithms feed on decisions so when used in a l inear equal izer they do not require the knowledge of the transmitted sequence to converge, nor should they ever crash!. If a l inear equal izer could be used on an HF channel these two techniques would el iminate many problems. We concluded in Section 4 though, that a l inear equal izer performs very poorly on an HF channel. Fortunately i t i s possible to adapt the constant modulus technique to use both the f a s t -adapting RLS algorithm and the DFE conf igurat ion , but with the loss of the above-mentioned advantages. Nonetheless, since the symbol-estimation modulus error i s never wrong, a DFE using t h i s error measure might not crash as o f ten . We invest igate 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 th i s sect ion , we test the performance of various equalizers in dec is ion -d i rected mode with per iodic r e t r a i n i n g . Our main interest 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 reference-directed mode. We w i l l only b r i e f l y t e s t a LE or experiment on one of the real channel tapes. An explanation is suggested fo r the surpr i s ing l y substant ial degradation in dec is ion -d i rected 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 pos te r io r i Kalman i s t r i e d . Since the FT DFE we use has 14 taps , we should choose a re t ra in ing block length (in symbols) of about twice t h i s so as to give the equal izer time to re-converge [28]. A Frank sequence was used for r e t r a i n i n g , for as discussed in 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 re t ra in ing sequence length of 32 symbols allows exact ly two f u l l Frank sequences in the re t ra in ing block. A l l per iodic ret ra in ing experiments were performed with reset t ing of internal var iables (£ and P) so as to be able to compare with other authors resu l ts [22,23] . The 3 path channel structure of Figure 4.1 (MPGAIN(0:4)=1,0,1,0,1) was investigated under 4 channel condi t ions : Eb/NQ=13 and 23 dB with B^=0.15 and 1.5 Hz. Considering the ret ra in ing sequence to be overhead, we tabulated the data error rate only, not including r e t r a i n i n g . We were also interested in the mean-t ime-to-crash. Both these performance f igures were measured as a function of re t ra in ing p e r i o d i c i t y because per iodic re t ra in ing can, in addit ion to recovering from crashes, also ease dec is ion -d i rected degradation, thus changing the error ra te . S imi lar 95 -experimental r e s u l t s , espec ia l l y as a function of re t ra in ing p e r i o d i c i t y , have never previously been presented. We comment that the net throughput in our pe r iod i c - re t ra in ing experiments i s not 2400 bps, but somewhat less due to the ret ra in ing overhead. Table 7.1 presents our dec is ion -d i rected resul ts as a function of packet frame composition for 60,000 symbol simulation runs. Optimum exponential - aging from reference-directed mode was used. A 32/64 designation indicates a 96 symbol frame in which the f i r s t 32 symbols (1 block) i s the re t ra in ing sequence, and the next 64 symbols (2 blocks) is the M-sequence data. The performance on channels with fading bandwidths of 1.5 Hz was very poor. Figure 7 . 1 , which p lots some of the resu l ts in comparison to the reference-directed and theoret i ca l performance, shows the error rate to be 5 to 10 times higher than in reference-directed mode. We surmise that when the equal izer adaptation rate i s taxed, even a few wrong decisions can cause i t to f a l l behind enough to crash. 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 br ief computations show that the b i t error rate shown in Table 7.1 is almost en t i re 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 f=0.15 Hz, 32/64 r e t r a i n i n g , and SNR=13 dB. Under the l a t t e r condi t ions , the chance of crashing i s low and the noise i s high, resu l t ing in about half the errors being random rather than end of frame bursts . This suggests that any data transmitted on our proposed modem should be r e l a t i v e l y to lerant to burst e r r o r s . General ly , the resu l ts on 0.15 Hz fading channels are not too bad in terms of absolute error r a t e , but they also show a 5-10 times increase in 32/32 32/64 32/256 32/1024 32/4096 Equal-i ze r Eb/No Blks Un-Crashed Data BER Blks Un-Crashed Data BER Blks Un-Crashed Data BER Blks Un-Crashed Data BER Blks Un-Crashed Data BER DFE 0.15Hz 23 dB 1.994/2 .0016 7.96/8 .002 31.52/32 .0068 122.6/128 .020 DFE 0.15 13 1.93 /2 .032 7.50/8 .038 28.78/32 .056 97.0/128 .123 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 .910/1 .069 1.61 /2 .108 3.44/8 .277 TABLE 7.1 Decision Directed Performance with Per iodic Retraining - 97 -Eb/No W&) Figure 7.1 Per iod ic -Ret ra in ing Decision-Directed Results error rate over s i m i l a r reference-directed r e s u l t s . This is cur ious, 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 error rate due to dec is ion -d i rected degradation was, in addit ive terms, much less than f o r the higher 1.5 Hz fad ing . 98 -It i s in terest ing to consider the trends as a funct ion of the re t ra in ing per iod ic i t y in Table 7.1 and Figure 7 . 1 . Notice that the 0.15 Hz resu l ts show r e l a t i v e l y l i t t l e degradation with decreased re t ra in ing p e r i o d i c i t y . The 0.15 Hz, 32/4096 resu l ts show the mean-time-to-crash i s approximately 100 b locks . In contrast , the 1.5 Hz resu l ts degrade s i g n i f i c a n t l y with decreased ret ra in ing and the mean-time-to-crash i s only 10 blocks. This indicates the equal izer w i l l only run 1/IOth as long at the higher fade rate before crashing. Periodic re t ra in ing was also t r i e d on Tape 11 to compare with the resu l ts of Falconer et al [22, 23] . Using an FT DFE (with tuning correct ion) on the f i r s t 300,000 M-sequence symbols, we achieved a b i t error rate of 0.010; t r i p l e the reference-directed 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 , t h i s b i t error rate is exact ly the same as he acheived on 30,000 symbols in an unspecified area of Tape 11. We have previously seen that a l inear equal izer general ly performs poor ly . A l inear equal izer however, i s not susceptible to error propagation from decision-feedback. Since our DFE r e s u l t s at high fade rates were so d isappoint ing , we postulated a LE might do as wel l or better under such condi t ions . A LE was t r i e d on the 3 path, 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 error rate of the LE only doubled from that under the equivalent reference-directed condit ions but s t i l l the absolute b i t - e r r o r - r a t e remained worse than fo r a DFE. In order to put the dec is ion -d i rected resu l ts in perspective, we simulated some of the channel condit ions 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 equal -- 99 -RMS-strength paths separated by 1 mi l l i second delay spread. We used a s l i g h t l y worse channel with 1.6 mi l l iseconds spread. Their spec i f i ca t ion of fading rate was based on twice the standard deviat ion of the Gaussian b e l l -shaped spectra, requir ing 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 ident ica 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 internal (25,16) er ror correct ion 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 retrained FT DFE i s shown i n Figure 7 .2 . Because most t ransmitters are peak power l imi ted and the p a r a l l e l modems have a p rac t i ca 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 . We see the DFE b o does very well against even the MX-190 modem when i t i s so penal ized, i f the fade rate i s less than about 0.5 Hz. Several comments should be made about t h i s comparison. 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 sens i t i ve to multipath spread above 1 m i l l i s e c o n d , i t s error rate increasing by a factor of 10 at 2 mi l l iseconds (see Figure 64 of [63]) . The DFE resu l t s presented herein used an equal izer which spanned 3.2 mi l l i seconds ; i t s performance would not l i k e l y degrade by more than a fac to 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 10 -10--x-3 10-3.0 Hz DFE NX. VS. v\ \ \ \ \ \ 1 \ \ >- 3.0 Hz MX-190 •* 0.6 Hz ACQ-6 * 0.12Hz ACQ- 6 > 0.6 Hz DFE \ \ V \ 2 PATH \ THEORETICAL'. \ N •^0.6 Hz MX-190 \ \ \ \ \ \ \ \ 1 • 0.12 Hz DFE to to 20 30 SNR (JB) HO 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 future . If the nature of the dec is ion -d i rected degradation could be ascertained, maybe fur ther performance gains could be made. Why were the dec is ion -d i rected mode resu l ts so disappointing? There are several unique features about the HF channel which could cause more degradation than say, telephone channels. Is i t the fad ing , the fade ra te , or the loca l i zed 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 s i g n i f i c a n t . This tap-weight structure could s i g n i f i c a n t l y disturb the equal izat ion of l a te r symbols i f wrong decisions are fed down the dec i s ion -feedback tapped delay l i n e . Unfortunately i t i s d i f f i c u l t to test th i s hypothesis in a s t ra ight 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 dec is ion -d i rected 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 -dec is ion -d i rec ted" mode. In one mode, which we w i l l c a l l "perfect-feedback" mode, the symbols sh i f ted into the decision-feedback tapped-delay-1ine are perfect reference symbols, while the e r ro r , e (k ) , used by the Kalman algorithm i s s t i l l formed using dec is ions . The other mode, c a l l e d "perfect e r r o r , " i s just the opposite. It uses error information calculated from a perfect reference, but feeds decisions down the decision-feedback part of the tapped-delay-1ine. By comparing the resu l ts of perfect-feedback, per fect -e r r o r , and f u l l dec is ion -d i rected 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 resu l ts of t h i s comparison on a 3 path, 20 dB, 1.5 Hz fading channel with 32/256 r e t r a i n i n g . The per fect -er ror mode is f a r b e t t e r . This indicates the accuracy of e(k) i s c r i t i c a l to the performance of the equal izer 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 d i s rup t i ve . In terest ing ly , t h i s shows that the loca l i zed nature of the HF impulse response is not as d is rupt ive as some of the other degrading ef fects of the HF channel. Since a l inear equal izer i s just 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 dec is ion -d i rected degradation too . Half dec is ion -d i rec t ion d e f i n i t e l y needs fur ther research, espec ia l l y under a wider var iety of channel condi t ions . Table 7.2 Half -Decis ion-Directed Mode Comparison Mode B i t Error Rate Mean # Blocks Uncrashed Perfect Feedback .138 5.6 / 8 Perfect Error .042 7.7 / 8 Fu l l Dec. Direct ion .154 5.5 / 8 In l i g h t of the ha l f - dec i s ion -d i rec ted r e s u l t s , why i s e(k) such a serious contr ibuter to dec is ion -d i rected degradation? Compared to some Kalman app l i ca t ions , the addit ive noise in the equal izer measurement equation i s quite smal l , -10 to -20 dB, and the rate of var iat ion 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 is general ly quite large. This gives each new measurement considerable weight compared to the past state and even one wrong decis ion af fect ing e(k) may cause s i g n i f i c a n t adaptation in the wrong d i rec t ion and star t an error propagation burst . This probably explains why even the 0.15 Hz r e s u l t s , where the equal izer should not be that fade-rate taxed, also show s i g n i f i c a n t - 103 -degradation when using the optimum reference-directed exponential -aging fac to r , but in dec is ion -d i rected mode. A very b r ie f invest igat ion showed the optimum dec is ion -d i rected aging to be approximately ha l f as fas t as that in reference-directed mode. Further research to determine the optimum exponential-aging factor in dec i s ion -directed mode as a funct ion of SNR, fading bandwidth, and re t ra in ing p e r i o d i c i t y i s ind icated . Because the resu l ts in Section 4.1 show considerable s e n s i t i v i t y to slowing the aging though, vast ly improved performance i s not expected. Much of the previous DFE research in dec is ion -d i rected mode has used the LMS algorithm which must, for s t a b i l i t y , use a very small step s i ze (or scalar gain) to even be s tab le . Researchers who have used the Kalman algorithm have used i t mainly for very fas t convergence, which is 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 factor are very s i g n i f i c a n t though, the gain w i l l automatical ly decrease to smaller values, possibly of the same order as the t yp ica l LMS scalar step s i z e , for the post-convergence dec is ion -d i rected operat ion. This ofcourse i s what i s desi rable 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 dec is ion -d i rected degradation. To q u a l i t a t i v e l y ve 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 predic tor . 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 izable RMS symbol error a f te r equal izat ion (see Table 4.1 fo r example), fa 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 poster io r i RMS symbol error 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 pos te r io r i RMS symbol error also seems to be of a more correct magnitude considering the channel SNR. In reference-directed mode the a pos te r io r i Kalman equal izer gives the i l l u s i o n of incred ib le performance, i f the Kalman gain i s kept large, because the i te ra t ion of the symbol estimate can p u l l a wrong estimate across a decision boundary. In dec is ion -d i rected mode however, the i te ra t ion only pu l l s bad symbol estimates toward wrong dec is ions . Consequently, the error rate i s exactly the same as the normal equal izer and in f a c t , the tap weight vector C{k) takes the same t ra jec to ry in t ime. In dec is ion -d i rected simulat ions, the a poster io r i RMS error remains very small u n t i l a decis ion error i s made. Then, the error increases greatly and the equal izer often crashes. Several conclusions can be drawn from these a poster ior i experiments. The low SNR's t yp i ca l to the equal izer 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 factor to be very short , resu l t ing in 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 t r a c k i n g . In one i t e r a t i o n the optimum C{k) changes so much that i t i s somewhat obsolete to use C_(k-1) in the symbol est imat ion. This could lead to another d e f i n i t i o n of slow f a d i n g . It also reinforces the suggestion in Section 5 that a better value than the ident i t y for the - 105 -equal izer 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 sens i t ive i f i t monitored the a poster io r i RMS e r ro r . In summary, we t r i e d the equal izer in dec is ion -d i rected mode using per iodic r e t r a i n i n g . The resu l ts were marginal fo r high fading bandwidth channels, though they did agree with Falconer 's resu l ts for 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 further performance gains might be uncovered, several reasons fo 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 tes ted . 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 fas t enough to fo l low 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 rect ion and lead to error propagation. The optimum exponential -aging i s l i k e l y s l i g h t l y d i f fe rent in dec is ion -d i rected mode than in reference-directed mode. In other appl icat ions such as control systems, the fast -adapt ing Kalman algorithms are not used in a dec is ion -d i rected manner. Previous equal izat ion work has used e i ther 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 automatical ly) decrease fo r the dec is ion -d i rected operat ion. Since we may be the f i r s t to use a large Kalman Gain in dec is ion-d i rected 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 di f ference in performance between reference-directed and dec is ion -d i rected mode. - 106 -7.3 Constant Modulus Algorithm Performance and Discussion In t h i s section we b r i e f l y invest igate the Constant Modulus Algorithm (CMA). It was developed from some general concepts of Godard [29] by T re ich le r , Larimore and Agee [36,60]. It i s novel in t h a t , in the LE conf igurat ion , i t needs no reference to converge and should never c rash . T re ich ler suggested i t could be modified to the DFE configurat ion and use an RLS algor i thm. 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 per fec t l y co r rec t , resu l ts in better performance in d e c i s i o n -directed mode than a two-dimensional error in which one dimension, the phase, can be corrupted by decis ion e r r o r s . This i s p a r t i c u l a r i l y relevant in l i g h t of the conclusions of the last sub-section that much of the error propagation tendancy or ig inates from decision errors corrupting e(k) . Because the HF channel i s rap id ly fad ing , we desire a fas t adapting RLS version of the CMA. In [36] , the CMA convergence performance is examined fo r the family of modulus cost funct ions : J a b =E[ (|d(k )| a - |x ' (k )C (k - l )| a ) b ] We see the least squares cost function i s resu l t ing in the state update: eCMA<k) = Cl-IJL' (k)c.(k-i) | D*CX' (k)C.(k-l )/| x.' (k)c.(k-l) | D C(k) = 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 fe rent e r ro r : a di rected modulus e r r o r . The f i r s t fac tor in e Q ^ ( k ) is the sca lar modulus e r r o r . The second i s a unit vector in the complex plane point ing i n the d i rec t ion of X . ' (k )C(k - l ) . We see e r M » ( k ) i s exactly that - 107 -complex number which when added to X ' ( k ) C ( k - l ) , the symbol estimate, lengthens i t to the correct modulus. This RLS-modified algorithm worked in the l inear equal izer configuration so we immediately t r i e d i t with the DFE. The CMA algorithm does not bas ica l l y change the funct ioning of any of the various equal izer configurations introduced in Section 3 . 1 , i t jus 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 per iodic re t ra in ing mode. During re t ra in ing i t was switched to the normal, reference-directed algorithm so as to receive the exact same r e t r a i n i n g . Table 7.3 presents some prel iminary resu l ts 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 Performance of Constant Modulus Algorithm Channel Condition A Equal izer Data BER Mean # Blocks Uncrashed 1.50 Hz 23 dB .83 DFE .130 5.8 / 8 CMA DFE .273 3.6 / 8 0.15 Hz 13 dB .03 DFE .054 7.3 / 8 CMA DFE .046 7.7 / 8 The CMA equal izer has a better error rate and mean-time-to-crash at slow fade rates while the regular algorithm is superior at 1.5 Hz. The CMA, because i t receives only rad ia l er ror information, depends heavi ly 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 fo r gathering the phase information. Its adaptation rate i s therefore slower than the regular DFE which, in sp i te of dec is ion -d i rected contamination of e (k) , does better at high fade r a t e s . - 108 -In summary, the CMA seems to o f fe r improved performance at slower fade ra tes . Further invest igat ion i s needed to determine where the advantage ends and whether t h i s endpoint i s dependent on various channel or equal izer f a c t o r s . Perhaps a superior hybrid technique could be devised that uses "one and a hal f dimensions" of error information: the modulus error and an f rac t ion of the phase er ror . A l so , 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 fo r the next re t ra in ing sequence. In l i g h t of our CMA r e s u l t s , some invest igat ion of the spectrum technique of [Morgan78] may be warranted i f i t generates a two-dimensional error and is not too computationally burdensome. - 109 -8. CONCLUSION 8.1 Summary And Major Conclusions The HF radio 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, rap id ly moving nu l l s in the frequency domain. Modems designed for use on HF frequencies must therefore be considerably d i f fe rent from those used on other channels. Recently, d igi tal ly - implemented s e r i a l modems used on telephone or t roposcatter channels have been implemented using recurs ive - least -square (Kalman) algorithms to r e a l i z e very fas t converging equa l i za t ion . These algorithms have been suggested f o r HF modem equal izers because they adapt very qu ick ly . L i t t l e previous work s p e c i f i c to the appl icat ion of RLS algorithms to HF equal izat ion 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 ther a s t a t i c or very narrow var ie ty of channel condi t ions , was purposeful ly undetai led, presented unbelievable r e s u l t s , or never investigated equal izer performance in dec is ion -d i rec ted mode. This project set out to determine how well RLS algorithms could perform in the HF modem appl icat ion using a 2400 bps QPSK s e r i a l modulation. It was also hoped to invest igate some of the accompanying synchronization, protocol , software and hardware considerat ions, but i t quickly became obvious that even character iz ing the performance as a function of the very large var iety of possible of HF channel parameters was a formidable task . A lso , the - n o -performance of several equal izer configurations was of p r i o r i t y i n t e r e s t . This character izat ion as a function of both the channel and configuration is necessary to determine the implementation d e t a i l s of a s e r i a l HF modem. An extensive character izat ion of equal izer performance was performed as a funct ion of both equal izer and HF channel parameters. The equal izer parameters investigated were configuration ( l inear versus decision-feedback equa l i za t ion , f rac t iona l versus symbol-spaced taps) , mode (reference versus dec is ion -d i rec ted ) , sample t iming , and length. 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 resolut ion spacing). The major conclusions of the project were: a) The f rac t iona l l y - tapped decision-feedback equal izer (FT DFE) was d e f i n i t e l y the superior conf igurat ion . Performance within 2-3 dB of theoret ica l was found possible on very slow fading channels in reference-directed mode. The DFE was able to provide 1-2 orders of magnitude reduction in error rate compared to a l inear equal izer on a var iety of HF channels. In addi t ion , the f ract iona l l y - tapped DFE proved to be insens i t i ve to sampling phase and provided much better performance on channels with non-symbol-period-spaced mult ipath, channels often overlooked by other authors. b) The decision-feedback equal izers were fade-rate l imited to approximately 0.5 Hz. Even though the RLS (Kalman) algorithm i s fas te r adapting than the LMS algorithmm, and in some cases produced an error rate 2 orders of magnitude smal ler , i t i s not fas t enough to track the ent i re range of - i n -fade rates t yp i ca l of the HF channel. A new d e f i n i t i o n of slow fading which depends on the equal izer length was proposed for exponentially-aged RLS equa l i ze rs . This was q u a l i t a t i v e l y confirmed by the b i t - e r r o r - r a t e r e s u l t s , and by the fac t that the best performance on channels fading faster was obtained with a feedforward tapped-delay - l ine length which just spanned the multipath spread. The received-sample feedforward l ine 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 equal izers in dec is ion -d i rected 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 di f ference between reference and dec is ion -d i rected mode should be s m a l l . This does not seem to be the case i f operating in dec is ion -d i rected mode with a large Kalman Gain, as, i s required in the HF equal izer 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 iso lated wrong decisions can cause a serious disturbance in the adaptation, resu l t ing in further errors and posssibly a crash. This suggests wrong decisions fed back into a DFE symbol estimation are not a serious fac to r in the HF a p p l i c a t i o n . A lso , the error 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 bursts . In sp i te of the dec is ion -d i rected 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 imi ted t ransmit ters , 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) Ident i f ied 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 [24,59] . 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 inef fectua l af ter 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 reference-directed 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 ra te , number of equal izer taps , conf igurat ion (LE vs DFE), mode (decision vs reference-d i rected) , and re t ra in ing 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 dec is ion -d i rected mode. The overal l research project has been a success, espec ia l l y in showing the improved performance of the RLS FT DFE over e i ther LMS, symbol-spaced, or l i n e a r equal izers 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 wi th , for example, fading channels with cont inously -vary ing, channel-cor re lat ion -matr ix eigenvalue ra t ios could be conducted. F i n a l l y , important new resu l ts in dec is ion -d i rected mode were uncovered. - 113 -Though the project concentrated on the problems and advantages of various equal izer 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 resu l ts and by dec is ion -d i rected resu l ts often better than obtainable with current p a r a l l e l tone modems. The dec is ion -d i rected absolute error rates ava i lab le on e i ther type of modem are quite s i g n i f i c a n t , ind icat ing that some sort of ARQ scheme w i l l be required when transmitt ing data other than d i g i t i z e d speech (which i s quite to lerant to e r r o r s ) . Since the s e r i a l scheme resul ts in burst e r ro rs , fo r the same b i t - e r r o r - r a t e i t w i l l require less retransmissions that the p a r a l l e l modems. A l so , the equal izers 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 ra tes . Maybe equal strength paths are f a i r l y atyp ica l and our resu l ts over ly pess imis t i c . S i m i l a r l y , maybe fade rates over 0.15 Hz, say, are uncommon and the t racking rate l i m i t s of the Kalman DFE w i l l not be a problem. Ser ia 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, r e l i a b i l i t y . 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 c a r r i e r frequency acquis i t ion and t rack ing , and handling the computational burden in a compact economical way. - 114 -8.2 Suggestions for Further Research Many areas have been invest igated, 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 est imat ion, should ce r ta in l y be examined. A c r i t i c a l eye i s needed as many factors are dependent on many others , yet i t i s not p rac t i ca l to extensively check every dependency. The fol lowing i s a l i s t of areas considered e i ther p r a c t i c a l , important, or loose ends needing a t tent ion : a) Further quant i f i cat ion of equal izer performance in dec is ion -d i rected mode i s requi red. In order to obtain the optimum performance and aging fac to r , more complete resu l ts as a funct ion of fading bandwidth, SNR, re t ra in ing p e r i o d i c i t y , and aging fac tor are needed. b) To reduce dec is ion -d i rected degradation a thorough invest igat ion of Constant Modulus Algorithm performance as a funct ion of fade ra te , SNR, re t ra in ing p e r i o d i c i t y , and aging factor i s ind icated . This w i l l determine the condit ions under which the CMA performs better than the f u l l - e r r o r algor i thm, and i f the aging factor i s d i f fe rent for t h i s a lgor i thm. A "one and a hal f dimension of e r ror " hybrid CMA algorithm which uses the modulus error and a f r a c t i o n (perhaps hal f ) of the phase error might be better suited than e i ther 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 " fas t " Kalman algorithm whose complexity i s proportional to the number of taps is probably des i rable . Such algorithms are often numerically unstable but there are several new, even - 115 -fas te 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 fect of 16 b i t arithmetic on the choice of a lgor i thm; w i l l square-root normalization be necessary fo 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 ipe l ine of several TMS320 processor chips? Can more than one log ica l stage be put on a s ingle TMS320 pipe computational element? If not, the modem may have to wait about one year fo 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) of fers only marginally better performance than a DFE. Our dec is ion -d i rected Kalman DFE resu l t s though, show the degradation from isolated decis ion errors 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 appl icat ion perhaps MLSE, which makes sequence rather than symbol-by-symbol dec is ions , should not be ruled out. The problem of c a r r i e r frequency acquis i t ion and t rack ing , and frame s ta r t locat ion (for use in s ta r t ing the reference-directed convergence) needs invest igat ion under fading multipath condi t ions . More data from HF channels i s needed to estimate whether multipath channels with equal-RMS-strength paths are common. The p robab 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 in terest . We w i l l then be able to determing the length of equal izer required and whether the l imi ted t racking rates noted in the project w i l l be a commonly occuring hindrance. - 116 -h) Future work might also invest igate 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 equal izer performance, or channel estimator performance for MLSE, might be obtainable. As t h i s thesis i s being w r i t t e n , Glenayre Electronics is s ta r t ing 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 resu l ts of t h i s t h e s i s . 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 in the t h e s i s ) , and is included for in terest only . REFERENCE SEQUENCE GENERATOR "REFSEQ" -"CHDLY" "UNTX" DIFFERENTIAL ENCODING 7-"CHDLY" ( T A P E ^ . "ENTX" SIMULATED BASEBAND CHANNEL "LBBSCH" NOTES: 1) Values in quotation marks are actual symbolic names, e i the r var iables or subrout ines, used in the code. 2) Z " 1 represents a one symbol delay. Z -"EQDLY" "EQDLY" "DENTX" "GETSAM" and "ENRX" "GETSYM" r e f . d i r . "ENRX" "IROD" sim. ch . dec. d i r . « • EQUALIZER "FUNHSU" DIFFERENTIAL DECODING "EQDLY" z - "EqOLY" DECISION DEVICE "ENEQ" ERROR STATISTICS ANALYSIS "ERSTAT" "DUNTX" "DENTX" "ERRMAG" "PHERR" "NUMERR" "ENEQ" DIFFERENTIAL DECODING "DEEQ" "DDERX" "DENRX" Figure A . l Simulation Flow Diagram 00 I - 119 -APPENDIX B Square-Root-Kalman, Fractional ly -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 tab le , or use less numerical dynamic range than others. The algorithm used in t h i s project i s taken from Hsu [31] . It i s an exponentially-aged RLS algorithm based on the diagonal and upper t r iangu lar 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 matr ix , D(k) i s a diagonal matrix fac tor of P, U(k) i s an upper t r iangu lar matrix double factor of P, (with i m p l i c i t unit d iagonal ) , * indicates complex conjugate, and 1 indicates matrix transpose. If P becomes non-posit ive d e f i n i t e due to round-off error the algorithm becomes unstable. The stor ing and updating of D and U instead of P, prevents U*DU' from losing i t s pos i t i ve def ini teness 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 ar i thmet ic , the o r ig ina l Kalman algorithm implemented with Equation 3.9 f a i l s due to the loss of pos i t ive def ini teness of P after 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 t ime. 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 lso , 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 stor ing 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 in 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 inear and decision-feedback configuration w i th , or without f rac t iona l - tapp ing of the input sect ion . This i s a f a i r l y s t ra ight forward process fo r algorithms with computational burden proportional to the number of taps squared (as opposed to fas t Kalman's). It simply requires careful manipulation of the values in 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 dec i s ions . If f ract ional - tapped with spacing T/2, the f i r s t part i s twice as long as normal and the samples are sh i f ted two elements down the l i n e each symbol i t e r a t i o n . The FORTRAN implementation of Hsu's algorithm is presented in Appendix E. - 121 -Hsu's algorithm, as extracted d i r e c t l y from [31] , i s included at the end of t h i s appendix. The reader i s encouraged not to examine i t too c lose ly unless he reads [31] as i t i s optimized by Hsu for storage and computational speed. I t i s included here only for completeness. The reader i s advised that Hsu also suggests a revised version of his algorithm in [31] which i s more sui table fo r f i xed -po in t ar ithmetic implementation. Notes on Hsu's algor i thm: 1) U i s a s t r i c t l y upper t r iangu lar matrix and is 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 unit d iagonal ) . 2) D i s a diagonal matrix i n i t i a l i z e d to the ident i t y matrix and can ofcourse be stored as a vector . 3) £ i s the equal izer tap-weight vector (state vector ) . 4) r i s the tapped-delay-1ine vector (measurement matr ix ) . 5) I|< i s the reference symbol or dec is ion . 6) q = (1/X)-1 i s Hsu's exponential -aging f a c t o r . 7) $ = R/(l+q) = RX , where R is the measurement noise covariance. V I . C O M P U T A T I O N A L 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 factori-zation) formulation defined in Section I V are given by N 7=1 (6.1) (6.2) fj<= 2 PI,JX* + */*» y = 2,3,-•• ,7V, (6.3) gJ = dj(k-\)fJ, j = 1,2,-••,JV, (6.4) (6.5) a y = « , _ , + $,/}•, j = : 2 ,3 , - - - , JV , (6.6) (6.7) _ J _ (6.8) dt(k) = dx{k - l)h„ly, (6-9) (6.10) \ = -fa, (6.11) 1 (6.12) dj(k) = dj(k - l )A, j8y, (6.13) Pi = (6.14) M V = £ | + ft*\,. (6-15) ft = ft + g / l * - (6.16) A l l constants, except {, aJt j= 1,2,---,/Y, y, hq, q, dJt j = l,,---,N, and /3 defined in (6.1) to (6.16), are complex quantities. The quantities defined in (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 in order for i = 1,2,- • • J — 1. The equalizer tap weights C = (C , , C 2 , - • -,CN) are given by e = *(k)y, (6.17) Cj=Cj + gje, 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 in (3.1). u's Square-Root Kalman Equalizer Algorithm [31] - 123 -APPENDIX C - Deta i ls 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 avai lable from the l a t t i c e study [42] . The transmissions were of 1201.739 symbol/second d i f fe rent ia l l y -encoded QPSK on a 1505.107 Hz subcar r ie r . 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° twist on the QPSK conste l la t ion resu l t ing in 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 fe ren t ia l l y -encoded , the conste l la t ion would rotate at l/8th the symbol rate so the symbol rate might be locked onto. This mechanism was not used in our pro ject . The transmissions took place between Cobden, Ontario and the CRC labs at Sh i r ley Bay, Ontario, a distance of about 60 ki lometers. 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 no ise ) . Table C . l Tape Recording Conditions Tape Frequency Local Time Comments 6 7.5555 MHz 11:40 Steady, strong c lear s i g n a l . 9 11.623 MHz 15:00 Strong interference at 10 pulses/second. 11 5.2675 MHz 17:40 Strong signal with deep fading at 0.5 Hz. The test data tapes started with a preamble of noise and several unusual synchronizing sequences which were not of par t i cu la 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 reg is ter with taps on b i t s 7 and - 124 -10. The s ta r t ing phase was considered to be 10 s h i f t s past a l l V s . Note that when th is b i t -s t ream is encoded into 2 -b i t QPSK symbols, the resul t is 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 tapes. The records used a f i xed 512 byte length. Record 0 was a tape header of 512 bytes. The fol lowing records were composed of 256 words of 16 b i t s , each word stor ing the unsigned 12 b i t output from an analogue- to -d ig i ta l converter. There was a s l ight 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 fo r the maximum envelope of the 1505.107 Hz subcarr ier . With no multipath t h i s would give symbol center, and in fac 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 tar t of the M-sequence (needed to converge the equal izer in reference-directed mode) was located by looking at every eighth sample. Once the equal izer could be converged we varied the s tar t ing sample s l i g h t l y to f ind the optimum values shown i n Table C.2. Table C.2 Star t ing 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 exact ly 8, a sample s l ipp ing 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 af ter 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. In te rna l l y , i t runs at the (simulated) receiver sampling rate of 9600 Hz. The input symbols are repl icated 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 for e f f i c i e n c y is operated as a c i r c u l a r buf fer , thus has 49 taps. Only 5, whose posi t ions 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 s ignals tapped out are weighted by the respective products of the 5 independently-fading uni t -var iance processes G(0:4) and the 5 RMS path amplitudes MPGAIN(0:4), and then summed. Complex addit ive white Gaussian noise i s added to give the in f in i te -bandwidth , noisy fading-mult ipath 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 band- l imi t ing low-pass f i l t e r c o e f f i c i e n t s . The delay i s arranged so tha t , including the inherent LPF delay, the tota 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 integral number (DELAY) of symbol per iods. This band- l imited sample stream i s then sub-sampled at 2400 Hz, with the speci f ied sample t iming , resu l t ing in a T/2 FT sample stream ready to be equal ized. If TIMING=0, samples are taken near the beginning of a symbol received on the  f i r s t path. TIMING=7 i s near the end, and 3 or 4 i s near the middle of a symbol. The sample t iming of the other paths depend on both TIMING and MPDEL of course. Frequency mistuning i s simulated in the main program by rotat ing the input samples fed to the equal izer a few degrees more ( in the complex plane) each successive symbol. - 126 -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 SUBROUTINE CDTPCH (SYMBIN,SYMOUT,MPGAIN,MPDEL,SNR,FCNORM, * TIMING,FT,GTAFS) ADAFTED 21/2/84 BY RUSS TRONT -FROM LBBSCH TO USE A CRITICALLY-DAMPED, RATHER THAN BUTTERWORTH, TWO-POLE FILTER TO BANDWIDTH RESTRICT THE FADING PROCESSES. A CRITICALLY-DAMPED FILTER BETTER APPROXIMATES THE GAUSSIAN-BELL SHAPED SPECTRUM OF HF FADING AS COMPARED TO A BUTTERWORTH WHICH IS TOO FLAT TOPPED. CRITICALLY-DAMPED TWO-POLE CHANNEL: THIS ROUTINE PROVIDES A SAMPLE-SPACED (T/8 OF SYMBOL) COMPLEX BASEBAND HF CHANNEL MODEL (FINITE BANDWIDTH, 5 COMPLEX FADING PATHS WITH INDIVIDUAL RELATIVE PROPAGATION DELAYS AND MEAN SQUARE GAINS, AND ADDITIVE WHITE GAUSSIAN NOISE. THE FINITE BANDWIDTH IS PROVIDED BY A LOW-PASS FILTER WHOSE (ODD LENGTH) IMPULSE RESPONSE IS READ IN FROM UNIT 7. THE 5 COMPLEX RAYLEIGH FADING MULTIPATHS ARE PLACABLE TO WITHIN 1/8TH OF A SYMBOL PERIOD RELATIVE PROPAGATION DELAY WITH MPDEL(0:4), AND EACH HAVE INDIVIDUALLY SELECTABLE RMS AMPLITUDES, MPGAIN(0:4). THE RMS PATH GAINS ARE ASSUMED TO BE NORMALIZED BEFORE BEING PASSED TO THIS SUBROUTINE (IE. SUM OF MPGAIN(I)**2 SHOULD BE 1.) FCNORM IS THE -3DB CUT-OFF FREQUENCY, NORMALIZED BY THE SYMBOL RATE, OF THE CRITICALLY-DAMPED TWO-POLE FILTERS CONTROLLING THE FADING BANDWIDTH OF THE INDIVIDUAL PATH GAINS. THE PROPAGATION DELAY OF THIS CHANNEL IS CHDLY (ASSUMING MPGAIN(O).NE.O AND MPDEL(0)=0.). SPECIAL SPECIFICATIONS: IMPLICIT NONE -VAX EXTENTION TO FORTRAN-77 TO FORCE DECLARATION OF ALL VARIABLES. 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> -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 ARGUEMENT DECLARATIONS: COMPLEX SYMBIN 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 c c c c c c c c c c REAL MPGAIN(0:NECHO) INTEGER MPDEL(0:NECHO) REAL SNR REAL FCNORM INTEGER TIMING LOGICAL FT COMPLEX GTAPS(0:NECHO) 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. -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. -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 C C C C C OTHER DECLARATIONS: 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 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. -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 BUFFER "DELAYD". C COMPLEX CHOUT C -THE CHANNEL OUTFUT BEFORE LPF, C AT ANY SAMPLE TIME. C COMPLEX LFFLIN(0:MAXSP7) DATA LPFLIN/MAXSP8*(0.,0.)/ C LFFLIN IS A CIRCULAR DELAY LINE OF C SAMPLE SPACED CHANNEL OUTPUTS C (BEFORE LPF) USED C IK THE LPF CONVOLUTION. IT C IS AT LEAST THE LENGTH OF THE LPF C ROUNDED UP TO THE NEAREST 8 SAMPLES C (NEAREST INTEGER NUMBER OF SYMBOLS)I C REAL LPF(0:MAXSM1) C -THE LPF IMPULSE RESPONSE C OF LENGTH LPFSIZ (ODD1), READ C IN FROM UNIT 7 USING C F0RMAT(I15/(E15.7)). THE 115 C IS THE LPFSIZ, THE RESPONSE IN C E15.7 FOLLOWS ONE PER LINE. C INTEGER DELAY C -THE LPF DELAY ROUNDED UP TO THE C NEAREST NUMBER OF SYMBOLS C INTEGER LPFSIZ,LPFSM1 C -THE SIZE (AND SIZE-1) OF THE LPF C READ IN FROM UNIT 7. C INTEGER LPFLSZ,LPFLM1 C -THE SIZE AND SIZE-1 OF THE PART C OF LPFLIN ACTUALLY USED. C INTEGER LPFPTR DATA LPFPTR/0/ C -THE POINTER TO THE PRESENT LOCATION C IN CIRCULAR DELAY LINE "LPFLIN". C COMPLEX LPFOUT C -THE RESULT OF THE LPF CONVOLUTION. C COMPLEX E(0:NECHO) DATA E/NPATH*(0. , 0.) / C -THE DIFFERENCE BETWEEN THE LAST AND SECOND C TO LAST FADING FILTER OUTPUTS. THESE C ARE USED IN THE FILTERS C RATHER THAN THE SECOND-TO-LAST FILTER C OUTPUTS FOR BETTER NUMERICAL PRECISION C AT VERY LOW Kl/1000) FCNORM. C REAL A0,B1PRIM,B2 C -THE CRITICALLY-DAMPED TWO-POLE DISCRETE C TIME FILTER COEFFICIENTS. REAL AT C -THE CRITICALLY-DAMPED FILTER "A" TIMES THE C SYMBOL PERIOD, T, WHERE "A" IS FROM THE C FILTER TRANSFER FUNCTION: C H(S)=A**2/((S+A)A*2>. C REAL STUFF REAL RATIO INTEGER I,J,SCRTCH REAL R00T2 REAL PI DATA PI/3.1415927/ LOGICAL FCALL DATA FCALL/.TRUE./ C C c IF (FCALL) THEN C -THEN THIS IS THE FIRST CALL. FCALL-.FALSE. - 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). STUFF=0. DO 20 I=0,LFFSM1 STUFF=STUFF+LPF(I)AA2 CONTINUE RATI0=(10.*A(SNR/20.))ASQRT(STUFF) -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. 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 UNIT-VARIANCE 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) C C ENDIF -ITERATE THE "NPATH" FADING PROCESSES. - 130 -DO 100 I=0,NECHO C C -NGAUSS IS A ZERO MEAN, UNIT VARIANCE C GAUSSIAN RANDOM NUMBER GENERATOR. C NORMALIZE COMPLEX NOISE TO MEAN C SQUARE MAGNITUDE OF UNITY. NOISE(I)=CMPLX(NGAUSS(),NGAUSS())/R00T2 C C -THIS IS THE CRITICALLY-DAMPED 2 POLE C FILTER ITERATION SPECIALLY ADAPTED C FOR NUMERICAL PRECISION AT VERY C LOW FCNORM. AS EXPLAINED IN J.K.CAVERS' C MEMO OF 12/12/83, THE TECHNIQUE INVOLVES C ITERATING THE CHANGE, E(I), IN THE LAST C 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 -COMPUTE PATH GAINS FOR USE WHEN CALLED C BY CHANNEL ESTIMATOR SOFTWARE SYSTEM. GTAPS(I)=G(I)*MPGAIN(I) C 100 CONTINUE C C -CREATE 8 SAMPLES FOR EACH SYMBOL. C RUN THE DELAY LINE AHEAD 8 SYMBOLS C WITH THE SAME INPUT. DO 200 1=0,7 POINTR=POINTR-l IF(POINTR.LT.0)POINTR=MAXDEL DELAYD(POINTR)=SYMBIN C C -DO THE CONVOLUTION OF THE CHANNEL C RESPONSE WITH THE SAMPLE SEQUENCE. CHOUT=(0.,0.) DO 150 J=0,NECHO CHOUT=CHOUT+MPGAIN(J)*G(J)* * DELAYD(MOD(POINTR+MPDELCJ),BUFSIZ)) 150 CONTINUE C C -ADD NOISE WITH THE C PROPER SNR TO EACH OUTPUT CHOUT=CHOUT+CMPLX(NGAUSS(),NGAUSS())/(R00T2*RATI0) C C -UPDATE THE LPF DELAY LINE OF C CHOUT'S. LFFPTR=LFFPTR-1 IF (LPFFTR.LT.O) LPFPTR=LPFLM1 LPFLIN(LPFPTR)=CHOUT C C -IF BIT TIMING IS RIGHT, THEN DO C THE LPF CONVOLUTION. C THE FOLLOWING CODE DOES NOT C PRODUCE A CORRECT SYM0UT(2) C FOR TIMING)=4 IF((I.EQ.TIMING) 1 .OR. (FT.AND.(I.EQ.MOD(TIMING+4,8))) 2 ) THEN LPFOUT=(0.,0.) SCRTCH=LPFPTR+LPFLSZ-LPFSIZ DO 170 J=0,LPFSM1 LPFOUT=LPFOUT+LPF(J)* A LPFLIN(MOD(SCRTCH+J,LPFLSZ)) 170 CONTINUE C IF (I.EQ.TIMING) THEN SYMOUTd) =LPF0UT ELSE SYMOUT(2)=LPFOUT ENDIF ENDIF C 200 CONTINUE RETURN END - 131 -APPENDIX E - Fractionally -Tapped Decision-Feedback Equal izer Subroutine See Appendix B fo r discussion of the algorithm. SUBROUTINE FUNHSU (YIN,IREF,IEST,IDEC,EPSIL, 1 NIFFOR,NIFBCK,NDFBCK, 2 DECDIR,TWIST,Q,R,FT) C C WRITTEN BY RUSS TRONT C LAST MODIFICATION 1/11/81 TO ADD ENTRY FUNSET C C THIS ROUTINE IMPLEMENTS A FRACTIONALLY (T/2) TAPPED VERSION OF C THE "UNREVISED" SQUARE ROOT KALMAN C EQUALIZER ALGORITHM DESCRIBED BY F.M. HSU IN IEEE TRANS. C ON INFO. THEORY, VOL. IT-25(5),SEPT. 82, P. 753-763 C (FOR REVISED HSU, SEE SUBROUTINE 'HSU'). C MORE SPECIFICALLY, I T IMPLEMENTS HSU'S "REVISED" C EQUATIONS 6.1-6.7, 6.19-6.21, 6.11, 6.22, AND 6.13-6.18, C BUT UNREVISES THEM BY SETTING HT=0l C I HAVE USED ESSENTIALLY THE SAME NOTATION AS HSU, EXCEPT I C USE R FOR XI'. C THIS IS A SHORT BUT VERY COMPLICATED ALGORITHM, OPTIMIZED C BY HSU FOR SPEED, STORAGE, AND A*ASNEAKY*AA EFFICIENT C CALCULATION OF UPPER TRIANGULAR AND DIAGONAL MATRICES. C C THE EQUALIZER TAKES A NEW FRACTIONALLY TAPPED VECTOR OF C INPUTS 'YIN', AND AN OLD FEEDBACK REFERENCE OR DECISION 'IROD' C (DEPENDING ON WHETHER I T IS IN REFERENCE OR C DECISION DIRECTED MODE), AND SHIFTS THEM INTO THE C TAPPED DELAY LINE 'X' (WHICH IS REALLY TWO SEPARATE TDL'S C CONCATENATED, ONE OF INPUT DATA 'YIN' , C AND ONE OF FED BACK REFERENCE OR DECISION SYMBOLS C 'IROD'). I T THEN CALCULATES THE NEW C OUTPUT ESTIMATE 'IEST' AS THE INNER PRODUCT OF THE C OLD TAP WEIGHT VECTOR ' C AND TAPS 'X', AND QUANTIZES I T TO THE C NEAREST SYMBOL IN ' I D E C . I T THEN CALCULATES A NEW C VALUE FOR 'IROD' DEPENDING ON THE MODE, AND THENCE THE C ERROR 'EPSIL' IN THE ESTIMATE USING EITHER THE REFERENCE C OR THE DECISION. C THE NEW KALMAN GAIN 'G' IS THEN CALCULATED USING THE ABOVE C REFERENCED ALGORITHM. FINALLY, THE ERROR AND THE NEW C KALMAN GAIN ARE USED TO UPDATE THE TAP WEIGHT VECTOR ' C . C c C SPECIAL SPECIFICATIONS: IMPLICIT NONE C -VAX EXTENTION TO FORTRAN-77 TO C FORCE DECLARATION OF ALL VARIABLES. C SAVE C -FORCES RETENTION OF SUBROUTINE C LOCAL VARIABLES BETWEEN CALLS, C LIKE GOOD OLD FORTRAN IVI 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 SET ARRAY SIZES TO C CORRESPOND TO MAXIMUM EXPECTED C NUMBER OF TAPS. C C ARGUEMENT DECLARATIONS: C COMPLEX YIN(2) C -THE FRACTIONALLY TAPPED INPUT SYMBOLS. C I F TIMING IN CHANNEL IS SET FOR MID-C SYMBOL, THEN C YIN(l) IS AT OPTIMAL SAMPLE TIME. C YIN(2) IS AT SUB-OPTIMAL SAMPLE TIME. COMPLEX IREF C -THE INPUT REFERENCE SYMBOL. C I T IS USED IN REFERENCE DIRECTED C MODE ONLY, TO CALCULATE THE C THE ERROR. AND IN THE FOLLOWING CALL C AS A FEEDBACK SYMBOL FOR THE TDL. - 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 c c c COMPLEX IEST COMPLEX IDEC COMPLEX EPSIL INTEGER NIFFOR INTEGER NIFBCK INTEGER NDFBCK LOGICAL DECDIR LOGICAL TWIST REAL Q REAL R LOGICAL FT -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 FEED-NEUTRAL 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) OTHER DECLARATIONS: REAL XI INTEGER NTAPS INTEGER NN,NN1,NN2 -XI=R/(1+Q) -TOTAL NO. OF TAPS. NTAPS=NIFFOR+NIFBCK+NDFBCK. - 134 -C -NIFFOR+NIFBCK, NIFFOR+NIFBCK+1, C NIFFOR+NIFBCK+2. C LOGICAL TOGGLE DATA TOGGLE/.FALSE./ C -FLIPS BACK AND FORTH FROM CALL TO CALL C IF TWIST=.TRUE. CONTROLS WHICH C SIGNAL CONSTELLATION THE QUANTIZER C QUANTIZES TO! C COMPLEX IROD DATA IROD/(0.,0.)/ C -LOCAL STORAGE FOR THE REFERENCE C OR DECISION USED C IN CALCULATING THE ERROR, AND FED-BACK C INTO THE TDL ON THE NEXT CALL. C IN REF. DIRECTED MODE, IT IS 'IREF'. C IN DECISION DIRECTED MODE, IT IS 'IDEC. C COMPLEX X(NTAPMX) DATA X/NTAPMX*(0.,0.)/ C -THE TAPPED DELAY LINE COMPOSED OF C (NIFFOR) FED-FORWARD INPUT SYMBOLS, C FOLLOWED BY (NIFBCK) INPUT SYMBOLS, C FOLLOWED BY (NDFBCK) FEED-BACK C DECISIONS (OR REFERENCE) C SYMBOLS, FOLLOWED BY (NTAPMX-NTAPS) C UNUSED VALUES. IN THE KALMAN FORMULATION C OF AN EQUALIZER, X ALSO PLAYS THE ROLE C OF THE MEASUREMENT MATRIX, COMMONLY C REFERRED TO IN THE LITERATURE AS H(K). C COMPLEX C(NTAPMX) DATA C/NTAPMX*(0.,0.)/ C -THE TAP WEIGHT VECTOR. IN THE KALMAN C FORMULATION OF AN EQUALIZER, C IS THE C STATE VECTOR BEING ESTIMATED. C COMPLEX G(NTAPMX) C -THE KALMAN GAIN MATRIX (IN THE EQUALIZER C CASE, A VECTOR). INITIALLY, G IS USED FOR C SCRATCH STUFF, THEN MODIFIED TO BE C KALMAN GAIN/LAST GAMMA. MULTIPLICATION C BY GAMMA OCCURS IN TAP UPDATE. C COMPLEX U(NTAPMX,NTAPMX) DATA U/NTAPM2*(0.,0.)/ C -AN UPPER TRIANGULAR MATRIX THAT IS ONE OF C THE FACTORS OF P=CONJG(U)*D*TRANSPOSE(U) C DECOMPOSITION. C THE DIAGONAL AND LOWER TRIANGLE ARE NOT C USED. (THE THEORY IMPLICITLY ASSUME THE C DIAGONAL IS ALL (l.,0.)) C REAL D(NTAPMX) DATA D/NTAPMX*(1. ,0.)/ C -A REAL DIAGONAL MATRIX (ONE OF THE C FACTORS OF P), STORED AS A VECTOR. C COMPLEX F(NTAPMX) C -A VECTOR DEFINED BY HSU AS C TRANSPOSE(U)*CONJG(X>, USED FOR C INTERMEDIATE CALCULATIONS. C COMPLEX LAMBDA(NTAPMX) C -A VECTOR DEFINED BY HSU AS F*GAMMA, C AND USED FOR INTERMEDIATE CALCULATIONS. C COMPLEX BETA1 C -USED BY HSU TO REDUCE ARRAY REFERENCES C AND STORAGE. C REAL BETA C -DEFINED IN EQN. 6.21 C REAL ALPHA(NTAPMX) - 135 -c c c c c c c c c c c c c REAL HT REAL HQ REAL GAMMA COMPLEX E INTEGER I,J -USED BY HSU IN INTERMEDIATE CALCULATIONS. ALPHA(NTAPS) CORRESPONDS TO THE ALPHA DEFINED AFTER EQN. 4.4 -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 CAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA C COMPLEX Z45 DATA Z45/(.7071068,.7071068)/ C -USED IN QUANTIZATION TO C "UNTWISTED" CONSTELLATION. COMPLEX QUANT,ZDUMMY QUANT(ZDUMMY)=CMPLX(SIGN(.7071068,REAL(ZDUMMY)), 1 SIGN(.7071068,AIMAG(ZDUMMY))) C -STATEMENT FUNCTION TO QUANTIZE C ZDUMMY TO THE NEAREST SYMBOL IN C THE "TWISTED" CONSTELLATION. C************************************************************************** c C -CONVERT MEASUREMENT NOISE COVARIANCE C INTO XI FOR USE IN UNHSU ALGORITHM. XI=R/(1+Q) C C -CALC AND CHECK TOTAL NO. OF TAPS. NTAPS=NIFFOR+NIFBCK+NDFBCK NN=NIFFOR+NIFBCK NN1=NN+1 NN2=NN+2 C IF (NTAPS.GT.NTAPMX) THEN WRITE(6,99999) FORMAT(' ***** ERROR ***** - TOO MANY EQUALIZER TAPS'/ 99999 STOP ENDIF C C C C C C C c C C C C c C 100 C - INCREASE SIZE OF PARAMETER', "NTAPMX"'/ IN EQUALIZER SUBROUTINE HSU. -UPDATE THE TAPPED DELAY LINE. FOR NON-FRACTIONALLY TAPPED, SHIFT THE WHOLE LINE TO THE RIGHT, ADD THE NEW INPUT TO THE BEGINNING, AND OVERWRITE THE APPROPRIATE ELEMENT WITH THE FEED-BACK SYMBOL 'IROD', WHICH IS EITHER 'IREF' OR 'IDEC' DEPENDING ON MODE. IF NO FEEDBACK TAPS, BE CAREFULI IF FRACTIONALLY TAPPED BE EVEN MORE CAREFUL! IF (.NOT.FT) THEN DO 100 I=NTAPS,2,-1 X(I)=X(I-1) CONTINUE X(1)=YIN(1) IF (NDFBCK.GT.O) X(NN1)=IR0D - 136 -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 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 .NOT. IF (TWIST) TOGGLE= IF (TOGGLE) THEN IDEC=QUANT(IEST) ELSE IDEC=QUANT(Z45*IEST > *CONJG(Z45 > ENDIF -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 TOGGLE -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 EPSIL=IROD-IEST -CALCULATE THE ERROR IN IEST (IF IN DECISION DIRECTED MODE, CALCULATE THE SUSPECTED ERROR). -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)) 190 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)) 200 CONTINUE C HQ=1+Q C -THIS SMALL MODIFICATION IS A QUICK C AND DIRTY CONVERSION OF THE REVISED C HSU ALGORITHM, TO THE UNREVISED ONE CAAAAAHT=ALPHA(NTAPS)AQ HT=0. CAAAAA GAMMA=1./(ALPHA(1)+HT) C 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) 290 CONTINUE 300 CONTINUE C C -NOW UPDATE THE TAP WEIGHTS WITH C THE KALMAN GAIN A ERROR A GAMMA. E=EPSILAGAMMA DO 400 J=l,NTAPS C(J)=C(J)+G(J)AE 400 CONTINUE C RETURN C C C CAAA*AAAAAA**AAAAAAA*AAAAAA**A*AA*AAAA*A*AAAA*AAAAAAA*AAA*A****AA*AAA ENTRY FUNSET C C THIS ENTRY RESETS MATRIX P, VECTOR C, AND SCALAR IROD. C NOTE P IS REALLY CONJGfU)ADATRANSPOSE(U), SO RESETTING IT TO AN C IDENTITY MATRIX INVOLVES CLEARING U, AND SETTING D TO ALL ONES. CAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA C C -ONLY RESET C THE USED ENTRIES. 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.) 5000 CONTINUE 5001 CONTINUE C -NOW RESET IROD IROD'=(0. ,0.) C C RETURN END 138 APPENDIX F - Symbol-Spaced Channel Estimator Subroutine See also Section 5.1 f o r discussion of the channel est imator. - 139 -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 SUBROUTINE CHEST(REF,REC, EST,DECAY,LAMBDA, * MFGSQ,R,NTAPS,C) . WRITTEN 20/6/83 BY RUSS TRONT IMPLEMENTS A KALMAN CHANNEL ESTIMATOR. USES AN INTERNAL MODEL OF AN HF RADIO CHANNEL BASED ON PATH GAINS DERIVED FROM COMFLEX GAUSSIAN NOISE FILTERED BY A 1ST ORDER AUTO-REGRESSIVE PROCESS, AND SCALED BY AN RMS AMPLITUDE MULTIPATH GAIN. EXPONENTIAL-AGING IS ALSO AVAILABLE. THE ALGORITHM IS IMPLEMENTED AS SIMPLE STRAIGHT FORWARD BRUTE FORCE (IE. ORDER NTAPS**2) KALMAN (AS OPPOSED TO FAST KALMAN OR SQUARE-ROOT NORMALIZED KALMAN). SEE FOR INSTANCE: A.GELB, "APPLIED OPTIMAL ESTIMATION", MIT PRESS, 1974, EQUATIONS 8.1-16 AND 8.4-4. GELB'S NOTATION IS USED, EXCEPT HIS S=l/LAMBDA. SFECIAL DECLARATIONS: IMPLICIT NONE SAVE -VAX EXTENSION TO FORTRAN-77 TO FORCE DECLARATION OF ALL VARIABLES. -SAVES ALL VARIABLE BETWEEN CALLS, LIKE GOOD OLD FORTRAN IVI INTEGER NTAPMX PARAMETER (NTAPMX=20) -MAX NUMBER OF TAPS IN THE TAPPED DELAY LINE CHANNEL MODEL. ARGUEMENT DECLARATIONS: COMFLEX REF COMPLEX REC COMPLEX EST REAL DECAY REAL LAMBDA INTEGER NTAPS -THE CURRENT SYMBOL BEING.TRANSMITTED. -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. REAL MPGSQ(O-.NTAPS-l) REAL R COMPLEX C(NTAPS) -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. OTHER DECLARATIONS: COMPLEX ERROR -THE ESTIMATION ERROR IN THE CURRENT SYMBOL. - 140 -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 INTEGER I,J,K -IS INITIALIZATION DONE ON FIRST CALL YET? - DECAY**2 AND 1-DECAY**2 C THE CODE: C IF (.NOT.INIT) THEN IF (NTAPS.GT.NTAPMX) THEN WRTTE(6,9999)NTAPS FORMAT(' ***** ERROR ***** IN SUBROUTINE CHEST:'/ * ' NTAPS=',G14.7,' TOO LARGE!', * ' CHANGE PARAMETER NTAPMX.') STOP ENDIF 9999 10 C C C 100 DSQ=DECAY**2 OMDSC=l.-DSQ DO 10 1=1,NTAPS P(I,I)=(1.,0) CONTINUE INIT=.TRUE. ENDIF DO 100 I=NTAPS,2,-1 X(I)=X(I-1) CONTINUE X(1)=REF -SHIFT THE TRANSMITTED SYMBOLS DOWN THE TDL. -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.) - 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)) 1100 CONTINUE ALPHA=ALPHA+X(I)*G(I) 1200 CONTINUE C C -FORM ERROR=THE ESTIMATION ERROR, C THE KALMAN GAIN=NUMERATOR/REAL(ALPHA), C UPDATE THE STATE VECTOR (IE. PATH TAP WEIGHTS), C 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) 1250 CONTINUE 1300 CONTINUE C C -CALCULATE T1=TEMP*P C =CI-G*TRANSP(X)3*P DO 1600 1=1,NTAPS DO 1500 K=l,NTAPS T1(I,K)=(0.,0.) 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