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Space-time coding for frequency-selective fading channels Chen, Harry Zhi Bing 2003

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S p a c e - T i m e C o d i n g f o r F r e q u e n c y - S e l e c t i v e F a d i n g C h a n n e l s by Harry Zhi Bing Chen B.Eng., Huazhong University of Science and Technology, 1990 M.A.Sc, University of Science and Technology of China, 2000 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Applied Science in THE FACULTY OF GRADUATE STUDIES (Department of Electrical and Computer Engineering) We accept this thesis as conforming to the required standard The Univers i ty of B r i t i s h C o l u m b i a December 2003 © Harry Zhi Bing Chen, 2003 Library Authorization In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Name of Author (please print) Date (dd/mm/yyyy) Title of Thesis: Degree: Year: Department of \~ C £T The University of British Columbia Vancouver, BC Canada Abstract This thesis studies space-time coded transmissions over frequency-selective channels. For this, the performance of maximum likelihood sequence estimation (MLSE) decoding is analyzed, taking into account channel estimation errors. Furthermore, reduced complexity suboptimum decoding schemes are investigated. To analyze M L S E decoding performance, three lower bounds, namely, the matched filter bound (MFB), the improved M F B (IMFB), and the IMFB I, are derived. The M F B assumes no intersymbol interference (ISI) in the received symbols, while IMFB takes into account the effect of ISI of neighboring symbols, and thus provides a tighter bound. IMFB JL, which is the tightest lower bound for time-reversal and space-time block coding (TR-STBC), in addition, considers the decoupling errors of the symbols from the other transmit antenna. Our numerical results for delay diversity (DD), T R - S T B C , and maximum ratio combining (MRC) show that the IMFB, and especially the IMFB JL, match well with simulation results. For reduced complexity decoding, we present three different decision-feedback sequence estimation (DFSE) schemes for T R - S T B C and' DD. The first scheme, called unwhitened D F S E (U-DFSE), performs reduced-state sequence estimation based on the output of the spatial-temporal matched filter (MF) typically employed in T R - S T B C . The second ap-proach improves upon U-DFSE by subtracting a bias term caused by anti-causal interference from the U-DFSE metric. In the third scheme, the noise component in the output of the spatial-temporal M F is first whitened using a prediction-error filter that can be efficiently ii Ill computed using the Levinson-Durbin algorithm. Subsequently, whitened D F S E (W-DFSE) is performed. Our results show that for binary modulation, U-DFSE and its improved ver-sion can approach the performance of W-DFSE for the full range of delay spreads relevant for the global system of mobile communication (GSM) and enhanced data rates for G S M evolution (EDGE). On the other hand, for high-level modulation, only W - D F S E gives a satisfactory performance. Contents Abstract i i List of Tables v i i List of Figures x Acknowledgments x i 1 Introduction 1 1.1 Motivation 2 1.2 Literature Review 3 1.3 Contributions 4 1.4 Thesis Organization 5 2 Background 6 2.1 Wireless System Transmission Model 6 2.2 Wireless Channel Model 8 2.2.1 Frequency-Selective Fading Channel 9 2.2.2 COST 207 G S M Channel Profiles 10 2.3 Maximum Likelihood Sequence Estimation 10 2.4 STC for Frequency Selective Channels 12 2.4.1 T R - S T B C 12 iv CONTENTS v 2.4.2 Delay Diversity 15 3 Channel Estimation 17 3.1 LSSE Channel Estimation 17 3.1.1 SISO Channel Estimation . 17 3.1.2 Multiple-Input Single-Output (MISO) Channel Estimation 19 3.2 Construction of Training Sequences 20 3.2.1 Single Training Sequence 20 3.2.2 Multiple Training Sequences 21 3.2.3 G S M Training Sequences 21 3.3 Performance Analysis for M L S E with Channel Estimation Errors 22 3.3.1 Performance Analysis for M L S E with Known Channel Length . . . . 23 3.3.2 Performance Analysis for M L S E with Over-Estimation 28 3.3.3 Performance Analysis for M L S E with Under-Estimation 32 3.4 Comparison with Simulations 35 3.5 Summary 40 4 Performance Analysis of MLSE Decoding for STCs 42 4.1 Performance Analysis of Maximum Ratio Combining 42 4.1.1 M F B for M R C ' 43 4.1.2 I M F B for M R C 44 4.2 Performance Analysis of DD 45 4.3 Performance Analysis of T R - S T B C 47 4.3.1 M F B for T R - S T B C 50 4.3.2 I M F B for T R - S T B C 52 4.3.3 I M F B I for T R - S T B C 54 CONTENTS vi 4.4 Simulation Results 56 4.5 Summary 61 5 Reduced. Complexity Decoding for STCs 63 5.1 DFSE 63 5.2 W-DFSE 64 5.3 U-DFSE and Modified U-DFSE 65 5.3.1 U-DFSE 65 5.3.2 Modified U-DFSE(M-UDFSE) and Multistage MU-DFSE (MMU-DFSE) 66 5.4 DFSE Applied to STCs 67 5.4.1 DFSE Applied to TR-STBC 67 5.4.2 DFSE Applied to DD 69 5.4.3 Complexity Consideration 70 5.5 Simulation Results 71 5.6 Summary 79 6 Conclusions and Future Research 80 Glossary 85 Bibliography 88 List of Tables 3.1 GSM training sequences 22 3.2 Some good training sequences. Sequences with * also have low cross-correlation. 36 vii List of Figures 2.1 Discrete-time model of a wireless communication system with S T C over frequency-selective channels 7 2.2 Block diagram of the MIMO system 8 2.3 Tapped delay line model of the SISO frequency-selective channel 9 2.4 Forney receiver structure 11 2.5 Ungerbeock receiver structure , 12 2.6 Block diagram of T R - S T B C scheme 13 2.7 Bock diagram for delay diversity 16 3.1 The relation between the various vectors for over-estimation 31 3.2 The interference of data symbols before training symbols 32 3.3 The relation between the various matrices for under-estimation 32 3.4 The relation between the various channel vectors for under-estimation. . . . 33 3.5 B E R performance for BPSK transmission with M L S E for a single transmit antenna and an independent three-tap channel. The M F B , the IMFB, and simulation results with different estimated channel lengths are plotted to-gether, (a) V = 2, (b) V = 3, (c) JJ = 5, and (d) L' = 7 37 3.6 Performance loss vs. estimated channel length for BPSK transmission with M L S E reception over an independent three-path channel at BER=1CT 2 . . . . 38 viii LIST OF FIGURES ix 3.7 performance (IMFB) loss vs. estimated channel length for B P S K transmis-sion with M L S E reception over correlated three-path channels (p = 0,0.5,1) at BER=10~ 2 39 3.8 B E R vs. lOlogio^b/No) for B P S K transmission with M L S E reception for a single transmit and receive antenna system over the independent three-tap channel. Three training sequences with different lengths (Nt = 15, 20, 26) are used for channel estimation 40 4.1 B E R vs. 101ogio(^6/iVo) for M L S E decoding for DD and perfectly known and estimated (1/ = L) three-tap channels. Three different delays D = 1, 2, 3 are considered, (a) D = 1, (b) D = 2, and (c) D = 3 57 4.2 B E R vs. 10\og1Q(Eb/N0) for M L S E decoding for M R C and T R - S T B C , and perfectly known and estimated (V = L) three-tap channels 58 4.3 B E R vs. lOlogir^-Efi/iVo) for M L S E reception for various schemes over per-fectly known (L = 6) and estimated (L' = L) HT channels 60 4.4 B E R vs. l'01ogio(£^b/A^o) for M L S E reception for various schemes over per-fectly known (L = 4) and estimated (1/ = L) T U channels 61 5.1 Block diagram of W-DFSE 65 5.2 Block diagram of W-DFSE decoding for T R - S T B C 69 5.3 Block diagram of U-DFSE decoding for T R - S T B C 70 5.4 Block diagram of W-DFSE decoding for DD 70 5.5 Block diagram of U-DFSE decoding for DD 71 5.6 Comparison of B E R performance for DD with different delays for B P S K transmission over perfectly known independent three-tap channels. D = 1,2,3 are considered. Solid line: B E R for Z = 1 (DFE) and Dashed line: B E R for Z = 8 73 LIST OF FIGURES x 5.7 Comparison of B E R performance for DD with the same complexity for BPSK transmission over estimated T U channels. Four different delay intervals D = 1,2,3,4 are considered. Solid line: B E R for Z = 1 (DFE) and Dashed line: B E R for Z = 4 74 5.8 B E R vs. 101ogio(£ViVo) for GSM and T U profile with L = 4 for different D F S E schemes for T R - S T B C . Solid lines: B E R for NT — 1 transmit antenna and, respectively, W-DFSE with Z = 1 state and M L S E 75 5.9 B E R vs. 101ogio(^6/iVo) for GSM and H T profile with L = 6 for different D F S E schemes for TR-STBC. Solid line: B E R for W - D F S E (Z = 1) with M M S E - D F E F F F as prefilter 76 5.10 B E R vs. lOlogio^t/iVo) for E D G E and T U profile with L = 4 for different D F S E schemes for TR-STBC. Solid line: B E R for W - D F S E (Z = 1) with M M S E - D F E F F F as prefilter 77 5.11 B E R vs. l O l o g i o ^ / A W f o r E D G E and H T profile with L = 6 for different D F S E schemes for T R - S T B C . Solid lines: B E R for NT = 1 transmit antenna and W-DFSE with, respectively, Z = 1 and Z = 8 states 78 Acknowledgments I would like to thank my supervisor, Dr. Robert Schober, for his constant guidance, encouragement, and inspiration. I would also like to thank all the colleagues with whom I have had the pleasure of working, especially those in Communications research group at U B C . I appreciate all the help they have given me; I enjoyed their company and our many interesting discussions on a variety of topics. xi Chapter 1 Introduction Driven by a growing demand for wideband services, wireless communication is now marching towards its third generation (3G) and beyond (>3G). However, high rate transmission over wireless communication links is severely restricted by channel fading. The most effective technique to combat fading relies on the exploitation of diversity by providing the receiver with more than one replica of the signals that have been transmitted. Several diversity schemes, such as time diversity, frequency diversity, and space diversity, can be employed [!]• Space-time coding (STC), is a novel and effective approach to increasing data rate over wireless channels, which employs coding techniques appropriate for multiple transmit anten-nas. Space-time codes introduce temporal and spatial correlation into signals transmitted over different antennas, in order to provide a diversity gain and a coding gain at the receiver without sacrificing precious bandwidth. The best-known STC is the space-time block code (STBC), proposed by Alamouti [2] for transmission with two transmit antennas over frequency non-selective channels. This scheme enables maximum likelihood (ML) detection, based on linear processing at the re-ceiver with complexity linear in the number of transmit antennas, and achieves full diversity 1 1.1 Motivation 2 at full transmission rate for any^signal constellation. The delay diversity (DD) scheme [3], [4] can be viewed as a special case of S T C . It transmits the same information over multiple antennas, thereby creating an artificial mult ipath distortion. 1.1 Motivation A s the transmission rate increases beyond the coherence bandwidth of the channel, frequency-selective fading becomes a major performance-limiting factor. Most S T C schemes init ia l ly proposed for transmission over flat fading channels are not suitable for frequency-selective channels. There are two important S T B C schemes for transmission over frequency-selective channels. The first scheme is D D , which can achieve full diversity order. The second scheme, time-reversal space-time block coding ( T R - S T B C ) , introduced i n [5], is an extension of Alamouti 's S T B C to frequency-selective channels by imposing the A l a m o u t i orthogonal structure at a block level. To mitigate intersymbol interference (ISI) in the received signals, equalization has to be incorporated into the receiver. The optimum equalization algorithm is maximum likelihood sequence estimation ( M L S E ) , the complexity of which increases exponentially the channel memory. For the long channels and/or higher modulation schemes, a very high complexity results for M L S E [6]. Therefore, we resort to suboptimum means, such as decision feedback sequence estimation ( D F S E ) , to achieve a tradeoff between complexity and performance. The goals of this thesis are to analyze the performance of S T C schemes w i t h opt imum detection ( M L S E ) , taking into account channel estimation errors, and to explore reduced complexity decoding schemes for S T C s . 1.2 Literature Review 3 1.2 Literature Review When judging the capabilities of a noisy channel, one of the simplest quantities one can consider is the so-called matched filter bound (MFB). MFB is a bound on the optimal performance over a communication channel that assumes that the transmitted pulses are sufficiently separated so that no ISI occurs. Thus the MFB is a lower bound, which may or may not be achievable for specific implementations. Nevertheless, it has theoretical and practical importance for estimating the limiting performance of receivers under considera-tion. The MFB for a perfectly known Rayleigh fading channel with two independent paths has been addressed in [7]. This bound is a special case of the analysis in [8]. To reduce the complexity of equalization for single antenna transmission, several sub-optimal schemes can be considered, e.g., decision-feedback equalization (DFE) [9], delay decision feedback sequence estimation (DDFSE) [10], and reduced-sate sequence estimation (RRSE) [11]. For any suboptimal trellis-based equalizer, a discrete-time minimum-phase overall impulse response is essential for high performance [10] [11]. A discrete-time prefilter is usually introduced in front of the equalizer. For prefilter calculation, the minimum phase equivalent channel can be first determined corresponding to the given channel, and then the transfer function of the prefilter is obtained [6]. However computational complexity of the available algorithms for calculation the minimum-phase channel is quite high or regularity is not sufficient. Recently, an even more efficient algorithm for prefilter calculation based on linear prediction (LP) has been proposed in [6]. In addition, DFSE can operates directly on the discrete-time unwhitened statistic obtained from conventional matched filtering for ISI channels [12]. 1.3 Contributions 4 1.3 Contributions The main contributions of this thesis are as follow: • The effect of mismatched-memory channel estimation on the performance of M L S E detection for single antenna transmission, is analyzed. Our analysis shows that the performance loss of over-estimation (the estimated channel is longer than the real one) generally increases with the estimated channel length; on the other hand, under-estimation (the estimated channel is shorter than the real one) leads to an error floor. • Three lower bounds, namely, the M F B , the improved M F B (IMFB), and the IMFB H, are derived to analyze the performance of M L S E decoding for DD, maximum ratio combining (MRC), and T R - S T B C . The M F B is the lowest bound which assumes data transmission without ISI in an addictive white Gaussian noise (AWGN) environment. Its improved version, IMFB, considers ISI of neighboring symbols and thus provides a tighter bound. IMFB I is a special lower bound for T R - S T B C and takes into account the decoupling errors from the output of the spatial-temporal matched filter, in addition to the effect of ISI and AWGN. This bound is tighter than the M F B and the IMFB. Numerical and simulation results show that, if the channel is perfectly known, the performance of DD improves with increasing the delay and T R - S T B C has the same diversity order as DD with full delay; if the channel is unknown, the performance of DD and T R - S T B C are degraded by the channel estimation errors. Nevertheless, DD with full delay, and T R - S T B C , both perform 3 dB worse than M R C over known channels and the gap is even larger for the estimated channels. • For reduced complexity decoding of STCs whitened DFSE (W-DFSE) and unwhitened D F S E (U-DFSE) have been adapted to DD and T R - S T B C . In W-DFSE, we employ a linear prediction method to calculate the prefilter for T R - S T B C , without knowl-edge of the equivalent single-input-single-output channel impulse response (CIR). We 1.4 Thesis Organization 5 compare the performance of various schemes for binary phase shift keying (BPSK) and 8-PSK transmission, over global system of mobile communication (GSM) and enhanced data rates for GSM evolution (EDGE) channels, via computer simulations. Our results show that U-DFSE can achieve an acceptable performance only for binary modulation and channels with small delay spreads. However, for binary modulation the multistage version of modified U-DFSE, M M U - D F S E , can approach the perfor-mance of W-DFSE, even for channels with large delay spreads and a small number of trellis states. For E D G E and low trellis complexity only W - D F S E leads to a sat-isfactory performance, since all U-DFSE based schemes suffer from metric bias and unreliable preliminary decisions. 1.4 Thesis Organization The remainder of this thesis is organized as follows: Chapter 2 describes the basic concept of wireless systems with S T C and reviews two STC schemes: DD and T R - S T B C . Chapter 3 discusses channel estimation and its effect on the M L S E detection. Chapter 4 analyzes the performance of M L S E decoding for STCs. Chapter 5 deals with reduced complexity decoding for STCs. Finally, in Chapter 6 the thesis is summarized and some conclusions are drawn. Chapter 2 Background In this chapter, several concepts of wireless communications systems are discussed. A generic communication system model, for wireless systems with space time processing over frequency-selective channels, is first presented. Then some wireless channel models are introduced. The M L S E receiver, which is optimum in the sense of sequence error probability, is also discussed. In the last section, two STC schemes suitable for frequency-selective channels are reviewed. 2.1 Wireless System Transmission Model Figure 2.1 depicts the discrete-time model of a wireless communication system with S T C over frequency-selective channels. All signals are presented by their complex-valued base-band equivalents. At the transmitter, the information sequence b[k] = {0, l},k € Z, is first passed to a mapper, which maps every R (transmission rate) bits onto a phase shift keying (PSK) or quadrature amplitude modulation (QAM) signal. The space-time encoder then groups N mapped signals x[A;],0 < k < N — 1, and performs space-time encoding, resulting in a vector of encoded symbols x[k] = [xi[k],X2[k], • • • ,XNT[k]]T, where NT is the number of 6 2.1 Wireless System Transmission Model 7 ( Message > b[k] Mapper x[k] Space-Time x[k] Source V J £{0,1} Encoder Message Demapper jefjfc] Space-Time y[k] Sink V J Decoder Channel | Estimation Figure 2.1: Discrete-time model of a wireless communication system with S T C over frequency-selective channels. transmit antennas. Each of the transmit antennas transmits one code symbol during the symbol period T. The discrete-time channel is a sampled version of the overall impulse response compriz-ing the cascade of the continuous-time transmit filter, propagation channel, and the receive filter. We only consider slowly time-varying channels which remain constant during one block of data here (block fading model). At the receiver, the received signals yjfc],l < i < NR, on the NR receive antennas are collected as vector y[k] = [yi[k],y2[k], • • • ,yNR[k]]T and are the sum of transmitted signals from the NT antennas, corrupted by AWGN and ISI. The space-time decoder, which 2.2 Wireless Channel Model 8 performs equalization, determines the estimates x[k] of the transmitted signals x[k]. Finally, the demapper performs the inverse operation of the mapper and passes the estimates b[k] of b[k] to the sink. Usually, pilot tones or training sequences are inserted between the data blocks to estimate the channel. 2.2 Wireless Channel Model AL A i M ,, hl2[kr'-"/ fhNT[k)/{\ Figure 2.2: Block diagram of the MIMO system. For the multiple-input multiple-output (MIMO) system with NT transmit antennas and NR receive antennas as shown in Fig. 2.2, the discrete-time channel can be completely characterized by a matrix form as H[k] = hn[k] h12[k] h2i[k] h22[k] hiNT[k] h2NT[k] , 0 < k < L - 1. (2.1) -hNRi[k) hNR2[k] ••• hNRNT[k]j H[k] is the kth. tap of the MIMO channel CIR matrix with its element hij[k], 1 < i < 2.2 Wireless Channel Model 9 NR, 1 < j < NT, denoting the CIR between the iih receive antenna and jth transmit antenna. The baseband input-output relationship can be expressed as y[k] = H[k] * x[k] + n[k], (2.2) where x[k] = [xx[k],x2[k], • • • ,xNT[k]]T and y[k] = [y1[k},y2[k], • • • ,yNR[k]]T. n[k] is the A W G N vector added at the receive antennas. " * " denotes the convolution operation. If the signal bandwidth is sufficiently narrow so that the channel can be treated as ap-proximately constant over frequency, the corresponding input-output relationship simplifies to y[k] = H • x[k] + n[k]. (2.3) 2.2.1 Frequency-Selective Fading Channel A channel is referred to as frequency-selective if the coherence bandwidth ( A / ) c of the chan-nel is less than the signal bandwidth W. Frequency-selective distortion occurs whenever a signal's spectral components are not all affected equally by the channel. S[k-1] S[k-2] d\k-L+l] —Kx) MI]—Kx) h[2]—Kx) h[L-l]—Kx) — " — ' 1 L-\ h[k] = YJh[i]S[k-i] ' i=0 Figure 2.3: Tapped delay line model of the SISO frequency-selective channel. Each element in the M IM O CIR matrix H[k] can be viewed as a single-input single-output (SISO) channel, which can be modelled by a tapped-delay line. The tapped-delay 2.3 Maximum Likelihood Sequence Estimation 10 line is usually truncated to several taps, cf. Fig. 2.3, and expressed in the form L - l h[k] = Y,h[i]5[k-i], (2.4) i=0 where L is the number of taps, 8[k] is the unit impulse and h[i],0 < i < L — 1, is the amplitude of the channel tap. 2.2.2 C O S T 207 G S M Channel Profiles The European COST 207 project has defined several SISO frequency-selective fading chan-nels for evaluation of equalization, adaptive antennas, and other processing algorithms in GSM. Several important channel power-delay profiles in COST 207 include Typical Urban (TU), Hilly Terrain (HT), Rural Area (RA), and Equalizer Test (EQ). Since E D G E uses the same frequency band as GSM, the same channel models can be used for system simulations and numerical analysis [13]. In this thesis, the test channel is the overall CIR of the the transmit filter, the channel specified by COST 207, and the receive filter, sampled with period T. For the transmit filter, the linearized impulse corresponding to GMSK with time-bandwidth product 0.3, is employed. The transmit filter impulse response is given in [13]. For the continuous-time receive filter, it is easier to implement a fixed square-root raised cosine (SRC) filter than the optimal whitened matched filter [6]. SRC, using a roll-off factor of 0.35, causes only a negligible loss in performance. 2.3 Maximum Likelihood Sequence Estimation The channel distortion of frequency-selective channels results in ISI, which causes a high error rate if left uncompensated. The solution to the ISI problem is to use an equalizer to mitigate the effect of ISI in the received signal. The optimum receiver is based on M L S E . 2.3 Maximum Likelihood Sequence Estimation 11 Forney [14] and Ungerboeck [15] have each developed MLSE receivers for SISO channels with ISI. These two receiver structures and corresponding metrics can be extended to MIMO systems. In Forney's receiver, the received signal is first filtered by a square-root Nyquist input filter and then sampled to produce a statistic for optimum detection. Fig. 2.4 illustrates the baseband portion of the receiver, which consists of a receive filter, a sampling operation, and a sequence estimation algorithm, such as Viterbi algorithm (VA). The sequence estimation algorithm employs the standard Euclidean distance metric. The metrics used in the VA may be expressed as A[k + 1] = A[fc] •+ X[k] = A[k] L-l y[k] — Y^ x[k — i]h[i] i=0 (2.5) where /&[&], 0 < k < L — 1, is the overall CIR with L taps. x[k] and y[k] are the trial symbols and the received symbols, respectively. X[k] is the branch metric, and A[k] is the accumulated metric up to time k. Sampling t = kT r(t) Square-Root Nyquist Filter yM Sequence Estimation Figure 2.4: Forney receiver structure. In the Ungerboeck receiver, the received signal is filtered by a square-root Nyquist receive filter, sampled, and filtered again by a matched filter (MF), which is matched to the overall CIR of the transmit filter, the physical channel, and the receive filter, as shown in Fig. 2.5. The baseband portion of the receiver comprises a receive filter, a sampling operation, a MF, and a sequence estimation algorithm, which employs a modified metric. 2.4 STC for Frequency Selective Channels 12 Sampling t = kT Kt) Square-Root Nyquist Filter Matched Filter y[k] Sequence Estimation m > Figure 2.5: Ungerbeock receiver structure. This modified metric can be expressed as A[fc + 1] = A[k] + \[k] = A [A;] + ^ix*[k\{2y\k] - j[0]x[k] - 2 ^ 2~f[k]x[k - i]) L-l i=l (2.6) with 7[fe] = h[k] *h*[-k] (2.7) where (•)* is the conjugate operation and denotes the real part of a complex number. 2.4 STC for Frequency Selective Channels Most STCs are originally designed for fiat fading channels. While there is some effort in code design for frequency-selective channels, most of these codes are restricted to orthogonal frequency division multiplexing (OFDM). T R - S T B C [5] is proposed for combating fading over frequency-selective channels. Gore [16] shows that a generalized DD code achieves full diversity over frequency-selective channels. We concentrate on the two known STCs suitable for frequency-selective channels in this thesis. 2.4.1 T R - S T B C T R - S T B C can be viewed as an extension of Alamouti's scheme from flat fading channels to frequency-selective channels. Fig. 2.6 shows a block diagram of T R - S T B C using two transmit antennas and one receive antenna. 2.4 STC for Frequency Selective Channels 13 A sequence of 2N symbols £[&;], 0 < k < 2N — 1, is divided info two halves, Xi[k] and x2[k) 0 < k < N—l. Furthermore, the transmission frame is divided into two halves. During the first half of the frame Xi[k] is transmitted from the first antenna and x2[k] is transmitted from the second antenna. During the second half of the frame, the sequence x2[k] is time reversed, complex conjugated, and negated to generate the sequence —x^N — 1 — k] before it is transmitted from the first antenna. The sequence xi[k] is time reversed and complex conjugated to generate the sequence x\[N — 1 — k], which is transmitted over the second antenna. TX S pace - Time Encoder x{[k] yjEJ2 x[k]< x2[k] yjEs/2 Time Reversal Complex Conjugate 4 Time Reversal! Complex Conjugate Mk] n[k] fhlk] RX S pace - Time Decoder r.[k] i 1 Time Complex * Reversal Conjugate r2[k] H\-k] y2m MLSE HMLSE -+xi[k] -*-X2 [k] Figure 2.6: Block diagram of T R - S T B C scheme. At the receiver, during the first half of the frame, the received signals are collected to form the sequence r\\k]. During the second half of the frame, the received signals are collected as a sequence and this sequence is time-reversed and complex-conjugated to form the sequence r2[k]. The sequences ri[k] and r2[k] are then fed into a MIMO matched filter 2.4 STC for Frequency Selective Channels 14 HH[—k] to produce the decoupled output y[k] = [yi[k], y^k]]2 The transmitted symbol sequence x[k] and and the received symbol sequence y[k] can be represented by vectors x[k] = [xi[k],X2[k]]T and r[k] = [ri[/c], r2[/V]]T, respectively. The first half of the frame of the received signals is given by n[k] = [hx[k] h*[k]]*x[k] + nx[kl (2.8) where h\[k] and h2[k] are the CIRs of the two channels between the two transmit antennas and the one receive antenna. Both hi[k] and h2[k] have length L. ni[k] is the AWGN. Similarly, the second half of the frame is given by r2[k] = [h*2[-k] - hll-k}} * x[k] + n2[k], [2.9) where hl[—k] and h2[—k] are the matched filters of hi[k] and h2[k], respectively. We stack the received two halves of the frame and obtain r[k] = H[k]*x[k] + n[k], (2-10) with H[k) = (2-11) h*2[-k] -h*x[-k]_ where n[k] = [n-i[k],n2[k]]T is the noise vector, assumed to be spatially and temporally white with spatial-temporal auto-covariance matrix -Rn[A] = E{n[k + \}nH[k]} = cr2I2. Note that the channels h2[—k] and hl[—k] have complex-conjugated coefficients and are time-reversed, and thus are anti-causal. To proceed, we need to exploit the following key property of U[/c]: HH[-k] * H[k] ht[-k) h2[k] h*2[-k] -h±[k] h[k] h2[k] h*2[-k) -h\[-k\ l[k] 0 0 7[fc] (2.12) 2.4 STC for Frequency Selective Channels 15 with 7[A;] = hi[k] * hl[—k] + h2[k] * h2[—k]. Eq.(2.12) can be seen as an extension of the orthogonal property of Alamouti's code. By using Eq.(2.12) and considering that each antenna transmits half of the total energy Es, we can show that y[k] = HH[—k] * r[k] = y/Es/2 HH[-k) * H[k] * x[k] + HH[-k) * n[k] ' = y/Ea/2-'y[k]*x[k] + v[k] • (2.13) where v[k] = [vi[k] v2[k]]T = HH[—k] * n[k]. The noise terms V\[k] and V2[k] are mutually uncorrelated [5]. The output from the matched filter is exactly the signal to be processed by a M L S E utilizing Ungerboeck's metric. The branch metric is given by K[k] = K |x:[fc](2^[fc] - i[0]xv[k] -2Y^1[^xv[k - , v = {1,2}. (2.14) The estimates of X\[k\ and x2[k] can be found using the VA. 2.4.2 Delay Diversity In DD [3], [4], [16], the same information is transmitted from several antennas simultane-ously but with different delays, hereby creating an artificial multipath distortion. DD can be viewed as a special case of STC as shown in Fig. 2.7. The mapped signal x[k] is passed to a filter bank which is composed of NT filters di[k], 1 < i < NT- The filter bank performs delay operations with di[k] = 6[(i - l)D], l < i < N T , (2.15) where D is the delay interval. The total transmit energy Es is equally divided among all antennas, i.e., the transmit power for each antenna is given by Ei = ES/NT, 1 < i < NT-2.4 STC for Frequency Selective Channels 16 x[k] Space-Time Encoder — ( X ) -dm d2[k] Km • • • dNTm KT[k] x[k] Figure 2.7: Bock diagram for delay diversity. hi[k], 1 < i < NT, are the channel CIRs. The received signal can be expressed as y[k] = ^Es/NTx[k](dr[k] * h[k] + d2[k] * h2[k] + • • • + dNr * hNr[k]) + n[k] = yjEs/NTx[k] * h'[k] + n[k] - (2.16) with NT NT hf[k] = di[k] * hi[k] = ^  hi[k - { i - 1)D] (2.17) i=l i=l where h'[k] is the equivalent channel impulse response. The received signals can be viewed as being transmitted over the equivalent channel h'[k] in a single antenna system. For standard DD [4] D = 1 is valid. However, if the channel suffers from frequency-selective fading, a longer delay is required. When D is chosen to the channel length L, consecutive symbols do not interfere with each other. In this case, the full diversity order can be achieved [17]. Note that the computational load increases with increasing delay, if we employ an M L S E receiver. Chapter 3 Channel Estimation The space-time decoder relies on channel estimates to perform equalization and to make a decision. In this chapter, a commonly used least sum of squared error (LSSE) chan-nel estimation approach and sequence construction methods are reviewed. The effect of mismatched channel memory on channel estimation is also investigated. 3.1 LSSE Channel Estimation 3.1.1 SISO Channel Estimation Consider a single antenna transmitting Nt training signals s = [s[0], s[l], • • • , s[Nt — 1]]T, \s[k]\ = 1, where |-| denotes absolute value, over a frequency-selective channel h = [h[0],h[l], • • • , h[L — 1]]T with L taps. We assume that the channel h remains constant over the trans-mission of a block and varies independently from block to block. The received symbols r = [r[L — l],r[L},--- ,r[Nt — L + 1}]T, which does not have interference from data symbols, can be expressed as r = Sh + nc, (3.1). 17 3.1 LSSE Channel Estimation 18 where S is an (Nt — L + 1) x L Toeplitz matrix defined by S = s[L - 1] s[L] ls[Nt - 1] s[l) s[2] s[0] s[l] s[Nt-L + l] s[Nt-L], (3.2) and n c is the AWGN vector of dimension Nt — L + 1 with auto-correlation matrix Rn = E{ncncH} = o2nINt_L+1, (3.3) where (-)H is the complex-conjugate (Hermitian) transpose, E{-} denotes expectation, and lNt-L+i is the (Nt — L + 1) x (Nt — L + 1) identity matrix. The estimated channel can be calculated [18] as h = (SHS)-1SHr, (3.4) assuming that S has full column rank. Here, (-) - 1 denotes the inverse. The channel estimation mean square error (MSE) is given by MSE = E{(h — h)H(h - h)} = ontr((SHS)-'), (3.5) where tr(-) denotes the trace of a matrix. The minimum mean-square error (MMSE) of the channel estimation is equal to [19] M M S E (Nt-L + 1) (3.6) and is achieved if, and only if SHS = (Nt-L + l)IL, (3.7) where II is an L x L identity matrix. A sequence s that satisfies Eq.(3.7) is called an optimal sequence and calculation of h can be simplified, without computing the inverse matrix, cf. Eqs. (3.4), (3.7). 3.1 LSSE Channel Estimation 19 3.1.2 Mult iple-Input Single-Output (MISO) Channel Est imation In a MISO system, training symbols S j = [sJO], sjl], • • • ,Si[NT — 1]]T, 1 < i < NT, are simultaneously, transmitted over NT frequency-selective channels, hi = [hi[0], hi[l], • • •, hi[L — 1]]T, 1 < i < NT. Each channel is modelled as a finite impulse response (FIR) filter with L taps. For NT channels, the receiver uses NTNT known, training symbols to estimate NTL unknown channel coefficients. The received symbols can be expressed as r — Sh + nc — S1S2 • • • SNT] h2 \-h>NT. + nc, (3.8) (3-9) where Si, 1 < i < NT is a (NT — L + 1) x L Toeplitz matrix with Si = Si[L - 1] Si[L - 2] Si[L] Si[L - 1] Si[0] (3.10) _Si[Nt-i\ Si[Nt-2] ••• Si[Nt-L]_ and nc is a A W G N vector with NT — L + 1 elements. The linear least square channel estimates are [19] h = h2 h = {SHS)-lSHr (3-11) if S has full column rank. To achieve the M M S E as in Eq.(3.6), the optimal sequences Si, 1 < i < NT, have to 3.2 Construction of Training Sequences 20 satisfy SHS = is? SNT ^2 &NT (NT — L + 1)1 N t L . (3.12) Eq.(3.12) implies that the optimal sequences have an impulse-like autocorrelation and zero cross-correlation. This requirement makes the channel estimation problem for multiple antenna systems different from single antenna systems. The channels corresponding to different receive antennas is estimated separately if the received signals are considered uncorrelated at different receive antennas. 3.2 Construction of Training Sequences 3.2.1 Single Training Sequence For implementation purpose, it is desirable to use training sequences (TSs) with constant amplitude [19]. The TS with constant amplitude can be classified into two main categories according to the training symbol alphabet size. One category uses perfect root-of-unity sequences (PRUS). Given an arbitrary training length Nt, an optimal training sequence is constructed from an Nf1 root-of-unity alphabet ANt = exp(j^) ,0 < k < Nt - 1 [20]. The TS best suited to a particular application depends on the TS length Nt, the number of channel taps L to estimate, and the signal constellation used. A PRUS of a predetermined length maybe not belong to a standard constellation. In the second category of TSs, training symbols are constrained to a specific constel-3.2 Construction of Training Sequences 21 lation, typically PSK. The optimal TSs are found either through an exhaustive search, or a clever search [21]. For BPSK signaling, however, the optimal TS is constructed from maximal-length sequences [18]. An optimal sequence does not exist for all TS lengths. Instead suboptimal sequences are found by extending optimal sequences [18], [19], or using binary almost perfect sequences (BAPSs) [21]. 3.2.2 Mul t ip le Training Sequences A straightforward method to design multiple optimal sequences Si, 1 < i < NT, of length NT to estimate NT channels each of L taps, is to design a single TS s of length iVt' = NT + Lx NT to estimate a single channel with L ' = L x NT taps [20]. The sequences S j , 1 < i < NT, for multiple transmit antennas are constructed as ai = [s[0],s[l],--- ,s[Nt-l}}, s2 = [s[L],s[L+ !},••• ,s[L + Nt-l]], S N t = [s[L{NT-l)],s[L(NT-l) + 1] • • • , s[L(NT-l) + Nt - 1]]. Since the optimal sequence s has an impulse-like autocorrelation, the subsequences s;, 1 < i < NT, have zero cross-correlation. Another method uses block coding to construct two optimal TSs from one optimal TS [20]. In addition to the described methods, optimal TSs can be constructed by an iterative method [21]. Shifts of BAPSs can also be used to construct suboptimal TSs as well [21]. 3.2.3 G S M Training Sequences In GSM, a normal burst contains 2 packets of 57 data bits plus 1 signaling bit. These packets are separated by a TS.' The TS is placed in the middle to minimize the distance to 3.3 Performance Analysis for MLSE with Channel Estimation Errors 22 Training Sequence No. Training Sequence Bits 1 10111 00001 00010 0 2 10111 01111 00010 1 3 O H I O 11101 00100 0 4 11110 11010 00100 0 5 01011 10010 00001 1 6 11010 11000 00100 1 7 11111 0110 0 01010 0 8 11100 01001 01110 1 Table 3.1: GSM training sequences. the data bits. These training signals are used to estimate the channel. Eight different TSs are defined as shown in Table 3.1. Only the 16 bits in the middle of the sequence are given. The complete TSs is obtained by periodic repetition of the 5 bits at the beginning and the end. These training sequences are all suboptimal and have similar auto-correlation. Never-theless, the third and the fifth sequences have the best (lowest) cross-correlation. 3.3 Performance Analysis for MLSE with Channel Es-timation Errors In this section, we investigate the impact of channel estimation errors on the quality of M L S E over SISO frequency-selective fading channels. For mobile communications, the actual length of the channel is unknown to the receiver. Since the length of the channel estimator in a receiver is usually fixed, channel-memory mismatch inevitably occurs. When the length of the channel estimator is shorter than the actual length, we call this under-estimation. On the other hand, when the length of the channel estimator is longer than 3.3 Performance Analysis for MLSE with Channel Estimation Errors 23 the actual length, we refer to it as over-estimation [22]. For simplicity, BPSK modulation is assumed, even though some results may be extended to MPSK signals. 3.3.1 Performance Analysis for M L S E with Known Channel Length Consider a simple case where the estimated channel has the same length as the real channel. A sequence of BPSK symbols x = [x[0], x[l], • • • ,x[N — 1]]T is transmitted over channel h = [h[0], h[l], • • • , h[L — 1]]T with L taps. The corresponding transmission matrix X is given by X = x[L - 1] x[L] x[l] x[2] x[0] x[l] (3.13) _x[Nt-l] ••• x[Nt-L + l] x[Nt-L}_ The received symbol sequence y = [y[L — 1], y[L], • • • , y[N — 1]]T can be expressed as y = Xh + n, (3-14) where n is an (N — L + 1) x 1 AWGN vector. The estimated channel h = [h[0], h[l], • • •, h[L - 1]]T is h~ (S"S)-]S"r = (SHS)-1SH{Sh + nc) = h + Mnc, (3.15) with M ± (SHS)~1SH. n and nc are independent AWGN vectors whose elements have variance on respectively. The probability density function (pdf) of y conditioned on X is given by 3.3 Performance Analysis for MLSE with Channel Estimation Errors 24 Therefore, the maximum-likelihood detector determines the estimates x, according to the decision rule X = argmin| |?/-Xri | | 2 , (3.17) x where X and X are the matrices corresponding to the estimated symbol vector x = [x[0],x[l], • • •, x[N — 1]]T and the trial symbol vector x = [x[0],x[l], • • • ,x[N — 1]]T, re-spectively. The pairwise error probability (PEP) Pe(a, (3) is the probability that x = x@ is detected if x = xa. Here xa and x@ are two sequences that differ just in one transmitted symbol, xa in xa and x@ in xp, respectively. It is easy to see that PEP is given by Pe(a,0) = Pr {||y - Xah\\2 > \\y - Xpfi\\2} , (3.18) where Xa and Xp are the two transmission matrices corresponding to xa and Xp. Matched Filter Bound Since the MFB does not take into account ISI, only one transmitted symbol is considered. From Eq.(3.18), we get Pe(a,3) = Pr [\\y - xah\\2 > \\y - xph\\2} = Pr[^{yHh(xa-xp)} <0] (3.19) = Pr ^ { ( h x a + n)Hh{xa - xp)} < o} . By introducing the vector I = [(h(xa — Xp))T (hxa + n ) T ] T , Eq. (3.19) can be further simplified to Pe(a,(3) = Pr{A(a,/3)<0}, (3.20) 3.3 Performance Analysis for MLSE with Channel Estimation Errors 25 with A(a, p) = lHFl, \oLxL IL IL 0 L X L (3.21) (3.22) where OLXL is a L X L zero matrix. Since A (a, (3) is a quadratic form of Gaussian random variables, the two-sided Laplace transform of its pdf can be expressed as [23] /CO PA(a,f3)(X)e~SXdx •oo exp ( - s Z ^ F ^ + s*,,)-1!) det(I 2 i + s VuF) where det(-) denotes the determinant of a matrix and l = E{l}, * u ±E{(l-l)(l-l)H}, (3.23) (3.24) (3.25) Considering I = 0 2 L for Rayleigh fading channels, where 0 2 L is a 2L x 1 vector, we obtain 1 A(a,/3) with det(I2L + s*uFy (3.26) (x* ~ xp) + MMHa2n) xa(xa-xp)yhh xa(xa - x0)^hh x2atyhh + o2NIL Vhh±E{(h-h)(h-h)H}. (3.27) (3.28) The P E P can be calculated directly from [24] 7+joo 7 - joo (3.29) for 0 < 7 < 3?{si}, and si is the pole with minimum positive real part. When the poles are simple, a closed-form solution for the integral in Eq.(3.29) may be obtained using the residue 3.3 Performance Analysis for MLSE with Channel Estimation Errors 26 method [24]. However, when $A(a,p)/s contains multiple poles or essential singularities, the actual calculation of Eq.(3.29) may be too difficult. One could use the Chernoff bound or saddle-point integration to get an approximation, but numerical evaluation using Gauss quadrature rules provides a more accurate alternative [24]. Pe(a, P) can be obtained with arbitrarily accuracy from where ^ {•} denotes the imaginary part of a complex number, NQ is even and zn = tan((2n— 1)TV/(2NG)). The error term E^G vanishes as NQ —> oo. In practice, even if high accuracy is desired, it suffices to use NQ < 50. For BPSK modulation over symmetric channels, the bit error rate (BER) Pb is identical to the PEP, i.e., Improved Matched Filter Bound An improved matched bound can be obtained if we take into account the effect of ISI. Assume that a symbol vector xa = [• • • ,x[—2],x[—1],xa,x[l],x[2]]T is transmitted. All symbols x[k],k ^ 0, are assumed to be correctly detected except x[0] = xp, xa ^ xp. The detected symbol vector is denoted as xp. We expand the received symbols y[k],k = Pb = Pe(a,p). (3.31) 2 , - l , 0 , l , 2 - - - as y[0] = h[0]xa + h[l]x[-l] H h h[L - l]x[—(L - 1)] + n[0] y[l] = h[0]x[l] + h[l]xa + ... + h[L- l]x[-(L - 2)] + n[l] y[L - 2] V[L -1] = h[0]x[L - 2] + • • • + h[L - 2]xa + h[L - l]x[-l] + n[L - 2], = h[0]x[L -l] + -- - + h[L- 2]x[l] + h[L - l]xa + n[L - 1]. 3.3 Performance Analysis for MLSE with Channel Estimation Errors 27 Note that only L received symbols are affected by xa. We do not need to consider other received symbols since they do not affect the detection of x[0]. The received L-symbol vector y — [y[0], y[l], • • • , y[L — 1]]T, can be expressed as y = Xah + n, where n is the A W G N vector and the transmission matrix Xa is given by (3.32) Xr xa x[-l] x[l] xa x[-(L-l)] x[-(L-2)] Xr (3.33) \x[L - 1] x[L-2] ••• We can decompose Xa into Xa = X + xaIL, (3.34) where X depends only on a;[A;], k ^ 0. Similarly, the transmission matrix corresponding to the sequence x@ can be expressed as X/3 = X + XfjIL = Xa + (xp - xa)IL. (3.35) Xa spans 2L - 1 symbols including xa, x[±l], x[±2], • • • , x[±(L - 1)]. Note that for the M F B we only considered the terms including xa. Therefore, the P E P using I M F B is given by Pe(a, /3, X) = Pr {\\y - Xah\\2 > \\y - Xph\\} = Pr J25R{ (y - Xah)Hh(xa - xp)} + \\(xa - xp)h\\2 < o} = Pr ^2M{(Xa(h -h) + ^(xa- xp)h + n)H (xa - x0)h} < o | . (3.36) Define I = [((xa - xp)h)T (Xa(h - h) + \(xa - Xp)h + n)T]T', and simplify the above equation to Pe(a, 0, X) = Pr {A(a, (3, X) < 0} , (3.37) 3.3 Performance Analysis for MLSE with Channel Estimation Errors 28 with A{a,B,X) = lHFl, \oLxL IL IL 0 L X L (3.38) (3.39) By using the fact that I = 0 2 L , the Laplace transform of the pdf of A(cv, B, X) becomes 1 $ A W ) - d e t ( / 2 i + S ^ F ) ' and the covariance matrix of the random variable I is given by [*21 *22 * n = (xa - Xp)2(Vhh + MMHa2n), * i 2 = \(xa - x p ) 2 * h h + MMHVHa2n(xa - xp), * 2 i = \{xa - x p ) 2 ^ h h + VMMHal(xa - xp), * 2 2 = \{xa - x p ) 2 * h h + (VMMHVH + IL)a2, v £\ -Xa~^XpiL - X. Again we use Eq.(3.30) to calculate the PEP. (3.40) (3.41) (3.42) (3.43) (3.44) (3.45) (3.46) Xa spans 2L—1 symbols including xa, and thus has 22^L ^ different combinations. To calculate B E R , we need to average over all the possible PEPs. Ph = X 2 2 ( L - 1 ) (3.47) 3.3.2 Performance Analysis for M L S E with Over-Estimation Over-estimation occurs when the estimated channel length L' > L. In this case, the channel estimation matrix SQ is different from the optimum matrix S with 3.3 Performance Analysis for MLSE with Channel Estimation Errors 29 S0 = •s[L'-l] s[L'] s[Nt-l] s[L-l] s[L] 8[l] s[2] s[0] s[l] s[Nt-L' + 2] s[Nt-L' + l}, (3.48) The received signals r = [r[L' — 1], r[L'], • • • , r[Nt — l]]T ave used to estimate the channel. The LSSE over-estimated channel h0 = [h[0], h[l], • • • , h[L' — 1]}T can be expressed as h0 = (S?S0)-'S»r = (SH:S0)-1S0i(SHh + nc) = (S?S0)-1S?(S?h0 + nc) = h0 + {S?S0)-1S?nc = h0 + M0nc, with hn = h M0 4 (5 f S 0 ) - X 5 f . (3.49) (3.50) (3.51) If we assume the original channel is hD of length L' instead of h, then the channel over-estimation can be viewed as channel estimation with known channel length. Calculation of the MFB and the IMFB in Section 3.3.1 still holds here, except that h and are replaced by h0 and ^ ° h h , which are given in following subsections. Matched Filter Bound PEP can be simply expressed as, cf. Eq.(3.19), Pe(a,/3) = Pr{A(a,/?)<0}, (3.52) 3.3 Performance Analysis for MLSE with Channel Estimation Errors 30 with A(a,P) = l"Fl, h0(xa - xp) OL'XL' IU Iv OL'XU The Laplace transform of the pdf of A(a, P) becomes 1 A(a,/3) det(I2L> + s^uF) with (xa - xpf (V°hh + M 0 M f 0-2) x a(xQ - z^tf £ where Sl/^ is given by ^hh Oix(L'-L) 0(Li-L)x(L'-L) It is then trivial to calculate the PEP and B E R to obtain the M F B . 0(L'-L)xl (3.53) (3.54) (3.55) (3.56) (3.57) (3.58) Improved Matched Filter Bound Similar to the IMFB for the case of known channel length, the P E P for over-estimation is given by Pe(a, P, X) = Pr {A(a, p, X) < 0} , (3.59) with A(a,P,X) = lHFl h0(xa - x^ Xa(h0 - ha) + \(xa - xp)h0 + n OL'XL' II1 Iv OyxV Now, the Laplace transform of the pdf of A (a, /?, X ) becomes 1 A(a,/3,X) det(I 2 L, + s * H F ) (3.60) (3.61) (3.62) (3.63) 3.3 Performance Analysis for MLSE with Channel Estimation Errors 31 with *11 *12 *21 *22 H „2\ nil * i 2 = g(xa - xpf^°hh + M0M^VHa2(xa - x0), * 2 i = \{xa - xp)2V°hh + VM0M^a2(xa - xp), * 2 2 = -^xa - xpY*°hh + (VM0M»VH + IL,) a2n, = ~ 7 / / - A > (3.64) (3.65) (3.66) (3.67) (3.68) (3.69) where ^ ° h h is given by Eq.(3.58). V h0 ^ Random K D e t e r m i n i s t i c e [ |Zero Figure 3.1: The relation between the various vectors for over-estimation. By using Eq.(3.30), it is easy to calculate the PEP. To get BER, we again use Eq.(3.47) to average over all possibilities of X. Define the error vector e 4 E[h0 - hD], (3.70) which is depicted in Fig. 3.1. One can see that both h and e are random for length L'. h0 tends to ha when the noise vector n —> Ojr/. Therefore, over-estimation does not cause an error floor. 3.3 Performance Analysis for MLSE with Channel Estimation Errors 32 3.3.3 Performance Analysis for M L S E with Under-Est imat ion In case of under-estimation, the estimated channel length L' is shorter than the real channel length L. In Section 3.3.1, we estimate the channel using the received symbols r[k], L' — 1 < k < Nt — 1, L' = L. For under-estimation, if we use the received symbols r[k], L' — 1 < k < Nt — 1, L' < L, we need to consider the effect of the data symbols prior to the training sequence, namely, x[— (L — L')], • • • , x[—2], x[— 1], on the symbols from r[L' — 1] to r[L — 1] as shown in Fig. 3.2. ••• L Y | - 2 | 4-1] *[0] ^ ^ ^ r[L'-l] r[L-l] r[tff-l] Figure 3.2: The interference of data symbols before training symbols. Fig. 3.3 illustrates the relationship between various matrices assuming L — L' = 2. The transmission matrix S, which includes the data symbols before training symbols, can be decomposed into the under-estimation matrix Su and an auxiliary matrix s[L'-l] s[L'- 2] •• v[l] s[0] 4-1] 4 - 2 ] s[L'] s[L{-•1] •• s[2] 5[1] s[0] 4-1 ] tfL'+l] s[L'1 ... s[3] s[2] s[l] .5[0] s[N,-2]s[Nt- -3] -s[N,- -L'+l] s[N, -L'] s[N,- L + l] s[N, -L] s[N,-iys[N,- -2] -s[Nt- L'+2] 5fAf--L'+l] s[N,- L + 2] s[Nr -L+l] Figure 3.3: The relation between the various matrices for under-estimation. 3.3 Performance Analysis for MLSE with Channel Estimation Errors 33 Now, the estimated channel hu = [h[0], h[l], • • • , h[L' — 1]]T becomes hu — (Su Su) Sur = Mu(Sh + nc) = MU([SU SA hu hA + nc) = hu + MuSAhA + Munc = h - h A + MuSAhA + Munc = h + he + Munc, (3.71) with ~h[L' - 1] hA = hp = Vh[L-l] MASU -IL-L> hA. (3.72) (3.73) (3.74) The relationship between the estimated channel hu and the actual channel response h are shown in Fig. 3.4. The channel estimation error can be expressed as hu — h = he + e. It is easy to see that hu^h even without any noise, therefore, an error floor inevitably occurs for under-estimation. 1 I h.. + L' 1 h % Random Deterministic Q Zero Figure 3.4: The relation between the various channel vectors for under-estimation. 3.3 Performance Analysis for MLSE with Channel Estimation Errors 34 Matched Filter Bound The P E P for under-estimation can be expressed as with Pe(a,p,S) = Pr{A(a,p,S)<0} A(a,p, S) = lMFl, hu(xa - x0) hxa + n oLxL IL IL 0 L X L (3.75) (3.76) (3.77) (3.78) The Laplace transform of the pdf of A(a, P, S) is calculated by exp (-s^iF-' + sVn)-1!)^ A(a,/3 ,S) = det(I2L + sVuF) with (xa ~ xp)2 {Vhh + heh? + MMHa2n) xa(xa - xp)V hh Xa. {Xa. - X p ) ^ h h x2Vhh + alii M M , , Or, L'x(L-L') (3.79) (3.80) (3.81) 0(L-L')xL' O(L-L') x {L-L1) Using the numerical method, we can easily calculate PEP. There are 2L~L' different combi-nations for S. We average over the PEPs for different S. Therefore, the B E R for the M F B is expressed as ZPe(a,P,S) Ph = 2L-U (3.82) Improved Matched Filter Bound To compute the I M F B in case of under-estimation, we take into account the combined effect of under-estimation matrix S and data transmission matrix X , cf. Eq.(3.35). The P E P can be expressed as Pe(a, p, S, X) = Pr {A(a , B, S, X) < 0} (3.83) 3.4 Comparison with Simulations 35 with A(a,p, S,X)^lnFl hu(xa - xp) Xa(h - h) + \{xa - Xp)hu + n OLXL II IL OLXL The Laplace transform of the pdf of A (a, /?, S, X) is with A(a,(3,S,X) expl-s^iF-' + sVu)-1!) det(I2L + s * „ F ) [*21 * 2 2 j * n = (xa - x p f { ^ h h + heh?MMHo2), * i 2 = \ { x a - x p ) 2 * h h + (heh? + MMHo2n)VH{xa - xp), * 2 i - \ ( x « - xpf^hH + V(heh? + MMHo2n)(xa - xp), * 2 2 = \{x«~ xp?*hh + V(heh? + MMHo2)VH + o2JL, V - lL - A . (3.84) (3.85) (3.86) (3.87) (3.88) (3.89) (3.90) (3.91) (3.92) (3.93) Again we calculate the PEP with the numerical method. After averaging over all possible PEPs, the B E R for the IMFB is given by ZPe(a,P,S,X) J2Pe(a,0,S,X) s,x s,x 2L-L'22(L-1) 2 3 £ - £ / - 2 (3.94) 3.4 Comparison with Simulations In this section, we study the effect of channel estimation on the performance of M L S E via simulations, and compare the M F B and the IMFB with the simulation results. BPSK 3.4 Comparison with Simulations 36 Nt Training Sequence Bits 15 11101 00011 10111 11110 11010 00111 20 00000 10100 11011 10000 10000 01001 11001 01000 * 10110 00101 00000 11011 * 25 00010 11110 01101 00001 00010 * 11110 01101 00001 00010 00010 * Table 3.2: Some good training sequences. Sequences with * also have low cross-correlation. modulation is adopted over three-tap channels. The correlation matrix of the channel taps is defined as = E{hhH} = I 1 p p2 p 1 p P2 P 1 0,0.5,1. (3.95) In most cases, we assume training sequences with 26 symbols (Nt = 26), and GSM TSs can be directly applied. The first GSM TS is used to estimate the channel. Some short TSs with good autocorrelation and low cross-correlation, constructed by extending the optimum sequences in [21], are tabulated in Table 3.2. Fig. 3.5 shows BER vs. 10logw(Eb/N0) for BPSK transmission with MLSE reception for a system with a single transmit antenna, over an independent three-tap channel (p = 0). We observe that the performance degrades with increasing estimated channel length L'. At a BER of 10~3, the performance loss caused by the channel estimation is only about 0.5 dB for V = L = 3, and it increases to over 2 dB when L' = 7. On the other hand, the performance becomes unacceptable with an error floor of BER = 8 x 10 - 2 in case of under-estimation (1/ = 2). Compared to numerical results, the IMFB is tighter than the MFB in all situations, especially for over-estimation. The gap between the MFB and the simulation 3.4 Comparison with Simulations 37 i r Figure 3.5: B E R performance for BPSK transmission with M L S E for a single transmit antenna and an independent three-tap channel. The M F B , the IMFB, and simulation results with different estimated channel lengths are plotted together, (a) L' = 2, (b) L' = 3, (c) 11 = 5, and (d) L' = 7. 3.4 Comparison with Simulations 38 result is 2.3 dB for L'=7. The IMFB significantly narrows the gap to 0.5 dB in that case. Both bounds are not tight in case of under-estimation (L'=2). In Fig. 3.6, the performance loss of the M F B , the IMFB, and simulation results, with respect to the estimated channel length, is plotted at BER=10~ 2. The performance loss of the IMFB matches much better with simulation results than that for the M F B . 3 2 0 1 -e- Sim - 1 - MFB - * - IMFB 1 3 4 5 6 7 Estimated Channel length Figure 3.6: Performance loss vs. estimated channel length for BPSK transmission with M L S E reception over an independent three-path channel at B E R = 1 0 - 2 . To investigate the effect of correlation between the channel taps on the M L S E perfor-mance for a single transmit antenna system, we calculate the IMFB for correlated three-tap channels with p = 0,0.5,1, respectively. Fig. 3.7 shows the performance loss with respect to the estimated channel length. We see that, with a moderate correlation of p = 0.5, the loss compared to the uncorrelated case (p = 0) is relatively small, about 0.6 dB for the specific channel. On the other hand, for full correlation (p = 1) between the channel taps, 3.4 Comparison with Simulations 39 8 -e- p=o p=0.5 " f - * - p=i ,l I I I I 3 4 5 6 7 Estimated Channel length Figure 3.7: performance (IMFB) loss vs. estimated channel length for BPSK transmission with M L S E reception over correlated three-path channels (p = 0,0.5,1) at B E R = 1 0 - 2 . the performance suffers a lot, and is almost 7 dB worse than the uncorrelated channel. In Fig. 3.8, simulation results for BPSK transmission with M L S E reception and different TSs over independent three-tap channels are shown. Three TSs of different lengths (Nt = 15,20,26) are considered. We observe that, as expected, the longer the TS the better the performance in general. If the over-estimated channel length is long, the performance degradation for the short TS is larger than for the longer TSs. However, the error floor for under-estimation (1/ = 2) is similar for all sequence lengths. For comparison, the performance for a perfectly known channel is also shown. 3.5 Summary 40 10" l o -ir LU CQ 10" 10" L'=3 -a- Perfect • - A - Nt=26 - e - Nt=20 - v - Nt=15 i N 10 12 10log 1 0(Eb/No) 14 16 Figure 3.8: B E R vs. 10log10(Eb/N0) for BPSK transmission with M L S E reception for a single transmit and receive antenna system over the independent three-tap channel. Three training sequences with different lengths (Nt = 15, 20, 26) are used for channel estimation. i 3.5 Summary We have discussed LSSE channel estimation and different TS construction methods. To analyze the effect of mismatched-memory channel estimation on the M L S E receiver, two an-alytical bounds are derived. For the single transmit antenna case, TSs have an impulse-like autocorrelation while for the multiple transmit antennas, the optimal sequences, in addi-tion, have zero cross-correlation property. Nevertheless, the optimal sequences do not exist for all channel lengths, given a fixed TS length. Several methods (alternative to exhaus-3.5 Summary 41 tive search) can be used to identify good suboptimal sequences. The mismatched-memory channel estimation has significant impact on the M L S E receiver. In case of over-estimation, receiver performance degrades as the over-estimated channel length increases. On the other hand, there is a high error floor for under-estimation. In addition, the correlation between the channel taps has a negative effect on the performance of reception. Chapter 4 Performance Analysis of MLSE Decoding for STCs In this chapter, we analyze the performance of M L S E decoding for two S T C schemes, namely DD and T R - S T B C . For comparison we also derive performance bounds for M R C with two receive antennas. 4.1 Performance Analysis of Maximum Ratio Com-bining The classical maximum ratio combining (MRC) technique has been shown to be optimum if the channel state is know perfectly at the receiver [25]. This technique gives the best statistical reduction of fading among all known linear diversity combiners. Assume that an iV-symbol sequence x = [x[0], x[l],..., x[N — 1]]T, with associated transmission matrix X, cf. Eq.('3.13), is sent from the transmitter. The uncorrelated channels between the transmit antenna and the two receive antennas are denoted by hu = [/ii/[0], hu[l],..., hv[L — 1]]T, v = {1,2}. The received signals at two receive antennas 42 4.1 Performance Analysis of Maximum Ratio Combining 43 are yv = [yv[L - l],.yv[L}} ••• ,y„[N - 1]]T given by yv = Xhv + n„ , ^ = {1,2}, (4.1) where nv,is = {1,2}, is the AWGN vectors. The maximum-likelihood detector determines the estimates x according to the decision rule. X = argmin j l ^ - Xh^2 + \\y2 - Xh2\\2^, (4.2) where X and X are the matrices associated with the estimated sequence x = [x[0], x[l],... , x[N — 1]]T and the trial symbol sequence x = [x[0],x[l],... ,x[N — 1]]T, respectively. The channel estimates hv = [^[0], hu[l], • • •, hv[L — 1]]T, v = {1, 2}, can be obtained by trans-mitting one TS and calculating the estimates from the received symbols at two receive antennas using the LSSE method separately. 4.1.1 M F B for M R C A single symbol xa is transmitted and detected as xp. The P E P can be expressed as f I I ~ n2 I I ~ n2 I I A I I 2 I I A 112*1 Pe(a,P) = Pr || |3/ x - xahi\\ + \\y2-xah2\\ >\\y1-xph1\\ + \\y2- xph2\\ j (4.3) and may be simplified to Pe(a,B) = Pr^{y?h1(xa-xp) + y%h2(xa-xp)} < oJ. (4.4) We can rewrite the above expression as quadratic form Pe(a,/3) = Pr{A(a , /3 )<0} , (4.5) 4.1 Performance Analysis of Maximum Ratio Combining 44 with A(a,0)£\l»Fl, hi{xa - xp) hixa + n x h2(xa - xp) h2xa + n2 oLxL IL IL 0 L X L (4.6) (4.7) (4.8) where (g> denotes Kronecker product. Since A(a, 0) is a quadratic form of Gaussian random variables, the Laplace transform of its pdf can be calculated by A(a,/3) exp (-slH{F~l -rs^f)-H det(J 4 L + s * f F ) (4.9) where Z and \1/^ 2 are the mean and covariance matrix of vector Z, respectively, with (4.10) where SI//z is given by Eq.(3.27). The PEP is calculated by using Gauss quadrature rules as given in Eq.(3.30). In the above derivation, a known channel length is assumed. In case of over-estimation or under-estimation, we substitute, respectively, in Eq.(4.10) with Eq.(3.57) and Eq.(3.80), re-spectively. For the known channel case and over-estimation, the BER is equal to the PEP. For under-estimation, Eq.(3.82) is used to calculate the BER for the MFB. 4.1.2 I M F B for M R C Taking into account ISI of neighboring symbols, the PEP can be expressed as Pe(a,0,X) = Pr{||y 1 - XM* + \\y2 - Xah2\\2 > y1-Xph1 2+ y2 - Xph2\\2}. (4.11) 4.2 Performance Analysis of DD 45 By using Eq.(3.35), we can simplify the above expression to Pe(a,P,X) = Pr{23?{(y 1 - Xahx)Hh^xa - xp) + (y2 - Xah2)Hfi2(xa - xp)} + \\(xa - ^ ) / i i | | 2 + ||(a;a - xp)h2\\2 < o | , which can be rewritten as (4.12) Pe(a,P,X) = Pv{A{a,P,X)<0} (4.13) with the quadratic form A{a,P,X) = lHFl, ki(xa - xp) Xa(hi - hi) + \{xa - xp)hx + n i h2(xa - xp) Xa(h2 - h2) + \{xa- xp)h2 + n2 OLXL II I L OLXL F ^ I (4.14) (4.15) (4.16) The Laplace transform of the pdf of A(a, P, X) still has the same form as Eq.(4.9) with (4.17) where is given by Eq.(3.27). The remaining procedure for calculation of the P E P and B E R for over-estimation and under-estimation is the same as that in the single antenna case. 4.2 Performance Analysis of DD For DD with two transmit antennas an iV-symbol stream x = [x[0],a;[l],... ,x[N — 1]]T, is transmitted over the first antenna, and the same stream delayed by D symbol intervals is transmitted via the second antenna. Both channels have length L. The received signal 4.2 Performance Analysis of DD 46 vector y = [y[L + D — l],y[L + £>],••• , y[N — 1]]T, is the superposition of the transmitted signals from one transmit antenna and the delayed version from the other transmit antennas, corrupted by ISI and AWGN. The two channels are assumed to be mutually uncorrelated and can be denoted as hv — [hu\0], hv\\],..., hv[L — 1]]T, v = {1, 2}. From Eq.(2.16), the elements of the equivalent CIR h' = h'[k], 0 < k < L + D - 1, are defined by where D < L is implicitly assumed. The equivalent channel is estimated by transmitting the same training sequence from one antenna and the delayed version from the other antenna. The receiver estimates the equivalent channel with the LSSE method. The estimate is the transmitted signals are scaled by the factor y/l/2. For simplicity, we do not include the factor in the following expressions, but deduct 3 dB from the calculated performance without scaling the transmitted signals. The received signal can be expressed as (4.18) denoted by h = [h'[0},h'[l] • • • , h'[L + D — 1]]T. Since each antenna emits half the energy, y = Xti + n, (4.19) where X is the transmission matrix of size (Nt — (L + D) + 1) x (L + D) with X = x[L+D-l] x[L+D] x[l] x[2] x[0] x[l] (4.20) L x[Nt-l] x[Nt-(L+D)+2] x[Nt-(L+D) + l\ The receiver detects the received signal just as for the single antenna case using the equivalent channel estimate h!. Therefore, we can use the analysis results of Section 3.3.1 4.3 Performance Analysis of TR-STBC 47 and substitute h and by hi and ^ww,respectively, ^h'h1 is given by ^hh oLxD OLXD ODXD ODXD OLXD OLXD &hh (4.21) Note that the equivalent channel response h' (L + D taps) is longer than hi and h2 (L taps). As a result, the receiver complexity which is directly related to the equivalent channel memory is much higher for DD than for a single antenna system if M L S E reception is adopted. 4.3 Performance Analysis of TR-STBC In T R - S T B C , the transmitted symbol stream is divided into two sequences X\[k] and x2[k], 0 < k < N — 1. In general, the transmitted symbols Xi[k] and x2[k] belong to a PSK or Q A M signal alphabet A of size M. x\[k] and x2[k] are space-time encoded in such a way that after spatial-temporal matched filtering the two received signal sequences are given by Eq.(2.13) if the channels are known. Similar to DD, each antenna transmits half the energy, however, we do not scale the signals by y/l/2 in the expressions, instead, 3 dB is deducted from the final calculated performance. To estimate the channels hv = [hu[0], hu[l], • • • ,hu[L — 1]]T, v = {1,2}, two train-ing sequences, S i and s2, with low cross-correlation, are transmitted from two anten-nas simultaneously. Eq.(3.11) can be used to calculate two estimated channels hu = [hu[0], fo„[l], • • • , hv\L — l]] r, v = {1, 2}. In this section, we only consider the case L' = L for simplicity of presentation, and the results can easily be extended to under-estimation and over-estimation. The estimated channels can be expressed as h„ = hu + pu, u£{l,2}, (4.22) 4.3 Performance Analysis of TR-STBC 48 with the channel estimation error term pv defined as P^iSfS^Sfa, «/ = {!, 2}, (4.23) where Su,v — {1,2}, is the transmission matrix of the training sequence su,v = {1,2}, and nc is the AWGN at the receive antenna during the channel estimation period. The transmitted sequence xv[k],v € {1,2}, and the received signals yv[k],u G {1,2}, after the matched filter H [—A;], are stacked as x[k] = xi[k] x2[k] y[k) = yx[k] V2[k] From Eq.(2.13), we obtain y[k] = HH[-k) * H[k] * x[k] + HH[-k] * n[k] h\[-k] h2[k] h*[-k] -h^k] h[k] h2[k] h*[-k] -K[-k} x[k] + n[k] hi[-k] h2[k] h*[-k] = (A + B) * x[k] + (C + D)* n[k] (4.24) (4.25) with */[k] 0 0 B Pll -k] * hi[k]+P2[k] *h*[ -k] Pl[~ -k] * h2[k] -p2[k] * h\\ -k] -k] * h[k] - Pi[k] *h*2[ -k] Pl\- -k] *h2[k] +pi[k] -k] -k] h2[k] ' -k] -h![k]_ J P i t -k] P2[k] Pll -k] -Pi[k] ) 7 [ fc ] ^ h{[-k] * hx[k] + h*2[-k] * h2[k]. (4.26) (4.27) (4.28) (4.29) (4.30) 4.3 Performance Analysis of TR-STBC 49 Since the error rate for Xi[k] and X2[k] are identical for symmetry reasons, we only need to consider yi[k] and X\[k] for performance analysis. Therefore, we have yi[k] = (7[fc] + £[k]) * xdk] + u[k] * x2[k] + vx[k] * ni[k] + u2[k] * n2[k], (4.31) where £[k] = h^k] *pl[-k] +p2[k] * h*2[-k], (4.32) ^ ^ ^ [ - f c ] * ^ ^ ] - ^ ] * ^ ^ ] , • (4.33) v^^hH-tf+pli-k], (4.34) i/ 2[fe]=Mfc]+p 2[fc]. (4.35) The M L S E decides for that trial symbol X\[k] which yields the minimal accumulated Ungerboeck metric. The accumulated metric is given by A[k + 1] = A[k] + X[k] = A[k] + m(xl[k](2yi[k] - 7[0]xx[fc] - 2^2mxi[k (4.36) ^ i=i ^ where 7[fc] = ^[fc] * &[[-fc] + h2[k] * ^ [-fc] - (4.37) = (hl[k] + Pl[k}) * (hl[-k] + pl[-k]) + (h2[k]+p2[k})*(h*2[-k]+p*2[-k}) = 7^]+ £[*]+#4+ (4-38) with i;[k}APi{k}*hl[-k} + h2[k}*p*2[-k}, (4.39) e[k]±Pl[k] *pl[-k]+p2[k]*p*2[-k]. (4.40) 4.3 Performance Analysis of TR-STBC 50 4.3.1 M F B for T R - S T B C For the M F B , we assume that only one transmitted symbol xa in the sequence x at time k = 0, i.e., xi[0] = xa, X\[k] = 0, k ^ 0, and x2[k] = 0 for all k. The accumulated metric at decision time k = L — 1 (cf. Eq.(4.36)) is simplified to A[L- 1] = »J25;[0]y[0] -7[0]|5i[0]|2}. (4.41) For PSK signals, | £ i [ 0 ] | 2 is identical for all trial symbols, and can be deleted from the metric. Hence we have A [ L - l ] = »|£;[0]y[0]}. (4.42) In this case, y[0], cf. Eq.(4.31), is simplified to y[0] = ((7[fc] + £[fc]) * Xl[k] + ux[k] * ni[k] + u2[k] * n2[k}) . (4.43) \ / k=0 For BPSK modulation, assuming that xa = 1 is transmitted but detected as xp = — 1, the P E P is given by Pe(a,/?) = P r | » { < y [ 0 ] j < fc{a>[0]} = P r | » { ( ^ - ^ ) y [ 0 ] } < 0 = Prjsfc{y[0]} < o| . (4.44) Now we substitute y[0] with Eq.(4.43) where xa — 1 and obtain Pe(a,8) = Pr |K{ 7[0] + £[0] + h f m + p f m + n f / i 2 + n?p 2 } < o | . (4.45) By defining I = [ri^ r i 2 nj n2 p{* p2 ]T, Eq. (4.45) is simplified to Pe(a,P) = Pr{A(a,P)<o), (4.46) 4.3 Performance Analysis of TR-STBC 51 with A(a,p)^lHFl, I2L \^2L \J-2L \^2L 02Lx2L 2^2L \l2L \^2L 02Lx2L (4.47) (4.48) By using I = OQL for Rayleigh fading channels, the Laplace transform of the pdf of A(a , /?) can be calculated to with * A ( a , / 3 ) = £ { e SX}= / PA(aMX)e ^ d x J—00 exp (-slH{F-1 + s*ll)-1l)') . dot(/,;,, + . s * „ F ) • 1 det(J 6 L + 5 * H F ) ' hh hh o P P 2 * h h = £ ; { ( / i -h ) ( / i -^ ) H } , * p p v = E{(pv - pu){pu - p„)H}, = (SuHS„)-1o2n, !/ = {!, 2}. (4.49) (4.50) (4.51) (4.52) (4.53) where h and p are the mean values of h and p, respectively. Eq.(3.30) is used to calculate the PEP and thus obtain B E R for the M F B . 4.3 Performance Analysis of TR-STBC 52 4.3.2 I M F B for T R - S T B C The IMFB takes into account ISI from neighboring symbols Xi[k], k ^ 0, without consider-ing the influence of decoupling errors from x2[k}. Now, y\[k] becomes yi{k] = (-y[k] + £[k]) * xx[k] + ut[k] * nx[k] + u2[k] * n2[k] L-l i=-L+l and Vl[0] = 7[0] + £[0] + / i f rn + p f m + 7$ h2 + n^p2 (4.55) L-l + + 7 [ ^ i H + £ H ] * i H + # i H ] } . (4-56) i=i Since we assume that the symbol Xi[0] = x a is transmitted and detected as xp. and the other symbols are detected correctly, the accumulated metric prior to k = 0 does not affect the detection of xi[0]. Considering that the channel has L taps, we obtain the following accumulated metric at time k = L — 1, cf. Eq.(4.36), A[L - 1} = X[L - 1] + X[L - 2] + • • • + A[0] L-l , L-l N = J > *l[k]{2Vi[k] - 7[0]xi[/c] -2^\%]Xl[k - i]) [. (4.57) k=0 ^ i=l ' Some terms in Eq.(4.57) are identical for all trial symbols and can be deleted, resulting in /• L-l L-l N A[L - 1] = 3?| 2xt[0]z/[0] - 2 ^ # 1 [ - i ] i ! [ 0 ] - 2^ 7 [^I [ i ]5 i [0] [. (4.58) i=i i=i ^ Consequently, the PEP for the IMFB is given by L-l L-l 1=1 1=1 L-l L-l PeCa.^Xx) = Prj K{2<y[0] - 2 ^ 7 [ $ i [ - » K - 2 ^ f l a f t z K ^ i=i i=i1,-1 L - l< ^{2x;y[0} -2^[i\x1[-i]x} -2j2l[i\xl\i]xp} I. (4.59) i=l i=l ^ 4.3 Performance Analysis of TR-STBC 53 Without loss of generality for BPSK, let xa = +1, xp — — 1, and Eq.(4.59) can be written as Pe(a,B,X1) = Prl $t{y[0} - ^ 7[i]( X l[-i] + X l } < 0 \ = Pr jsft^O] + £[0] + / i f n x + p f m + n f / i 2 + n f p 2 z,-i + S (7H£i [ i ] + 7[^iH] + + £ [ ^ i H ] i=i • - m++#]+^w)(*i[-^]+5i w))} < o|. (4.60) A comparison with Eq.(4.45) shows that the last sum term in Eq.(4.60) reflects the ISI from neighboring symbols. Note that 7*[-fc] = i[k], C[-k] = ip[k] and £*[£:] = ib[-k\. Therefore, Eq.(4.60) can be simplified to Pe(a, (3, Xx) = Pr|sR{7[0] + f [0] + hf m + p f m + n f / i 2 + n f p 2 } -^E{(^] + ^ H + <?W + <?H])5i[«] i=l + ^[-i}+ip[i] + 6[i\+6[-i})x1[-i}^ <.o|. (4.61) By introducing Z = [Zif / i f n f n f pf pf ] T , the PEP can be written as Pe(a, p, X1) = Pr {A(a, 8, X x ) < 0} , with (4.62) A(a,P,X1)^lHFl, I2L \^2L F 13 7 j J 2 L 0 2 L x 2 L 2 - t 2 L 31 ^33 (4.63) (4.64) 4.3 Performance Analysis of TR-STBC 54 13 31 33 w oLxL pLXL WT _A ~WT oLxL oLxL w V oLxL oLxL V (4.65) (4.66) (4.67) where W and V are Toeplitz matrices. Their first columns and rows Cw, Tw, cy, and r y are defined by c w = | [ l -£ i [2] . . . - X x f L - l j f , r f f = | [ l - x i [ - l ] - £ i [ - 2 ] . . . -xxf-L+1]], c y ^ ± [ 0 - (^[lj+xxt-l]) . . . - (x^L-^+x.i-L+l})]-A T rv = cv. (4.68) (4.69) (4.70) (4.71) The Laplace transform of the pdf of A(cv, 3, Xx) has the same form as Eq.(4.49) but with different F. It is then trivial to calculate PEP. Since £i[±z],z = {1, 2, . . . , L — 1} have 2 2( L _ 1) different combinations, we average over all possible PEPs. The final B E R is given by Ph = Xi 22( i"1) (4.72) 4.3.3 I M F B H for T R - S T B C Further considering the influence of the decoupling errors from X2[k] in calculation of the M F B leads to a tighter bound, the IMFB I. The PEP, assuming that xi[0] = xp is transmitted while £i[0] = xp is detected, can be 4.3 Performance Analysis of TR-STBC 55 L - l L - l expressed as P e(a, B, XUX2) = Pr|$R{2x;y[0] - 2 £ 7 [ ^ i H K - 2 7H* i [^} ^ i = l i = l L - l L - l N < u{2x0y[O] -2^7[»]iiH]^ - 2 ^ 7 [ ^ i f e } [, (4-73) i = l i = l ^ where X i and X2 are the transmission matrices corresponding to X\ and x2, respectively. Substituting y[0] with Eq.(4.31) at k = 0, and let x a = 1 and x^ = —1, Eq.(4.73) becomes F e (a, X i , X 2 ) = Pr |K{ 7 [0] + £[0] + ro[0] + / i f m + p f m + n f h2 + n2Hp2 • - & { m + 1>[-i]+m+8[-i])*i\i] i=i + (SH] + #] + o[{[ + e[-i])Xl[-i]}} < o where zu[0] reflects the effect of x2[k] on detecting x[0] with ™[0] = {u[k]*x2[k])k=Q i=-L By introducing I = [h\ hT nj n2 pj p2 ] T , the PEP can be written as Pe(a, 8, XUX2) = Pr {A(a, B, XUX2) < 0} , with (4.74) (4.75) (4.76) A(a,B,X1,X2)±lMFl J-2L \I^L F13 \^2L O21, \J-2L F31 \l2L F$3 13 31 — -F33 — W -U u WT WT u -u w ' v oLxL oLxL V (4.77) (4.78) (4.79) (4.80) (4.81) 4.4 Simulation Results 56 where W, U and V are Teoplitz matrices. Their first columns and rows cw, rw, cu,ru,cv, and ry are given by c w = §[l - z i [ l ] -Xl[2) ... -Xl[L-l]]T, (4.82) r w ± \ [ l -Xl[-l] -x i [ -2] . . . - X x b L + l ] ] , (4.83) ^ i[x2[0] x2[l] x2[2] ... x2[L-l]]T, (4.84) rv 4 |[x2[0] z 2 [ - l ] x2[-2] . . . z 2 [-L+l]] , (4.85) cy = |[0 - ^ [ l j + x ^ - 1 ] ) . . . - ( ^ [ L - l J + ^ - L + l])] 7 , (4.86) r y ^ cv. (4.87) The Laplace transform of the pdf of A(a,P, Xi, X2) is calculated by Eq.(4.49). Again we use a numerical method to obtain the PEP. Now, we take into account the decoupling errors of x2[k], k = 0, ± 1 , . . . , ±L — 1, which span (2L — 1) symbols with 2 2 L _ 1 combinations. The total number of different combinations of Xx and X2 is 2^L-^+2L~1 = 24L~\ The final BER averages over all possible PEPs E Pe(a,P,XlyX2) Pb=XuX" 2 4 , - 3 • ( 4 - 8 8 ) 4.4 Simulation Results The MRC, DD, and TR-STBC schemes described in this chapter have been applied to the three-tap channel, and TU and HT channels. BPSK modulation is employed for simulation. The third and the fifth GSM TSs are used for channel estimation. Fig. 4.1 shows the simulation results for DD for the perfectly known and the estimated (JJ = L) independent three-tap channel with MLSE reception. The corresponding MFBs and IMFBs are also calculated and shown. We see that the performance of DD improves 4.4 Simulation Results 57 Figure 4.1: BER vs. 10\ogio(Eb/N0) for MLSE decoding for DD and perfectly known and estimated {JJ = L) three-tap channels. Three different delays D = 1,2,3 are considered, (a) D = 1, (b) D = 2, and (c) D = 3. 4.4 Simulation Results 58 with increasing the delay for both the known and the estimated channels. In addition, the performance gap between the perfectly known case and the estimated channel case becomes wider with increasing delay interval. This is reasonable since the equivalent channel of DD has L + D+l taps, and a longer delay D means a longer equivalent channel. Given a fixed TS length, the quality of channel estimation becomes worse with increasing channel length. As expected, the IMFB matches better with the simulation results than the MFB in all cases. — - Simulation Est. Channel IMFB II Est. Channel • • - • IMFB Est. Channel - - MFB Est. Channel Simulation Known Channel -o- MFB Known Channel DC LU m 6 8 10log 1 Q(Eb/No) Figure 4.2: BER vs. 101ogi0(£b/JVo) for MLSE decoding for MRC and TR-STBC, and perfectly known and estimated {JJ — L) three-tap channels. In Fig. 4.2 we compare the simulation results for MRC and TR-STBC with MLSE 4.4 Simulation Results 59 decoding and perfectly known and estimated (1/ = L) independent three-tap channels. The performance of M R C is 3 dB better than that of T R - S T B C over the perfectly known channel. This can be observed from both the simulation and numerical results. From the simulation results, we see that the performance degradation owing to channel estimation error is worse for T R - S T B C than for M R C . For M R C at a B E R of IO"3, the performance loss of the estimated channel, compared to the known channel, is about 0.6 dB, however, the performance loss for T R - S T B C is to 1.3 dB. This may be attributed to fact that in T R - S T B C , each antenna only radiates half the energy as in M R C , which leads to poorer quality of channel estimation for T R - S T B C . Furthermore, for the estimated channel T R -S T B C suffers from decoupling errors in addition to ISI and AWGN. For T R - S T B C , the IMFB I is closest to the simulation results for M L S E decoding. Fig. 4.3 shows B E R vs. mogw{Eb/N0) for the H T (6 taps) channel profile and M L S E decoding for various STCs over known and estimated channels (1/ — L). If the channel is known, the diversity gain of DD is obvious. With one symbol delay, it yields a performance gain of 3 dB over single antenna transmission at a B E R = 10~3. However, the additional performance gain for a further delay (D = 2) is quite small. For DD we only consider small delay intervals (D = 1, 2) since the complexity of VA for M L S E is directly related to the channel memory and the longer delays mean high complexity. T R - S T B C yields better performance than DD (D = 1,2) for the known channel. If the channel is unknown, DD with two symbol delay performs even worse than with one symbol delay, and loses about 2.2 dB compared to the known case. This may be attributed to a negative effect of the longer equivalent channel. The quality of channel estimation degrades with the channel length, cf. Eq.(3.6). T R - S T B C yields a performance 3 dB worse at a B E R of 10~3 than the perfectly known channel case. Nevertheless, the diversity order for T R - S T B C is higher than that for DD with D = 1,2. For comparison, B E R for single antenna transmission and M R C are also shown in Fig. 4.3. 4.4 Simulation Results 60 Figure 4.3: B E R vs. 101ogi0(-Eb/./Vo) for M L S E reception for various schemes over perfectly known (L = 6) and estimated (1/ = L) HT channels. In Fig. 4.4, we make a similar comparison as in Fig. 4.3 for T U (4 taps) channel profile. For DD, three delay intervals (D = 1,2,3) are considered. If the channel is known, the perforamance gains for DD with D = 1,2,3 at a B E R = 10~3, compared to the single antenna transmission system are, 2.3 dB, 3.5 dB, and 3.8 dB respectively. If the channel is unknown, DD with a longer delay achieves a performance gain at a higher Eb/N0. For example, DD with D = 2 outperforms DD with D = 1 at 101og10(.E6/iVo) = 10 dB, while DD with D = 3 yields better performance than DD with D = 2 at 101ogi0(i?fc/./Vo) = 16.5 dB. For both known and estimated channels, T R - S T B C always outperforms DD within the Eb/N0 range of interests. Simulations for M R C and a system with a single transmit 4.5 Summary 61 antenna are also plotted in In Fig. 4.4 for comparison. 10log10(Eb/No) Figure 4.4: BER vs. 10logw(Eb/N0) for MLSE reception for various schemes over perfectly known (L = 4) and estimated (1/ = L) TU channels. 4.5 Summary We analyze the performance of MLSE decoding for DD, MRC, and TR-STBC. Three per-formance bounds, the MFB, the IMFB, and the IMFB I are derived. The IMFB matches better with the simulation results than the MFB. For TR-STBC, IMFB I is the tighteset bound and is very close to the simulation results. 4.5 Summary 62 The performance of these S T C s are compared for both perfectly known and estimated channels. If the channel is known, the performance of M R C is 3 d B better than that achieved by D D and T R - S T B C , since each transmit antenna for D D and T R - S T B C radiates half the power compared to M R C . D D achieves a performance gain as a result of the artificially induced multipath, and its performance improves w i t h increasing delay, at the price of increased computational complexity. T R - S T B C is a promising technique, since its computational complexity is practically in the same order as that of a single antenna system, while providing better performance than D D . O n the other hand, if the channel is unknown, The performance gains of D D and T R - S T B C are channel dependent, the performance of D D is a tradeoff between the diversity gain from the artificially induced mult ipath and the channel estimation error. T R - S T B C achieves the same diversity order as D D w i t h full delay, nevertheless, it does not always yield a better performance than D D . M R C outperforms D D and T R - S T B C by more than 3 d B for the estimated channels. Chapter 5 Reduced Complexity Decoding for S T C s For channels with long channel memory and/or nonbinary signal alphabets, the complexity of MLSE decoding becomes impractically high. In this case, reduced complexity decoding schemes must be considered [13]. In this chapter, one category of the reduced complexity decoding schemes-DFSE is discussed. For simplicity, we restrict our discussion to perfectly known channels, but the results obtained can be extended to estimated channels as well. 5.1 D F S E DFSE is based on truncating the CIR to reduce the number of state of trellis searched by the VA. Thereby, DFSE feeds back previous decisions in order to reduce the effects of residual ISI resulting from the truncation process. With this arrangement, the DFSE algorithm combines the VA with decision feedback equalization (DFE). For DFSE schemes, the accumulated metric can be obtained recursively as A[fc + l] = A[fc] + ADFSE[fc], (5.1) where ADFSE[&] is the branch metric. 63 5.2 W-DFSE 64 The performance of DFSE highly depends on the structure of the channel under consid-eration. Specifically, DFSE is well suited for channels having most of their energy concen-trated at the beginning of the CIR. For channels that do not satisfy this condition, such as nonminimum phase channels, the performance of D F S E may become poor [10]. Therefore, it maybe be necessary to process the received signal by a-prefilter prior to DFSE, the aim of the prefilter is to shape the original CIR into one that has more energy concentration at the beginning of the impulse response, to make it more suitable for DFSE. D F S E can be divided into two types according to the metric used for detection. The first type is whitened D F S E (W-DFSE), which usually includes a prefilter to convert the channel into a minimum phase channel and the Forney metric is adopted for detection. The other type is unwhitened D F S E (U-DFSE), which directly uses the Ungerboeck metric to make a decision. 5.2 W-DFSE Fig. 5.1 shows the block diagram of W-DFSE. The CIR is divided into a leading part and and a tail. An M L S E equalizer is constructed, based on the leading part, and the interfering effect of the CIR tail is cancelled by feedback using previous decisions (assumed to be correct). At time k, the branch metric X[k] of the W-DFSE decoder/equalizer is given by -W-DFSE [&] — K L-l 2 (5.2) y'M ~ ^2 hmin[i]x[k -i]- ^ hmin[i]x[k - i] i=0 i=K+l where K, 0 < K < L — l i s a design parameter that determines the number of W - D F S E trellis states, /imjn[/c] is the minimum phase CIR, x[k] are trial symbols, x[k] are the tentative decisions taken from the surviving path, and y'[k] is the received signal after the prefilter. Note that if K = L — 1, the algorithm reduces to the VA, if K = 0, the algorithm is 5.3 U-DFSE and Modified U-DFSE 65 n[k] 4k] ^ H(z) P i y[k] ) *^ Hp(z) Reduced-State Equalization x[k] Figure 5.1: Block diagram of W-DFSE. equivalent to the zero-forcing decision feedback equalization (ZF-DFE). The prefilter can be calculated by MMSE-DFE or other methods [6]. 5.3 U-DFSE and Modified U-DFSE 5.3.1 U - D F S E Similar to W-DFSE, U-DFSE can be achieved by reducing the number of states in the Ungerboeck metric. In that case, the branch metric is given [12] by AU-DFSEM = M \x*[k] (2y[k\ - >y[0]x[k] - 2 ^ 7 [ i ] £ [ A : - i] - 2^ 7 [z ]x[A; - * ] ) } , (5-3) I ^ i=0 i=K ' J where x[k] are the tentative decisions, taken from the surviving path terminating in the corresponding state. x[k] are trial symbols, y[k] is the received signal filtered by the matched filter, j[k] is the CIR of the cascade of the channel and the matched filter, and K, 0 < K < L — 1, determines the number of U-DFSE trellis states. When K = L — 1, the metric in Eq.(5.3) is a full-state Ungerboeck metric. However, for K < L — 1, the reduced-state Ungerboeck metric does not perform as well as W-DFSE on most channels owing to its untreated interference components. The bias term, bias [A;], is sequence-dependent and given by [12] k=K ( L - l ^ ~ " (5-4) bias[A;]= Y, ^ 2 x * [ i ] £ .-y[-j]x[i +j] : j=k—i+l i=k—L Obviously, the bias term may significantly reduce the distance between different can-didate sequences, and, since it is independent of the noise variance, may even cause an 5.3 U-DFSE and Modified U-DFSE 66 irreducible bit error floor. As will be shown in Section 5.5, this problem is more severe for high-level modulation formats, since the minimum distance between candidate sequences is smaller than for binary modulation. In addition, Eq.(5.3) suggests that, on average, the bias increases with increasing channel delay spreads and decreasing K. 5.3.2 Modif ied U - D F S E ( M - U D F S E ) and Mult istage M U - D F S E ( M M U - D F S E ) An obvious way to improve the performance of U-DFSE is to subtract the bias term bias [A:] from A[fc + 1] [12]. Since biasffc] depends on unknown future symbols x[k + i], 1 < i < L — K, preliminary tentative decisions x[k + i] = dec{y[k + 1]}, 1 < i < L — K, have to used to calculate biasffc], where dec{.} refers to a symbol-by-symbol decision, i.e., x[k + i] € A is the signal point with minimum Euclidean distance to y[k + i] [26]. The same accumulated and branch metrics as for U-DFSE are used for MU-DFSE. However, the selection of the surviving path is based on the metric A[k] + A U - D F S E M — bias[fc]. Furthermore, the bias term can be simplified to include only the leading term to keep the complexity at minimum bias[A;] ~ 9fc hx^k - K] ^ l[-j]x[k - K + 1] 1. (5.5) I j=K+l J M U - D F S E can be further improved in a multistage configuration [12]. The first stage of M M U - D F S E is identical to MU-DFSE. In the second stage, however, the more reliable final decision b[k] of the first stage are used for calculation of the bias instead of b[k]. Further iterations are possible, however, the resulting additional performance gains are negligible. 5.4 DFSE Applied to STCs 67 5.4 DFSE Applied t o STCs 5.4.1 D F S E A p p l i e d to T R - S T B C In contrast to the DFSE schemes proposed in [27], [28] which adopt the feedforward filter (FFF) of MMSE-DFE, the W-DFSE scheme proposed here relies on the FFF of ZF-DFE. Note that the fast algorithms for calculation of the MMSE-DFE FFF reported e.g. in [29], [30] cannot be directly applied here, since only the MF output is available and the equivalent CIR h[k] is not explicitly known at the receiver. However, as we will show in the following, adapting the approach proposed in [6] to the problem at hand, we are able to compute a whitening filter F\y{z) in a very efficient and elegant way. The cascade of MF H*(l/z*) and Fw(z) constitutes an allpass filter that transforms h[k] into its minimum-phase equivalent hmin[k],0 < k < L — 1, which is favorable for W-DFSE. The resulting overall channel transfer function Hmm {%) = -2 { hmin [k]} = H(z)H*(l/z*)Fw(z) (5.6) has all zeros inside and on the unit circle. Using the identity H(z)H*(l/z*) = S(z) we observe that Hmm(z) can be calculated from Obviously, the noise-whitening filter Fw(z) is an infinite impulse response (IIR) filter with transfer function ™*>-Hdi7?j- ( 5- 8 ) Fortunately, Fvv{z) can be approximated (up to a constant) by an FIR prediction-error filter (PEF) E(z) of length Le for a process with power spectral density S(ei2nfT). It is 5.4 DFSE Applied to STCs 68 well known that if Le is chosen sufficiently large, the relation E(z)E*(l/z*)S(z) = constant (5.9) holds, i.e., E(z) is a whitening filter. From Eq.(5.9) it can be observed that E(z) may be either a causal, minimum-phase forward prediction-error filter or an anti-causal, maximum-phase backward prediction-error filter. However, since H^in(l/z*) is anti-causal and maximum-phase, E(z) has to be a backward prediction-error filter. Therefore, E(z) is given by [31] Le-i = (5.10) k=l where the backward predictor coefficient p[k] can be calculated from the Yule-Walker equa-tion [31] Sp = s, (5.11) with s[0] s[l] s[-l) s[0] [_s[Le-2] s[Le-3] ... P=b[l]p[2] ••• p[Le-l]]T, s±[s[-l]s[-2] ... s[-Le + l][ s[-Le + 2]' l[-Le + 3] ' s[0] . (5.12) (5.13) (5.14) Fortunately, Eq.(5.11) can be efficiently solved by using the Levins on-Durbin algorithm requiring q2 + 0(qp) operations [6]. Note that the proposed algorithm does not require knowledge of the equivalent CIR h[k]. An approximation of the scaled version of the minimum phase CIR ^ m i n M c a n be calculated from (5.15) Hgin(z) 4 % M « E(z)H(z)H*(l/z*), C 5.4 DFSE Applied to STCs 69 where C is an irrelevant non-zero constant and H^in(z) = ,2{/i£ in(£;)}. The branch metric for W-DFSE is given by K L-l X W-DFSE [k] i=K+l i=0 (5.16) where y'[k] is obtained by filtering of y[k] with E(z). For practical implementation, an additional delay has to be introduced for a causal realization of E(z). The receiver structure for TR-STBC with W-DFSE detection scheme is depicted in Fig. 5.2. The output signals of the spatial-temporal matched filter HH[—k] are divided into two sequences, which are filtered by the noise-whitening filter separately before being detected by W-DFSE. „ Time Complex Reversal Conjugate Channel Estimation Figure 5.2: Block diagram of W-DFSE decoding for TR-STBC. The application of U-DFSE for TR-STBC is more straightforward since, after matched filtering, the two signal sequences can be directly processed by the U-DFSE using the Ungerboeck metric without additional processing. Fig. 5.3 shows the block diagram at the receiver side. 5.4.2 D F S E A p p l i e d to D D As indicated in the previous chapters, DD with two transmit antennas can be viewed as a single antenna transmission with the equivalent CIR h'[k] given by Eq.(4.19). Fig. 5.4 shows the block diagram for the DD receiver using W-DFSE detection scheme. The received 5.4 DFSE Applied to STCs 70 Time Complex Reversal Conjugate Channel Estimation xi[k] A U-DFSE »x2[k] Figure 5.3: Block diagram of U-DFSE decoding for TR-STBC. signals are filtered by a prefilter and then passed to the W-DFSE detector for detection. In each block, the equivalent channel CIR h'[k] is estimated and the prefilter transforms the equivalent channel to the minimum phase channel according to the estimated channel information. y M I „ , J y'[k] | | x[k] Hp(z) W-DFSE Channel Estimation Figure 5.4: Block diagram of W-DFSE decoding for DD. For the U-DFSE detection scheme, the received signals are filtered by a matched filter. The output symbols of the matched filter are then fed to the U-DFSE detector for detection using Ungerboeck metric. The modified versions of U-DFSE algorithm can also be used to achieve a higher performance. The block diagram for the DD scheme using U-DFSE detection is shown in Fig. 5.5. 5.4.3 Complexity Consideration Assuming the same K for all presented DFSE schemes, it is clear that their complexity is in the same order since all considered schemes operate on a trellis with Z = MK states. Also 5.5 Simulation Results 71 MF v'M U-DFSE IK Channel Estimation m Figure 5.5: Block diagram of U-DFSE decoding for DD. for metric calculation all schemes require only additions and multiplications. Nevertheless, the lowest complexity can be attributed to U-DFSE since it requires only the MF output. MU-DFSE has a slightly higher complexity since the bias term is calculated for each branch metric. The complexity of MMU-DFSE is approximately twice as high as that of MU-DFSE, assuming two iterations. Finally, the metric computation of W-DFSE requires approximately the same number of operations as that of U-DFSE. However, for W-DFSE the additional prefilter has to be calculated once per data burst, and the MF output has to be filtered by the prefilter. Given the same i f and using the same decoding scheme, the complexity of DD and TR-STBC are in the same order. For W-DFSE, DD uses a prefilter with all-pass trans-fer function while TR-STBC needs two prefilters, actually noise-whitening filters, after a spatial-temporal filter. For U-DFSE, DD and TR-STBC, both operating on the output of the matched filter, have almost the same complexity. 5.5 Simulation Results In the previous chapter, we analyze the performance of MLSE decoding for STCs. However, doing a similar analysis for W-DFSE decoding is much involved, and therefore, we resort to computer simulations. The GSM/EDGE system parameters, and the independent three-tap, the TU, and the HT channels are considered. For W-DFSE, the prefilter length is 5.5 Simulation Results 72 chosen to Le = AL. In Fig. 5.6, BPSK simulation results of DD with W-DFSE detection over the per-fectly known independent three-tap channel are shown. Three different delay intervals (D = 1, 2, 3) for DD are compared with Z = 1 (solid line) and Z = 8 (dashed line), respec-tively. For Z = 1 state, i.e., DFE, the performance for D = 1 is worst. When D — 2, the performance improves by 0.75 dB at a BER = IO - 3 . However, for D — 3, the additional improvement is only 0.46 dB. For Z = 8 states, the performance also improves with in-creasing delay. Compared to Z = 1 case, W-DFSE for Z = 8 yields a higher performance, improved by 2.0 ~ 2.4 dB for different delays at a BER = 10~3. For the same complexity, DD with different delay intervals yields similar performance at low SNRs and the advantage of DD with a longer delay over a shorter one becomes more obvious with increasing Eb/N0, especially for lOlogic^-E^/iVo) > 10 dB for this channel. In Fig. 5.7, we consider DD for the TU delay profile. The performance of DD with differ-ent delay intervals (D = 1, 2,3, 4), but with the same complexity is compared. The channels are estimated using the third and the fifth GSM training sequences. For Z = 1 state, DD with one symbol delay yields the best performance at low SNRs (101ogi0(jBfe/A/0) < 14 dB). At high SNRs, DD with three-symbol delay yields the best performance. For Z = 4 states, DD with one symbol delay still achieves the best performance at low SNRs, however, DD with two-symbol delay D = 2 outperforms the other schemes at high SNRs in this case. An interesting fact is that the performance with full delay is the worst in both cases for Z = 1 and Z = 4. This may be attributed to the fact that, after processed by the prefilter, the taps at the end of the equivalent channel h' have little energy for average TU chan-nels. Furthermore, the channel estimation quality degrades with increased channel length, given a fixed training sequence length, cf. Eq.(3.6). The performance gain obtained from multi-paths of full delay is counteracted by the performance degradation resulting from the 5.5 Simulation Results 73 DC LU m 8 10 10log 1 Q(Eb/No) Figure 5.6: Comparison of BER performance for DD with different delays for BPSK trans-mission over perfectly known independent three-tap channels. D = 1,2,3 are considered. Solid line: BER for Z = 1 (DFE) and Dashed line: BER for Z = 8. channel estimation error. Fig. 5.8 shows BER vs. 101ogio(i?&/iVo) for the TU channel profile and the proposed DFSE schemes for TR-STBC. For Z = 1 state, only W-DFSE, which is identical to ZF-DFE in that case, gives a satisfactory performance. The modified U-DFSE schemes (MU-DFSE and MMU-DFSE) actually yield worse performances than the original U-DFSE scheme. On the other hand, for Z = 2 states the performance of all four DFSE schemes is close to that of optimal MLSE. MMU-DFSE even slightly outperforms W-DFSE for high SNRs. At a 5.5 Simulation Results 74 10" io"'h DC LU m 1 0 " > 10" -ex. o D=1 A D=2 V D=3 • D=4 Known Channel , Estimated Channel s ® "V S • v v \ \ \ : N • v: : :NV...Q. . . . . Y V . ^ , N ^ 10 10log 1 0(Eb/No) 12 14 16 18 20 Figure 5.7: Comparison of BER performance for DD with the same complexity for BPSK transmission over estimated TU channels. Four different delay intervals D = 1,2,3,4 are considered. Solid line: BER for Z = 1 (DFE) and Dashed line: BER for Z = 4. BER of IO - 3 the performance loss of U-DFSE compared to W-DFSE is only 0.23 dB. For Z = 4 (not shown in Fig. 5.8), all considered DFSE schemes achieve practically the same performance as MLSE. In order to illustrate the potential performance gain of TR-STBC in GSM, in Fig. 5.8, we have also included the BER for a system with only Nt = 1 transmit antenna for, respectively, W-DFSE with Z = 1 state and MLSE. In Fig. 5.9, we consider GSM with an HT power delay profile. Again, for Z = 1 only W-DFSE yields an acceptable performance. Since the delay spread of HT is larger than 5.5 Simulation Results 75 6 8 10 12 14 16 10log10(Eb/No) Figure 5.8: BER vs. 101ogi0(£'6/Aro) for GSM and TU profile with L = 4 for different DFSE schemes for TR-STBC. Solid lines: BER for NT = 1 transmit antenna and, respectively, W-DFSE with Z = 1 state and MLSE. that of TU, even for Z — 2 U-DFSE and MU-DFSE suffer from a considerable performance loss compared to W-DFSE. Note that a large delay spread means that, on average, the bias in Eq.(5.9) is larger and the preliminary tentative decisions necessary for estimation of the bias are less reliable. Nevertheless, for the considered EB/N0 range MMU-DFSE yields a similar performance as W-DFSE. For comparison, for Z = 1 we also include the performance of W-DFSE, using an MMSE-DFE feedforward filter (FFF) as a prefilter. The performance loss of the proposed W-DFSE scheme is less than 0.25 dB. For Z > 2 the performance of both W-DFSE schemes is practically identical, and the corresponding 5.5 Simulation Results 76 6 8 10 12 14 16 18 10log10(Eb/No) Figure 5.9: BER vs. lO\ogw(Eb/N0) for GSM and HT profile with L = 6 for different DFSE schemes for TR-STBC. Solid line: BER for W-DFSE (Z = 1) with MMSE-DFE FFF as prefilter. curves for the MMSE-DFE FFF are omitted for clarity of presentation. In Fig. 5.10 the BER of the proposed DFSE schemes for EDGE and the TU channel are depicted. As expected, for Z = 1 only W-DFSE yields a high performance. In contrast to the GSM case, cf. Fig. 5.8, for EDGE U-DFSE results in a high error floor. This can be attributed to the negative effect of the bias, which is more pronounced for high-level modulation formats. Even for Z = 8 U-DFSE and MU-DFSE suffer from a large loss in performance compared to W-DFSE. Although iterative MMU-DFSE yields a similar BER 5.5 Simulation Results 77 10" 10" LU m 10" 10" : : - . - . - T . - . - . - . T - . H ! ! ! ,v...._..„,„,;_..._.. y • • • -a-M M S E - D F E . . prefilter.... r x N > N X . , V U - D F S E X M U - D F S E • M M U - D F S E A W - D F S E - • Z=1 Z=8 Z=64 • -.v \ \y '- \ \ \ N _1_ 10 12 14 10log 1 Q(Eb/No) 16 18 20 Figure 5.10: BER vs. lO\ogw{Eb/N0) for EDGE and TU profile with L = 4 for different DFSE schemes for TR-STBC. Solid line: BER for W-DFSE (Z = 1) with MMSE-DFE FFF as prefilter. as W-DFSE for 101ogi0(£6/./Vo) < 16 dB, for high SNR it is negatively affected by the poor performance of MU-DFSE. Since MLSE is computationally too expensive, we have included the performance of W-DFSE with Z = 64 states for comparison, which should be close to that of MLSE. Note, however, that for a practical implementation Z < 8 is relevant. Fig. 5.10 also shows that for Z = 1 the proposed W-DFSE scheme yields a similar performance to that with an MMSE-DFE FFF. Finally, we consider EDGE with an HT profile in Fig. 5.11. For both Z = 1 and 5.5 Simulation Results 78 LU CD 10 V U - D F S E X M U - D F S E • M M U - D F S E A W - D F S E - • Z=1 Z=8 Z=64 10 12 10log 1 0(Eb/No) 14 16 18 20 Figure 5.11: BER vs. 101ogi0(£&/iVo) for EDGE and HT profile with L = 6 for different DFSE schemes for TR-STBC. Solid lines: BER for NT = 1 transmit antenna and W-DFSE with, respectively, Z = 1 and Z = 8 states. Z = 8 states only W-DFSE results in an acceptable performance. As expected, U-DFSE and MU-DFSE yield a high error floor, which also negatively influences the performance of MMU-DFSE. The comparison with a single transmit antenna system using W-DFSE confirms the large potential performance gains of TR-STBC also for EDGE and the HT profiles. 5.6 Summary 79 5.6 Summary In this chapter, three different DFSE schemes, namely, W-DFSE, U-DFSE, and improved versions of U-DFSE (MU-DFSE and MMU-DFSE), for DD and TR-STBC, are discussed and compared, by means of simulations over GSM and EDGE channel profiles as well as the three-tap channel. U-DFSE has the lowest complexity since it performs DFSE directly using the MF output. However simulation results indicate that it can achieve acceptable performance only for binary modulation (as in GSM) and channels with small delay spreads. From our results it appears that for most channels MU-DFSE does not yield a higher performance than U-DFSE. However, for binary modulation MMU-DFSE can approach the performance of W-DFSE even for channels with large delay spreads and a small number of trellis states. W-DFSE always yields the best performance in all simulations, especially for EDGE, where all U-DFSE schemes suffer from the metric bias and unreliable tentative decisions. For W-DFSE, MMSE-DFE based prefilter only outperforms the proposed linear prediction based prefilter slightly at the cost of a high complexity. Assuming the same complexity, for DD and perfectly known channels, the performance improves with increasing the delay interval up to full delay. However, for the estimated channel, the performance gain obtained from a longer delay is counteracted by the higher channel estimation error given a fixed TS length. Chapter 6 Conclusions and Future Research In this thesis we have investigated several issues pertaining to STCs for frequency-selective fading channels. The LSSE channel estimation and TS construction methods are reviewed. Optimal TSs do not exist for all channel lengths given a fixed TS length. In addition, the effect of mismatched-memory channel estimation on MLSE reception is analyzed. Over-estimation degrades the performance while under-estimation causes an error floor in the performance. To analyze and benchmark the performance of MLSE for STCs, three lower bounds are derived, namely, the MFB, the IMFB, and the IMFB I for DD, STBC, and MRC. Al l these bounds take into the effect of mismatched-memory channel estimation. The IMFB, especially the IMFB I, matches well with the simulation results. Furthermore, the performance of different STCs are compared for both perfectly known and estimated channels. If the channel is known, DD achieves a performance gain owing to the artificially induced multipath, and its performance improves with increasing delay at the" price of increased computational complexity. TR-STBC yields almost the same performance as DD with full delay with a computational complexity practically in the same order as that 80 81 of a single antenna system. If the channel is unknown, the performance gains of DD and TR-STBC are channel dependent. The performance of DD is a tradeoff between the diversity gain from the multipath and the channel estimation error. TR-STBC achieves the same diversity order as DD with full delay; nevertheless, it does not always yield better performance than DD. Reduced complexity decoding schemes for STCs are discussed. Three different DFSE schemes, W-DFSE, U-DFSE, and MU-DFSE, are adapted to DD and TR-STBC. For W-DFSE and TR-STBC, an efficient algorithm, based on linear prediction for prefilter calcu-lation, is proposed. The performances of different schemes are compared via simulations. Our results show that for binary modulation, U-DFSE and its improved version can ap-proach the performance of W-DFSE for the full range of delay spreads relevant for the GSM/EDGE. On the other hand, for high-level modulation, only W-DFSE gives a satis-factory performance. An interesting direction for future research is to combine different STC structures to provide a higher diversity order. For example, we can combine the structure of generalized DD (GDD) [17] and TR-STBC to improve the performance. In the meantime, other reduced complexity equalization algorithms, such as M-BCJR [32], can be considered. Glossary Operators argmax{-} Argument Maximizing Operator argmin{-} Argument Minimizing Operator * Convolution Kronecker Product (0* Complex Conjugate (•)» Conjugate Transpose ( . ) T Transpose l - l Abstract Value II' 2 Frobenius Norm A Alphabet E{-} Expectation det(-) Determinant 3(0 Imaginary Part PbO Probability Pel'} Error Probability Real Part tr(-) Trace 82 GLOSSARY 83 Sets A Signal Alphabet Z Integer Number Constants j Imaginary Unit: j2 = — 1 TT The Number pi TT = 3.14159265358979 e Euler number e = 2.718281828 J n n x n Identity Matrix 0 n n x l Zero Vector O M X N m x n Zero Matrix Acronyms AWGN Addictive White Gaussian Noise BER Bit Error Rate CIR Channel Impulse Response COST European Cooperation in the Field of Science and Technical Research DD Delay Diversity DFE Decision-Feedback Estimation DFSE Decision-Feedback Estimation EDGE Enhanced Data Rates for GSM Evolution EQ Equalization Test FEC Forward Error Correction FIR Finite Impulse Response GSM Global System for Mobile Communication HT Hilly Terrain GLOSSARY 84 IIR Infinite Impulse Response IMFB Improved Matched Filter Bound IMFB I Improved Matched.Filter Bound I ' . ISI Intersymbol Interference LSSE Least Sum of Squared Error MF Matched Filter MFB Matched Filter Bound MIMO Multiple-Input Multiple-Output ML Maximum Likelihood MLSE Maximum Likelihood Sequence Estimation MMSE Minimum-Mean-Square Error MMU-DFSE Multistage Modified Unwhitened Decision Feedback Sequence Estimation MRC Maximal Ratio Combining MSE Mean-Square Error MU-DFSE Modified Unwhitened Decision Feedback Sequence Esimation OFDM Orthogonal Frequency Division Multiplexing PRUS Perfect Root of Unity PSK Phase Shift Keying QAM Quadrature Amplitude Modulation SISO Single-Input Single-Output SIMO Single-Input Multiple-Output SNR Signal-to-Noise Ratio SRC Square-Root Raised Cosine STBC Space-Time Block Code STTC Space-Time Trellis Code STC Space-Time Coding TR-STBC Time Reversal Space-time Block Coding TU Typical Urban U-DFSE Unwhitened Decision Feedback Sequence Estimation W-DFSE Whitened Decision Feedback Sequence Estimation ZF-DFE Zero-Forcing Decision Feedback Equalization Bibliography [1] V. 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