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Equalization for DS-UWB systems Parihar, Ambuj 2006

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EQUALIZATION FOR DS-UWB SYSTEMS by Ambuj Parihar B.Eng., Malaviya National Institute of Technology, India, 1999  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS F O R T H E D E G R E E OF M A S T E R OF A P P L I E D SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Electrical and Computer Engineering)  T H E UNIVERSITY OF BRITISH C O L U M B I A November 2005 © Ambuj Parihar, 2005  Abstract Ultra-wideband wireless transmission has attracted considerable attention both in academia and industry. For high-rate transmission, direct sequence based ultra-wideband (DS-UWB) systems are a strong contender for standardization by the IEEE 802.15.3a Wireless Personal Area Networks (WPAN) committee. The DS-UWB proposal envisages two modulation formats: binary phase shift keying (BPSK) and 4-ary bi-orthogonal shift keying (4BOK). Due to the large transmission bandwidth, the U W B channel is characterized by a long rootmean-square delay spread and the R A K E receiver cannot always overcome the resulting intersymbol interference. We therefore study equalization for DS-UWB systems employing B P S K and 4BOK modulation. In the first part of this work, we consider equalization for DS-UWB with B P S K modulation, which is mandatory for standard-proposal compliant DS-UWB devices. Assuming R A K E preprocessing at the receiver, we analyze the performance limits applicable to any equalizer, taking into account practical constraints such as receiver filtering, sampling, and the number of R A K E fingers. Our results show that chip-rate sampling is sufficient for close-to-optimum performance. For analysis of suboptimum equalization strategies, we further study the distribution of the zeros of the channel transfer function including R A K E combining. Our findings suggest that linear equalization is well suited for the lower data rate modes of DS-UWB systems, whereas nonlinear equalization is required for high-data rate modes. Moreover, we devise equalization schemes with widely linear processing, which ii  Ill  improves performance without increasing equalizer complexity. Simulation and numerical results confirm the significance of our analysis and equalizer designs and show that low-complexity (widely) linear and nonlinear equalizers perform close to the pertinent theoretical limit. In the second part, we investigate equalization for DS-UWB with 4BOK. To this end, we first derive expressions for the bit-error rate according to the matched-filter bound for 4BOK DS-UWB, which serve as theoretical performance limits for equalization. We then devise structures and methods for filter optimization for low-complexity linear and nonlinear equalization schemes. In this context, we develop a new equivalent multiple-input multiple-output (MIMO) description of 4BOK DS-UWB, which facilitates the design of efficient equalizers using MIMO filter optimization techniques. Furthermore, we propose the application of widely linear processing to these equalizers. Simulation and semi-analytical results show that (a) MIMO equalization is greatly advantageous over more obvious nonMIMO schemes, and (b) the proposed MIMO equalizers allow for power-efficient 4BOK DSU W B transmission close to the theoretical limits with moderate computational complexity.  Contents Abstract  ii  Contents  iv  List of Tables  viii  List of Figures  ix  List of Abbreviations and Symbols  xiii  Acknowledgments  xvi  1 Introduction  1  1.1  Ultra-wideband Technology  2  1.2  DS-UWB Proposal for IEEE 802.15.3 W P A N Physical Layer Standard  1.3  Challenges and Motivation  6  1.4  Contributions  8  1.4.1  Contributions for BPSK DS-UWB Systems  8  1.4.2  Contributions for 4BOK DS-UWB Systems  9  iv  ...  5  CONTENTS 1.5 2  3  '  .  Thesis Organization  v 11  Transmission System  13  2.1  Introduction  13  2.2  Transmission Model  13  2.2.1  B P S K Modulation .  14  2.2.2  4BOK Modulation  15  2.3  Channel Model  16  2.4  Receiver  20  2.4.1  Receiver for BPSK  21  2.4.2  Receiver for 4BOK  22  Equalization for B P S K  25  3.1  Performance Measures for Equalization  26  3.1.1  Matched Filter Bound (MFB)  26  3.1.1.1  MFB I  27  3.1.1.2  M F B II  27  3.1.1.3  M F B III  28  3.1.2  Densities of Zeros of the Overall Transfer Function  29  3.1.2.1  Overall Transfer Function  29  3.1.2.2  Density of Zeros  30  3.1.2.3  Marginal Density and Cumulative Distribution  31  CONTENTS  vi 3.1.2.4  3.2  31  Equalization Strategies  32  3.2.1  Classical Equalization Schemes  32  3.2.1.1  Linear Equalization  32  Decision Feedback Equalization (DFE)  33  •3.2.1.2 3.2.1.3 3.2.2  4  Application to Equalizer Design for DS-UWB  Delayed Decision-Feedback Sequence Estimation (DDFSE) . 34  Equalization Schemes with W L Processing  36  3.2.2.1  Widely Linear Equalization (WLE)  37  3.2.2.2  Widely Linear Decision Feedback Equalization (WDFE)  3.2.2.3  Widely Linear DDFSE (WDDFSE)  . . 38 39  Equalization for 4 B O K  40  4.1  Performance Bound for 4BOK Equalization  41  4.1.1  MFB I  41  4.1.2  M F B II •  4.2  •;' •  43  Equalization Strategies  45  4.2.1  4BOK Equalization with SIMO Filter Optimization '.' ."  45  4.2.1.1  Filter Design I  48  4.2.1.2  Filter Design II  49  4.2.1.3  L E based on Filter I and Filter II  51  4.2.1.4  DDFSE-based on Filter I and Filter II  52  4.2.2  4BOK Equalization with MIMO Filter Optimization  53  CONTENTS  • 4.2.2.1  4.2.3  5  MIMO Optimization based L E and DDFSE  vii 57  4BOK Equalization with W L - M I M O Filter Optimization  58  4.2.3.1  60  W L - M I M O - L E and WL-MIMO-DDFSE  Results and Discussion  61  5.1  Performance Results and Discussion for B P S K DS-UWB Systems  62  5.1.1  Simulation Parameters  62  5.1.2  Application of Distribution of Zeros to Equalizer Design for DS-UWB  63  5.1.3  Simulation Results for B P S K DS-UWB Systems  65  5.1.3.1  Equalization Strategies and MFBs  66  5.1.3.2  Linear vs. W L Processing  70  5.1.3.3  Number of R A K E Fingers  71  5.2  6  .  Performance Results and Discussion for 4BOK DS-UWB Systems  73  5.2.1  Simulation Parameters  74  5.2.2  Simulation Results  75  5.2.2.1  Filter Optimization  75  5.2.2.2  Equalization for 4BOK and MFBs  80  Conclusions  Bibliography  84 87  List of Tables 2.1  U W B channel parameters for different scenario models  18  5.1  Parameters for the considered BPSK DS-UWB systems  63  5.2  Parameters for the considered 4BOK DS-UWB systems  75  viii  List of Figures 1.1  U W B spectral mask for indoor and outdoor communications systems  3  2.1  Block diagram of B P S K DS-UWB transmission system  2.2  Block diagram of 4BOK DS-UWB transmission system  2.3  Impulse response of a sample realization for channel scenarios, a) CM1 b)CM4. 19  4.1  Block diagram of D F E with SIMO filter optimization for 4BOK DS-UWB.  4.2  SIMO Filter Design I: Magnitudes of the impulse responses z„[fc] = fF,v[k] *  14 '  16  . 46  h \ [k] at the F F F outputs (circles) and of the F B F coefficients fs,vi [k] (stars) v  (see also Figs. 4.1 and 4.4) for an exemplary CM4 channel realization and spreading with code length N — 6 4.3  50  SIMO Filter Design II: Magnitudes of the impulse responses i [k] = fF,v\k\ * u  h \ [fc] at the F F F outputs (circles) and of the F B F coefficients v  JB,UI  [fc] (stars)  (see also Figs. 4.1 and 4.4) for an exemplary CM4 channel realization and  4.4  spreading with code length N = 6  51  Block diagram of D F E with MIMO filter optimization for 4BOK DS-UWB.  54  ix  LIST OF FIGURES 4.5  x  MIMO Filter Design: Magnitudes of the impulse responses i [k] = v  *  h \ [k] at the F F F outputs (circles) and of the F B F coefficients /_ ,„i [k] (stars) v  (see also Figs. 4.1 and 4.4) for an exemplary CM4 channel realization and spreading with code length N = 6 5.1  57  U W B CM1 and N = 24. z = x + ]y. (a) Normalized density f (z)/(L z  - 1)  of zeros of effective transfer function H(z). (b) Normalized marginal density f (r) j{L — 1) of zeros of H(z). (c) Normalized average number n(R)/(L — 1) r  of zeros of H(z) inside the disc \z\ < R 5.2  U W B CM4 and N = 24. z = x+jy.  64  (a) Normalized density f (z)/(L z  - 1)  of zeros of effective transfer function H(z). (b) Normalized marginal density f (r)/(L r  — 1) of zeros of H(z). (c) Normalized average number n(R)/(L — 1)  of zeros of H(z) inside the disc \z\ < R 5.3  U W B CM4 and N = 6. z = x+jy.  65  (a) Normalized density f {z)/{L z  - 1)  of zeros of effective transfer function H(z). (b) Normalized marginal density f (r)/(L r  — 1) of zeros of H(z). (c) Normalized average number n(R)/(L — 1)  of zeros of H(z) inside the disc \z\ < R 5.4  B E R versus 10log {E /N ) w  b  0  66  for M M S E - L E and M M S E - D F E for CM1 and  TV = 24. Also shown: R A K E combining without equalization and MFBs I-III (cf. Section 3.1.1) 5.5  67  B E R versus 10 log (E /N ) 10  b  0  for M M S E - L E , M M S E - D F E , and MMSE-DDFSE  for CM4 and N = 24. Also shown: R A K E combining without equalization and MFBs I-III (cf. Section 3.1.1)  68  LIST OF FIGURES 5.6  xi  B E R versus 10 log (£ /7V~o) for M M S E - L E , M M S E - D F E , and MMSE-DDFSE 10  b  with different filter lengths for CM4 and N = 6. Also shown: R A K E combining without equalization and MFBs I-III (cf. Section 3.1.1) 5.7  69  B E R versus 101og (E /A ) for MMSE-(W)LE and MMSE-(W)DFE for r  10  6  0  CM4 and N = 24. Also shown: M M S E - W D D F S E , R A K E combining without equalization, and MFBs I-III (cf. Section 3.1.1) 5.8  71  B E R versus 10 log (£; /7Vo) for M M S E - L E and M M S E - W L E with different 10  6  filter lengths for CM4 and N = 6. Also shown: M M S E - D F E with q = 50 F  and qs = 23, R A K E combining without equalization, and MFBs I-III (cf. Section 3.1.1) 5.9  72  B E R versus 10 \og (E /N ) 1Q  b  0  for M M S E - D F E and M M S E - W D F E with dif-  ferent filter lengths for CM4 and N = 6. Also shown: R A K E combining without equalization, and MFBs I-III (cf. Section 3.1.1)  73  5.10 Average SNR loss A S N R ( F ) as function of the number of R A K E fingers F for the three scenarios (CM1, N = 24), (CM4, TV = 24), and (CM4, N = 6).  74  5.11 Performance comparison for equalization with SIMO and MIMO filter optimization.- CM4 and N = 6. : .. :  76  5.12 Performance comparison for equalization for CM4 and N — 24. (a) Between SIMO and MIMO filter optimization. (b)Between MIMO and W L - M I M O filter optimization  78  5.13 Performance comparison for equalization with MIMO and W L - M I M O filter optimization. CM4 and N = 6  79  LIST OF FIGURES  xii  5.14 B E R versus 101og (f? /iVo) for CM1 and N = 24. R A K E combining with10  6  out equalization, W L E , and W D F E (MIMO equalization). B E R  MFB-I>  BER'  M F B  _ , according to M F B I and B E R ^  F B  _  n )  BER'  Also shown: M F B  _  M  accord-  ing to M F B II from Section 4.1  80  5.15 B E R versus 101og (£7 /iVo) for CM4 and N = 24. R A K E combining with10  6  out equalization, W L E , W D F E , and W D D F S E (MIMO equalization). Also shown: B E R ^  F B  _  h  B E R  'MFB-I  according to M F B I and B E R | J , _ „ , B E R ' FB  M F B  _  M  according to M F B II from Section 4.1 5.16 B E R versus 10 \og (E /N ) 1Q  b  0  81  for CM4 and N = 6. R A K E combining without  equalization, W L E and W D F E with different filter lengths, and W D D F S E and D F E for q = 50, q = 35 (MIMO equalization). Also shown: B E R ^ _ | , F  BER'  M F B  B  _ i according to M F B I and B E R ^  from Section 4.1  F B  F B  _ , BER' M  M F B  _ „ according to M F B II  83  List of Abbreviations and Symbols Acronyms AWGN  Additive White Gaussian Noise  BER  Bit Error Rate  BPSK  Binary Phase Shift Keying  cdf  Cumulative Distribution Function  CDMA  Code Division Multiple Access  DDFSE  Delayed Decision Feedback Sequence Estimation  DFE  Decision-Feedback Equalization  DS-UWB  Direct Sequence Ultra Wideband  FBF  Feedback Filter  FFF  Feed-Forward Filter  GPS  Global Positioning System  ISI  Intersymbol Interference  LE  Linear Equalization  LOS  Line-of-Sight  NLOS  Non-Line-of-Sight  BOK  Bi-Orthogonal Keying  MFB  Matched Filter Bound  MIMO  Multiple-Input Multiple-Output  xiv  LIST OF FIGURES MLSE  Maximum-likelihood sequence estimation  MMSE  Minimum Mean-Square Error  MSE  Mean-Square Error  OFDM  Orthogonal Frequency Division Multiplexing  pdf  Probability Density Function  PEP  Pairwise Error Probability  SIMO  Single-Input Multiple-Output  SISO  Single-Input Single-Output  SNR  Signal-to-Noise Ratio'  SRRC  Square-Root Raised Cosine  UWB  Ultra-Wideband  UMTS  Universal Mobile Telecommunication System  VA  Viterbi Algorithm  WDFE  Widely-Linear Decision-Feedback Equalization  WDDFSE  Widely-Linear Delayed Decision Feedback Sequence Estimation  WL  Widely Linear  WLE  Widely-Linear Linear Equalization  WPAN  Wireless Personal Area Network  WLAN  Wireless Local Area Network  Operators and Notation argmax{-}  Argument maximizing the expression in the bracket  argmin{-}  Argument minimizing the expression in the bracket  |•| •*•  Absolute value of a complex number Convolution  )  LIST OF FIGURES  X V  In x  Natural logarithm of x  Re{-}, Im{-}  Real and Imaginary part of a complex number  z{-}  ^-transform  c5(-)  Dirac delta function  1 1 •1 1  Frobenius norm  sign{a;}  Sign of  E{-}  Expectation  [•]*  Complex conjugate  f  x G  IR  Matrix or vector transposition Matrix or vector Hermitian transposition  0m J  m  Zero vector with m elements Identity matrix with dimension m x m  Acknowledgments I would like to express my gratitude to Dr. Lutz Lampe for his invaluable guidance and continuous support at every stage of my research. His suggestions and constant encouragement helped me complete this challenging work. I am also indebted to Dr. Cyril Leung for providing much of the initial motivation to pursue this research work and for providing invaluable feedback that helped in improving the quality of this work. I would also like to extend my sincere thanks to Dr. Robert Schober for his invaluable inputs and suggestions that were very helpful throughout this work. Furthermore, I am thankful to my colleagues in the Communications Theory group for creating a stimulating and friendly environment. This work was partially supported by the National Sciences and Engineering Research Council (NSERC) (Grants OGP0001731 and C R D P J 320552) and by Bell University Laboratories.  xvi  Chapter 1 Introduction Advances in wireless communications in the last decade have revolutionized our lives. The market for wireless devices such as mobile phones, laptops and personal digital assistants has grown extremely rapidly in the past several years. Wireless technologies such as wireless local area networks (WLANs) and Bluetooth, a wireless personal area network (WPAN) technology, have enabled high speed data transmission over shorter distances, whereas wireless metropolitan area network (commonly known as W i M A X ) technology has enabled high-throughput broadband connections over longer distances.  In the W P A N domain,  Bluetooth technology promises data rates of up to 1 Mbps using the 2.4 GHz Industrial, Scientific and Medical (ISM) band [Blu03]. However, the growing demand for much higher data rates and the rising importance of WPANs, give rise to the need for new reliable wireless technologies enabling robust, low-cost, high-rate and low-power W P A N devices. With the approval of its application for commercial purposes in the U.S. and with similar trends being anticipated elsewhere in the world, the ultra-wideband (UWB) technology is well positioned to meet the evolving demands in wireless networking and to dominate the market place.  1  1.1 Ultra-wideband Technology  1.1  2  Ultra-wideband Technology  Traditionally, the term ' U W B signals' refers to very short duration pulses of the order of a few nanoseconds that do not require a carrier modulation. Until the late 1980s this technology was mainly known as 'impulse radio', 'baseband' or 'carrier free' technology. U W B pulses were first produced, in the late 1890s, using spark gap transmitters by Gugliemo Marconi, and were used to transmit Morse code across the Atlantic in 1901 [Fon04]. The potential of pulse based U W B technology for use in radars and communications was realized in the 1960s and 70s from the study of electromagnetic-wave propagation as viewed from the time-domain perspective [Fon04, BR78]. The term 'Ultra-wideband' was first used in 1989 by the Defense Advanced Research Projects Agency (DARPA), for differentiating impulse based radars from conventional ones [Fon04]. Until recently, this technology was restricted for use in defense applications such as radars and covert communications. The advances in semiconductors and microprocessors in the last few years, along with a more mature understanding of U W B system characteristics, have made it possible to build commercial applications based on U W B technology. In the U.S., the Federal Communications Commission (FCC) issued First Report and Order [FCC02] in February 2002, approving commercial use of U W B devices in the 3.1-10.6 GHz band with strict limits on power emission levels to allow co-existence of U W B systems with other wireless communication systems such as IEEE 802.11a W L A N , Universal Mobile Telecommunication System (UMTS) and Global Positioning System (GPS). The F C C spectral mask, shown in Fig. 1.1, allows communication devices to use 7.5 GHz of bandwidth between 3.1 GHz and 10.6 GHz, with power spectral density not exceeding -41.3 dBm/MHz (75 nW/MHz) [FCC02]. The F C C defines a U W B signal as any transmission with fractional bandwidth equal to or greater than 20 percent of the center frequency (/ ), or with -10 dB bandwidth occupying c  500 MHz or more of the spectrum in 3.1 to 10.6 GHz band at all times of transmission. The  1.1 Ultra-wideband Technology  3  -40 E -45  I  -50  c-55  ('/) D_  •  -65h  C0  1-70  — Indoor limit • - P a r t 15 limitl  - Outoor limit - Part 15 limitl  -75 Frequency in G H z  Frequency in G H z  Figure 1.1: UWB spectral mask for indoor and outdoor communications systems. fractional bandwidth (Bf) is defined as the ratio of -10 dB bandwidth (B) to the center frequency (/ ). Mathematically, c  B = B/f f  c  = 2(f -f )/(f h  l  h  + f) l  (1.1)  where /; and fh are the lowest and highest frequencies with signal 10 dB below the peak emission, respectively. The center frequency f  c  is given by (fh + fi)/2 and the -10 dB  bandwidth is (f - fi). h  Given its extremely wide bandwidth U W B , promises improved channel capacity. The channel capacity (C), defined as the maximum achievable data rate over an additive white Gaussian noise (AWGN) channel is given by Shannon's formula [Sha48] C = #{log (l + 2  SNR)}  (1.2)  where, SNR is the signal to noise ratio. From (1.2), it is evident that the channel capacity increases linearly with B, whereas it increases logarithmically with the SNR.  Thus, the  wide bandwidth of U W B allows for the possibility of very high data rates even at relatively low SNR.  However, due to F C C power restrictions the high data rate U W B devices can  operate only over short ranges (< 10m). U W B devices are intended to operate in the 3.110.6 GHz band, on a license free basis, without causing harmful interference to existing  4  1.1 Ultra-wideband Technology  radio systems like IEEE 802.11a W L A N , UMTS and GPS, thereby increasing the spectrum utilization. Along with the aforementioned advantages, U W B has many unique properties such as robustness to fading, multipath diversity, low probability of detection and accurate positioning capabilities [PH03], that make it a very attractive technology for short range high rate WPANs and high precision positioning/tracking systems [YG04, OSR 04]. +  In the last few years U W B has received enormous attention from industry. Technology leaders like Motorola, Texas Instruments, Mitsubishi, Freescale and General Atomics have already started development of U W B based W P A N devices capable of providing data rates up to 500 Mbps. A large group of companies under the Zigbee Alliance [Zig] is working towards the development of UWB based devices that provide lower data rates (up to 250 kbps) and consume very low power, and so are characterized by long battery life. Market researchers are predicting that usage of U W B based systems will exceed 150 million devices by the end of 2008 [Sur05]. In order to harmonize interoperability of U W B devices, the IEEE 802.15.3a working group [WPA] is pursuing standardization of an alternative physical layer for WPANs based on U W B technology. In particular, the IEEE 802.15.3a committee is considering two standard proposals: direct-sequence spreading based U W B (DS-UWB) systems [FKMW05] and U W B systems based on multiband orthogonal frequency-division multiplexing (MB-OFDM) [P8004]. Both these proposals are different from the original U W B 'impulse radio' technology in the sense that these proposals use carrier based systems. In this research work we focus on DS-UWB systems. In the following section, the DS-UWB standard is briefly reviewed. Section 1.3 describes the specific challenges faced by DS-UWB systems that have been addressed in this research work. Section 1.4 provides a brief summary of contributions made in this work. Finally, Section 1.5 outlines the thesis organization.  1.2 DS-UWB Proposal for IEEE 802.15.3 WPAN Physical Layer Standard  1.2  5  D S - U W B Proposal for I E E E 802.15.3 W P A N Physical Layer Standard  The DS-UWB standard proposal [FKMW05] envisages two bands of operation: the lower band occupies 1.75 GHz of spectrum from 3.1 GHz to 4.85 GHz and the higher band occupies 3.5 GHz of spectrum from 6.2 GHz to 9.7 GHz. Each band supports up to six piconets. Any DS-UWB compliant device is required to support piconet channels 1-4, that lie in the lower band. Support for piconets 5-6 in the lower band and 7-12 in the upper band is optional. For each of these 12 piconets, the proposal specifies a fixed chip rate and a fixed center frequency. In order to support data rates ranging from 28 Mbps to 2 Gbps (depending on the code rate), the standard specifies spreading codes of lengths varying from 1 to 24 chips for each of the two modulation schemes (cf. [FKMW05, Table 6-8] for a list of spreading codes). The DS-UWB physical layer proposal aims at utilizing the unique properties of UWB by using the wide bandwidth of the two bands and at the same time exploiting the multiple access capabilities offered by the standard code division multiple access(CDMA) technique. The DS-UWB technology, however is different from conventional spread spectrum in the sense that the purpose of employing the spreading codes is not to increase the signal bandwidth, but to reduce the interference from adjacent piconets by using the orthogonal property of these spreading codes [WMLM02]. The chip rate, which determines the bandwidth of the DS-UWB signal, remains fixed regardless of the length of the spreading code used. The DS-UWB standard proposal specifies the use of binary phase-shift keying (BPSK) and 4-ary bi-orthogonal keying (4BOK) [ProOl] to modulate data symbols. Any DS-UWB compliant device is required to be able to transmit and receive B P S K signals, and to transmit 4BOK signals. Support for the ability to receive and demodulate 4BOK modulated signals is optional (for more details on BPSK and 4BOK see Sections 2.2.1 and 2.2.2, re-  1.3 Challenges and Motivation  6  spectively).The proposal specifies use of square root raised cosine (SRRC) pulse shaping. Use of convolutional codes of ratse 1/2 and 3/4 is specified, to provide forward error correction. In order to reduce the sensitivity of convolutional codes to error bursts, convolutional interleaving is used in conjunction with the encoder. In this work we do not consider the effect of convolutional coding and interleaving.  1.3  Challenges and Motivation  The multipath resolution of a signal that propagates over a channel is defined as the minimum delay between two paths that can be distinguished at the receiver as individual multipath components, and is given by the inverse of the bandwidth occupied by the transmitted signal. All the multipath components separated by more than the pulse duration can be resolved as individual paths with distinct propagation delays, whereas multipath components arriving within the resolution time of received signal cannot be resolved as individual paths and hence interfere constructively/destructively, resulting in fading. The very fine resolution provided by U W B due to the short duration of pulses (0.7 ns for DSU W B systems in the lower band) offers, ad vantages of robustness to fading and enhanced multipath diversity. Due to this very short duration of U W B pulses, fewer multipath components arrive within the resolution period, thereby reducing destructive interference and resulting in reduced signal fading. Moreover, a larger number of multipaths are resolvable, resulting in significant multipath diversity. R A K E receivers have been suggested for DS-UWB systems in order to exploit this diversity by coherently combining the multipaths [RMMW03, CWVM02, RSF03]. In indoor U W B channels signal propagation typically results in long root-mean-squared (rms) delay spreads, whereas the symbol durations in DS-UWB are comparatively very short. As an example, to achieve an uncoded data rate of 220 Mbps, the symbol duration  7  1.3 Challenges and Motivation  for DS-UWB is about 5.3 ns, whereas the typical rms delay spread at a distance of 10 m is between 14-26 ns. This results in significant ISI that spans a number of symbols. Thus, low-complexity equalizers are needed in order to effectively mitigate ISI at the output of R A K E combiner. Equalization for DS-UWB has been addressed in a number of recent publications. In [MWS03] fractionally-spaced R A K E receivers have been analyzed and in [ED05] the performance of decision-feedback and linear equalization techniques for DSUWB systems employing Gaussian monocycles, a derivative of Gaussian pulse, is discussed. R A K E performance for carrierless pulse-based U W B transmission has been investigated in [RSF03]. However, most of these works either use pulse position modulation or Gaussian monocycle (which do not require a carrier), and hence do not provide a real evaluation of these equalization schemes when applied to DS-UWB systems that follow standard specifications (cf. Section 1.2, [FKMW05]). In the 4BOK modulation scheme, a set of 4 bi-orthogonal signals is constructed by mapping two bits of information onto one of the four spreading sequences that include two orthogonal codewords and their negatives (Section 2.2.2, [ProOl]). Although the 4BOK modulation scheme has been specified in the DS-UWB standard proposal [FKMW05], equalization for 4BOK DS-UWB systems has received only little attention. Ishiyama and Ohtsuki [IO04] study frequency-domain equalization for 4BOK DS-UWB. Equalization is performed at the chip rate and relies on the use of a non standard-compliant cyclic prefix. Takizawa and Kohno [TK04] propose a suboptimum delayed decision feedback sequence estimation (DDFSE) [DHH89] scheme. However, since no prefllter is applied, state reduction will be problematic for non-minimum phase channels (cf. Section 3.1.2.4). j  Another issue with 4BOK DS-UWB systems is the interference due to cross-correlation between spreading sequences. When the 4BOK modulated signal propagates through the multipath U W B channel the orthogonality of spreading codes is lost, resulting in severe interference due to cross-correlation between spreading sequences (cf. Chapter 4). This  1.4 Contributions  8  interference degrades the performance of 4BOK DS-UWB systems even when long spreading sequences (24 chips) are used. In DS-UWB systems, higher data rates are achieved by reducing the length of spreading sequences while keeping the chip rate constant [FKMW05]. Therefore, at higher data rates the problem of interference due to the cross-correlation of the spreading sequences further intensifies, thereby seriously limiting the performance of 4BOK DS-UWB systems. Conventional equalization schemes such as linear equalization (LE) and decision feedback equalization (DFE) that could effectively reduce ISI in B P S K systems cannot be applied to 4BOK DS-UWB systems in their current form, since they ignore the aforementioned cross-correlation effects (cf. Section 4.2). Therefore, in this work we aim at developing equalization schemes specific to 4BOK DS-UWB systems that can effectively eliminate the interference and at the same time provide a good performance-complexity tradeoff.  1.4  Contributions  Since B P S K and 4BOK modulation schemes require a conceptually different analysis and lead to essentially different equalizer structures, the contributions of this research work are described in two parts.  1.4.1  Contributions for B P S K D S - U W B Systems  For B P S K DS-UWB systems we make the following contributions. • We derive different versions of the M F B for DS-UWB. More specifically, besides the conventional M F B corresponding to the channel seen at the receiver input, which yields the absolute performance limit for any equalization scheme [ProOl], we consider MFBs which take into account the effect of (a) chip-matched filtering and chip-rate sampling and (b) R A K E combining with a limited number of R A K E fingers. The  1.4 Contributions  9  evaluation of these MFBs for the IEEE 802.15.3a standard channel model for W P A N systems [P8002] shows that (a) a R A K E receiver combined with low-complexity equalization can well approach the performance of optimum equalization, (b) chip-matched filtering and chip-rate sampling yields a close-to-optimum performance and not much additional gain could be achievable with fractionally-spaced sampling, and (c) the number of R A K E fingers to sufficiently capture the useful received energy varies between about 8 and 30 depending on the underlying UWB channel. • We develop an analytical expression for and investigate the distribution of the zeros of the channel transfer function effective at the R A K E combiner output. The distribution of the zeros provides useful information on the design of suboptimum equalizers, cf. [SG01, SG02]. In particular, we compare linear equalization (LE) and nonlinear equalization, where we focus on low-complexity decision-feedback equalization (DFE) [BP79] and delayed decision feedback sequence estimation (DDFSE) [DHH89] schemes. It is shown that L E is well suited for lower data rate modes (long spreading sequences), whereas D F E is favorably applied for high-data rate modes (short spreading sequences). • DS-UWB with SRRC pulse shaping and carrier modulation is found to yield a secondorder noncircular received signal [PB97]. We therefore propose the application of widely linear (WL) processing [PB97, PC95], i.e., W L equalization [GSL03] for DSUWB. Requiring the same or even lower computational complexity than their "linear" counterparts, W L equalization schemes perform in the vicinity of the appropriate MFB.  1.4.2  Contributions for 4 B O K D S - U W B Systems  For 4BOK DS-UWB systems we make the following contributions.  10  1.4 Contributions  • We derive expressions for the bit-error rate (BER) limits according to the matchedfilter bound (MFB) for 4BOK DS-UWB. In particular, we consider two variants of the M F B , which (a) take chip-matched filtering and chip-rate sampling and (b) subsequent R A K E combining into account. The B E R expressions allow us to compare the performances of the devised low-complexity equalizers with the theoretically achievable limits. • We develop new structures and methods for filter optimization for linear equalization (LE), decision feedback equalization (DFE) [BP79], and DDFSE [DHH89] for 4BOK DS-UWB. Different from conventional DS systems (e.g. B P S K DS-UWB), two spreading sequences are used in 4BOK DS-UWB, and one data bit determines which sequence is selected. The first considered equalizer design employs two separate feedforward filters, each of which is optimized for only one of the spreading sequences. The second design is improved in that filter optimization takes the data-dependence of the spreading sequence into account.  Due to the presence of two feedback fil-  ters in the case of D F E / D D F S E , we refer to these two approaches as single-input multiple-output (SIMO) filter optimization. Based on the insights from the second SIMO design, we develop an equivalent multiple-input multiple-output (MIMO) channel model for 4BOK DS-UWB, which leads to an efficient, third equalizer structure based on MIMO filter optimization techniques. • As for the B P S K case, 4BOK DS-UWB is found to yield a second-order norncircular received signal [PC95]. We therefore propose the application of W L processing to equalization of 4BOK DS-UWB resulting in W L - M I M O equalization. It was found that the W L - M I M O equalization schemes offer considerable performance improvements.  1.5 Thesis Organization  1.5  11  Thesis Organization  The remainder of this thesis is organized as follows. In Chapter 2, the DS-UWB transmission model is introduced. In particular, the two modulation schemes, B P S K and 4 B O K , are briefly reviewed and the IEEE 802.15.3 channel model [P8002] is described. Finally, the receiver structures for B P S K and 4BOK DS-UWB are discussed in this chapter. In Chapter 3, equalization for BPSK DS-UWB systems is considered. Different M F B versions are derived and the distribution of the zeros of the effective transfer function is studied and analyzed for BPSK modulation. The L E , D F E , and DDFSE equalization schemes are then briefly reviewed, and the application of W L techniques to L E , D F E and DDFSE equalization techniques for B P S K DS-UWB is introduced. In Chapter 4, equalization for 4BOK DS-UWB systems is discussed. First, equalization schemes based on SIMO filter optimization are derived. Subsequently, MIMO based L E , D F E and DDFSE are developed. Furthermore, the "widely linear" counterparts of MIMO L E , D F E and DDFSE are devised. This chapter also presents the three M F B versions derived, in order to compare the performance of the aforementioned equalizers with the theoretically achievable limits. In Chapter 5, we present the simulation results for various equalization schemes considered. First, the results for BPSK DS-UWB systems are discussed. In particular, the distribution of zeros for different channel scenarios and data rates is presented and predictions are made for equalizer design. Subsequently, performance results for L E , D F E and DDFSE equalization schemes for BPSK DS-UWB are presented and compared with the predictions made from the study of distribution of zeros. Furthermore, the performances of "linear" equalization schemes are compared with the performances of their "widely linear" counterparts. Second, we discuss the results for 4BOK DS-UWB systems. More specifi-  1.5 Thesis Organization  12  cally, we present the results for performance differences between SIMO and MIMO filter optimization based equalization schemes, followed by the results for performance differences between the "linear" MIMO and "widely linear" MIMO schemes. Finally, the performance results obtained for the W L - M I M O equalization schemes are evaluated with respect to the different versions of M F B . Finally, in Chapter 6, we summarize this work and draw some conclusions. We also make some suggestions for further extensions.  Chapter 2 Transmission System 2.1  Introduction  In this chapter, the transmission model for DS-UWB systems, including the modulation schemes, the pulse shaping filter, the channel model, the receive input filter and the demodulator is discussed.  In this work, an equivalent baseband transmission model, i.e.,  complex-valued signals and systems [ProOl], has been adopted. The transmission model follows the specifications in the DS-UWB physical layer proposal [FKMW05]. In Section 2.2, the two proposed modulation schemes, B P S K and 4BOK, are briefly reviewed. Subsequently, the IEEE 802.15.3a channel model [P8002] for U W B based W P A N systems is described in Section 2.3. Finally, in Section 2.4 the receiver structures for B P S K and 4BOK modulation schemes are discussed.  2.2  Transmission Model  As mentioned in Section 1.2, the DS-UWB proposal [FKMW05] envisages two modulation formats: B P S K and 4BOK. This section briefly reviews these modulation schemes.  13  2.2 Transmission Model  14 White Gaussian Noise  Spreading code  n (t)  c[j]  a[k]  a[k]  T N  c  | s(t) »($) , Transmit Filter >{£) » 9r(t)  Equalizer  Channel  h (t) c  KT  y[k]  T[K\  RAKE  C  /  j  |  Receive Filter  9n(t)  Figure 2.1: Block diagram of BPSK DS-UWB transmission system.  2.2.1  B P S K Modulation  The standard proposal [FKMW05] requires all DS-UWB systems to be able to transmit and receive B P S K modulated signals. Data rates ranging from 28 Mbps to 2 Gbps (depending on the code rate) are achievable by direct sequence spreading of B P S K symbols, using spreading sequences of lengths varying from 1 to 24 chips (for more details see [FKMW05, Table 6-7]). The block diagram of the equivalent baseband system model for B P S K DS-UWB system is shown in Fig. 2.1. At the transmitter, BPSK symbols a[k] are spread and modulated with chip pulses gr(t). The pulse shape is defined as J V - l  g(t) = J2 ^9T(t-jT ,  (2.1)  c  c  j=o  where c[j] denotes the jth chip of the spreading code of length 7Y and T is the chip c  duration. The pulse shaping filter g (t) is SRRC [Rap96] with roll-off factor a = 0.3 T  2.2 Transmission Model  15  [FKMW05]. Finally, the transmit signal s(t) can be written as OO  s(t) = J_ a[k]g(t - kT )  (2.2)  s  k=—oo  with symbol duration T = NT . s  2.2.2  C  4BOK M o d u l a t i o n  In the M-ary bi-orthogonal keying modulation ( M B O K ) scheme [ProOl, LS75] K = log M 2  bits of information are mapped onto one of the M spreading sequences chosen from the code set C = {ci, c , . . . CM/2, —C\, — C , . . . - c /2}, consisting of M / 2 orthogonal codewords 2  2  M  Cm, 1 < m ^ M / 2 , and their negatives.  For M = 2, M B O K reduces to BPSK, i.e.,  C = {ci, -ci}. The standard proposal uses a 4BOK ( M = 4) modulation scheme and requires all compliant systems to allow for transmission of 4BOK modulated signals. However, the reception of these signals is optional [FKMW05]. In 4BOK signaling, the 4-ary data symbol at discrete-time k can be represented by the pair (a[k], b[k]), where b[k] € {1,2} is an index which chooses one of the two spreading codes c = [Q,[0] . . . Cb[N — 1]] of length N, T  b  b = {1,2}, 0 < j < N - 1, and a[k] G {±1} is a B P S K symbol which modulates the spreading code c . The block diagram of this equivalent baseband system model is shown b  in Fig. 2.2. The two pulse shapes can be defined as N-l  g (t) = Y, ^\9 {t-jT ),  (2.3)  c  h  T  c  i=o which correspond to the two spreading codes c . As in the B P S K case, the pulse shaping b  filter gr(t) is SRRC with a = 0.3. The transmit signal s(t) can then be written as oo  s(t)=  a{k}g (t - kT ) , b[k]  s  (2.4)  k=—oo  where T = NT denotes the 4BOK symbol duration. 4BOK DS-UWB systems can achieve s  C  2.3 Channel Model  b[k]  16  White Gaussian Noise  Spreading code selection  ncit)  b[k}[j] /]  c  H8H  ,  a[k] b[k]  m  Transmit Filter  1  s(t)  9r(t)  yi[k]  Channel c  RAKE Combining, C\  I r[ ] K  Equalizer  " V2[k)  J  RAKE Combining, c  V  h (t)  J  K  C  Receive Filter  9R(t) 2  Figure 2.2: Block diagram of 4BOK DS-UWB transmission system. data rates ranging from 110 Mbps to 2 Gbps using spreading sequences of lengths varying from 1 to 24 chips (depending on the code rate).  2.3  Channel Model  The channel model considered is the IEEE 802.15.3a model for U W B W P A N systems [P8002, MFP03]. This channel model is based on the Saleh-Valenzuela model [SV87] with some modifications to account for the properties of measured U W B channels. Multipath arrivals are grouped into two categories: cluster arrivals and ray arrivals within each cluster. The interarrival times between clusters or rays within a cluster are exponentially distributed. Lognormal fading is associated with clusters and also with rays within a cluster. The cluster and ray decay factors are based on a given power profile. Finally, the entire impulse response undergoes lognormal shadowing.  2.3 Channel Model  17  The impulse response of the multipath channel consists of L clusters of K rays and c  r  can be expressed as (5(t) denotes the Dirac-delta function)  ak f Ti Tfc J2 ^ ~ .') 1=1 fc=i L  c  h' (t) = x c  ( 2- 5)  K  r  where • Ti is the delay of the Zth cluster. • Tk i is the delay of the kth multipath component relative to the Ith cluster arrival time t  • a ,i is the multipath gain coefficient, where a ,i = p ,i£,iP ,i- The phase of a j is given k  k  k  k  k  by the term p i which can be +1 or —1 with equal probability to account for signal kt  inversion.  reflects the lognormal fading associated with the Ith cluster and f3 ,i is k  the lognormal fading associated with the A;th ray of the Ith cluster. The variables £i and B i are characterized as: 20iog10(£//?fc];) oc Normal(/Xfc,j,o'i + cr\) , where a\ k>  and a\ are the standard deviations of the cluster and the ray lognormal fading term respectively. The term u-k,i is given by  =  lOTn(flo) -  lOTj/r  hlfio)  - 10r  fc|/  /  (a? + a )ln(10) 2  7  20 2  '  (2  '  6)  where Qo is the mean energy of the first path of the first cluster. T and 7 denote the cluster and ray decay factors, respectively. The total energy contained in the multipath gain coefficients for each realization is normalized to one. • X is the lognormal shadowing and is characterized as 201og (X) oc Normal(0,al), 10  where al is the standard deviation of the lognormal fading term for the total multipath realization. There are four different channel models (CMs) specified [P8002] with parameters designed to fit four different usage scenarios: CM1 for 0-4 m Line-of-Sight (LOS), CM2 for  2.3 Channel  18  Model  Table 2.1: UWB channel parameters for different scenario models CM1  CM2  CM3  CM4  0.0233 2.5 7.1 4.3 3.4  0.4 0.5 5.5 6.7 3.4  0.0667 2.1 14.0 7.9 3.4  0.0667 2.1 24.00 12 3.4  2  3.4  3.4  3.4  3.4  2  3  3  3  3  Model Parameters A [1/ns] (cluster arrival rate) A [1/ns] (ray arrival rate) T (cluster decay factor) 7 (ray decay factor) <7i [dB] (stan. dev. of cluster lognormal fading term in [dB]) LT [dB] (stan. dev. of ray lognormal fading term in [dB]) cr [dB] (stan. dev. of lognormal fading term for total multipath realization in [dB])  0-4 m non-LOS (NLOS), CM3 for 4-10 m NLOS, and CM4 for NLOS with an extreme root-mean-square (rms) delay spread of 25 ns. For a particular C M , 7V channel realizar  tions are randomly generated according to the parameter set. The channel model parameters for the different channel scenarios are described in Table 2.1 (for further details see [P8002, MFP03]). Fig. 2.3 shows the impulse response h' (t) of a sample realization for c  channel CM1 and CM4. The typical rms delay values range from 5 ns for channel CM1 to 25 ns for channel CM4 [P8002, MFP03], whereas the duration of transmitted DS-UWB signals, i.e, the chip period, is approximately 0.76 ns in the lower band and 0.38 ns in the higher band. This implies that the indoor U W B channel behaves in a highly frequency selective manner for DS-UWB systems. Due to the very fine multipath resolution of DS-UWB signals, resulting from the short duration of DS-UWB signals, and longer delay spreads associated with the U W B channel, a large number of multipaths can be resolved, thereby resulting in high multipath diversity. Since few multipath components arrive within one resolution period, DS-UWB signals do not undergo significant fading. Moreover, this results in the total received energy  2.3 Channel Model  19  Figure 2.3: Impulse response of a sample realization for channel scenarios, a) CM1 b)CM4being distributed among a large number of paths, however, not every resolvable multipath contains a significant amount of energy [MFP03]. A l l these characteristics significantly impact the overall receiver design of DS-UWB systems as will be discussed in the following sections. In the following, the equivalent baseband representation of h' (t) (2.5), which is denoted c  by hcit) (see Figs. 2.1 and 2.2) is considered.  20  2.4 Receiver  2.4  Receiver  As mentioned previously, the high resolution characteristic of U W B signals results in enhanced multipath diversity. A R A K E combiner [ProOl] can be employed at the receiver in order to capture and coherently combine the received energy. The total number of resolved multipaths (N ) is roughly proportional to the ratio of transmitted signal duration (chip p  duration T ) to the excess delay r , i.e, N = r / T c  m  p  m  c  [Rap96]. Due to large excess delays  for U W B indoor channels compared to the short chip duration (T ) the value of N can c  p  be very high. A R A K E receiver employing a sufficiently large number of fingers to collect and coherently process all the multipath energy is equivalent to a matched filter receiver. Such a receiver provides an upper bound on the performance and serves as a measure to evaluate the limiting performance of DS-UWB receivers (see Section 3.1.1) but cannot be realized in practice. Therefore, a R A K E receiver that captures a reasonable fraction of energy by selectively combining a smaller number of paths needs to be employed. Selective R A K E receivers, which select the F paths with the largest amplitudes and then use maximum ratio combining for these paths, have been proposed and studied for UWB systems [CWVM02, RSF03, RSZ+04, WCS00]. Selective R A K E provides a good performance versus complexity trade-off by collecting a large amount of energy while employing relatively few fingers. However, due to the frequency selective nature of the U W B channel, significant ISI occurs even at the R A K E output. Thus, post-RAKE equalization is necessary in order to effectively mitigate this ISI. In this research work, a R A K E with F fingers is considered, where F is a parameter chosen to trade-off performance and complexity (as will be shown in Section 5.1). The fingers are assigned to the F resolvable paths with the largest magnitudes. In the following, the demodulator architectures for B P S K and 4BOK DS-UWB systems are described. Equalization schemes for BPSK and 4BOK DS-UWB systems are developed and analyzed  2.4 Receiver  21  in Chapters 3 and 4.  2.4.1  Receiver for B P S K  The equivalent complex baseband received BPSK signal r(t) can be written as r{t) = s(t) * h {t) + nc(t)  (2.7)  c  where s(t) is the transmitted B P S K signal, hc(t) is the equivalent baseband channel impulse response, nc{t) is an equivalent baseband noise process and is modelled as additive white Gaussian noise (AWGN) with two-sided power spectral density N , and "*" denotes 0  convolution. As shown in Fig. 2.1, the receiver front-end consists of a chip-matched filter g (t) = R  9T(~~followed  by chip-rate sampling ((•)* denotes complex conjugation). The received  signal after the matched filter (r(t)) is given by N-1  oo  •  r(t) = J_ a[k] J2 b1M* - kT, - jT ) + n (t) c  c  R  (2.8)  j=0  fc=—oo  where n {t) = n {t) * g (t) R  is the filtered A W G N n (t) c  c  (2.9)  R  and . h (t) = gr(t) * hc(t) * g (t) a  (2.10)  R  is the continuous-time overall impulse response including the effects of transmitter and receiver filters. Introducing the discrete-time overall impulse response h [n] = K{KT ) 0  C  ,  (2.11)  the discrete-time received signal after sampling with chip rate 1/T can be expressed as C  oo  r[ ] = J_ K  fc=—oo  N-1 a[k]  c\j] K[K -kNj=0  j) + n [n] . R  (2.12)  22  2.4 Receiver where n [K] = n {KT ) R  R  (2.13)  c  is the sampled noise, which is AWGN. As mentioned previously, an F finger R A K E is considered and the fingers are assigned to the F resolvable paths with largest amplitudes. The discrete-time receive signal r[n] is then fed into the R A K E combiner. The delayed versions of r[«] are despread and then maximum-ratio combined. The received sequence is delayed by di chip intervals by the i  th  finger. The corresponding R A K E output sequence can be written as oo  [k] = ]T  y  a[v\h'[k -u)+  n'[k] ,  (2.14)  v=—oo  where n'[k] is the noise sequence at the R A K E output and the following symbol-time impulse response h'[.] is introduced F  N-l N-l  h'[k] = __; K[di) J_ J_ cL7']c[m] h [kN + m + di - j] . 0  i—l  j=0  (2.15)  m=0  Finally, the R A K E output y[k] is input to an equalizer to obtain the data estimate a[k] as shown in Fig. 2.1.  2.4.2  Receiver for 4 B O K  The equivalent complex baseband received 4BOK signal r(t) is given by r(t) == s(t) * h (t) + ncit) c  (2.16)  where s(t) is the transmitted 4BOK signal as in (2.4), hcit) is the equivalent baseband channel impulse response and nc{t) is AWGN with two-sided power spectral density A^ 0  Similar to the B P S K case, the 4BOK receiver front-end consists of a chip-matched filter g (t) = g (—t) followed by chip-rate sampling. The received signal after the matched filter R  T  23  2.4 Receiver (r(t)) is given by oo  N - 1  r{t) = J _ W Yl ^UKit  - kT - jT ) + n (t)  a  s  c  (2.17)  R  j=0  fe=—oo  where the continuous-time overall impulse response h (t) and noise n (t) are defined as in 0  R  (2.9) and (2.10). The discrete-time received signal after sampling with chip rate 1/T is C  oo  N - 1  r[«] = J _  a  ^ S  M« -kN-  j] + n [K] .  (2.18)  R  j=0  fc=—oo  where the definitions in (2.11) and (2.13) have been used. It should be noted that the sampled noise n [n] = n (KT ) is still AWGN. R  R  c  Different from BPSK, in 4BOK two spreading sequences (and their negatives) are used as explained in Section 2.2.2. The receiver has no a priori information about the spreading code used at the transmitter to spread the data bits. Therefore, the received signal needs to be correlated separately by each of the two spreading sequences C\ and c . 2  This is  implemented by employing two R A K E combiners, as shown in Fig. 2.2, at the receiver wherein the received signal is correlated with spreading sequence C\ in one R A K E and with spreading code c in the other. Both the R A K E s maximum-ratio combine (matched filters) 2  the F strongest resolvable signal paths. One R A K E combiner selects the F strongest paths assuming C i is transmitted and finger i delays the received sequence by d\ , while the other ti  R A K E combiner assumes transmission with c for finger assignment and applies delays d ,i2  2  The despread signals can be written as oo  y„[k] = J _ a[i]K [k-i] m  l=  + n [k] , u  i/ € {1,2} ,  (2.19)  —oo  where n [k\ is the noise sequence after despreading with c„ and the following definition for u  symbol-time impulse response is introduced F N-1 N-1 [k] = i=l  E M^H c  *oW  h  j=0  m=0  o[kN  h  + m + d„ - j] . ti  (2.20)  24  2.4 Receiver  Finally, the despreader outputs y [k\ are input to an equalizer to obtain the data estimate u  (a[fc],6[fc]) (see Fig. 2.2) as will be discussed in Section 4.2.  Chapter 3 Equalization for BPSK In this chapter various suboptimum equalization schemes for B P S K DS-UWB systems are discussed. In order to evaluate the performance of various equalization schemes and also the performance of the overall receiver, different versions of the matched filter bound are derived in Section 3.1, taking into account practical constraints such as receiver filtering, sampling, and the number of R A K E fingers when R A K E preprocessing is applied at the receiver. The distribution of the zeros of the overall transfer function provides important information, with implications about the equalizer design [SG02]. A n expression for the overall channel transfer function, which takes into account the effects of the spreading code, the SRRC transmit filter, the UWB channel, the receiver filter and the F finger R A K E , is derived and analytical expressions for density, marginal density and cumulative distribution of zeros are reviewed in Section 3.1.2. In Section 3.2.1, application of suboptimum equalization schemes such as L E , D F E and DDFSE to BPSK DS-UWB systems is discussed.  Subsequently,  equalization schemes with widely linear processing, which improves performance while not increasing equalizer complexity, are devised in Section 3.2.2.  25  3.1 Performance Measures for Equalization  3.1  26  Performance Measures for Equalization  Whether or not a certain equalization scheme achieves a favorable performance-complexity tradeoff depends strongly on the nature of the ISI channel. In particular, for suboptimum equalization techniques, such as L E , D F E , and DDFSE, the location of the zeros of the channel transfer function plays a crucial role [ProOl, SG01, SG02]. Therefore, we study and discuss the distribution of the zeros of the transfer function of the effective symboltime channel (2.15) as an appropriate measure for the design of suboptimum equalizers in Section 3.1.2. Prior to that, the M F B , which is a performance bound for any equalization scheme, (cf., e.g., [Lin95]) is derived in Section 3.1.1.  3.1.1  Matched Filter Bound (MFB)  For the DS-UWB system under consideration, three versions of the M F B are developed, which, to different extents, account for suboptimalities of the receiver structure shown in Fig. 2.1. To obtain expressions for the bit-error rate (BER) for the respective MFBs, the signal-to-noise ratio is defined as  where E {r) is the received energy per bit for the rth U W B channel realization after the b  respective matched filter. The corresponding bit-error rate BER(7 ) for B P S K and one r  particular channel realization is given by (3.2) The average B E R is obtained semi-analytically by averaging over N channel realizations r  (3.3) r = l  3.1 Performance Measures for Equalization  3.1.1.1  27  MFB I  The first M F B , referred to as M F B I, is based on the continuous-time channel JV-1  h (t) = g(t) * h (t) = TC  c\j]gr(t - jT ) *h (t)  c  c  (3.4)  c  j=o including spreading, transmit filter, and channel impulse response. The optimum receiver input filter is the matched filter h (—t), which maximizes the SNR (3.1) and for which TC  symbol-rate sampling provides a sufficient statistic. The received energy per bit for the rth channel is given by E (r) = h (t)*h (-t)\ . b  TC  TC  (3.5)  t=Q  It should be noted that M F B I is the ultimate performance bound, as optimum preprocessing (matched filtering) is assumed. More specifically, this bound shows the performance of a matched filter which assumes that all multipath energy can be captured, that the time duration between transmitted symbols is large enough such that no ISI occurs, and that the channel is perfectly known to the receiver. No realizable equalization scheme or receiver design can exceed this performance bound [Lin95].  3.1.1.2  M F B II  The second M F B , referred to as M F B II, takes the suboptimality of chip-spaced sampling after chip-matched filtering with g (t) into account. This means that it is based on the R  discrete-time channel impulse response hrcR[K] = h {t) *g {t)\ TC  R  t=KTc  ,  (3.6)  and the normalized received energy per bit after matched filtering with respect to h cR[n] T  follows as oo  •  £  \h RcH  2  T  3.1 Performance Measures for  28  Equalization  where the denominator term represents the normalization factor for received energy per bit since the noise variance is still considered to be N (see (3.1)). 0  The B E R of M F B II is a lower bound on the performance of any receiver that employs a front end filter with chip rate sampling. A comparison of M F B I and M F B II also provides information about the potential performance gain that could be achieved by sampling at rates higher than chip rate sampling. Based on comparison of M F B I and M F B II, it was found that for BPSK DS-UWB systems chip rate sampling does not result in significant performance degradation, as will be shown in Chapter 5. 3.1.1.3  M F B III  The third M F B , referred to as M F B III, also considers the discarding of energy of the useful received signal by using a finite number F of R A K E fingers, where, for complexity reasons, F is chosen smaller than the total number of resolvable channel paths. In general, n'[k) in (2.19) is correlated Gaussian noise, which is directly accounted for in the equalizer design. However, for derivation of M F B III and analysis of distribution of zeros in Section 3.1 it is convenient to apply a noise whitening filter f [k] and to consider w  the effective impulse response h[k] = h'[k) * f [k] . w  (3.8)  The coefficients f [k] of the whitening filter are normalized such that w  n[k] = n'[k] * f [k] w  (3.9)  is A W G N with variance N . The effective impulse response h[k] allows us to fairly compare 0  the requirements for the equalizer design for different system parameters and channel models. The total received energy per bit for the rth U W B channel realization at the output  29  3.1 Performance Measures for Equalization of the F finger R A K E is given by oo  IM H fc  E (r) = b  2  •  (3-10)  fc=—oo  Using (3.10) in (3.1), (3.2) and (3.3) gives the bit error rate corresponding to M F B III. For the BPSK DS-UWB receiver that employs a chip-matched filter as the receiver front end, followed by an F finger R A K E , the B E R of M F B III is the lower bound on the B E R achievable by any post-RAKE equalization scheme. Moreover, a comparison of performance of M F B III with that of M F B II serves as an indicator of the performance loss that occurs on account of the use of a finite number F of R A K E fingers.  3.1.2  Densities of Zeros of the Overall Transfer Function  The application of the distribution of zeros of the channel transfer function to equalizer design was introduced by Schober and Gerstacker in [SG01, SG02]. Based on the theory developed in [SG01, SG02], in the following, we study the distribution of zeros of the effective transfer function H(z) = Z{h[k]} for DS-UWB with R A K E combining. 3.1.2.1  Overall Transfer Function  The effective discrete-time impulse response h[k] (3.8) depends on the spreading code, the transmit filter, the UWB channel, the receiver input filter, and the R A K E combiner. For the following considerations, it is appropriate to limit the length of h[k] to some finite value L. In particular, a window k < k < k + L is applied such that the L taps contain the s  s  fraction n = 0.99 of the total energy of the impulse response of all the N realizations. For r  ease of notation, the truncated version of the effective impulse response is still denoted by h[k].  3.1 Performance Measures for Equalization  30  The transfer function H(z) is then obtained as ks+L-l  H(z) = Z{h[k]} = £  h[k]z~ = z-^ -^h v{z) +L  k  where the definitions h = [h[k + L - 1] h[k + L - 2] . . . s  ,  T  (3.11)  and v(z) = [1 z ...  s  z'] 1 1  are used. 3.1.2.2  Density of Zeros  In order to derive an analytical expression for the density of the zeros of H(z) in the complex plane, it is assumed that the elements of h are non-zero mean Gaussian distributed with (a) uncorrelated real and imaginary parts, (b) identical covariance matrices for real and imaginary parts, (c) mean fi  h  =  E{h},  (d) correlation matrix  covariance matrix C [0] — {&hh — M/iM/Di nh  $>  hh  =  E{hh }, H  and (e)  [SG02].  Although assumptions (a) and (b) hold only approximately for the U W B channel models, the performance results (shown in Section 5.1.3) are in perfect agreement with the equalizer design implications from the analysis of distribution of zeros obtained from simulating the density (shown in Section 5.1.2). With the above assumptions, the density of zeros of H(z) in the complex plane is given by [SG02]  (3.12) with the componentwise derivative v'(z) — [0 1 . . . (L — l)z ~ ] L  k(z) = v (z)C [£]  v(z*),  T  hh  2  T  Z e {0,1} .  and (3.13)  For £ = 1 the elements c^[l] of matrix C [ l ] are given by c^[l] = /zc^fO], where c [0] h h  are the elements of the covariance matrix C/j/JO].  M1/  7  31  3.1 Performance Measures for Equalization 3.1.2.3  M a r g i n a l Density and Cumulative D i s t r i b u t i o n  For channel equalization, the important figure of merit is the density of the magnitude of the zeros. Therefore, the density of zeros is expressed in polar coordinates r, <p, i.e., z = r • e ^ , 0 < r < oo, 0 < ip < 2TT. The marginal density f (r) is given by r  (3.14) o  From (3.14), the expected number of zeros inside the disc \z\ = r < R is given by R (3.15) o  As the total number of zeros is L — 1, we have lirnR_ 3.1.2.4  >00  n(i?) = L — 1.  A p p l i c a t i o n to Equalizer Design for D S - U W B  The distribution of zeros of the overall transfer function in the complex plane gives important information useful for equalizer design. If many zeros lie close to the unit circle, L E  1  will perform poorly, since the presence of zeros on unit circle leads to enhancement of noise power in the case of L E . Otherwise, L E is recommended as low-complexity equalization scheme. In the case of non-linear equalization, the expected number of zeros outside the unit circle x(oo) = n(oo) — n(l) is important. If x(oo) > 0, the impulse response h[k] is not minimum phase with some probability. The absence of a minimum phase equivalent of the impulse response h[k] results in performance degradation for non-linear equalization techniques, since the energy of the minimum phase channel h [k] min  first few taps, i.e., Ylk=ohmin[k]  ^ 2fc=o M^]  is concentrated in the  for 1 ^ rj ^ L, which leads to a increase in  minimum euclidean distance in the case of DDFSE [GH96]. Therefore, allpass pre-filtering, It should be noted that when studying zeros of the channel transfer function, we consider classical linear and non-linear equalization without widely linear processing. :  3.2 Equalization Strategies  32  which is approximated by an M M S E feedforward filter (FFF), is required to transform h[k] into its minimum-phase equivalent h i [k]. m n  Results for the distribution of zeros of h[k] for combinations of different channel scenarios and spreading lengths and their application to the equalizer design for the BPSK DS-UWB systems will be discussed in Section 5.1.  3.2  Equalization Strategies  In the following, the classical L E , D F E , and DDFSE are briefly reviewed in the context of DS-UWB. The minimum mean-square error (MMSE) criterion is applied for filter optimization [ProOl]. Subsequently, equalization with W L processing for BPSK DS-UWB is introduced and the respective schemes are referred to as W L E , W D F E , and WDDFSE.  3.2.1  Classical Equalization Schemes  3.2.1.1  Linear Equalization  L E is a suboptimum equalization scheme that finds widespread application on account of its low implementation complexity. The L E employed for B P S K DS-UWB systems consists of a linear filter with coefficients f[k] optimized according to the M M S E criterion. Introducing the following definition for filter vector / /  =  /[0]  /[I]  ...  /[?/]  (3.16)  with order qf, and taking into account the correlated Gaussian noise n'[k] in (2.19), the optimized filter coefficients are given by /=  (AA  H  + (Qn'n')- h! 1  (3-17)  3.2 Equalization Strategies  33  where C denotes the SNR at the R A K E output,  denotes the autocorrelation of noise  n'[k] given by *  = E[n'[k]n' [k}} ,  (3.18)  H  n  W  and the following definitions are used  ti =  h'[k ] h'[k -l] 0  h'[0] h'[l] A  =  0  ...  0  ...  h'[L]  0  0  0  h'[0] h'[l]  f  . .  h'[L] •  0  (3.19)  h'[k -q ] 0 0  . 0 h'[l] . . h'[L]  h'[0]  (3.20)  In the above k denotes decision delay. The output d[k] of the M M S E linear equalizer is 0  given by d[k] = f y [k]  ,  H  qf  (3.21)  where VqM  =  [y[k] •••  y[k-q ]]' f  (3.22)  is a vector for R A K E output sequence y[] given by (2.19). Since the modulation scheme considered here is BPSK, the data estimate a[k — k ] can be obtained simply by considering 0  the sign of the real part of the L E output d[k). Mathematically, a[k — k ] = s\gn{Re{d[k}}} 0  3.2.1.2  (3.23)  Decision Feedback Equalization ( D F E )  The D F E equalizer consists of a feedforward filter (FFF) and a feedback filter (FBF) with taps spaced T (symbol duration) time apart. The R A K E output y[k], as in (2.19), is fed to s  the F F F . The F B F eliminates the post-cursor interference, i.e., the interference caused by previous data symbols to the currently estimated symbol. The filters are jointly optimized  34  3.2 Equalization Strategies  according to the M M S E criterion [BP79]. For M M S E D F E , the F F F coefficients f [k] and F  the F B F coefficients f [k] are given by B  f F  =  ((AA  f B  =  H" f  — HH )  H  + C^n'n')'  H  1  hf ,  (3.24)  ,  B  where A, h' are defined in (3.20), (3.19), respectively,  is the autocorrelation of noise  n'[k] given by (3.18), and the following definitions are used  =  fp  F  F  h[ko + 1 -  h'[k + q ] h'[ko + q - 1]  h'[k + 2] h'[k + 1]  0  0  0  Q  h[k + 2 -  q]  0  F  (3.26)  MQi  /B[2]  B  h'[k + 1] h'[ko]  H  H  / [l]  SB =  (3.25)  ... MQF]  f [o] f [i]  B  (3.27)  B  ••• h[k + q  q]  0  F  B  - qF]  where q and q denote the order of F F F and F B F , respectively and k is the decision F  B  Q  delay which has to be optimized (cf. [VLC96]). The equalizer output d[k] can be expressed  as d[k] = f y [k] H  F  qF  - f  H B  a [k] . B  (3.28)  Data decisions a [A; — k ] are obtained as in (3.23), and fed back to the F B F as vector 0  a [k] = [a[k - k - 1] . . . a[k - k B  0  0  —  q ]] • T  B  The output of the F B F is a weighted sum of  these past decisions a [k) and the F B F coefficients f , which results in total elimination B  B  of ISI caused by previous symbols to the currently estimated symbol (assuming that all the past decisions were correct). 3.2.1.3  Delayed Decision-Feedback Sequence E s t i m a t i o n ( D D F S E )  The optimum equalization scheme for any ISI wireless channel is maximum likelihood sequence estimation (MLSE), which is implemented using a Viterbi algorithm (VA). The  3.2 Equalization Strategies  35  complexity of the V A is dependent on the number of states employed by the VA, which increases exponentially with the channel memory. For a channel with finite memory length L, the number of states employed in the V A performing M L S E is exponential in L. The long excess delay in U W B channels compared to the short symbol duration results in long channel memory. Moreover, as data rates are increased in the DS-UWB system, the symbol duration decreases, further increasing the L. In particular, for higher data rate DS-UWB systems L could be as high as 30. Then, for M L S E , the number of states in VA would be 2 , which is not feasible for practical implementation. Therefore, we consider DDFSE 30  [DHH89], a hybrid algorithm between M L S E and D F E , which provides a direct trade-off between complexity and' performance. DDFSE is based on a trellis with a reduced number of states as compared to MLSE. More specifically, for a channel with memory length L, DDFSE employes a V A that performs M L S E for the initial q  tr  0 < q  tr  past symbols where  < L, and at the same time performs D F E to remove ISI due to the remaining  L — qtr past symbols, where the feedback information provided by the best path leading to a state is used. For the B P S K DS-UWB system, the DDFSE uses a VA with 2  qtr  states s[k] = [a[k-  k — 1]. .. a[k — k — q ]] - A n M M S E - F F F is applied prior to the reduced state VA in order T  Q  0  tr  to change the channel to its minimum phase equivalent, thereby avoiding any performance losses that could occur due to the non-minimum phase channel, see Section 3.1.2.4, [GH96], Defining the decision-feedback symbol vector for state s[k] by d_ [fc] as )S  o-B, [k\ =  r  s  a[k - k - 2 - q ] 0  tr  a[k - k - 1 - q 0  tr  .. • a[k - k - q ] 0  B  i  T  (3.29)  and introducing the following definitions fB 9  fT  qtT  =  [/s[l]  /B[2] . . .  =  [/flfer +  1]  f [qtr]]  H  B  /ilfer +  2] . . .  , / B M ] " ,  (3-30)  where fs[k] is given by (3.24) and (3.26), the DDFSE branch metric for state s[k] and trial  3.2 Equalization Strategies  36  symbol d[k — k ] is given by 0  X(d[k - k ], s[k]) = \f y {k] - a[k - ko) - ( / _ ' ) " s[k] - [fT ) H  0  qtr  F  qF  H  "B,s[k]\ . 2  (3.31)  It should be noted that for q = 0 DDFSE reduces to D F E . Also, DDFSE approximates tr  M L S E as q —> L. tT  3.2.2  Equalization Schemes with W L Processing  In order to completely describe the second-order statistics of a complex signal x[k] — xi[k] + jxQ[k] where xj{k\ = Re{x[k}} and XQ[k] — lm{x[k]}, it is necessary to consider the pseudocorrelation E{x[k + m]x[k]} in addition to the correlation E{x[k + ra]£.*[&]} [PB97, NM93]. The complex signal x[k] is said to be second-order noncircular (or improper) if its pseudo2  autocorrelation is not zero. Mathematically, E{x[k + m]x[k}} ^ 0  for some m G Z  (3.32)  For complex valued signals a widely linear (WL) approach to mean square estimation (MSE) has been proposed in [PC95], wherein the estimation of a random process w[k] is done by considering not only the original observed signal x[k] but also the complex conjugate of the observed signal x*[k]. If the observed signal x[k] is non-circular (as characterized by (3.32)) or if the observed signal is circular x[k] but its pseudo-crosscorrelation with the estimated process w[k] is non-zero, i.e., both E{x[/c +ra]a;[fc]}= 0 V ra G Z, and E{x[k + m]w[k]} ^ 0 for some m e Z, are valid, then, "widely linear" M S E results in improved estimation of the random process w[k] as compared to the conventional "linear" MSE [PC95]. When both the pseudo-autocorrelation and pseudo-crosscorrelation are zero, i.e., E{x[k + ra]z[fc]} = E{x[k + m)vj[k]} = 0 V ra e Z, then "widely linear" MSE reduces to conventional "linear" MSE. In the following, we will use the term 'non-circular' to denote 'second-order non-circular' for the sake of conciseness. 2  3.2 Equalization Strategies  37  From the fact that BPSK signaling and carrier modulation are applied, i.e., complex y[k] (2.14), we observe that the pseudo-autocorrelation function oo  E{y[k + m}y[k}} = J_  h'{u]h'[u + m\  (3.33)  u=—oo  for the R A K E output signal y[k] (2.19) is nonzero in general. Hence, y[k] is noncircular. As a result, W L processing, i.e., joint processing of y[k] and its complex-conjugated version y*[k], should be applied. It should be noted that for carrierless U W B modulation (often considered with Gaussian monocycles, cf. e.g. [MWS03, ED05, RSF03, IO04]), W L processing is identical to conventional "linear" processing since no equivalent complex baseband signal is generated. W L equalization schemes for frequency-selective channels have recently been proposed in [GSL03]. In particular, M M S E - W L E and M M S E - W D F E schemes have been developed as extensions to conventional M M S E - L E and M M S E - D F E , respectively. In the following section, we discuss W L processing based equalization schemes for B P S K DS-UWB systems. In the remainder of this chapter, the correlated noise at the output of R A K E n'[k] is considered to be circular i.e., E{n'[k + m]n'[k]} — 0. As in the Section 3.2.1, the noise autocorrelation is denoted by «&„'„'• 3.2.2.1  W i d e l y Linear Equalization ( W L E )  For M M S E - W L E the R A K E output signal y[k] and its complex conjugate y*[k] are filtered using linear filters fwL\[k] and <7w£,[fc] and their outputs are linearly combined. Therefore, the final output d[k] can be expressed as d[k} = K y {k]+g^ y* [k}, L  where f  qf  = [/VL[0] . . . /WL[<?/]] and g  L  H  WL  order g/ and y  WL  qf  qf  (3.34)  = [g L[0} • •. gwdqf]]" are filter vectors of W  is given by (3.22). Modifying the optimization presented in [GSL03] in  3.2 Equalization Strategies  38  order to directly account for correlation of the noise n'[k] in (2.19), we obtain fv/L  = [(AA + C*„'„0 - (AA ) • (A*A H  T  •[/»' - AA • (A*A T  T  +  T  + CK'n')'  1  • (A*A")]'  1  C*k0-^'1 ,  (3-35)  = f*wL >  9WL  (3-36)  where the definitions (3.19), (3.20) and (3.27) are used. The W L E output d[k] can now be expressed as d[k] = 2Re{f% y [k]}. L  (3.37)  q/  It should be noted that the W L E output d[k] is real valued. The data estimate a[k — k ] is 0  given by a[k - k ] = s\gn{d[k}} .  (3.38)  0  3.2.2.2  W i d e l y Linear Decision Feedback Equalization ( W D F E )  In M M S E - W D F E in addition to W L processing of the R A K E output signal y[k], feedback filtering is applied. The W D F E output d[k] can be expressed as d[k] = f?, V [k] WL  where f  qf  = [/>,WL[0] • • • fF,WL[qF\] and g H  F>WL  vectors of order q  F)  and /  B  W  L  - flWLaB[k]  + g$,wLV* [k]  v  ,  (3.39)  = [0B,WL[O] ... gB,WL[qF]] are the F F F H  F y v L  = [/B,WL[0] • • • /B,WL[<7B]P is the F B F vector with order  q . The filters are jointly optimized according to the W L M M S E criterion as in [GSL03]. B  Taking into account the noise autocorrelation 3>//, the filter coefficients are given by n n  fF,WL  = [(AA - HH H  •(A*A  T  + C*n>n>) - (AA -  H  - H*H  T  T  + CK'n')'  • (A*A  1  \h - (AA - HH ) • (A*A T  9F,WL  =  f*F,WL .  SB,WL  =  H SF,WL H  T  -  H  T  - H*H  T  HH ) T  H*H )Y H  l  + C#;, ,) /i'*] , n  _1  (3.40) (3-41)  + H f*F,WL = 2Re{H f' 1  }  H  FWL  .  (3.42)  3.2 Equalization Strategies  39  Substituting (3.40) and (3.41) in (3.39), the equalizer output can be expressed as d[k] = 2Re{f^ y [k}} - f^ a [k] WL  qF  WL  .  B  (3.43)  Data decisions a[k — ko] are obtained as in (3.38), and fed back to the F B F as vector d [k] B  — [d[k - k  3.2.2.3  0  -  1] . . . a[k-  k-  q ]] '• T  Q  W i d e l y Linear D D F S E  B  (WDDFSE)  The W D D F S E scheme is an extension of the conventional D D F S E (described in Section 3.2.1.3) with the difference that the F F F is optimized according to the W L M M S E criterion. Observing that the minimum phase channel coefficients f ,WL[k] B  (see (3.42)) are  real, the W D D F S E branch metric X(a[k — k ],s[k]) can be expressed as 0  X(a[k - k ],s[k]) = (2Re{f^y [k}} - a[k - k ] - (/_>_)*' a[k] - (f^t)" 0  qF  0  ^[k])  ,  2  (3.44) where f%  WL  .= [/B.WLW  • • • fB,WL[qtr]}  H  and f ^ B  B  = [[f ,w [qtr B  L  + 1] • • •  IB^L^B]]"-  It should be noted that the FBFs used in W D F E and W D D F S E are real-valued, which yields a small complexity advantage of W L equalization over conventional equalization.  Chapter 4 Equalization for 4BOK In the previous chapter, we discussed equalization for B P S K DS-UWB systems. As explained in Chapter 2, in 4BOK DS-UWB systems, different from the B P S K case, two spreading sequences are employed at the transmitter, and at the receiver two R A K E s are used in order to despread the received data separately with each of the two spreading sequences. Since the corresponding demodulator structure for 4BOK is conceptually different from that of BPSK, the equalization schemes discussed for B P S K DS-UWB systems, in their current form, cannot be applied to the 4BOK case. Therefore, in this chapter, we discuss the equalizer designs developed specifically for 4BOK DS-UWB systems. In particular, we first derive the MFBs in Section 4.1. In Section 4.2.1, two SIMO filter optimization methods are described and the corresponding equalizer structures are discussed. Subsequently, in Section 4.2.2, MIMO filter optimization is discussed and L E , D F E and DDFSE based on this optimization method are described. Furthermore, widely linear processing is applied to the MIMO filter optimization and W L - M I M O equalization schemes are developed in Section 4.2.3.  40  4.1 Performance Bound for 4B0K Equalization  4.1  41  Performance Bound for 4 B O K Equalization  Before considering particular equalizer implementations, it is useful to determine the performance limit for any equalizer. It is well known that this performance limit is given by the M F B , cf. e.g. [Lin95]. Therefore, we apply the M F B concept to 4BOK transmission and consider the error probability when a single symbol pair (a, b) is transmitted. Since four different pairs (a,b) , 1 < m < 4, are possible, we will first determine the pairwise m  error probability (PEP) P (m,n) that (a,b) is transmitted and (a,b) with n / m is dee  m  n  tected. A n upper bound for the B E R is then obtained from the union bound over the PEPs, whereas a lower bound, and thus a true performance limit, results from only considering the dominant PEP. In order to also analyze the potential performance losses due to the R A K E as equalizer front end, two versions of the M F B are considered. First, the M F B based on the chipmatched filtered and chip-rate sampled received signal r[«] is determined. This M F B is referred to as M F B I in the following. Subsequently, we obtain the M F B given the R A K E outputs y„[k], v e {1,2}, which is referred to as M F B II. Since for BPSK DS-UWB in Section 3.1.1.2 it was found that chip-matched filtering and chip-rate sampling cause only negligible performance degradations compared to optimum matched filtering, we concentrate on the corresponding MFBs for the 4BOK case.  4.1.1  MFB I  If only a single symbol pair (a, b) is transmitted, the sampled received signal is (see (2.18)) N-l r[n] =  c \j] h [n - j] + n [K] b  0  R  ,  (4.1)  where n^ft] is AWGN with variance o . Furthermore, it is appropriate to consider a window 2  K < n < K + L of L taps of the impulse response h [«], such that this window captures the s  S  0  42  4.1 Performance Bound for 4B0K Equalization  fraction q of the total energy of h [n], where rj is chosen (arbitrarily) close to one. Defining 0  the corresponding vectors h = [/i [«s] • • • h {n + L — 1]] , r = [r[n ] ... r[K + N + L — 1]] , T  0  and n =  ...  +  UR[K  S  N  4-  0  T  s  s  S  L — l ] ] , (4.1) can be compactly written as r  (4.2)  r = C(a, b)h + n where the data dependent (N + L — 1) x L circulant matrix C(a, b) is defined as ac {0]  0  0  0  :  ac [0]  0  0  ac [N - 1]  :  0  ac [N - 1]  b  b  b  C(a,b) =  0  b  ac [0]  0  fc  0  0  0 0  0 0  (4.3)  ac [0] b  ac [N - 1] ! 0 ac [N - 1] b  b  Since the elements of n are independent complex zero-mean Gaussian random variables, the (optimum) maximum-likelihood (ML) decision rule is given by (4.4)  (a,6) = argmin{||r-C(a,6)/ || } 2  l  (a,b)  Based on (4.4), the P E P for channel realization h can be written as (we use C(x) as short-hand for C((a,b) )) x  P (m,n\h) e  = Pv{\\r-C{m)h\\ >\\r-C(n)h\\ } = Pr{0 > \\(C(n) - C(m))h\\ - 2Re{n {C{n) - C(m))h}} , 2  2  2  H  (4.5)  which immediately leads to P (m,n\h) = Q(\\(C(n) e  C{m))h\\/a ) n  (4,6)  It should be noted that since the calculation of actual M F B is not possible, so we consider a lower bound and an upper bound on the actual M F B . Applying the union bound  4.1 Performance Bound for 4B0K Equalization  43  to P (m,n\h) in (4.6), an upper bound B E R ^ ( / i ) on the M F B is obtained e  F B  BER" (h) = \}2}2  £ &  MFB  m=l  _^eKn\h) ,  (4.7)  «=i  where g(m, n) G {1,2} denotes the number of bit errors if (a, b) is transmitted and (a, b) m  is detected. A lower B E R bound BER'  MFB  n  (h) results if only the dominant error event is  considered:  BERUW = m a x { ^ ^ P ( m , n | / i ) l .  (4.8)  e  Finally, the average BERs are obtained semi-analytically by averaging over N channel r  realizations BER  X M F B  = ^-^  B E R  MFBW  !  (4.9)  xG{l,u}.  As for the B P S K case, the performance of this M F B is an upper bound on the performance of any 4BOK DS-UWB receiver that employs a front end filter with chip rate sampling.  4.1.2  M F B II  For the second M F B based on y„[k] (2.19) it is convenient to define, the N x N matrix F H [k] with element £ h* [d^i]h [kN + d + m — j] in row m and column j, 1 < j, m < N. v  t=i  0  0  Vti  Then, the symbol-time impulse response from (2.20) can be rewritten as h„ [k]=clH [k]c , ll  n,ve{l,2}.  fi  v  (4.10)  Using (4.10) and assuming a single symbol pair (a, b) is transmitted, the R A K E output y„[k] (2.19) is given by y [k\ = oclH [k]c v  v  b  +  n [k], v  u e {1,2} .  (4.11)  We again consider an appropriately chosen window k < k < k + L of length L , such s  s  that this window captures a sufficient fraction n of the total energy of the impulse responses  44  4.1 Performance Bound for 4B0K Equalization  hnu[n], A*, v S {1,2}. It is important to note that L is chosen such that L = max{L „}, M  H, v e {1,2} where L^ is the length that captures a fraction 77 of total energy of the impulse v  responses h^n], u.,v G {1,2}. Defining the following 2L-dimensional vectors H  [H^[k ]  =  a  ...  Cl  H"[% + L - l]ci Hl[k ]c s  + L - l]c ] ,  ... Hl\ks  2  T  2  V  = [Vi[k ] • • • V2[k + L - 1] y [k ] ... y [k + L - l ] ] ,  n  = [m[k ] ... n [k + L- l}n [k ] ... n [k + L - 1]] ,  r  s  s  2  s  2  s  (4.12)  T  a  2  s  2  s  2  s  we obtain the compact expression y  = aHc  +  b  n .  (4.13)  The noise n is jointly Gaussian with zero mean and covariance matrix R , which depends nn  on the R A K E finger delays d i and the auto- and cross-correlation functions of the spreading Ui  sequences C\ and c . 2  From (4.13) and using the Cholesky decomposition R  nn  = L L  H  , we have the M L  decision rule (a, 6) = argmin{||L- (y 1  aHc )\\ } 2  b  .  (4.14)  (a,6)  Finally, writing c(m) = ac for the pair (a,b) , the P E P P (rri,n\h) follows as b  m  P (m,n\h) = Q (\\L- H{c{n) l  e  e  - c(m))\\) .  (4.15)  Substituting P (m,n\h) from (4.15) into (4.7) and (4.8) gives the corresponding upper and e  lower B E R bound, respectively. For the 4BOK DS-UWB receiver that employs chip rate sampling and F R A K E fingers for each of the two R A K E s (see Fig.2.2), the performance of this lower B E R bound is the upper limit on the performance achievable by any post-rake equalization scheme.  4.2 Equalization Strategies  4.2  45  Equalization Strategies  In this section the design of equalizers, which process the despread signals y [k] (2.19), v 6 u  {1, 2} is discussed, to obtain decisions (d[k), b[k]). We aim at low-complexity solutions and 1  therefore consider L E , D F E , and DDFSE. First, in Section 4.2.1, equalization strategies with separate feedforward filtering for yi[k] and 2/2[&] are devised. These strategies apply two feedback filters per despreading branch and are therefore referred to as equalization with SIMO filter optimization. It turns out, however, that 4BOK transmission is conveniently modelled by an equivalent MIMO system, for which a MIMO equalization strategy is devised in Section 4.2.2. Finally, by recognizing the non-circularity of the signals y„[k], W L processing for DS-UWB MIMO equalization is introduced in Section 4.2.3. Throughout this section, we focus on D F E for the development of equalizer structures and filter optimization, and we briefly describe the necessary modifications for L E and DDFSE. We exclusively consider equalizer filter optimization according to the minimum mean-square error (MMSE) criterion, and, for the sake of conciseness, the prefix "MMSE" is omitted, when referring to the respective equalizers.  4.2.1  4BOK Equalization with SIMO Filter Optimization  Fig. 4.1 illustrates the straightforward approach to D F E for 4BOK. Assuming strong selfinterference suppression, i.e., signal components spread with  are largely suppressed after  despreading with c , u ^ fi, fi,u £ {1,2}, the two despreader outputs y \k] are processed u  u  by two separate feedforward filters (FFF) fF,v[k]. Since the appropriate feedback filter (FBF) depends on the effective channel seen at the F F F output, i.e., it depends on Cb[k\, two feedback filters / a ^ f e ] per branch v are needed. Therefore, this structure is referred to as D F E with SIMO filter optimization. It should be noted that, different from conventional throughout this section, indices fj, and v are from the set {1,2} and for the sake of readability, we omit repeated stating of \x 6 {1,2} and v g {1,2}, respectively.  46  4.2 Equalization Strategies impulse response i\[k] fF,i[k)  f ,n[k}  1 oi [A;] L  B  \  1, 02[fc] fB,12[k]  r  -rt\ «0'  :  select  a [k — ko]  a[k—ko] b[k - k ]  select  0  6[fc-.*o]  ri  r ] /B,22W  2/2 [fc]  r  &2[fc]  -"0"  r  fr,2[k] d [k} 2  impulse response ^[fc]  Figure 4-1-' Block diagram of DFE with SIMO filter optimization for 4BOK DS-UWB. D F E where the effective channel is known, a switch is required that selects the appropriate feedback filter according to the assumed effective channel depending on the estimated spreading code cgj j. fc  In order to appropriately model the data-dependent spreading codes of 4BOK, it is convenient to introduce a ternary data symbol  I 0  otherwise  which (a) accounts both for BPSK modulation and spreading code selection and (b) leads to a time-invariant overall channel impulse response. More specifically, the transmission channel between input a^k] and v  th  R A K E output y„[k] can be formulated as (cf. (2.19)  4.2 Equalization Strategies  47  and (2.20)) y [k) = ai[fcj v  *  (4.17)  h„x[k) + a [k] * h [k] + n[k] 2  v2  The R A K E outputs y\[k] and y [k] are fed to FFFs / F , I M and 2  /F,2[&],  respectively.  The feedback part of the equalizer is used to eliminate ISI caused by previous symbols to the currently estimated symbols. For each equalizer branch, to cancel interference caused by the i  th  previous symbol, one of the two FBFs is chosen based on the previous decision  a[k — i], assuming that the previous decision was correct. For example, if the previous symbol detected was ai[k — i] then for branch 1, F B F /s,n[fc] is used and for branch two, F B F fs,2i[k] is used to cancel ISI due to this symbol. On the other hand, if the previous symbol detected was a [k — i] then for branch 1, F B F fB,n[k] is used and for branch two, 2  F B F /B,22[&] is used. For each branch, the output of the feedback part, which is a weighted linear combination of the previous symbol decisions, is subtracted from the output of the F F F filter to give decision variable d [k]. u  Defining the FFFs f  F v  and FBFs f  as  Bu  f' F,u  /F,„[0] /F,„[1] •  IFAQF] H  S''B,vn  •  (4.18)  f B , v t M B  where q and qs denote the order of FFFs and FBFs, respectively, the output of the D F E F  equalizer branch v can be compactly written as (see Fig. 4.1) (4.19) where the following definitions have been used yu[k]  y [k  -  v  K^[ko]  1]  y [k v  h {k -\] vtl  ...  Q  -  qF]  K^ko-qp]  d [k - k - 1] d [k - k - 2] . . . a [k - k - q ] u  0  u  0  v  0  B  (4.20)  4.2 Equalization Strategies  48  It should be noted that the estimates d^k] are the outputs of the switch in the feedback path (see Fig. 4.1). The data estimates for b[k] and a[k] are obtained as b[k — k ]  =  a[k-ko]  = sign{Re{d _ [fc]}} ,  0  argmax{|Re{<4[/c]}|}  fc=i.2 6[fe  (4.21) fco]  where ko is the decision delay, which has to be optimized, as was done for conventional D F E (cf. [VLC96]). For the optimization of FFFs and FBFs according to the M M S E criterion, the equalizer error signal e„[fc] = d„[k] - a[k - k ] '  (4.22)  0  is considered. Two different approaches are pursued, which are referred to as Filter Design I and II. 4.2.1.1  Filter Design I  In Filter Design I for optimization of F F F f „ we make the following assumption: due to F>  the orthogonality between spreading sequence c and c„, signal components spread with M  are largely suppressed after despreading with c„, v 7^ fj,, i.e., h [k] w O , y ^ f i . Therefore, Uil  ignoring the term a [k] * h^^k], v 7^ //, in (4.17), the received signal in equalizer branch v u  is well approximated by y„[fc] = a„[k] * h [k] + n[k] .  (4.23)  vv  — fB,uv^-B,v[k] ~ [k — ko] is  This means that the simplified error signal e [k] = f y [k\ v  Fv  a  v  used for filter optimization. Assuming, as usual, correct feedback decisions a[k] — a[k] [ProOl] and taking into account the correlation *  = E{n[fc]n [rc]} of noise n[k] — H  n  n  [n[ko] ... n[k — q ]], the optimized FFFs are obtained as 0  F  f F>u  =  {•^•vu-^-vv  H  V  V  H  v v  + <^$ ) nn  h  v v  1  (4.24)  49  4.2 Equalization Strategies  where ( denotes the SNR at the R A K E output and the following definitions are used K^ko + 2]  h [ko + 1] Ufi  M o] fc  Kn[ko +I-qF]  Kti[ko + qB\ h [k + q ~ 1]  K^ko + 1] K [k fl  Q  Ufi  + 2-q ]  ...  F  Mo] M ! ••• 0 M°] M ] 1  0  h [k + q ~ QF] ull  h „[L]  0  B  0  0  v  0  1  A  Uil  —  Ufi  (4.26)  0  o The matrix H  (4.25)  B  and A  Vii  ...  o MO] M ] ••• M^] 1  J  has dimensions (q + 1) x q and (q + 1) x (q + L + 1). F  B  Since, different from the design assumption for f  F  F  F  , the impulse responses h [k], fi ^ v, ulx  are non-zero in general, the two feedback filters fBw = H&Fr  (- ) 4  27  are used to achieve postcursor cancellation (cf. (4.19) and Fig. 4.1). Assuming b[k] = 1, Fig. 4.2 shows the magnitudes of the impulse responses i„[k] = f ,„[k] * h \[k\ at the F F F outputs (circles) and of the corresponding the F B F coefficients F  u  fB,v\[k] (stars) for an exemplary CM4 channel realization and spreading with code length N — 6 (cf. also Fig. 4.1). It can be seen that i\{k) is minimum phase with negligible precursors due to the M M S E optimization. For v = 2 ^ b[k] we observe that (a) i [k] is 2  non-zero, i.e., orthogonality of the spreading codes is lost due to the U W B ISI channel, and (b) stronger precursors occur as b[k] — 1 is not accounted for in the design of f 2[k). Fi  Postcursors are fully cancelled through the use of the two feedback filters fB,v\[k\ = iv[k] for k > k . 0  4.2.1.2  F i l t e r Design II  As illustrated in Fig. 4.2, a shortcoming of Filter Design I is the conceptual separation of the 4BOK transmission (4.17) into two parallel B P S K transmissions (4.23) for optimization  4.2 Equalization Strategies  50  Figure 4-2: SIMO Filter Design I: Magnitudes of the impulse responses i [k\ = fF,v[k\*h,,\[k\ at the FFF outputs (circles) and of the FBF coefficients fB,vi[k] (stars) (see also Figs. 4-1 and 4-4) f exemplary CM4 channel realization and spreading with code length N = 6 . v  or  a  of the FFFs f  n  Fv  due to the assumption that h [k] « 0 , i / / / i . To overcome this problem, vll  Filter Design II is directly based on (4.17), i.e, the error signal in (4.22) with d„[k] from (4.19) is used. Noting that E {[a [k + K] a [k + K]} [ [k} a [k}}*} = T  x  2  ai  2  i<5[«] 0 0 ±8[K]  (4.28)  the optimized F F F coefficients are given by SF,V = {^(AviA^  + A A" v2  - HH  R  2  The FBFs are given by (4.27) with f  vl  F>u  x  — H H" ) v2  +  2  C*nn)  h  1  vv  (4.29)  from (4.29).  Fig. 4.3 shows the magnitudes of the impulse responses i [k] (circles) and of the F B F v  coefficients fB,vi\k] (stars) with the FFFs according to (4.29). We observe that, different  4.2 Equalization Strategies  30  40  50  k  51  60  >-  70  80  30  40  50  k  60  70 —>-  80  Figure 4-3: SIMO Filter Design II: Magnitudes of the impulse responses i [k] — fF,i/[k] * h \[k] at the FFF outputs (circles) and of the FBF coefficients fB,v\[k\ (stars) (see also Figs. 4-1 and 4-4) f exemplary CM4 channel realization and spreading with code length u  v  or  N  =  a  n  6.  from Filter Design I in Fig. 4.2, precursors of i [k] are much better suppressed at the ex2  pense of slightly larger precursors in i\[k]. This is due the fact that f  Fv  from (4.29) is  optimized such that both spreading sequences c are taken into account (compare with v  (4.24)). Clearly, the design of two separate FFFs cannot accomplish full precursor suppression in ii[k] and i [k] at the same time. It can also be seen that ii[k] is not minimum phase, 2  which causes increased error propagation for D F E and DDFSE. 4.2.1.3  L E based on Filter I and Filter II  In the case of L E , the decision variable reads d \k) = fl y [k\ v  v  v  ,  (4.30)  4.2 Equalization Strategies  where f ^  v  f  Fv  52  is optimized according to the M M S E criterion and follows as a special case of  for D F E with q = qj and q = 0. The optimized filter for SIMO filter I design can F  B  be expressed as /„  =  + C * n „ n „ )  _  ,  X „  (4.31)  and for SIMO filter II is given by /„ = {\(A A^  + AA )  + C*„,nJ  H  vl  v2  v2  _  1  /i  •  w  (4.32)  The data estimates for b[k] and a[k] are obtained as in (4.21). 4.2.1.4  D D F S E based on F i l t e r I and F i l t e r II  In the case of DDFSE, we employ the Viterbi algorithm with 4  (?(r  states s[k] = [a~i[k —  k — 1] d [k — k — 1 ] . . . di[k — k — q ] d [k — k — q ]]. We apply the filters obtained from 0  2  0  Q  tr  2  0  tr  the optimization for D F E (cf. e.g. [GH96] for single-input single-output DDFSE), and the Viterbi branch metric variables are (|di[fc] — di[k — ko]\ + \d [k] — d [k — fco]| ) with d [k] 2  2  2  2  u  from (4.19) and the 2q feedback symbols a„ [k — k — m], 1 < m < q , are replaced by the tr  0  tr  trial symbols in s[k]. Introducing the following definitions f B,^ Q  f ^ B  tT  B  =  /B,^[l]  =  [ / W « r +  •••  IBM ) 2  1]  W f t r  fB.vMA +  ••• fB,M\  2]  ,  (4-33)  The state metric \(a[k — ko], s[k]) can be obtained as X(d[k - k } s[k}) 0  :  =  \f? [k]  -~ [k-  tiyi  -(f B n ) q  B  qtr  2  n  u  )  H  (ft  2  s  s[k] -  (f n n *B,s[k] B  Q  H  B  2  ~~a [k- ko] -  F  B  -[ f %  B s  H  B  0  a , [k]\  H  + \f Mk] - ( f  k]  ai  2  H  a , [k]\  2  B s  .  )  H  s[k] -  ( f  B  B  2  r )  H  &B,.[k] (4.34)  53  4.2 Equalization Strategies  4.2.2  4 B O K Equalization with M I M O Filter Optimization  From (4.17) we see that 4BOK with R A K E combining can be regarded as a MIMO transmission system. Therefore, defining the rake output vector as y[k] = [yi[k] y [k]] and using T  2  equations (4.17) and (2.19) we arrive at the following expression y[k] = J_ H[m]a[k - m) + n[k] = H[k] * a[k] + n[k] ,  (4.35)  m  with the (2 x 2) channel matrix h [k] hi [k] h i[k] h [k]  H[k] =  n  (4.36)  2  2  22  and two-dimensional data vector a[k] = [ai[k] a [k]] and noise vector n[k] — [ni[k] n [k]] . T  T  2  2  Note that the components of a[k] are drawn from a ternary alphabet according to (4.16) with autocorrelation as given in (4.28). We therefore conclude that MIMO equalization strategies, cf. e.g. [DH92, YR94], can be applied to 4BOK systems. Fig. 4.4 shows the block diagram for MIMO-DFE. We define the vector y[k] of despread signal samples, the vector a [k] of data estimates, and the vector n[k] of noise samples as B  y[k] = a [k] B  =  n[k] =  [y [k]  y [k-l]  T  . . . y [k - q ] f  T  ,  T  F  a [k - k - 1] a [k - k - 2] . . . a [k — k — q ] T  T  T  0  [n [k]  0  n [k-l]  T  T  0  . . . n [k-q ]] T  (4.37)  ,  T  F  and the feedforward matrix filter F_p and feedback matrix filter F  B  F_p =  F»[0]  F^{1]  ...  F [q ]  F%\1\  F {2]  ...  F [q )  H  B  B  as  H  F  B  F  B  (4.38)  with (2 x 2) component matrices F [k] and F [k] given by F  F [k] F  F [k] B  =  B  fF,n[k] fF,2\[k]  fFM \ fF,22[k] k  fB,\\[k\ /B,12[fc] fB,2\[k] fB,22[k]  (4.39)  4.2 Equalization Strategies  54  impulse response i\[k]  yjk]  ajk — ko] /  >'yi[k}\  fc  I  I  !  I  I  I  i  i I  impulse response  \  ____ \ - o]  .  r.  Decision I .  •, '  r.  1  •aa[kj- ko]  i [k] 2  Figure 44: Block diagram of DFE with MIMO filter optimization for 4B0K  DS-UWB.  where fp,ij and fp,ij, 1 < i, j < 2 denote the F F F and F B F coefficients, respectively (see Fig. 4.4). The M I M O - D F E decision variable d[k] = [di[k] d [k]] can now be written as T  2  d[k) = F3y[k]-F%a [k\.  (4.40)  B  It should be noted that the matrix F F F F_ for M I M O - D F E contains four scalar filters F  with, compared to the SIMO case, two additional cross-FFFs between yi[k] and d [k] and 2  between y [k] and di[k], respectively (cf. Figs. 4.1 and 4.4). Since the complexity of D F E 2  is dominated by feedforward filtering, M I M O - D F E is approximately twice as complex as D F E with SIMO filter optimization from Section 4.2.1. For optimization of the matrix filters F_ and F_ , we again apply the M M S E criterion F  B  and follow the derivation in [ADS00]. Using the definitions in (4.37), the R A K E output y[k] can be expressed as yjk] =Ka[k]  + n[k] ,  (4.41)  4.2 Equalization Strategies  55  where I f is given by " i f [0] H[l] 0  H =  ...  H[L]  H[0] H[l]  ;  '•.  ••.  0  ...  0  0  ...  ...  0  H[L] ••.  0  (4.42)  ••. '•. o  '•.  H[0] H[l]  Then, the input-output cross-correlation matrix R  ... H[L]  and output auto-correlation matrix  ay  Ryy are given by  E{a[k}y [k]} = RaaH  R,ay R  H  H  = E{y[k}y [k}} = HR H H  vv  H  &&  + R»  (4.43)  where Raa denotes the input auto-correlation matrix Raa — E{a[k]a [k]} and a R H  nn  the noise auto-correlation matrix R  nn  =  is  E{ri\k\n [k\}. H  Defining the matrix R as  R = R~al + H R~^H H  (4.44)  and introducing the following definitions for matrix ip, and C 2fc x2(fc +l)  !  0  0  0 2x2fc  0  C  H  the optimized FBFs F  (4.45)  0 2x2(fe +l) 0  are obtained from [ADS00]  B  F» JTlH  As can be seen, F  =  =R ^ R ^ C _  0 2x2fco lo  FH  (4.46)  is obtained from F ^ after discarding a block of leading zeros and the 1  B  matrix I . 2  The optimized FFFs F  F  are then given by [ADS00], (4.47)  4.2 Equalization Strategies  56  Assuming correct previous decisions, the decision vector d[k] = [di[k] d2[k]] can be written T  as d[k] = a[k - ko] + e[k] .  (4.48)  The error vector e[k] has the autocorrelation matrix -R e,min given by (cf. [ADSOO]) e  Re, » M  = C {? mp)7 C. H  H  .  l  (4.49)  Assuming e[k] is approximately Gaussian distributed, the decision rule d[k - k ] = argmin {(d[k] ^ a[k - k ]) R; ] (d[k] - a[k - ko])}  (4.50)  H  0  0  e  min  a[k—kg]  is obtained. It should be noted that the optimization in (4.50) is performed for the 4-ary vector signal set a[k] G { Q , ft), (-°i) > (!) > (  s e e  ( - ))4  16  The decision metric (4.50) considerably simplifies if we assume that e\[k] and e [k] are 2  uncorrelated and have identical variance cr , i.e., -R , in = c f ^ - Then, we have 2  ee  m  d[k-ko] = argmax{Re{d [k]a[k-ko]}} . H  (4.51)  a\k—ko]  Considering the particular 4-ary signal set of a[k], (4.51) can be simplified to the two-stage decision in (4.21). Fig. 4.5 depicts the magnitudes of the impulse responses i [k] (circles) and F B F coefv  ficients fB,ui[k] (stars) with the MIMO FFFs. Compared to the SIMO filter optimization cases in Figs. 4.2 and 4.3, improved precursor suppression is achieved and the impulse responses more closely resemble minimum-phase responses, which is crucial for efficient 2  D F E and DDFSE. 2  The minimum-phase criterion for J[k] = ^2 F [m]H[k — m] does not imply that the scalar elements F  m=0  of J[k] are minimum-phase responses (cf. e.g. [GT04]).  4.2 Equalization Strategies  57  Figure 4-5: MIMO Filter Design: Magnitudes of the impulse responses i [k] — f [k] * h \ [k] at the FFF outputs (circles) and of the FBF coefficients fB,vi[k] (stars) (see also Figs. 4-1 and 4-4) f exemplary CM4 channel realization and spreading with code length N = 6. u  or  4.2.2.1  a  Fv  v  n  M I M O O p t i m i z a t i o n based L E and D D F S E  As for the SIMO filter optimization case, we use the D F E filters also for, respectively, L E and DDFSE with MIMO F F F . In case of L E , the decision variable reads d[k] = Ffyjk] where F  f  has 4  qtr  = F  F  for D F E with q = q and q F  s  B  ,  (4.52)  = 0. For D D F S E the Viterbi algorithm  states S[k] = [a [k — k — 1]... a [k — k — qt }] , and the Viterbi branch metric T  T  0  T  0  r  variables are ||d[A;] — a[k — ko]\\ from (4.40) with q 2  tr  feedback symbols a[k — kg — rn],  1 < m < qtr, replaced by the trial symbols in S[k]. Mathematically, the branch metric can be expressed as \(a[k - k ], S[k}) = \\F$y[k] - a[k - fc ] 0  0  S[k] - ( F ^ - ^ ) « B , s [ ] l l H  2  f c  (4-53)  58  4.2 Equalization Strategies  4.2.3  4BOK Equalization with W L - M I M O Filter Optimization  From (4.16) it is easy to see that the correlation between a[k] and a*[k] is non-zero: E{a[/c + K]a [k]} = J  ±6[K] 0  (4.54)  0  Furthermore, the pseudo-autocorrelation function [NM93] of the received sequence y[k] E{y[k + K]y [k)} = -H[K] * H[K] T  1  (4.55)  is also nonzero, implying that y[k] is non-circular. As explained in Section 3.2.2, for a noncircular y[k], W L processing can be applied to the M M S E based equalization schemes. W L M M S E equalization schemes, in particular W L E and W D F E schemes, for SISO channels have been developed by in [GSL03]. Furthermore, in [MPS05] these schemes have been extended to the MIMO channel case considering complex baseband transmission with a mixture of real- and complex-valued modulations.  In the following, we discuss 4BOK  equalization schemes based on the application of widely linear processing to the MIMO filter optimization discussed in the previous section. In order to cast the receiver processing and the filter optimization for 4BOK equalization with W L - M I M O - D F E in a form analogous to the "linear" case discussed in the previous section, it is useful to introduce the transformation X  X :  X =  [X  T  X] H  T  (4.56)  for a general matrix (vector) X. Applying this transformation to the variables in (4.35), we obtain the augmented MIMO channel description y\k] = H[k]*a\k]+n[k).  (4.57)  Substituting this MIMO channel description for (4.35), we can immediately use the results of [ADSOO] for W L - M I M O - D F E filter optimization. Using (4.56) and (4.41), we obtain y[k] = Ha{k] + n[k] ,  (4.58)  59  4.2 Equalization Strategies  where y[k], a[k], and n[k] are defined using (4.56) and (4.37) and the matrix H_ can be obtained by replacing H[k] by H[k] in (4.42). The definitions of noise autocorrelation matrix R ,  the input-output cross-correlation matrix R  ay  and output auto-correlation  ay  matrix R  are given by  yy  Rm.  =  R  M  0  R  rH  H  = • E{y[k)y [k]} = H H  y  0  Tin  = E{a[k}y [k}} = R^JI" ,  ay  R  R  E{n[k)n [k}}  y  R  e  B  H  H  +  (4.59)  R  :  It should be noted that the correlated noise at the R A K E output n[k] is circular i.e., E{n[A;]n [/i;]} r  =  0. Furthermore, defining the matrix RWL as R  WL  =  E_ ,  the F F F matrix F ^ ^ a n d F B F matrix by using variables F [k] fx,a, ,xe{F,  BWL  and f ,WL,ij„  XjWL  + EI  Raa  (4.60)  II  7  which can be defined by (4.38) and (4.39)  € {F,B},  %  x  H ,„  1< hj <  instead of F [k] and  2  x  B}, 1 < i, j < 2, is given'by (see (4.46)) H  RWLV{<P RWHP)~ C H  F  X  —B,WL H —B,WL  02x2fc  F  0  (4.61)  12 F-B,WL  where <p and C are defined in (4.45). Again, as in the "linear" MIM.O-DFE case,the F B F matrix F ^ H  WL  matrix  I. 2  is obtained from F ^  W  L  after discarding a block of leading zeros and the  Also, it should be noted that all F B F coefficients fs.wL.ij, 1 < h j < 2, are real  valued. The feedforward matrix filter F_F,WL = [ES,wL E-F,wL)  T  i s  given by [ADS00]  = F H r R'/l.ll - =B,WL izRR'*VV. '  ^F.WL  (4.62)  l  The W L - M I M O - D F E decision variable reads (compare with (4.40)) d[k] = F y[k] - FL a [k) = 2Re{F y[k)} H  r  F  H  B  B  F  F a [k] B  B  (4.63)  60  4.2 Equalization Strategies  The data estimates are obtained by following the decision rule in (4.21). It should be noted that the complexity of W L - M I M O - D F E is slightly lower than that of M I M O - D F E (4.40), since the coefficients of the feedback filter F^ are real valued. 4.2.3.1  W L - M I M O - L E and W L - M I M O - D D F S E  W L - M I M O - L E uses the same F F F as W L - M I M O - D F E assuming q = 0 and the decision B  variable is d[k] = Re{F^y[k]}.  WL-MIMO-DDFSE applies the W L - M I M O - D F E FFFs  and FBFs and the Viterbi algorithm with 4  qtr  states as described for MIMO-DDFSE. The  WL-MIMO-DDFSE branch metric X(a[k - k ], S[k]) is given by 0  X(a[k - ko],S[k)) = (2Re{F» y[k}} WL  - a[k - k ] - (F% ) S[k] H  0  WL  - {F% ^) a ) B  H  2  Bm  . (4.64)  From a comparison of (4.64) and (4.53), it can be concluded that the complexity of W L MIMO-DDFSE is slightly lower than that of MIMO-DDFSE (4.40), since the components of vectors F  r B WL  and  are real valued.  Chapter 5 Results and Discussion In this chapter, the simulation results for the various equalization schemes considered for BPSK and 4BOK DS-UWB systems are presented. First we present results for B P S K DSU W B systems in Section 5.1. In Chapter 3, the analysis of the distribution of zeros for DS-UWB systems was discussed. Based on this, in Section 5.1.2, we present the results for distribution of zeros for three different scenarios corresponding to different channels and data rates. Subsequently, based on information from these distribution results, we analyse the equalizer designs for BPSK DS-UWB systems. These predictions about the equalizer designs are then compared with the actual performance results obtained for L E , D F E and DDFSE equalization schemes for BPSK DS-UWB systems in Section 5.1.3. Furthermore, performance results obtained from the application of widely linear processing to L E , D F E and DDFSE are compared with the performance of the corresponding "linear" equalization schemes. Second, we present results for 4BOK DS-UWB systems in Section5.2. In Section 5.2.2 equalization techniques based on SIMO filter optimization are compared with those based on MIMO. Also, M I M O - L E and M I M O - D F E schemes are compared with M I M O - W L E and M I M O - W D F E , respectively. Finally, the absolute performance results for 4BOK W L - M I M O equalization schemes are presented.  61  5.1 Performance Results and Discussion for BPSK DS-UWB Systems  5.1  62  Performance Results and Discussion for B P S K DSU W B Systems  In this section we provide results obtained for the distribution of zeros of the overall channel transfer function and the performance results for the L E , D F E and DDFSE equalization schemes for BPSK DS-UWB systems. Furthermore, results for comparison of the performances of "linear" and "widely linear" schemes are also shown. First, we describe the system parameters used in simulations.  5.1.1  Simulation Parameters  We consider the lower U W B operating band from 3.1 to 4.85 GHz. The system parameters are as specified in Table 5.1, except for the case when we study the effect of varying the number of R A K E fingers F in Section 5.1.3.3. For the results showing the performance of the various equalization schemes, simulations are performed over A ,. = 100 channel 7  realizations of CM1 and CM4. The results show the performance averaged over the best 90 out of 100 channel realizations, cf. [P8002]. For all simulations it is assumed that perfect channel state information is available at the receiver. For a fair comparison of (W)LE and 1  (W)DFE, we choose q/ = q . This is because the feedback part of (W)DFE does not involve F  any multiplications, but only sign-changes and subtractions, and hence practically does not contribute to the overall complexity of (W)DFE. For all equalization schemes a favorable delay parameter k is found by testing various A; within a reasonable range [VLC96]. 0  0  The B E R results are plotted as functions of 10\og (E /N ), w  b  0  where E is the average b  received energy per data bit a[k] assuming optimum matched filtering, i.e., E = E{E (r)} b  b  with E (r) from (3.5). As discussed in Sections 3.2.1.3 and 3.2.2.3, (W)DDFSE uses the b  same F F F as. (W)DFE and the number of states is two, i.e., q — 1 is applied. tr  The notation ( W ) L E / D F E / D D F S E is used when we refer to both "linear" L E / D F E / D D F S E and "widely linear" L E / D F E / D D F S E . x  5.1 Performance Results and Discussion for BPSK DS-UWB Systems  63  Table 5.1: Parameters for the considered B P S K DS-UWB systems. BPSK  AT = 24:  Spreading  [-1,0,1,-1,-1,-1,1,1,0,1,1,1, 1,-1,1,-1,1,1,1,-1,1,-1,-1,1]  Codes [FKMW05]  AT = 6:  Channel  CM1 and CM4, lower band from 3.1 to 4.85 GHz  Models  (see Section 2.3 and [P8002])  Pulse shape  Square root-raised cosine (cn = 0.3) [FKMW05]  Data rates  57 Mbps (N = 24) and 220 Mbps (N = 6) [FKMW05]  Number of R A K E fingers  F = 16 [TK04] (except in Section 5.1.3.3)  5.1.2  [1,0,0,0,0,0]  Application of Distribution of Zeros to Equalizer Design for DS-UWB  In section 3.1.2, an expression for the overall channel transfer function for BPSK DS-UWB systems, which takes into account the effect of the spreading code, the RRC transmit filter, the U W B channel, the receiver filter and the F finger R A K E , is derived and analytical expressions for probability density function and cumulative distribution function for the zeros were reviewed. In the following, the results for the distribution of zeros for different channel scenarios and spreading sequence lengths are discussed. In particular, we have evaluated the expressions (3.12), (3.14), (3.15) for the DS-UWB system with spreading codes of lengths N = 24 (lower data-rate mode) and N = 6 (high data-rate mode) and channel models CM1 and CM4, which constitute the two extreme cases in terms of rms delay spread [P8002]. For DS-UWB with N = 24 and CM1, Figs. 5.1(a)-(c) show the normalized density f (z)/(L z  — 1), the normalized marginal density f (r)/(L r  — 1), and the normalized number  of zeros n(R)/(L — 1) inside the disc \z\ < R, respectively. It can be seen that all the zeros are located inside \z\ < 0.3. Based on the discussion in Section 3.1.2.4, we conclude that  64  5.1 Performance Results and Discussion for BPSK DS- UWB Systems  <> a  (b)  r  — ' ( c )  R  —>  Figure 5.1: UWB CM1 and N = 24. z = x + jy. (a,) Normalized density f (z)/(L - 1) o/ zeros of effective transfer function H(z). (b) Normalized marginal density f (r)/(L — 1) of zeros of H(z). (c) Normalized average number n(R)/(L — 1) of zeros of H(z) inside the disc \z\ < R. z  r  L E should perform well for CM1 and N = 24, and that D F E / D D F S E will yield only very small improvements. Figs. 5.2(a)-(c) depict the mentioned functions considering the same spreading code length N = 24 but CM4. We observe that all zeros lie inside \z\ < 0.6. Thus, for CM4 with significant multipath components and relatively large delays, we expect that L E after R A K E combining works well and that D F E / D D F S E lead to only small performance improvements. However, we expect larger gains with D F E / D D F S E than for the CM1, N = 24 case. For higher data-rate DS-UWB systems, shorter spreading code lengths are used. For N = 6 and CM4, Figs. 5.3(a)-(c) show the distribution of zeros. As can be seen, the distribution is quite different from those for the long spreading code with N = 24. For  5.1 Performance Results and Discussion for BPSK DS-UWB Systems  65  Figure 5.2: UWB CM4 and N = 24. z = x + ]y. (a) Normalized density f {z)/(L - 1) of zeros of effective transfer function H(z). (b) Normalized marginal density f (r)/(L — 1) of zeros of H(z). (c) Normalized average number n(R)/(L — 1) of zeros of H(z) inside the disc \z\ < R. e  r  this high-rate (220 Mbps) DS-UWB system, a considerable fraction of zeros lies close to and outside the unit circle. In particular, from Fig 5.3(b) we observe that about x(oo) « 0.35(L — 1) zeros lie outside the unit circle. We thus anticipate that a F F F has to be applied for D F E and DDFSE to avoid a considerable performance loss due to the nonminimum-phase impulse response. Since there are also zeros very close to the unit circle, L E is expected to suffer from a significant performance degradation.  5.1.3  Simulation Results for B P S K D S - U W B Systems  In this section, the performance results for the proposed equalization schemes for DS-UWB systems using BPSK modulation are presented and compared with (a) the three MFBs  5.1 Performance Results and Discussion for BPSK DS- UWB Systems  66  Figure 5.3: UWB CM4 and N = 6. z = x + \y. (a) Normalized density f (z)/(L - 1) of zeros of effective transfer function H(z). (b) Normalized marginal density f (r)/(L — 1) of zeros of H(z). (c) Normalized average number n(R)/(L — 1) of zeros of H(z) inside the disc \z\ < R. z  r  developed in Section 3.1.1, and (b) the findings from studying the distribution of the zeros of H(z) in Section 5.1.2. First, the B E R results for the different equalization strategies without W L processing and with F = 16 R A K E fingers are presented. Subsequently, the improvement due to W L processing (Section 5.1.3.2) and the effect of a finite number F of R A K E fingers (Section 5.1.3.3) are considered. 5.1.3.1  Equalization Strategies and M F B s  Fig. 5.4 shows the B E R performance of the DS-UWB system for spreading code length N = 24 and CM1 as function of 10Tog (.Ey./Vo). We compare M M S E - L E with a two-tap 10  filter / and M M S E - D F E also with a two-tap F F F f  F  and a single-tap F B F f . B  Although  5.1 Performance Results and Discussion for BPSK DS-UWB Systems  R A K E ._>_.LE(q=1) _  *  :::;::::::::::::::::::::::::::::::::: > \ \ * ; \ ::::::::::::::::::::::::: :::;:::::::::::::::::::::::::::::::::::^&::::::;::::::::::::::::  :. \ >V \ > .\\ \\. . . :  _DFE(q =1,q =1) F  MFB MFB MFB  67  B  III II I  ;  N>&\  1 0 l o g ( E / N ) [dB] 1 0  b  0  Figure 54: BER versus 101og (£ /7Vo) for MMSE-LE and MMSE-DFE for CM1 and N = 24. Also shown: RAKE combining without equalization and MFBs I-III (cf. Section 3.1.1). 10  6  the effective channels are minimum phase (cf. Section 3.1.2.4), the short F F F is beneficial for D F E with this short F B F length. The B E R curves without equalization (labeled as R A K E ) and the MFBs I-III are also plotted in Fig. 5.4. As anticipated from the analysis in Section 5.1.2 (cf. Figs. 5.1(a)-(c)), L E and D F E achieve practically identical performances. The loss in power efficiency compared to M F B III is only about 0.15 dB at BER = 1 0  -5  and hence, DDFSE (not shown) does not yield a sig-  nificant performance improvement. In fact, simple R A K E combining without additional equalization already approaches M F B III within 0.5 dB at BER = 10~ . This can be at5  tributed to the facts that (a) the delay spread for CM1 is relatively short, and (b) a long spreading code is applied. M F B II and M F B III are almost identical, which means that 16 R A K E fingers are (more than) sufficient for CM1. Interestingly, the gap between M F B I  5.1 Performance Results and Discussion for BPSK DS- UWB Systems  68  10log (E- /N )[dB] 10  b  0  Figure 5.5: BER versus 10log (E /N ) for MMSE-LE, MMSE-DFE, and MMSE-DDFSE for CM4 and N = 24. Also shown: RAKE combining without equalization and MFBs I-III (cf. Section 3.1.1). w  b  0  and M F B II is about 0.25 dB at BER = 1 0 , which indicates that only little can be gained -5  by sampling faster than the chip rate. Fig. 5.5 presents the B E R curves for the DS-UWB system with AT = 24 and CM4 channels. Filters of order 6 are used for M M S E - L E and as F F F of M M S E - D F E and MMSEDDFSE. Due to the longer impulse responses h'[k) for CM4, the performance with pure R A K E combining deteriorates for low error rates. Different from that, the B E R curves for M M S E - L E and M M S E - D F E are almost parallel to that of M F B III even for this short filter length. In accordance with the findings in Section 5.1.2, L E and D F E show a similar performance, but the gap in power efficiency between L E and D F E is larger than for CM1 and N = 24 in Fig. 5.4 (cf. Figs. 5.2(a)-(c) and 5.1(a)-(c)). Since D F E already approaches M F B III closely, the additional gain with DDFSE is rather small. On the other hand, we  5.1 Performance Results and Discussion for BPSK DS-UWB Systems  69  Figure 5.6: BER versus 10log {E /N ) for MMSE-LE, MMSE-DFE, and MMSE-DDFSE with different filter lengths for CM4 and N = 6. Also shown: RAKE combining without equalization and MFBs I-III (cf. Section 3.1.1). 10  b  Q  observe a gap between M F B III and M F B II of about 1.6 dB, which can only be bridged by using more R A K E fingers (cf. Section 5.1.3.3). Again, the curves for M F B I and M F B II are relatively close. As third scenario, we consider the simulation results for spreading with iV = 6 and CM4 channels in Fig. 5.6. Equalizers of different filter lengths are chosen to illustrate the performance-complexity tradeoff. Clearly, R A K E combining without further equalization is not a viable solution in this case. We observe that M M S E - D F E significantly outperforms M M S E - L E for the same filter orders <j/ = q as was (qualitatively) to be expected from the F  results in Section 5.1.2 (cf. Figs. 5.3(a)-(c)). DDFSE provides only small gains over DFE, as the latter already approaches M F B III within 0.9 dB for q = 50 and q — 23. Compared F  B  to the low-rate case with N = 24 in Fig. 5.5, much longer filters are needed for efficient  70  5.1 Performance Results and Discussion for BPSK DS-UWB Systems  equalization, which is in accordance with the predictions in Section 5.1.2. The gap of about 1.4 dB between M F B III and M F B II indicates that the use of more than F = 16 R A K E fingers could yield further performance improvement. We further observe that, as for the other two scenarios discussed above, the curves for M F B I and M F B II are very close. This leads to the general conclusion that almost no additional gain can be achieved by sampling faster than the chip rate, which renders chip-rate sampling an attractive alternative when aiming at low receiver complexity. It is interesting to note that an oversampling factor of 10 was found necessary for DS-UWB systems employing Gaussian monocycles in [MWS03]. Finally, the comparison of performance results in Fig. 5.6 with those of [ED05, Fig. 2] leads to the conclusion that DS-UWB with R R C pulse shaping and carrier modulation is more power efficient than carrierless U W B with Gaussian monocycles for high-rate transmission. 5.1.3.2  Linear vs. W L Processing  From Fig. 5.7, which corresponds to the lower data rate (57 Mbps) scenario with N = 24, it can be seen that W L E provides a performance gain of around 0.2 dB and W D F E , with a reduced complexity F B F design, also performs better than D F E . For the more important high data rate scenario corresponding to CM4 and N = 6 (as in Fig: 5.6) the benefits of W L processing over conventional "linear" processing are illustrated in Figs. 5.8 and 5.9, where B E R curves for L E and D F E are compared with those for W L E and W D F E , respectively. For a further comparison, the curve for D F E with q = 50 and F  qs = 23 is also included. As can be seen, W L E consistently outperforms L E for low BERs by about 1 dB, whereas the gains of W D F E over D F E are in the order of 0.3 dB. It should be noted that these improvements in power efficiency come at no cost in complexity (for L E vs. WLE) or even  5.1 Performance Results and Discussion for BPSK DS- UWB Systems  71  Figure 5.7: BER versus lOlog (E /N ) for MMSE-(W)LE and MMSE-(W)DFE for CM4 and N = 24. Also shown: MMSE-WDDFSE, RAKE combining without equalization, and MFBs I-III (cf. Section 3.1.1). w  b  0  with reduced complexity (for D F E vs. W D F E ) . Moreover, we found that W L E achieves similar BERs as D F E with identical F F F order. As an example, Fig. 5.8 shows the corresponding BERs for qf = qp = 50. Hence, for fixed power efficiency, W L processing achieves a similar gain over L E as decision-feedback processing, but it is less complex and, different from DFE, it can easily be combined with error-correcting decoders as no decision feedback is required. 5.1.3.3  N u m b e r of R A K E Fingers  In order to assess the effect of the number F of R A K E fingers independent of a possibly applied equalization strategy, we study the loss in average SNR for F < FOQ, where F^ denotes the number of taps of the overall impulse response h [K] (2.11) after chip-rate 0  5.1 Performance Results and Discussion for BPSK DS-UWB Systems  72  Figure 5.8: BER versus 10\og (E /N ) for MMSE-LE and MMSE-WLE with different filter lengths for CM4 and N = 6. Also shown: MMSE-DFE with q = 50 and q = 23, RAKE combining without equalization, and MFBs I-III (cf. Section 3.1.1). 10  b  Q  F  B  sampling. This average SNR loss is defined as  ASNIKF)-10^(^1^),  (5,)  where y (F) is the SNR (3.1) with E (r) from (3.10) for F R A K E fingers. We note that T  b  the average SNR loss A S N R ( F ) is closely related to but not identical with the difference in E /N b  Q  between M F B II and M F B III, since BERs are averaged to obtain MFBs but SNR  ratios are averaged to obtain A S N R ( F ) . Fig. 5.10 shows A S N R ( F ) as function of F for the three scenarios (CM1, N = 24), (CM4, N = 24), and (CM4, N = 6). We observe that for CM1, F = 16 R A K E fingers incur only a very small SNR loss of ASNR(16) = 0.3 dB, which is in accordance with the B E R simulations in Fig. 5.4. From this curve we also conclude that for CM1 a good capture of the useful received energy is accomplished with F > 8 fingers.  5.2 Performance Results and Discussion for 4B0K DS-UWB Systems  73  Figure 5.9: BER versus 10\og (E /N ) for MMSE-DFE and MMSE-WDFE with different filter lengths for CM4 and N = 6. Also shown: RAKE combining without equalization, and MFBs I-III (cf. Section 3.1.1). w  b  Q  In the case of CM4, on the other hand, non-negligible SNR losses occur for small to moderately large numbers of fingers F < 20. This result, which has already been observed in Figs. 5.5-5.9 for F = 16, is due to the long delay spread associated with CM4 and it is (almost) independent of the spreading code length N. A rather large number of R A K E fingers (F > 30) would be required to approach the optimum performance, which seems impractical for implementation.  5.2  Performance Results and Discussion for 4 B O K DSU W B Systems  In this section, we provide a performance comparison of the equalization strategies proposed in Section 4.2 based on simulated B E R results. We also compare the simulation results with  5.2 Performance Results and Discussion for 4B0K DS-UWB Systems  74  - « - CM1 N=24 - o - CM4 N=24 - a - CM4 N=6  '  1  •z. CO  <  <  2  x •—o 20  30  40  50  —•_  60  N u m b e r of R A K E F i n g e r s (F)  Figure 5.10: Average SNR loss A S N R ( F ) as function of the number of RAKE fingers F for the three scenarios (CM1, N — 24), (CM4, N = 24), and (CM4, N = 6). the B E R bounds from Section 4.1.  5.2.1  Simulation Parameters  As for the B P S K case, we consider the lower U W B operating band from 3.1 to 4.85 GHz. Simulations are performed over N = 100 channel realizations of CM1 and CM4, which r  constitute the extreme cases of the U W B channel model. The B E R results are obtained as an average over the best 90 out of 100 channel realizations, cf. [P8002]. 4BOK DSU W B transmission with spreading codes of length N = 24 (corresponding to a data rate of 110 Mbps) and N = 6 (corresponding to data rates 440 Mbps), respectively, are exemplarily chosen [FKMW05]. At the receiver side, R A K E combining with F = 16 fingers is applied (cf. e.g. [TK04]) and we assume that the channel realizations are perfectly estimated at the receiver. For the MIMO equalization case we consider the simplified decision rule (4.21).  5.2 Performance Results and Discussion for 4B0K DS-UWB Systems  75  Table 5.2: Parameters for the considered 4 B 0 K DS-UWB systems. 4B0K Spreading Codes [FKMW05]  Channel Models Pulse shape Data rates Number of R A K E fingers  TV = 24:  [-1,1,-1,-1,1,-1,-1,1,-1,0,-1,0, -1,-1,1,1,1,-1,1,1,1,-1,-1,-1] AT = 24: [ 0 , - 1 , - 1 , 0 , 1 , - 1 , - 1 , 1 , - 1 , - 1 , 1 , 1 , 1,1,-1,-1,1,-1,1,-1,1,1,1,1] N = 6: [1,0,0,0,0,0] AT = 6: [0,0,0,1,0,0] CM1 and CM4, lower band from 3.1 to 4.85 GHz (see Section 2.3 and [P8002]) Square root-raised cosine (a — 0.3) [FKMW05] 110 Mbps (N = 24) and 440 Mbps (N = 6) [FKMW05] F = 16 [TK04]  Since the complexity of L E and D F E is dominated by feedforward filtering, we choose If — Q.F- D D F S E uses the same F F F as D F E and the number of states is four, i.e., q = 1 tr  is considered. For all equalization schemes a favorable delay parameter k is found by 0  testing various k within a reasonable range. The system parameters are summarized in Q  Table 5.2.  5.2.2  Simulation Results  First, the performance differences due to (a) SIMO vs. MIMO filter optimization, and (b) MIMO vs. W L - M I M O filter optimization, are discussed in Section 5.2.2.1. Then, in Section 5.2.2.2, we consider the absolute performance of 4BOK equalization for different transmission scenarios and compare these with the respective MFBs. 5.2.2.1  Filter O p t i m i z a t i o n  We consider L E and D F E , and, for the sake of brevity, we use SIMO-x-Y, MIMO-Y, and WL-MIMO-Y, x G {I, II}, Y G {LE, DFE}, to refer to L E and D F E with SIMO Filter De-  5.2 Performance Results and Discussion for 4B0K DS-UWB Systems  i :i > i' - l I ri  l,q =16 (  10 I1 1  :  1  l,q,=40  /  '  . _ A - • I. q,=50 —  ll,q,=16  —  -B  —  H  II, q,=40  -  e  - II, q,=50  •  i '  ,  I  {  ''  1.5  (a)  1 0  (BER)  x  '+•  I, q = 1 6 , q = 1 5 p  B  ( l  I, q =40, q =30 F  ; ?  i :  B  8  i -  ' /  .A  ! ,  r  9  MIMO-LE 1  2.5  2  -log  _ _. 9  p  t-•  SIMO-x-LE (x € {1,11}) vs; 1  /'  1 1  ' i  i  ../.... i  :'  76  1.5  (b)  -  2  2.5  -lo  6 l 0  3  (BER)  3.5  4  4.5  5  — ,  Figure 5.11: Performance comparison for equalization with SIMO and MIMO filter optimization. CM4 and N = 6. (a) LE: A S N R = 101og (SNR(5/MO-a;-L£)) - 101og (SNR(M/MO-L£)) XG{I,II}, (b) DFE: A S N R B = 10log (SNR{SIMO-x-DFE)) - 10log {SNR(MIMO-DFE)), xe{I,II}. Filter orders qf — 16, qf = 40, and qj = 50 for LE and (qp = 16, q = 15), (qF = 40, q = 30j and (q = 50, q = 35) for DFE. d B  10  10  D  ;  1Q  10  B  B  ;  F  B  sign I and II, and MIMO and W L - M I M O filter optimization, respectively. SIMO vs. MIMO Filter Optimization: Fig. 5.11 compares the performances of different filter optimization techniques for L E (Fig. 5.11(a)) and D F E ((Fig. 5.11(b)) assuming different filter orders and the example of CM4 and N = 6. More specifically, the signal-to-noise ratio (SNR) difference A S N R B = 101og (SNR(SIMO-x-Y)) - 101og (SNR(MIMO-Y)) , x E {1,11} , D  10  10  Ye{LE,DFE},  (5.2)  between SIMO and MIMO equalization required to achieve a certain B E R assuming identical filter orders (qf,q ,q ) F  B  is plotted vs. — log (BER). Since A S N R 10  d B  is always positive  5.2 Performance Results and Discussion for 4B0K DS-UWB Systems  77  for L E (Fig. 5.11(a)) and for B E R > 1CT - in case of D F E (Fig. 5.11(b)), we conclude that 1 5  MIMO equalization is practically always superior to SIMO equalization, which was anticipated from the findings in Section 4.2 (cf. also the impulse responses in Figs. 4.2, 4.5 and 4.5). The SNR gains due to MIMO filter optimization are more pronounced for L E , where SIMO-I-LE and SIMO-II-LE show a relatively high error floor at B E R « I O depending on the filter order qf, i.e., A S N R  - 2 0  . . . 10 ' -2  5  grows unboundedly for L E and B E R < 10~ . 25  d B  It should be mentioned that no error floor was observed for M I M O - L E with qf = 50 and B E R > 1 0 . Also, for D F E the differences in power efficiency increase for lower target BER, -5  e.g., M I M O - D F E with (q - 40,q = 30) outperforms SIMO-II-DFE with the same filter F  B  orders by 0.9 dB and 3.7 dB for B E R = IO" and IO" , respectively. Furthermore, SIMO3  5  II-DFE with shorter filters and SIMO-I-DFE cause a higher error floor than M I M O - D F E with identical filter orders. When comparing the two SIMO filter optimization schemes, we note that SIMO-II has considerable performance improvements over SIMO-I, especially for D F E and lower target BER, where the effect of error propagation is insignificant (cf. Section 4.2.1). Fig. 5.12 (a) compares the performance of different filter optimization techniques for L E and D F E for the lower data rate scenario of CM4 and N = 24. Interestingly, even for this low data rate scenario, MIMO equalization schemes perform better than the corresponding SIMO equalization schemes. More specifically, M I M O - L E outperforms SIMO-I-LE and SIMO-II-LE by 3.75 dB and 2.75 dB, respectively and M I M O - D F E outperforms SIMO-ID F E and SIMO-II-DFE by 1.1 dB and 1.25 dB, respectively, for B E R = I O . Moreover, -5  this performance gap increases for higher target BER. MIMO vs. WL-MIMO Filter Optimization: In Figs. 5.12(b) and 5.13 the performances for MIMO and W L - M I M O equalization are compared in terms of A S N R  d B  vs. - l o g ( B E R ) 1 0  for CM4 and N = 6 and N = 24, respectively. The SNR difference for a fixed target B E R  5.2 Performance Results and Discussion for 4B0K DS-UWB Systems  •e •  I.  4=6  _  I, q =6, q =6  •a-  l.q =6,q =6  A  78  F  F  B  B  2.5 SIMO-x-LE(xie  2h  {1,11})  VS-MIMO-LE; &4  SIMO-x-DFE(x  -0.5  vs M I M O - D F E  3  (a)  e  {1,11}) • : •  1  3  4  •log (BER) —  4  log (BER)  10  10  Figure 5.12: (a) Performance comparison for equalization with SIMO and MIMO filter optimization. CM4 and N = 24. LE: A S N R = 10\og (SNR(SIMO-x-LE)) 101og (SNR(M/MO-L£)), x G {I, II}, DFE: A S N R = 10 \og (SNR(SIMO-x-DFE)) 10\og (SNR(MIMO-DFE)), xe{I,II}. (b) Performance comparison for equalization with MIMO and WL-MIMO filter optimization. CM4 and N = 24. LE: A S N R B = 101og (SNR(M/MO-££)) - 10 log (SNR( WL-MIMO-LE)), DFE: A S N R = 10\og (SNR(MIMO-DFE)) - 101og (SNR( WL-MIMO-DFE)) d B  10  d B  10  1Q  1Q  d  10  1Q  10  d B  10  and identical filter orders (qf,qF,QB) is denned as A S N R B = 10log (SNR(MIMO-Y)) - 10log (SNR(WL-MIMO-Y)) , d  10  10  Y e {LE, DFE} . (5.3)  For this low data rate scenario with N = 24, as can be seen from Fig. 5.12(b) that W L E and W D F E perform better than the corresponding "linear" schemes i.e., L E and D F E , with same filter lengths. For the higher data rate scenario with N = 6, it can be seen that W L processing consistently improves performance of M I M O - L E (Fig. 5.13(a)) and M I M O - D F E (Fig. 5.13(b)).  5.2 Performance Results and Discussion for 4B0K DS-UWB Systems  7 9  10r  — Q — q =16, q =15  • — Q- q,=16 • — * -  -o-  8h  I  i 2  m a  F  q,=40  - * -  q,=50  -  e  B  q =40, q =30 F  B  - q =50, q =35 F  Q  m a  5 [  o? 2 co <  L 4|-  A  3^ li  y.  •«  mnMIMO-DFEivs. WL-MIMO-DFE  mnMIMO-LE vs. WL-MIMO-LE 1  2  3  4  1  mn—log (BER)  mn(a)  2  3  4  mn—log (BER)  mn(b)  10  10  Figure 5.13: Performance comparison for equalization with MIMO and WL-MIMO filter optimization. CM4 and N = 6. (a) LE: A S N R = 10 log (SNR(MJMO-L£)) 10\og {SNR(WL-MIMO-LE)), (b) DFE: A S N R = 101og (SNR(MIMO-DFE)) 10 log (SNR(WL-MIMO-DFE)). Filter orders q = 16, q = 40, and q = 50 for LE and (q = 16, q = 15), (q = 40, q = 30), and (q = 50, q = 35) for DFE. d B  10  10  d B  w  10  f  F  B  F  f  B  F  f  B  For relatively long filters with qj = 50 and (q = 50, q = 35) the gains in power efficiency F  B  are 0.6 dB and 3.4 dB at BER = 1 0 , respectively. W L processing becomes even more -5  beneficial for shorter filter lengths (qf = 16, and q = 16, q = 15), where A S N R F  B  d B  grows  steeply with decreasing target BER, i.e., W L processing effectively lowers the error floor of MIMO-LE and MIMO-DFE. It should be emphasized that these gains come at no cost or even reduced computational complexity. In the following, we therefore concentrate on WL-MIMO equalization and present absolute performance results for 4BOK transmission.  5.2 Performance Results and Discussion for 4B0K DS-UWB Systems  80  Figure 5.14: BER versus 10\og (E /N ) for CM1 and N = 24. RAKE combining without equalization, WLE, and WDFE (MIMO equalization). Also shown: B E R ^ _ B E R ' _ | according to MFB I and B E R ^ _ B E R ' _ according to MFB II from Section 4-1w  b  Q  F B  F B  5.2.2.2  M ;  M F B  H  M F B  M  Equalization for 4 B O K and M F B s  We show results for the three channel-rate pairs (CM1, N = 24), (CM4, N = 24), and (CM4, N = 6). For all three scenarios, we include the semi-analytical B E R limits according to the MFBs from Section 4.1. Upper and lower B E R bounds are denoted by and  BER'  M F B  _!  for M F B I and B E R M _ F B  m  and  BER'  M F B  _  M  BER^  F B  _,  for M F B II, respectively. For the  sake of conciseness, we use W L E , W D F E , W D D F S E when referring to W L - M I M O - L E , W L - M I M O - D F E , and WL-MIMO-DDFSE, respectively. CM1 and N = 24: Fig. 5.14 shows B E R results for CM1 and N = 24 as functions of 10log (E /N ), 10  b  0  where E is the average received energy per bit and JV is the one-sided b  0  noise power spectral density of the passband noise process. B E R curves for the R A K E  5.2 Performance Results and Discussion for 4B0K DS-UWB Systems  81  Figure 5.15: BER versus 10\og (E /N ) for CM4 and N = 24. RAKE combining without equalization, WLE, WDFE, and WDDFSE (MIMO equalization). Also shown: B E R ^ , _ B E R ' _ | according to MFB I and B E R ^ _ BER' _ according to MFB II from Section 4.1. 1Q  b  0  F B  M F B  F B  M ;  M F B  I ;  M  receiver without equalization (labeled as " R A K E " ) and for W L E and W D F E are plotted. FFFs of order qf = q = 1 and a single-tap F B F F F  B  = F [l] are employed. B  It can be seen that for CM1 with a relatively short delay spread [P8002] and transmission with long spreading codes, i.e., relatively lower data rate, Wthe performance of L E and W D F E is almost identical. The B E R bounds are closely approached by these schemes and more complex W D D F S E (not shown in Fig. 5.14) cannot further improve performance. Even the B E R curve for R A K E combining without additional equalization is within 1 dB of the M F B for this transmission scenario. We also observe that upper and lower MFBs converge for B E R < 1 0 . -3  CM4 and N = 24: The results for the second scenario are shown in Fig. 5.15. FFFs  5.2 Performance Results and Discussion for 4B0K DS-UWB Systems  82  of order 6 are used for W L E , W D F E , and W D D F S E . As can be seen, the B E R curves for R A K E combining without and with equalization diverge for error rates lower than 1 0 . -2  This is due to the longer effective impulse responses h^[k], //, v G {1,2}, for CM4, which require equalization. We also note that the B E R curves for R A K E combining with equalization run almost parallel to the B E R bounds even for the short filter lengths adopted. W D F E provides a gain of 0.5 dB over W L E and achieves practically the same performance as WDDFSE. In particular, the gap of about 2 dB between  BER[  V 1 F B  _|  and the B E R for W D F E at low BERs  cannot be reduced by using WDDFSE. Instead, comparing the curves for BER|  V ) F B  BER^  / ) F B  _  I  and  _ , it can be seen that R A K E combining with F = 16 does not sufficiently capture m  the received signal energy and that more fingers are required to close this gap. CM4 and N = 6: Fig. 5.16 gives simulation results for spreading with N = 6 and CM4 channels. W L - M I M O equalizers with different filter lengths are chosen to illustrate the performance-complexity tradeoff.  The curve for M I M O - D F E (not W L ) with (qp =  50, q = 35) is also included for comparison. B  Clearly, R A K E combining without further equalization is not a viable solution in this case. We observe that W D F E outperforms W L E for the same filter orders qf — qp. W L E achieves a very similar performance as D F E for qf = qp = 50. Hence, for fixed power efficiency, W L processing achieves a similar gain over L E as decision-feedback processing, but it is less complex. Different from D F E , which may suffer from error bursts due to feedback errors, W L E can easily be combined with error-correcting decoders. The application of more complex W D D F S E (only shown for qp = 50,qs = 35) is not beneficial as only marginal gains over W D F E are achieved. Again, the B E R curves for W D F E run almost parallel to the lower theoretical M F B limit. More specifically, W D F E performs within 1 dB of the  BER'  M F B  _  M  bound at  BER  = 10~ . As shown in Fig. 5.10 for 5  5.2 Performance Results and Discussion for 4B0K DS-UWB Systems  83  Figure 5.16: BER versus 101og (£ /No) for CM4 and N = 6. RAKE combining without equalization, WLE and WDFE with different filter lengths, and WDDFSE and DFE for q — 50 q — 35 (MIMO equalization). Also shown: B E R ^ _ B E R ' _ , according to MFB I and B E R ^ _ B E R ' _ according to MFB II from Section 4-110  F  ;  b  F B  B  FB  M;  MFB  h  MFB  M  B P S K DS-UWB that a relatively large number of F > 30 R A K E fingers would be required to close the gap of about 2-3 dB to BER^ _,. Since B P S K and 4BOK apply the same set FB  of spreading sequences, similar conclusions can be derived for the 4BOK case.  Chapter 6 Conclusions In this work, equalization for DS-UWB systems using two different modulation schemes, B P S K and 4BOK, has been studied. The receiver for DS-UWB systems employs R A K E combining in order to capture the multipath energy. Due to the highly frequency selective nature of U W B channels, severe ISI occurs at the R A K E output. Furthermore, DS-UWB systems are designed for providing high data rates and so the chip-rates used are very high, about 1.3 GHz for lower band and 2.6 GHz for higher band. Therefore, implementation complexity is an important consideration in the receiver design of these systems. In this work, we focused on developing equalization techniques for DS-UWB systems that can be employed after R A K E . Post-RAKE equalization is peformed at symbol level, resulting in reduced implementation complexity. In the first part of this work we have investigated equalization for DS-UWB systems with BPSK modulation as specified in the standard proposal [FKMW05]. To this end, we have (a) derived three versions of the M F B , which to a different extent take into account receiver suboptimalities, (b) considered the effective discrete-time impulse response at the R A K E combiner output and have analyzed the distribution of the zeros of the corresponding transfer function, and (c) studied several equalization strategies including the novel application of W L processing to DS-UWB. The results of our investigation can be  84  85 summarized as follows cf. [PLSL05a, PLSL05b]: • From the M F B analysis, carried out for the IEEE 802.15.3a standard channel model, it can be concluded that suboptimum chip-matched filtering and chip-rate sampling do not incur a significant performance degradation, and that R A K E combining with a practical number of fingers results in performance losses of about 1-2 dB for channels with long delay spread. • The analysis of the zeros of the channel transfer function suggests that L E is well suited for lower data-rate modes of DS-UWB, whereas non-linear equalization (DFE, DDFSE) is preferable for high data-rate modes. These findings have been confirmed by simulation results. • The proposed W L processing was found to provide gains of up to 1 dB, which come at no cost, or even slightly reduce equalizer complexity. More specifically, simple W L E and W D F E schemes were shown to perform in close vicinity of the pertinent M F B , whereas the negligible gains with W D D F S E do not justify its higher complexity. W L E and W D F E are thus recommended for implementation. In the second part of this work equalization for a 4BOK DS-UWB system is studied. To this end, we first have derived expressions for the B E R according to the M F B , which serve as theoretical performance limits. Second, equalizer designs based on SIMO optimization methods have been considered.  Further improving the design of these SIMO schemes,  MIMO filter optimization based equalization schemes have been devised. In this context, an equivalent MIMO channel model for 4BOK DS-UWB, which facilitates the design of efficient equalizers using MIMO filter optimization, has been developed. Third, we have applied W L processing to these MIMO equalization schemes. The results of our investigations for the 4BOK case can be summarized as, cf.[PLSL05c, PLSL06]:  86 • R A K E combining without equalization is not a viable solution for 4BOK DS-UWB systems when aiming at higher data rates. Both SIMO and MIMO equalization improve the performance compared to R A K E combining without equalization. • MIMO equalization with joint feedforward filtering of the two R A K E outputs for 4BOK DS-UWB is clearly advantageous over SIMO equalization with separate feedforward filtering. More specifically, even at lower data rates (such as 57 Mbps) MIMO equalization provides performance gains varying from 1 dB to 4 dB over SIMO equalization. At higher data rates SIMO based equalization techniques either cause error floors or provide high perfomance losses as compared to MIMO techniques, which perform close to the M F B limit. • Low-complexity M I M O - L E and M I M O - D F E with W L processing provide performance gains over their "linear" counterparts and achieve power efficiencies close to the theoretical M F B limit. More specifically, M I M O - W L E outperforms M I M O - L E and achieves a very similar performance as MIMO-DFE. Hence, for fixed power efficiency, W L processing achieves a similar gain over L E as decision-feedback processing, but is less complex. The application of more complex W D D F S E is not beneficial as only marginal gains over W D F E are achieved. Therefore, MIMO based (W)DFE and W L E are recommended for implementation. 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