Iterative Techniques Based on Energy Spreading Transform for Wireless Communications. Taewon Hwang

Transcription

1 Iterative Techiques Based o Eergy Spreadig Trasform for Wireless Commuicatios A Thesis Preseted to The Academic Faculty by Taewo Hwag I Partial Fulfillmet of the Requiremets for the Degree Doctor of Philosophy School of Electrical ad Computer Egieerig Georgia Istitute of Techology December 2005

2 Iterative Techiques Based o Eergy Spreadig Trasform for Wireless Commuicatios Approved by: Professor Ye (Geoffrey) Li, Advisor School of Electrical ad Computer Egieerig Georgia Istitute of Techology Professor Gordo L. Stüber School of Electrical ad Computer Egieerig Georgia Istitute of Techology Professor Gregory D. Durgi School of Electrical ad Computer Egieerig Georgia Istitute of Techology Professor Xigxig Yu School of Mathematics Georgia Istitute of Techology Professor Joh R. Barry School of Electrical ad Computer Egieerig Georgia Istitute of Techology Date Approved: October 24, 2005

3 To my parets, for their edless support iii

4 ACKNOWLEDGEMENTS I would like to express my sicere gratitude to my advisor, Prof. Ye (Geoffrey) Li. Durig my Ph.D. study, he has always bee motivatig, guidig, ad ecouragig me to be more fruitful i my research. This thesis has sigificatly beefited from his guidace ad ecouragemet. I ca ever thak him eough for his time ad efforts devoted o my papers. I would like to thak my thesis committee members, Prof. Grodo L. Stüber, Prof. Joh R. Barry, Prof. Gregory D. Durgi, ad Prof. Xigxig Yu. Their remarks ad commets have made sigificat cotributios to improvig the quality of my thesis. I am very thakful to the former ad curret members of the Iformatio Trasmissio ad Processig (ITP) Laboratory, Jigog Yag, Goucog Sog, Hua Zhag, Jiaxua Du, Zet Zhu, Uzoma Aaso Oukwo, ad Ghurumuruha Gaesa for their help, ecouragemet, ad umerous discussios. They were great compaios i my log jourey to this thesis. Fially but ot the least, I would like to thak my parets for their edless support. This thesis is dedicated to them. iv

5 TABLE OF CONTENTS DEDICATION ACKNOWLEDGEMENTS iii iv LIST OF TABLES vii LIST OF FIGURES viii SUMMARY ix INTRODUCTION Chael Equalizatio Covetioal Methods Turbo Equalizatio MIMO sigal detectio Covetioal Methods Turbo BLAST ITERATIVE EQUALIZATION BASED ON EST System Descriptio Eergy Spreadig Trasform Complexity Performace Aalysis MMSE Equalizer Geie-Aided Equalizer Iterative Equalizer with Hard Decisio Iterative Equalizer with Soft Decisio Simulatio Results Performace for Proakis-B chael Performace for other challegig chaels IMPROVED SCHEME FOR ENERGY SPREADING TRANSFORM BASED EQUALIZATION Optimal Equalizatio Filter Desig Hard Decisio v

6 3..2 Soft Decisio Simulatio Results ITERATIVE MIMO SIGNAL DETECTION BASED ON EST: FLAT FADING CHANNELS System Descriptio ST-EST ad Its Impact Complexity Performace Aalysis MMSE Receiver Geie-Aided Receiver Iterative Receiver with Hard Decisio Iterative Receiver with Soft Decisio Asymptotic Property of Rayleigh Fadig Chaels Simulatio results ITERATIVE MIMO SIGNAL DETECTION BASED ON EST: FRE- QUENCY SELECTIVE FADING CHANNELS CONCLUSION APPENDIX A DERIVATION OF (25) APPENDIX B DERIVATION OF (52) APPENDIX C DERIVATION OF (70) AND (00) APPENDIX D PROOF OF (92) FOR SUFFICIENTLY HIGH SNR 75 APPENDIX E PROOF OF ID APPENDIX F DERIVATION OF (93) APPENDIX G DERIVATION OF (94) APPENDIX H DERIVATION OF (95) REFERENCES VITA vi

7 LIST OF TABLES Spreadig properties of some ESTs, N = Computatioal Complexity. F-filter = frequecy-domai filter, T-filter = time-domai filter, Y = required, N = ot required Computatioal Complexity. Y = required, N = ot required vii

8 LIST OF FIGURES Turbo equalizatio Turbo BLAST Low-complexity equalizatio Priciple of EST Equivalet system model Performace of the EST-based equalizer for Proakis-B chael BER performace verses iteratio with differet block sizes N at 0 db for Proakis-B chael ad hard decisio Performace of the EST-based equalizer for differet types of chaels Performace of the improved scheme with (a) hard decisio ad (b) soft decisio for Proakis-B chael Performace of the improved scheme with (a) hard decisio ad (b) soft decisio for Proakis-C chael (a) Iterative detectio for MIMO flat fadig chaels ad (b) its equivalet model Priciple of ST-EST Distributio of (a) K H ad (b) Q H for differet umbers of T = R Performace of the proposed iterative sigal-detectio approach with (a) hard decisio ad (b) soft decisio for MIMO flat fadig chaels whe T = R = Performace of the proposed iterative sigal-detectio approach with (a) hard decisio ad (b) soft decisio for MIMO flat fadig chaels whe T = R = Required SNR to achieve BER = 0 4 for differet umber of ateas Iterative detectio for MIMO frequecy-selective fadig chaels Performace of the proposed iterative sigal-detectio approach with (a) hard decisio ad (b) soft decisio for MIMO frequecy-selective fadig chaels whe T = R = L = viii

9 SUMMARY The objective of the proposed research is to devise high-performace ad lowcomplexity sigal-detectio algorithms for commuicatio systems over fadig chaels. They iclude chael equalizatio to combat itersymbol iterferece (ISI) ad multiple iput multiple output (MIMO) sigal detectio to deal with multiple access iterferece (MAI) from other trasmit ateas. As the demad for higher data-rate ad more efficiecy wireless commuicatios icreases, sigal detectio becomes more challegig. We propose ovel trasmissio ad iterative sigal-detectio techiques based o eergy spreadig trasform (EST). Differet from the existig iterative methods based o the turbo priciple, the proposed schemes are idepedet of chael codig. EST is a orthoormal that spreads a symbol eergy over the symbol block i time ad frequecy for chael equalizatio; space ad time for MIMO sigal detectio with flat fadig chaels; ad space, time, ad frequecy for MIMO sigal detectio with frequecy-selective fadig chaels. Due to the spreadig, EST obtais diversity i the available domais for the specific applicatio ad icreases the reliability of the feedback sigal. Moreover, it eables iterative sigal detectio that has ear iterferece-free performace oly at the complexity of liear detectors. Either a hard or soft decisio ca be fed back to the iterferece-cacellatio stage at the subsequet iteratio. The soft-decisio scheme prevets error propagatio of the harddecisio scheme for a low SNR ad improves the performace. We aalyze the performace of the proposed techiques. Aalytical ad simulatio results show that these schemes perform very close to the iterferece-free systems. ix

10 CHAPTER INTRODUCTION Cofrotig severe iterferece is ot uusual i moder wireless commuicatio. Cosider data trasmissio over terrestrial radio chaels, which is characterized by multipaths resultig from atural ad ma-made objects betwee the trasmitter ad receiver. Whe symbol duratio is shorter tha multipath spread (or delay spread), chael equalizatio [] is ecessary to combat itersymbol iterferece (ISI). As aother example, multiple iput multiple output (MIMO) systems aimed at obtaiig maximal efficiecy [2, 3] should deal with multiple access iterferece (MAI) from other ateas. As the demad for higher data-rate ad efficiecy wireless services icreases, those sigal-detectio problems i severe iterferece eviromets become more importat. To obtai optimal performace, maximal likelihood (ML) sigal detectio eeds to be used; however, its complexity icreases expoetially with chael memory for ISI chaels ad the umber of trasmit ateas for MIMO chaels. Liear detectors ad decisio feedback (DF) detectors have favorable complexities, but their performace is limited because of oise ehacemet or error propagatio. Recetly, iterative sigal-detectio schemes based o the turbo priciple [4] have draw much attetio. Those turbo-like approaches have sigificatly better performace tha liear ad o-iterative schemes. However, these schemes rely o joit sigal detectio ad soft-output chael decodig as i [5] or its variat [6], whose complexity is desired to be further reduced. The objective of the proposed research is to devise high-performace ad low-complexity sigal-detectio algorithms i severe iterferece eviromets. We propose ovel trasmissio ad iterative sigal-detectio techiques based o eergy spreadig trasform (EST). EST is a orthogoal trasform that spreads symbol eergy over the symbol block ad thereby icreases the reliability of the feedback sigal. Also, it eables iterative sigal detectio without chael codig; therefore, it saves the complexity of soft-output chael

11 decodig i turbo-like methods. Theoretical ad simulatio results show that these schemes perform very close to the iterferece-free systems oly at the complexity of liear detectors. I the rest of this itroductory chapter, we preset some relevat backgroud material o chael equalizatio ad MIMO sigal detectio.. Chael Equalizatio Chael equalizatio is ecessary to mitigate the effect of itersymbol iterferece (ISI). We review the existig techiques for chael equalizatio with emphasis o recetly devised turbo equalizatio. For the ISI chael, we use the followig received sigal model. r = M k=0 h k x k +, () where r is the received sigal, {h k } M k=0 is the chael, x is the trasmitted symbol with power σ 2 x, ad is the oise with power σ 2... Covetioal Methods Traditioal methods for chael equalizatio are maximal likelihood sequece detectio (MLSD), liear equalizatio (LE), ad decisio-feedback equalizatio (DFE) []. The MLSD uses Viterbi algorithm (VA) to fid the most likely trasmitted sequece. It is optimal, but it suffers from high complexity for chaels with large memory or large costellatio sizes. The LE ad the DFE use liear filters to mitigate ISI. To calculate the filter coefficiets, zero forcig (ZF) criterio tries to remove oly ISI, but miimum mea-square-error (MMSE) criterio miimizes the total power of ISI ad oise. They have favorable complexities, but their performaces are far from the matched filter boud (MFB), especially for chaels with high frequecy selectivity. Noise ehacemet ad error propagatio are the major causes of their performace loss...2 Turbo Equalizatio Figure shows the block diagram of turbo equalizatio, where chael decodig is icorporated ito a part of chael equalizatio. It is based o the iterative exchage of soft 2

12 iformatio betwee the outer soft-output chael decoder ad the ier soft-output symbol detector. Aalogous to turbo codes [4], this scheme has bee called turbo equalizatio. Softouput detector λ E λ D, E λ π π E λk E λ D,k Softouput decoder λ D, k Figure. Turbo equalizatio This techique was first proposed by Douillard [7] i 995. It cosists of a soft-output ML equalizer ad a soft-output chael decoder. The Bahl-Cocke-Jeliek-Raviv (BCJR) algorithm [5] or its variats ca be used i the soft-output ML equalizer ad the soft-output decoder. The BCJR algorithm produces the sequece of the most likely bits alog with their soft values i terms of a posteriori log likelihood ratio (LLR). However, oly the extrisic LLR (a posteriori LLR mius a priori LLR) is exchaged betwee the soft-output detector ad decoder. The performace of the origial turbo equalizer approaches the MFB. But its ier detector, which is soft-output MLSD, has prohibitive complexity for chaels with large delay spread or large costellatio sizes. The filter-based turbo equalizatio [8,9] replaces the soft-output MLSD by a soft-output liear filter, which sigificatly reduces the complexity. The soft-output filter produces y = c H (r HE{x } + E{x }s), (2) where the superscript H is the Hermitia operator, c = [c N2,..., c +N ] T is the filter with legth N = N + N 2 +, r = [r N2,..., r +N ] T is the received sigal vector, E{} deotes the expectatio, H is the N (N + M ) covolutio matrix defied as h M h M 2... h h M h M 2... h H =, (3) h M h M 2... h 0 ad s is the (M + N 2 )-th colum of H. The MMSE filter that miimizes the cost fuctio 3

13 E{ x y 2 } is ( ) σ 2 c = σx 2 I + HV H H + ( v )ss H s, (4) where v = E{ x E{x } 2 } ad V = diag{v M N2,..., v +N }. The statistics E{x } ad v ca be calculated from the extrisic LLR, λ E D,, obtaied from the soft-output decoder. For BSPK, E{x } = p{x = } + ( ) p{x = } = tah ( ) λ E D, v = E{x } 2. (6) 2 (5) At the first iteratio, sice there is o LLR available from the soft-output decoder, we set λ E D, = 0 ad this yields E{x } = 0 ad v =. The complexity of the MMSE filter is rather high because it requires a N N matrix iversio for each. Low-complexity approximatio of the MMSE filter ca be obtaied by assumig ) E{x } = 0 or 2) E{x } = x. Uder the former assumptio, (4) reduces to the MMSE LE solutio: c NA = ( σ2 σx 2 I + HH H ) s (7) ad uder the latter assumptio, (4) reduces to the matched filter: c MF = /(σ 2 /σ 2 x + s H s) s. (8) The output of the soft-output filter y, uder the coditio that x = x {+, } has bee set at the trasmitter, is assumed to be Gaussia with mea µ x, = E{y x = x} ad variace σ 2 x, = E{ y µ x, 2 x = x}. The soft-output detector passes the extrisic LLR, to the soft-output decoder. λ E = log p{y x = +} p{y x = } = 2y µ +, σ 2 +, (9).2 MIMO sigal detectio Multiple iput ad multiple output (MIMO) techique ca achieve high spectral efficiecy [2, 3] ad reliability [0]- [3] i wireless eviromets. We focus o the Bell Labs layered 4

14 space time (BLAST) architecture or the MIMO system aimed at achievig the maximal efficiecy i a flat fadig eviromet ad review the existig sigal-detectio techiques. With T trasmit ateas ad R receive ateas, the MIMO chael is modeled as r = Hx +, (0) where r = [r,..., r R ] is the chael output, H C R T is the chael matrix, x = [x,..., x T ] is the chael iput, ad = [,..., R ] is the oise..2. Covetioal Methods Maximal likelihood (ML) detectio chooses the decisio vector ˆx that miimizes r Hˆx 2. With a exhaustive search, the complexity is X T, where X is the symbol alphabet size. Sice it has a expoetially icreasig complexity with the umber of chael iputs, it is ofte prohibitively complex. Liear detectio ad decisio-feedback (DF) detectio are suboptimal schemes with reduced complexity. For those detectio methods, both ZF ad MMSE criteria ca be used. MMSE criterio miimizes total power of multiple access iterferece (MAI) ad oise, while ZF criterio removes oly MAI. The DF detectio ca be cosidered as a modificatio of the DFE for ISI chaels. However, i the DF detectio, symbol-detectio orderig is possible [4]- [6]. It is show i [4] that choosig the best SNR at each stage i the detectio process leads to optimum orderig. The performace of the liear ad DF detectio is limited because of oise ehacemet ad error-propagatio..2.2 Turbo BLAST Figure 2 shows the block diagram of turbo-blast [7]. Similar to turbo equalizatio, it has a cocateated structure of a ier soft-output detector ad a outer soft-output decoder separated by a iterleaver ad a deiterleaver. The soft-output detector calculates the decisio variable for the k-th symbol ( k R ), y k = wk H r u k, () 5

15 Data stream Demux : T... Ecoder Ecoder Ecoder π... Mod Mod Mod Trasmitter... π Soft-output Detector π... Soft-output Decoder Soft-output Decoder Receiver Figure 2. Turbo BLAST where w k C R ad u k are a liear filter ad estimated iterferece, respectively. The optimum ŵ k ad û k that miimize the cost fuctio E{ x k y k 2 } are ŵ k = ( h k h H k + H k[i (T ) diag{e{x k }E{x k } H }]H H k + σ2 /σ 2 xi R ) hk (2) û k = w H k H ke{x k }, (3) where h k is the k-th colum of H, H k [h,..., h k, h k+,..., h T ], x k [x,..., x k, x k+,..., x T ] T, ad diag{} is the operator applied to a L vector ad outputs L L diagoal matrix with the vector elemets alog the mai diagoal. For the first iteratio, E{x k } = 0 ad (2) reduces to a MMSE receiver y k = h H k ( ) HH H + σ2 σx 2 I R r. (4) As the iteratio proceeds, we assume E{x k } x k, ad (2) simplifies to a perfect iterferece caceller: y k = (h H k h h + σ/σ 2 x) 2 h H k (r H kx k ). (5) Similar to turbo equalizatio, E{x k } ca be calculated from the extrisic LLR of the 6

16 soft decoder. Also, y k is assumed to be Gaussia ad the extrisic LLR of x k calculated by the soft-output detector is iput to the soft-output decoder. 7

17 CHAPTER 2 ITERATIVE EQUALIZATION BASED ON EST 2. System Descriptio The proposed iterative equalizatio is show i Figure 3. A symbol block to be trasmitted, {x } N =0 with a average power σ2 x is mapped to { x } N =0 by a EST, where N is the block size. A cyclic prefix with legth ν is iserted betwee data blocks to prevet iterblock iterferece. x EST x~ CPI Trasmitter CPE r~ FFT R ~ ( ) k (i) A k IFFT IEST Decisio xˆ i Receiver ~ ( i) q (i) b ~ˆ ( i ) x EST Delay ( ) xˆ i CPI: Cyclic Prefix Isertio EST: Eergy Spreadig Trasform (i) (i) A k : Frequecy-Domai Filter b CPE: Cyclic Prefix Extractio IEST: Iverse Eergy Spreadig Trasform : Time-Domai Filter Figure 3. Low-complexity equalizatio The chael is modeled as a (L )-th order FIR filter with coefficiets, {h k } L k=0, ad additive white Gaussia oise (AWGN) with zero mea ad variace σ 2. With eough legth of cyclic prefix (ν L ), the received samples ca be expressed as the circular covolutio [9] of the chael ad the iput symbols: where (k) N L r = h k x ( k)n +, 0 N, (6) k=0 is the residue of k modulo N ad is the AWGN. It is assumed that the chael is static durig a block ad perfectly kow at the receiver. After applyig the fast Fourier trasform (FFT), (6) ca be represeted by R k = H k Xk + N k, 0 k N, (7) 8

18 where the uppercase letters represet the frequecy-domai couterparts of their timedomai otatios i lowercase letters. Without the EST, the structure is just a sigle-carrier system with frequecy-domai equalizatio (SC-FDE) [2]. With the EST, chael equalizatio ca be iteratively performed to improve sigal-detectio performace. After forward frequecy-domai equalizatio, the iverse EST (IEST) is performed for a hard or soft decisio. The decided symbols are trasformed by the EST ad the fed back through the time-domai filter, which performs a N-poit circular covolutio. At the first iteratio, miimum mea-square-error (MMSE) criterio is used to determie the coefficiets of the frequecy-domai filter. Therefore, its frequecy respose will be A () k = where the superscript,, deotes the complex cojugate. time-domai filter is set to be zero: filter: H k H k 2 + σ/σ 2 x 2, k = 0,,..., N, (8) b () = 0. The impulse respose of the From the secod iteratio (i 2), the frequecy-domai filter is chose as the matched A (i) k = H k. The time-domai filter cacels the residual iterferece after the frequecy-domai (matched) filter; therefore, where b (i) g ( 0), = b 0 ( = 0), (9) g l h l h l. After the secod iteratio, we oly iteratively process the same block of sigal obtaied from the iverse FFT (IFFT) at the secod iteratio. As idicated i Sectio 2.2, the EST spreads the eergy of each symbol to differet frequecies ad times. The purpose of frequecy-domai spreadig is to utilize the frequecy 9

19 diversity; the symbol decisio is based o the total eergy trasmitted over the whole badwidth. By time-domai spreadig, the eergy of icorrectly decided symbols is spread to differet times, which ca be illustrated by Figure 4. =3 b 0 b = b 0 b (a) (b) L ˆ ˆx ˆx ˆx ˆx ˆx L x N L ~ˆx ~ˆx 0 ~ˆx 2 ~ˆx ~ˆx L ~ˆ 3 4 x N = b xˆ 0 + b xˆ 2 q q ~ = b ~ˆ x + b ~ˆ x 0 2 : Correctly-decided symbol eergy : Icorrectly-decided symbol eergy b : Time-domai filter ( b 0 = 0) : Time-domai filter output q xˆ x ~ˆ x : Decisio for : EST of Figure 4. Priciple of EST. xˆ I Figure 4, we have assumed x 2 is icorrectly detected ad the time-domai filter has three taps {b, b 0, b }. Note that b 0 is always zero from (9). Without time-domai spreadig, the icorrectly detected symbol will affect the detectio of x ad x 3 through the time-domai filter as depicted i Figure 4 (a). If b ad b are sigificat, the decisio error of x 2 will cause large iterferece to the detectio of x ad x 3. Cosequetly, the overall performace improvemet will be limited. With time-domai spreadig, o the other had, the icorrectly detected symbol eergy is spread over the whole block. Eve though it affects detectio of all the symbols i the block, as depicted i Figure 4 (b), the erroeous symbol eergy captured by the time-domai filter for each symbol detectio is reduced by a factor of the block size, N. Therefore, the probability of symbol decisio error will be reduced compared with the equalizer without the EST if the iitial umber of icorrectly decided symbols is less tha a certai threshold. Furthermore, the umber of errors will keep o decreasig with iteratio util it reaches the MFB. 0

20 2.2 Eergy Spreadig Trasform A EST is a ormalized orthogoal trasform whose role is to spread a symbol eergy over the block i both the time ad frequecy domais. The ideal EST is a EST that has perfect spreadig i both the time ad frequecy domai, that is, (E) l, = (FE) l, = N (20) ad whose phase (E) l, is pseudo-radomly ad eve-symmetrically distributed i [ π, π] for 0 l, N, where E C N N is the EST matrix, (W) l, is the elemet of W C N N at the l-th row ad -th colum, ad F C N N is the ormalized Fouriertrasform matrix, i.e., (F) l, = N e j2πl/n. A EST ca be costructed by E = (P µ )U µ P µ U µ...p U, (2) where U l C N N is a ormalized uitary matrix, P l C N N is a pseudo-radom permutatio matrix for l µ, ad µ is the umber of uitary matrices. To quatify the degree of spreadig for a EST, we defie the time- ad frequecydespreadig factors of E, which will be show to be closely related to the performace i Sectio 2.4. The -th time-despreadig factor of E is defied as N s T (E H ; ) ( (E H ) l, 2 N )2 (22) l=0 for 0 N, where the superscript H deotes the Hermitia traspose. Similarly, the -th frequecy-despreadig factor of E is defied as N s F (E; ) s T (FE; ) = ( (FE) l, 2 N )2. (23) Because of the orthogoality ad ormality of E, the -th time- ad frequecy-despreadig factors of E are bouded by l=0 0 s T (E H ; ), s F (E; ) N N. (24) It is obvious that a EST, E has perfect time spreadig whe s T (E H ; ) = 0 for all 0 N ad perfect frequecy spreadig whe s F (E; ) = 0 for all 0 N.

21 I Table, we list six ESTs: E E 3 are based o Fourier trasform ad E 4 E 6 are based o Hadamard trasform. We use T to deote the ormalized Hadamard matrix. Whe E is used, the system is equivalet to orthogoal frequecy-divisio multiplexig (OFDM). Apparetly, E, E 2, E 4, ad E 5 have perfect time spreadig, while E 3 ad E 6 have perfect frequecy spreadig. It is show i Appedix A that s F (E ; 0) = s F (E 2 ; 0) = s T (E H 3 ; 0) = N N, (25) idepedet of the permutatio matrix P. Similarly, it ca also be show that s F (E 4 ; 0) = s F (E 5 ; 0) = s T (E H 6 ; 0) = N N. (26) Therefore, all the ESTs above have either the maximal time-despreadig factor or the maximal frequecy-despreadig factor for = 0. To compare the time- ad frequecydespreadig factors of differet ESTs for N, we have calculated E d {s T (E H ; )}, V d {s T (E H ; )}, E d {s F (E; )}, ad V d {s F (E; )}, where E d {} ad V d {} deote average ad variace calculated over the idex N (excludig = 0), respectively. For the measuremet of those parameters, we set N = 2048 ad radomly geerated the pseudoradom permutatio matrix. From Table, E ad E 4 have a poor frequecy spreadig property, therefore, they are ot good ESTs. We also see that the ESTs based o Fourier trasform (E 2 ad E 3 ) have better time- or frequecy-spreadig property tha those based o Hadamard trasform (E 5 ad E 6 ). Table. Spreadig properties of some ESTs, N = E i E d {s T (E H i ; )} V d{s T (E H i ; )} E d{s F (E i ; )} V d {s F (E i ; )} E = F H E 2 = P F H E 3 = F H P F H E 4 = T E 5 = P T E 6 = F H P T

22 2.3 Complexity Now, we discuss the complexity of the proposed scheme. I Table 2, listed are the required operatios ad the complexity (i umber of multiplicatios) per block for the hard-decisio receiver at each iteratio. The block-wise complexity of (I)FFT, (I)EST, frequecy-domai filter ad time-domai filter are Nlog 2 N, µnlog 2 N, N, ad (2L 2)N, respectively, where µ is the umber of the orthogoal matrices comprisig the EST as defied i (2). Therefore, the block-wise complexity of the receiver for the first, the secod, ad the i-th (i 3) iteratio are N((2 + µ)log 2 N + ), N(2( + µ)log 2 N + 2L ), ad N(2µlog 2 N + 2L 2), respectively. The block-wise complexity of the proposed scheme for each iteratio is comparable to LE or DFE that is MN, where M is the total umber of equalizer taps (feed forward ad feedback) of LE or DFE. Table 2. Computatioal Complexity. F-filter = frequecy-domai filter, T-filter = timedomai filter, Y = required, N = ot required. Iteratio (i) FFT F-filter IFFT IEST EST T-filter Complexity i = Y Y Y Y N N N((2 + µ)log 2 N + ) i = 2 Y Y Y Y Y Y N(2( + µ)log 2 N + 2L ) i 3 N N N Y Y Y N(2µlog 2 N + 2L 2) 2.4 Performace Aalysis We first summarize the otatios ad the matrix idetities that will be used i our aalysis. For ay square matrix W = (x ij ) N i,j=0, defie tr{w} = N =0 w, w w.. 0 D{W} = w N N, 3

23 ad D{W} = W D{W} = 0 w 0 w 0N w w N w N 0 w N 0. From the above defiitios, we have the followig matrix idetities, For 0 N, ad (D{W}D{W} H ), = (W), 2, (27) ( D{W} D{W} H ), = (WW H ), (W), 2 (28) = l (W),l 2. (29) For ay orthogoal matrix U, tr{u H D{W}U} = 0. (30) For ay orthogoal matrix U ad ay diagoal matrix W D, D{U H W D U} = m(w D )I + D (W D ; U), (3) where m(w D ) N (W D ) l,l (32) N is the average of the diagoal elemets of W D ad D (W D ; U) is a diagoal matrix whose -th diagoal elemet is Usig the Cauchy-Schwarz iequality, l=0 N ( D (W D ; U)), (W D ) l,l ( (U) l, 2 ). (33) N ( D (W D ; U)), 2 l=0 ( N l=0 (W D ) l,l 2 ) s T (U; ), (34) where s T (U; ) is the -th time-despreadig factor for U H as defied i (22). 4

24 x x~ E C(h) r~ F H A (i) D F ~ ( i) y ~ ( i) z H E C( b (i) (i) z ( i ~ˆ x ) Decisio ) E Delay ( ) ˆ i x ( ) ˆ i x Figure 5. Equivalet system model To facilitate our aalysis, we use a equivalet model of the system i Figure 3. The equivalet model is show i Figure 5. I the figure, h [h 0, h,..., h L, 0,..., 0 ] T (35) }{{} N L zeros is a N-dimesioal vector whose first L elemets are {h k } L k=0 ad the rest are zero. C(h) C N N is the circulat matrix [22] defied as h h L h. h h h hl C(h). h L.... h (36). 0 h... L h h h L h.. L 2 h 0 It is well-kow that a circulat matrix ca be diagoalized by the DFT matrix. For example, where ad H is the DFT of h: C(h) = F H H D F, (37) H D = diag(h), H = NFh. Also, b, C(b), B, ad B D are similarly defied. 5

25 From Figure 5, the decisio vector for the i-th iteratio is z (i) = E H F H A (i) D H DFEx E H C(b (i) )Eˆx (i ) +E H F H A (i) D F, (38) where E is the EST matrix defied i (2), ˆx (i ) is the hard- or soft-decisio vector for x at the (i )-th iteratio, ad A (i) D = diag(a(i) 0,, A(i) N ). At each iteratio, the -th decisio variable, z (i), cosists of the desired sigal, iterferece from other symbols, ad oise compoets, whose powers are deoted as P (i) si,, P(i) i,, ad P (i) o,, respectively. The, the symbol-error rate (SER) at the i-th iteratio, p (i), is the average of the SERs of each symbol, p (i) : where p (i) = N N =0 p (i) SINR (i) = = N N =0 P (i) si, P (i) i, + P(i) o, Ψ(SINR (i) ), (39) is the sigal-to-iterferece-oise ratio (SINR) at the -th symbol at the i-th iteratio, ad Ψ( ) is a fuctio that maps SINR ito SER for a give modulatio scheme. For the coveiece of our aalysis, we assume that the iterferece from other symbols be Gaussia. Cosequetly, for QPSK modulatio [], (40) Ψ(x) = [ Q( x)] 2. (4) 2.4. MMSE Equalizer At the first iteratio, a MMSE equalizer is used. Therefore, the decisio vector ca be writte as z () = D{E H F H M D FE}x + }{{} D{E H F H M D FE}x }{{} sigal iterferece + E H F H A () D }{{ F, } (42) oise 6

26 where M D A () D H D. From (28) ad (42), the desired sigal, iterferece, ad oise powers for the -th decisio variable are ad P () si, = σ2 x{(e H F H M D FE), } 2, P () i, = σ2 x[(e H F H M 2 DFE), {(E H F H M D FE), } 2 ], P () o, = σ 2 (E H F H A () D 2 FE),. As idicated before, E = F H correspods to a OFDM system. I this case, iterferece from other symbols is zero ad the SINR for the -th decisio variable is SINR ofdm = σ2 x (H D ), 2. O the other had, if E = I, the system will be SC-FDE with MMSE equalizatio. I this case, the SINR for the -th decisio variable is idepedet of ad ca be expressed as where m si, m i, ad m o are ad SINR sc mmse = σ 2 σ 2 xm 2 si σ 2 xm i + σ 2 m o, (43) m si = N (M D ) l,l, (44) N l=0 l=0 m i = N {(M D ) l,l } 2 { N (M D ) l,l } 2, (45) N N m o = N N l=0 l=0 (A () D ) l,l 2. (46) I geeral, the desired sigal, iterferece, ad oise powers for the -th decisio variable ca be writte as ad P () si, = σ2 x[m si + α ] 2, P () i, = σ2 x[m i + β 2m si α α 2 ], P () o, = σ 2 [m o + γ ], 7

27 where α = (E H D{F H M D F}E),, β = (E H D{F H M 2 DF}E),, ad γ = (E H D{F H A () D 2 F}E),. are perturbatios. From (30), the averages of the perturbatios are all zero: N Also, usig matrix idetity (3), α = N β = N γ = 0. α = D (M D ; FE),, (47) β = D (M 2 D; FE),, (48) ad γ = D ( A () D 2 ; FE),. (49) From (34) ad (47)-(49), whe s T (FE; ) or s F (E; ) is sufficietly small, α, β, ad γ ca be igored ad the SER ca be approximated by p () = N N =0 Ψ(SINR () ) Ψ(SINR () ), (50) where SINR () is the mea SINR obtaied by igorig the perturbatios i SINR (). For the EST with perfect frequecy spreadig, i.e., s F (E; ) = 0 for 0 N, α, β, ad γ are all zero ad (50) is a exact expressio of the SER Geie-Aided Equalizer For the geie-aided equalizer, the iterferece symbols are assumed to be kow whe detectig the desired symbol. Therefore, their effect o the desired symbol ca be completely cacelled. It is derived i Appedix B that the decisio vector will be z (g) = g 0 x + E H F H H H DF, (5) 8

28 where g 0 = (E H D{F H H D 2 F}E), = ( H D 2 ) l,l. N Cosequetly, the desired sigal ad oise power for the -th decisio variable are ad respectively, where P (g) si, = g2 0σ 2 x, P (g) o, = σ 2 (g 0 + ξ ), ξ = (E H D{F H H D 2 F}E),. As discussed before, the average of ξ is zero, ad for the EST with sufficietly small s F (E; ), ξ ca be eglected. Cosequetly, SER ca be approximated as p (g) = N N =0 l Ψ(SINR (g) ) Ψ(SINR (g) ), (52) where SINR (g) is the SINR obtaied by igorig the perturbatio ξ i SINR (g). Similar to Subsectio 2.4., for the EST with perfect frequecy spreadig, i.e., s F (E; ) = 0 for 0 N, the perturbatio ξ = 0 ad (52) is a exact expressio of the SER Iterative Equalizer with Hard Decisio We aalyze the performace of the iterative equalizer with a hard decisio. I this subsectio, ˆx is used to deote the hard decisio for x. We first preset a performace aalysis for a fiite block size ad the describe a simplified aalysis for a ifiite block size. It is show i Appedix B that the decisio vector for the iterative equalizer (i 2) is where z (i) = g 0 x + E H C(b)Ed (i ) + E H F H H H }{{}}{{} DF, (53) }{{} sigal iterferece oise d (i) = x ˆx (i) (54) is the hard-decisio-error vector at the i-th iteratio. Therefore, the desired sigal ad oise powers are same as those of the geie-aided receiver: P (i) si, = P(g) si,, (55) 9

29 ad P (i) o, = P (g) o,, (56) respectively. The power of iterferece from other symbols depeds o the umber ad locatio of errors at the previous iteratio. Deote D (i) to be the set of the idices of icorrectly detected symbols i a block after the i-th iteratio, whose cardiality is N (i). The, the power of iterferece from other symbols i the -th decisio is where P (i) i, (D(i ) ) = (E H C(b)EΩ(D (i ) )E H C(b) H E),, (57) Ω(D (i ) ) E{d (i ) d (i )H D (i ) } (58) is the coditioal error covariace matrix. Ω(D (i) ) is a diagoal matrix whose mai diagoal cosists of N (i) ozero elemets ad (N N (i) ) zero elemets, ad Ω(D (i) κ(n (i) /N)σx 2 if D (i) ), =, (59) 0 if / D (i) where κ(p) E{ d 2 D (i) } σ 2 x is a fuctio of SER that depeds o the modulatio scheme. It ca be show that for QPSK, where κ(p) 4 2 (6) 2p. By direct calculatio, the power of iterferece i (57) ca be decomposed ito P (i) i, (D(i ) ) = N (i ) N (60) κ(n(i ) N )σ2 xg 2 0(K + ɛ ), (62) ad K = (EH D{C(b)D{EΩ(D (i ) )E H }C(b) H }E), N (i ) N (i ) N κ( N )σ2 xg0 2 (63) ɛ = (EH D{C(b) D{EΩ(D (i ) )E H }C(b) H }E), N (i ) N (i ) N κ( N )σ2 xg0 2 + (EH D{C(b)EΩ(D (i ) )E H C(b) H }E),. (64) N (i ) N (i ) N κ( N )σ2 xg0 2 20

30 By matrix idetities (3) ad (34), if s T (E H ; ) is sufficietly small, K ca be well approximated as a costat: K h 2 L g0 2 g l 2, (65) which idicates the frequecy selectivity of the chael. However, the perturbatio term, ɛ, is comparable with K h ad it ca ot be igored uless N is large eough. The coditioal l= SINR for the -th decisio variable give D (i ) is SINR (i) (D (i ) ) = N (i ) N κ( N (i ) N )(K + ɛ ) + g 0+ξ g 2 0SNR. (66) The probability that the symbol errors occur oly at the idices i D (i) at the i-th iteratio is P{D (i) } = p (i) D (i) Thus, the SER for the -th symbol at the i-th iteratio is p (i) = Ψ(SINR (i) (D (i ) ))P{D (i ) }, D (i ) ( p (i) ). (67) / D (i) where P{ } stads for probability. From this, the the SER at each iteratio ca be calculated recursively. Previously, we obtaied a recursive formula to calculate the exact SER of the iterative equalizer for a fiite block size. Here, we derive a simplified SER expressio of the iterative equalizer for a ifiite block size. For simplicity, we assume the EST is ideal. I this case, s F (E; ) = 0 for all 0 N ; therefore, the perturbatio terms disappear from the SINR expressios for MMSE ad geie-aided equalizers: SINR () = SINR (), (68) SINR (g) = SINR (g), (69) which are idepedet of the symbol idex. Cosequetly, the SINR after MMSE equalizatio ca be expressed as SINR () = N N k=0 SNR H k 2 +. (70) 2

31 The correspodig SER ca be calculated by p () = Ψ(SINR () ). (7) Now, we show that the perturbatio term that depeds o i the coditioal iterferece power P (i) i, (D(i ) ) i (62) ca be igored for a large N. From the defiitio of the ideal EST, we ca write (E),l = N e jθ,l, where {θ,l } N,l=0 is pseudo-radomly ad eve-symmetrically distributed i [ π, π]. Defie v as the iterferece compoet i the -th decisio variable of z (i) i (53) give D (i ) : where v = l D (i ) c,l d (i ) l, (72) c,l (E H C(b)E),l = N N N m =0 m 2 =0 (C(b)) m,m 2 e j(θ m 2,l θ m,). (73) Cosiderig θ m,l as a idepedet radom variable with zero mea, c,l ca be treated as a Gaussia radom variable ivokig the cetral limit theorem [23]. I this case, the mea ad variace of c,l are ad m E{c,l } = 0, V{c,l } = N 2 (C(b)) m,m 2 2 = g2 0 K h N, m 2 respectively, where E{ } ad V{ } deote statistical expectatio ad variace, respectively. Also, we see that c,l ad c,l2 are idepedet for l l 2 sice they are Gaussia ad E{c,l c,l2 } = E{c,l }E{c,l2 } = 0. Therefore, v i (72) is the sum of N (i ) idepedet complex Gaussia radom variables. Cosequetly, the coditioal iterferece power for the -th decisio variable, Y P (i) i, (D(i ) ) = E{ v 2 D (i ) } (74) 22

32 is a χ 2 -distributed radom variable with 2N (i ) degrees of freedom, whose probability desity fuctio (PDF) is f Y (y D (i ) ) = σ 2 χ2 N (i ) Γ(N (i ) ) yn (i ) e y/2σ2 χ = f Y (y N (i ) ), (75) where σ 2 χ = N (i ) κ( N )σ2 xg0 2K h. 2N I (75) we chaged the coditio of f Y (y D (i ) ) to N (i ) sice it depeds oly o N (i ) rather tha D (i ). Hereafter, for this reaso, we chage the coditio from D (i ) to N (i ) i all the related coditioal fuctios, i.e., the power of iterferece, SINR, ad SER. The mea ad variace of Y are ad E{Y } = 2N (i ) σ 2 χ = N (i ) N V{Y } = 4N (i ) σ 4 χ = N (i ) N 2 κ(n(i ) N )σ2 xg 2 0K h, (76) κ( N (i ) N )2 σ 4 xg 4 0K 2 h, (77) respectively. Therefore, for a large N, V{Y } is small ad P (i) i, (N (i ) ) ca be approximated to its mea i (76), which is idepedet of the symbol idex. From the above discussio, for a iterative equalizer with the ideal EST ad a large N, the SINR ca be simplified to SINR (i) (N (i ) ) = N (i ) N N (i ) κ( N )K. (78) h + g 0 SNR Cosequetly, the coditioal SER is p (i) (N (i ) ) = Ψ(SINR (i) (N (i ) )). (79) I this case, SER is idepedet of ad the probability of havig k symbol-errors i a N-symbol block at the i-th iteratio is P{N (i) = k} = ( ) N (p (i) ) k ( p (i) ) N k, (80) k 23

33 for k = 0,,, N. Cosequetly, the SER at the i-th (i 2) iteratio is p (i) = N p (i) (N (i ) = )P{N (i ) = }. (8) =0 From Equatios (70), (7), ad (78) - (8), the SER after the i iteratio ca be calculated. To study the asymptotic property, we first defie the relative frequecy of symbol error i a block after the i-th iteratio by Therefore, (8) ca be expressed i terms of F (i ) as p (i) = N =0 Whe N, (83) ca be writte as p (i) = 0 F (i) N (i) N. (82) p (i) (F (i ) = N )P{F (i ) = }. (83) N Ψ(SINR (i) (F (i ) = x))f F (i )(x)dx, (84) where f F (i )(x) is the PDF of F (i ). From the DeMoivre-Laplace Theorem [23], f F (i )(x) ca be approximated as Gaussia PDF with mea p (i ) ad variace p(i ) ( p (i ) ) N. By the defiitio of the Dirac Delta fuctio δ(x) [24], Therefore, the effective SINR is lim f F (i )(x) = δ(x N p(i ) ). ad SINR (i) = SINR (i) (F (i ) = p (i ) ) (85) = K h p (i ) κ(p (i ) ) +. (86) g 0 SNR p (i) = Ψ(SINR (i) ). (87) The strog law of large umbers states that the relative frequecy F (i ) approaches p (i ) almost everywhere (AE) [23]. Let r(x) Ψ( K h xκ(x) + ). g 0 SNR 24

34 The r(x) is cotiuous i the iterval [0, ] ad differetiable i (0, ), ad r(f (i ) ) r(p (i ) ) R max F (i ) p (i ) where R max = max x (0,) dr dx is a fiite umber. Therefore, the covergece i (87) is also AE. From (86) ad (87), the SINR after the i-th iteratio is iversely proportioal to p (i ), ad p (i) mootoically decreases with SINR (i). Therefore, if the ˇp = lim i p (i) ad SINR () < SINR (2), (88) ˇ SINR = lim i SINR (i) exist ad satisfy the followig equatios ˇp = Ψ( SINR) ˇ (89) ad ˇ SINR = K h ˇpα(ˇp) +. (90) g 0 SNR I particular, from (4), (86) ad (89), for QPSK symbols, ˇp is determied by ( ) Q ˇp K h ˇp + g 0 SNR 2 = ˇp (9) It ca be see from (9) that for a sufficietly small ˇp, the iterferece term is egligible compared with /(g 0 SNR); cosequetly, ˇ SINR is very close to SNR, the MFB. From the above discussio, the iterative equalizatio coverges above a SNR threshold, SNR T that satisfies SINR () = SINR (2). Equivaletly, if SNR > SNR T, the followig iequality holds: N N k=0 SNR H k 2 + < α(p () s )K h p () + g 0 SNR. (92) Furthermore, we have proved i Appedix D that SNR T exists for all chaels. Therefore, the iterative approach always coverges to the MFB as log as SNR is large eough. 25

35 2.4.4 Iterative Equalizer with Soft Decisio By feedig back the soft decisio, we ca prevet error propagatio i the hard-decisio equalizer ad further ehace the performace, especially at low SNR. I this subsectio, we deote ˆx as the soft decisio for x. Also, for simplicity, we assume that QPSK ad the ideal EST are employed. We cosider the ormalized decisio vector at the i-th iteratio z (i) = x + e (i), (93) where e (i) is the pre-decisio-error vector that cosists of oise ad iterferece. the previous sectio, each elemet e (i) As i of e (i) is assumed to be idepedet for differet s ad complex Gaussia with power (σ (i) e,) 2. For QAM modulatio, all the variables i (93) are complex, they ca be decomposed ito real (i-phase) ad imagiary (quadrature) compoets, that is, z (i) z (i) I, +jz(i) Q,, x x I, +jx Q,, ad e (i) e (i) I, +je(i) Q,. Hereafter, to avoid repetitio, we will describe oly the i-phase compoet of a complex variable. The quadrature compoet ca be similarly defied. Log-likelihood ratio (LLR), which is widely kow i the turbo literature [4, 8, 9], is employed here for the soft decisio. The a posteriori LLR of x I, at the i-th iteratio is It ca be decomposed ito I, log P[x I, = + z (i) I, ] P[x I, = z (i) (94) ]. λ (i) I, λ (i) I, = λe,(i) I, + λp,(i) I,, (95) where ad λ E,(i) I, P[z(i) log I, xi, = +] P[z (i) I, x I, = ] = λ P,(i) I, log P[x I, = +] P[x I, = ] 2z (i) I, (σ (i) e,i, )2 (96) are the extrisic LLR ad the a priori LLR, respectively. We use the extrisic LLR from the previous iteratio as the a priori LLR: (97) λ P,(i) I, = λe,(i ) I,. (98) 26

36 For the first iteratio, sice there is o a priori LLR available, we set λ P,() I, = 0. Also, as show i Appedix C, pre-decisio-error power after MMSE equalizatio is (σ e,) () = σ H k 2 x 2 N k H k 2 +/SNR (99) (σ e () ) 2, (00) which is idepedet of. Each compoet has the same pre-detectio-error power after MMSE equalizatio, that is, (σ () e,i )2 = (σ () e,q )2 = (σ () e ) 2 /2. z (i) I, : The soft decisio for x I, is the coditioal expectatio of x I, give the observatio ˆx (i) I, E{x I, z (i) I, } = P[x I, = z (i) I, ] P[x I, = z (i) I, ], (0) which, i terms of LLR, ca be easily show to be tah(λ(i) I, = I, ). (02) 2 ˆx (i) The soft-decisio error for the -th symbol at the i-th iteratio is defied as d (i) x ˆx (i). (03) The power of its i-phase compoet is (i) E{( d I, )2 z (i) I, } = ( ˆx(i) I, )2 P{x I, = z (i) I, } + ( + ˆx(i) I, )2 P{x I, = z (i) I, } = (ˆx (i) I, )2. (04) Similar to the hard-decisio case i (B.5), the decisio variable (after ormalizatio) is z (i) = }{{} x + E H C(b (i) )E d (i ) + E H F H H H g 0 g DF, (05) sigal }{{}} 0 {{} iterferece oise where d (i) is the soft-decisio error vector at the i-th iteratio. The coditioal iterferece power give z (i ) is P i, (z (i ) ) = g 2 0 (E H C(b)E Ω(k) D E H C(b) H E),, (06) where Ω(k) D E{ d (i ) d(i )H z (i ) } (07) 27

37 is the coditioal soft-decisio-error covariace matrix. As i the previous sectio, for a fiite block size, P i, (z (i ) ) depeds o the idex ad the complexity of calculatig its value for each, 0 N, is high. However, for a ifiite block size ad employig the ideal EST, P i, (z (i ) ) is equal to its average value, P (i) i = K h N N =0 E{ (i ) d 2 z (i ) }, which does ot deped o. I this case, the pre-decisio error power at the i-th iteratio (i 2) is idepedet of ad give by where (σ (i) e,i )2 = P (i) i,i + σ2,i g 0, (08) P (i) i,i = K N h N =0 E{( (i ) d I, )2 z (i ) I, }. Fially, for a ifiite block legth, the SER at the i-th iteratio (i ) is where (σ e (i) ) 2 = (σ (i) e,i )2 + (σ (i) e,q ) Simulatio Results p (i) = Ψ( σ 2 x (σ e (i) ), (09) ) 2 I this sectio, we preset simulatio results usig QPSK modulatio. Sice the performace of a equalizer usually depeds o the characteristics of a chael, we preset our results for differet types of chaels Performace for Proakis-B chael I this subsectio, we preset the performace of the hard- ad soft-decisio equalizer with differet ESTs ad block sizes usig the Proakis-B chael [], whose impulse respose ad K h are h P roakis B = 0.407δ δ δ 2, K h = 0.94, 28

38 respectively. We cosider E, E 2, E 5 i Table i our simulatio. Sice each EST i Table has either the maximal time- or the maximal frequecy-despreadig factor for = 0, we sed a dummy symbol for x 0 ad trasmit iformatio through the rest of the N symbols. Figure 6 (a) compares the aalytical ad simulatio results for the hard-decisio equalizer. To calculate aalytical performace, we used the ideal EST ad a ifiite block size. For the simulatio results, we used E, E 2, ad E 5 for the EST ad set the block size at N = Note that the hard-decisio equalizer with E correspods to OFDM. From the 0 0 (a) Hard decisio 0 st & 2d iter. 0 2 st iter. BER 0 3 MFB 3rd iter. 2d iter. 0 4 Simulatio, E, N=2048 Simulatio, E 5, N=2048 Simulatio, E 2, N=2048 Aalysis, N = MFB 0th iter SNR per bit, γ (db) b 0 0 (b) Soft decisio 0 st iter d iter. BER 3rd iter. 0 3 DFE with 4th iter. perfect feedback 0 4 Simulatio, E, N= Simulatio, E 2, N=2048 DFE with prefect feedback MLSD MLSD MFB 0th iter SNR per bit, γ b (db) Figure 6. Performace of the EST-based equalizer for Proakis-B chael. 29

39 figure, the iterative equalizer with E as a EST has o performace improvemet with iteratio sice E has poor frequecy spreadig. The BER of the equalizer based o E 2 improves with the umber of iteratios whe SNR is above 7 db. After the teth iteratio, the required SNR for a 0 5 BER is about 9.8 db, which is oly 0.2 db from the MFB. The equalizer based o E 5 shows slightly worse performace tha that based o E 2. The aalysis for the MMSE equalizer (st iteratio) is very close to the simulatio result. There is a performace gap betwee the aalytical ad simulatio results for the other iteratios, which is due to fiite block legth ad imperfect eergy spreadig. Figure 6 (b) shows the performace of the soft-decisio equalizer compared with that of the MLSD [] ad the DFE assumig perfect feedback [], which are two covetioal schemes that do ot employ the EST. By feedig back the soft decisio, the performace is sigificatly improved over the hard-decisio case, especially at a low SNR. After the teth iteratio, there is almost a 2 db gai for the soft decisio at BER = 0 2 over the hard decisio. The soft-decisio equalizer based o E 2, after the third iteratio, outperforms the DFE with perfect feedback by 0.4 db at BER = 0 4. After the teth iteratio ad at BER = 0 4, its performace is 2.5 db better tha the MLSD ad is very close to the MFB. Figure 7 shows the relatioship betwee the performace of the hard-decisio equalizer at 0 db ad its block size. The performace degrades as block size decreases, which ca N=28 N=256 N=52 N=024 N=2048 N= BER Iteratio Figure 7. BER performace verses iteratio with differet block sizes N at 0 db for Proakis-B chael ad hard decisio. 30

40 be aticipated from the variace of Y P (i) i, (D(i ) ) i (77); the block performace, coditioed o D (i ), is domiated by the symbol detectio with the largest P (i) i, (D(i ) ), but decreasig block size icreases the variace of P (i) i, (D(i ) ) Performace for other challegig chaels I this subsectio, we preset the performace of the equalizer for other challegig chaels. I the simulatio, we use E 2 for the EST ad the block size N is assumed to be 2048 if ot explicitly stated. Figure 8 (a) shows the performace of the equalizer for the chael proposed by Porat ad Friedlader [26], whose ormalized impulse respose ad K h are P orat et al h = ( j)δ + ( j)δ δ 2 K h = 0.73, + ( j)δ 3 + ( j)δ 4, respectively. The SNR threshold for the hard-decisio equalizer occurs ear 3. db. Also, its performace after the teth iteratio ad above 6 db is similar to that of the soft-decisio equalizer ad very close to the MFB. Figure 8 (b) shows the performace of the equalizer for the Proakis-C chael, whose impulse respose ad K h are h P roakis C = 0.227δ δ δ δ δ 4, K h = 2.3, respectively. This chael has the severest frequecy selectivity amog the determiistic chaels used for the simulatio. With this chael, because of the high K h, N = 2048 is ot sufficietly large for the hard-decisio equalizer to be approximated as the ideal harddecisio equalizer with a ifiite block size. Therefore, we use N = 4096 for the simulatio with this chael. The SNR threshold occurs ear 22.6 db, which is much higher tha that of the previous chaels. However, the performace of the equalizer with the soft decisio, after the teth iteratio ad at BER = 0 5, has 8.5 db gai over the hard decisio ad 3

41 0 0 0 (a) Porat ad Friedlader chael Simulatio, hard decisio Aalysis, N =, hard decisio Simulatio, soft decisio DFE with perfect feedback MLSD MFB 0 2 BER 0 3 MFB st iter. 0 4 DFE with perfect feedback 0th iter. MLSD SNR per bit, γ b (db) 0 0 (b) Proakis C chael 0th iter. aalysis, hard decisio, N= 0th iter simulatio hard decisioi 0 st iter. BER MFB MLSD 0th iter. simulatio soft decisio DFE with perfect feedback SNR per bit, γ (db) b 0 0 (c) BU chael Simulatio, hard deciso Simulatio, soft deciso MFB BER 0 3 MFB st iter d iter. 0th iter SNR per bit, γ b (db) Figure 8. Performace of the EST-based equalizer for differet types of chaels. 32

42 outperforms the MLSD ad the DEF with perfect feedback by.5 db ad 3 db, respectively. From the slope of the BER curves of the equalizer ad the MFB, we ca estimate that the proposed equalizer, after the teth iteratio, will reach the MFB at a very low BER (far below 0 5 ) ear 5 db ad 23.3 db for a soft decisio ad a hard decisio, respectively. Figure 8 (c) shows the average performace of the equalizer for reduced bad-urba (BU) [29] chael. We use reduced BU power delay profile to geerate 000 fiite impulse respose (FIR) chael realizatios that have, assumig 0.95 µs symbol duratio, symbol-spaced taps. For those chael realizatios, the mea ad variace of K h are 0.63 ad 0.045, respectively. Also, the mea ad variace of g 0 are.0 ad 0.26, respectively. For the chael realizatios with low g 0 ad/or high K h, the SNR threshold will be high. Below 3 db, as show i the figure, the performace of the hard-decisio equalizer, after the teth iteratio, is domiated by error propagatio of those chaels with a high SNR threshold. However, the soft-decisio equalizer prevets error propagatio ad shows good performace, which is close to the MFB after the teth iteratio. 33