Filter Bank Transceivers for OFDM and DMT Systems

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FILTER BAN K TRANSCEIVER S FO R OFD M AN D DMT SYSTEM S Providing key background material together with advanced topics, this self-contained book i s writte n i n a n easy-to-rea d styl e an d i s idea l fo r newcomer s t o multicarrie r systems. Early chapters provide a review of basic digital communication, starting from th e equivalent discrete-time channel and including a detailed review of the MMSE receiver. Later chapters then provide extensive performance analysis of OFDM and DMT systems, with discussion s o f man y practica l issue s such a s implementation an d power spectrum considerations. Throughout, theoretical analysis is presented alongside practical design considerations, whilst the filter bank transceiver representation of OFDM and DMT systems opens up possibilities for further optimization such as minimum bit error rate, minimum transmission power, and higher spectral efficiency . With plenty of insightful real-world examples and carefully designed end-of-chapter problems, this is an ideal single-semester textbook for senior undergraduate and graduate students, as well as a self-study guide for researchers and professional engineers. YUAN-PEI LI N i s a Professor i n Electrical Engineering at the National Chiao Tung University, Hsinchu, Taiwan. She is a recipient of the Ta-You Wu Memorial Award, the Chinese Institute o f Electrica l Engineering’ s Outstandin g Yout h Electrical Enginee r Award, and of the Chinese Automatic Control Society’s Young Engineer in Automatic Control Award. SEE-MAY PHOON G i s

a Professor i n the Graduate Institute of Communication Engineering and the Department of Electrical Engineering at the National Taiwan University (NTU). He is a recipient o f th e Charles H. Wilts Prize for outstandin g independen t doctoral research in electrical engineering at the California Institute of Technology, and the Chinese Institute of Electrical Engineering’s Outstanding Youth Electrical Engineer Award. P . P . VAIDYANATHA N i

s a Professor i n Electrica l Engineerin g a t th e Californi a Institute o f Technology , where he has been a faculty membe r sinc e 1983. He is an IEEE Fellow and has authored over 400 technical papers, four books, and many invited chapters in leading journals, conferences, an d handbooks. He was a recipient o f th e Award for Excellence in Teaching at the California Institute of Technology three times, and he has received numerous other awards including the F. E. Terman Award of the American Society for Engineering Education and the Technical Achievement Award of the IEEE Signal Processing Society.

FILTER BAN K TRANSCEIVER S FO R OFDM AN D DM T SYSTEM S YUAN-PEI LI N National Chiao Tung University, Taiwan

SEE-MAY PHOON G National Taiwan University

P. P . VAIDYANATHA N California Institute of Technology

CAMBRIDGE UNIVERSIT Y PRES S

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information o n this title: www.cambridge.org/9781 107002739 © Cambridge University Press 2011 This publication is in copyright. Subject to statutory exceptio n and to the provisions of relevant collective licensing agreements , no reproduction of any part may take place without the written permission of Cambridge University Press. First published 201 1 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library ISBN 978-1-107-00273-9 Hardbac k

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred t o in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To our families Yuan-Pei Lin and See-May Phoong

To Usha, Vikram, Sagar, and my parents — P. P. Vaidyanathan

Contents Preface x

i

1 Introductio n 1 1.1 Notation s 7 2 Preliminarie s o f digita l communication s 9 2.1 Discrete-tim e channe l model s 9 2.2 Equalizatio n 1 2.3 Digita l modulatio n 1 2.3.1 Puls e amplitud e modulatio n (PAM ) 1 2.3.2 Quadratur e amplitud e modulatio n (QAM ) 2 2.4 Paralle l subchannel s 2 2.5 Furthe r readin g 3 2.6 Problem s 3

6 7 8 2 8 1 1

3 FI R equalizer s 3 3.1 Zero-forcin g equalizer s 3 3.2 Orthogonalit y principl e an d linea r estimatio n 3 3.2.1 Biase d an d unbiase d linea r estimate s 4 3.2.2 Estimatio n o f multipl e rando m variable s 4 3.3 MMS E equalizer s 4 3.3.1 FI R channel s 4 3.3.2 MIM O frequency-nonselectiv e channel s 4 3.3.3 Example s 5 3.4 Symbo l detectio n fo r MMS E receiver s 5 3.5 Channel-shortenin g equalizer s 5 3.6 Concludin g Remark s 6 3.7 Problem s 6

3 4 9 1 4 5 5 8 0 6 9 5 5

4 Fundamental s o f multirat e signa l processin g 7 4.1 Multirat e buildin g block s 7 4.1.1 Transfor m domai n formula s 7 4.1.2 Multirat e identitie s 7 4.1.3 Blockin g an d unblockin g 7 4.2 Decimatio n filter s 7 4.3 Interpolatio n filter s 8 4.3.1 Tim e domai n vie w o f interpolatio n filte r 8 4.3.2 Th e Nyquist(M ) propert y 8

1 1 3 5 6 9 0 2 2

vn

CONTENTS

Vlll

4.4 Polyphas e decompositio n 8 4.4.1 Decimatio n an d interpolatio n filter s 8 4.4.2 Synthesi s filte r bank s 8 4.4.3 Analysi s filte r bank s 9 4.5 Concludin g remark s 9 4.6 Problem s 9

4 7 9 0 1 1

Multirate formulatio n o f communicatio n system s 9 5.1 Filte r ban k transceiver s 9 5.1.1 Th e multiplexin g operatio n 9 5.1.2 Redundanc y i n filte r ban k transceiver s 9 5.1.3 Type s o f distortio n i n transceiver s 10 5.2 Analysi s o f filte r ban k transceiver s 10 5.2.1 ISI-fre e filte r ban k transceiver s 10 5.2.2 Polyphas e approac h 10 5.2.3 Channel-independen t ISI-fre e filte r ban k transceiver s 10 5.3 Pseudocirculan t an d circulan t matrice s 10 5.3.1 Pseudocirculant s an d blocke d version s o f scala r system s 10 5.3.2 Circulant s an d circula r convolution s 10 5.4 Redundanc y fo r IB I eliminatio n 11 5.4.1 Zero-padde d system s 11 5.4.2 Cyclic-prefixe d system s 11 5.4.3 Summar y an d compariso n 11 5.4.4 IBI-fre e system s wit h reduce d redundanc y 12 5.5 Fractionall y space d equalize r system s 12 5.5.1 Zero-forcin g F S E system s 12 5.5.2 Polyphas e approac h 12 5.6 Concludin g remark s 12 5.7 Problem s 12

5 5 7 7 0 1 1 3 5 6 6 8 1 2 5 9 1 2 4 5 9 9

D F T - b a s e d t r a n s c e i v e r s 13 6.1 O F D M system s 13 6.1.1 Nois e analysi s 14 6.1.2 Bi t erro r rat e 14 6.2 Zero-padde d O F D M system s 14 6.2.1 Zero-forcin g receiver s 14 6.2.2 Th e MMS E receive r 15 6.3 Single-carrie r system s wit h cycli c prefi x (SC-CP ) 15 6.3.1 Nois e analysis : zero-forcin g cas e 15 6.3.2 Th e MMS E receive r 15 6.3.3 Erro r analysis : MMS E cas e 15 6.4 Single-carrie r syste m wit h zero-paddin g (SC-ZP ) 16 6.5 Filte r ban k representatio n o f O F D M system s 16 6.5.1 Transmitte d powe r spectru m 16 6.5.2 Z P - O F D M system s 16 6.6 D M T system s 16 6.7 Channe l estimatio n an d carrie r frequencysynchronizatio n 17 6.7.1 Pilo t symbo l aide d modulatio n 17 6.7.2 Synchronizatio n o f carrie r frequenc y 17

5 6 0 2 7 7 0 2 5 6 7 0 3 6 8 8 8 8 9

CONTENTS 6.8 A historica l not e an d furthe r readin g 18 6.9 Problem s 18 7 P r e c o d e d O F D M s y s t e m s 19 7.1 Zero-forcin g precode d O F D M system s 19 7.2 Optima l precoder s fo r Q P S K modulatio n 19 7.3 Optima l precoders : othe r modulation s 20 7.4 MMS E precode d O F D M system s 20 7.4.1 MMS E receiver s 20 7.4.2 Optima l precoder s fo r Q P S K modulatio n 20 7.4.3 Othe r modulatio n scheme s 20 7.5 Simulatio n example s 21 7.6 Furthe r readin g 21 7.7 Problem s 22

IX

0 1 3 4 8 2 3 4 7 9 1 9 0

8 Transceive r desig n w i t h channe l informatio n a t th e transmitter22 3 8.1 Zero-forcin g bloc k transceiver s 22 3 8.1.1 Zero-forcin g Z P system s 22 5 8.1.2 Zero-forcin g Z J system s 22 6 8.2 Proble m formulatio n 22 8 8.3 Optima l bi t allocatio n 22 9 8.4 Optima l Z P transceiver s 24 0 8.4.1 Optima l G zp 24 0 8.4.2 Optima l A zp 24 1 8.4.3 Summar y an d discussion s 24 3 8.5 Optima l zero-jammin g (ZJ ) transceiver s 24 7 8.5.1 Optima l S^ - 24 7 8.5.2 Optima l A zj 24 9 8.5.3 Summar y an d discussion s 24 9 8.6 Furthe r readin g 25 3 8.7 Problem s 25 4 9 D M T s y s t e m s wit h improve d frequenc y characteristic s 25 9.1 Sidelobe s matter ! 26 9.2 Overal l transfe r matri x 26 9.3 Transmitter s wit h subfilter s 26 9.3.1 Choosin g th e subfilter s a s a D F T ban k 26 9.3.2 D F T ban k implementatio n 26 9.4 Desig n o f transmi t subfilter s 27 9.5 Receiver s wit h subfilter s 27 9.5.1 Choosin g subfilter s a s a D F T ban k 27 9.5.2 D F T ban k implementatio n 27 9.6 Desig n o f receive r subfilter s 28 9.7 Zero-padde d transceiver s 28 9.8 Furthe r readin g 28 9.9 Problem s 28

9 0 3 5 6 6 2 6 7 7 0 5 5 6

X

CONTENTS

10 Minimu m redundanc y FI R transceiver s 29 10.1 Polyphas e representatio n 29 10.2 Propertie s o f pseudocirculant s 29 10.2.1 Smit h for m decompositio n 29 10.2.2 DF T decompositio n 29 10.2.3 Propertie s derive d fro m th e tw o decomposition s 29 10.2.4 Congruou s zero s 29 10.3 Transceiver s wit h n o redundanc y 30 10.3.1 FI R minima l transceiver s 30 10.3.2 II R minima l transceiver s 30 10.4 Minimu m redundanc y 30 10.5 Smit h for m o f FI R pseudocirculant s 30 10.6 Proo f o f Theore m 10. 2 31 10.6.1 Identica l Smit h form s 31 10.6.2 Zero s fro m differen t Bi decoupl e 31 10.6.3 A n exampl e o f derivin g th e Smit h for m o f 5](z ) 31 10.6.4 Smit h for m o f £(z ) 31 10.7 Furthe r readin g 31 10.8 Problem s 31

1 2 3 4 5 6 7 1 1 1 3 8 1 2 3 3 6 9 9

A Mathematica l tool s 32

3

B Revie w o f rando m processe s 32 B.l Rando m variable s 32 B.2 Rando m processe s 32 B.3 Processin g o f rando m variable s an d rando m processe s 33 B.4 Continuous-tim e rando m processe s 33

7 7 9 2 6

References 34

1

Index

355

Preface Recent year s hav e see n th e grea t succes s o f O F D M (orthogona l frequenc y division multiplexing ) an d D M T (discret e multitone ) transceiver s i n man y applications. T h e O F D M syste m ha s foun d man y application s i n wireles s communications. I t ha s bee n adopte d i n I E E E 802.1 1 fo r wireles s loca l are a networks, DA B fo r digita l audi o broadcasting , an d DV B fo r digita l vide o broadcasting. T h e D M T syste m i s th e enablin g technolog y fo r high-spee d transmission ove r digita l subscribe r lines . I t i s use d i n ADS L (asymmetri c digital subscribe r lines ) an d VDS L (very-high-spee d digita l subscribe r lines) . T h e O F D M an d D M T system s ar e b o t h example s o f D F T transceiver s t h a t employ redundan t guar d interval s fo r equalization . Havin g a guar d interva l can greatl y simplif y th e tas k o f equalization a t th e receive r an d i t i s now on e o f the mos t effectiv e approache s fo r channe l equalization . I n thi s boo k w e wil l study th e O F D M an d D M T unde r th e framewor k o f filte r ban k transceivers . Under suc h a framework, ther e ar e numerou s possibl e extensions . T h e freedo m in th e filte r ban k transceiver s ca n b e exploite d t o bette r th e system s fo r variou s design criteria . Fo r example , transceiver s ca n b e optimize d fo r minimu m bi t error rate , fo r minimu m transmissio n power , o r fo r highe r spectra l efficiency . We wil l explor e al l thes e possibl e optimizatio n problem s i n thi s book . T h e firs t thre e chapter s describ e th e majo r buildin g block s relevan t fo r th e discussion o f signa l processin g fo r communicatio n an d giv e th e tool s usefu l fo r solving problem s i n thi s area . Chapter s 4- 5 introduc e th e multirat e buildin g blocks an d filte r ban k transceivers , an d th e basi c ide a o f guar d interval s fo r channel equalization . Chapte r 6 give s a detaile d discussio n o f O F D M an d D M T systems . Chapter s 7-1 0 conside r th e desig n o f filte r ban k transceiver s for differen t criteri a an d channe l environments . A detaile d outlin e i s give n a t the en d o f Chapte r 1 . Thi s boo k ha s bee n use d a s a textboo k fo r a first-year graduate cours e a t Nationa l Chia o Tun g University , Taiwan , an d a t Nationa l Taiwan University . Mos t o f th e chapter s ca n b e covere d i n 16-1 8 weeks . Homework problem s ar e give n fo r Chapter s 2-10 . It i s ou r pleasur e t o t h a n k ou r familie s fo r th e patienc e an d suppor t durin g all phase s o f thi s time-consumin g project . W e woul d lik e t o t h a n k ou r univer sities, Nationa l Chia o Tun g Universit y an d Nationa l Taiwa n University , an d the Nationa l Scienc e Counci l o f Taiwa n fo r thei r generou s suppor t durin g th e writing o f thi s book . W e woul d als o lik e t o t h a n k ou r student s Chien-Chan g Li, Chun-Li n Yang , Chen-Ch i Lo , an d Kuo-Ta i Chi u fo r generatin g som e o f the plots . P P V wishe s t o acknowledg e th e Californi a Institut e o f Technology , the Nationa l Scienc e Foundatio n (USA) , an d th e Offic e o f Nava l Researc h (USA), fo r al l th e suppor t an d encouragement .

XI

1 Introduction T h e goa l o f a communicatio n syste m i s t o transmi t informatio n efficientl y and accuratel y t o anothe r location . I n th e cas e o f digita l communications , the informatio n i s a sequenc e o f "ones " an d "zeros " calle d th e bi t stream . T h e transmitte r take s i n th e bi t strea m an d generate s a n information-bearin g continuous-time signa l x a(t), a s i n Fig . 1.1 . W h e n th e signa l propagate s through th e channel , suc h a s wirelines , atmosphere , etc. , distortio n i s in evitably introduce d int o th e transmitte d signa l x a(t). A s a result , th e receive d signal r a(t) a t th e receive r i s i n genera l differen t fro m th e transmitte d signa l xa(t). T h e tas k o f th e receive r i s t o mitigat e th e distortio n an d reproduc e a bit strea m wit h a s fe w error s a s possible . 01001100... bit stream

xa(t)

transmitter

ra(t)

channel

receiver

0110110 bit strea

Figure 1 . 1 . Digita l communicatio n system .

A digita l communicatio n syste m i n genera l consist s o f man y buildin g blocks. Figur e 1. 2 show s a bloc k diagra m consistin g o f th e majo r buildin g blocks t h a t ar e relevan t t o th e topi c o f signa l processin g fo r communications . At th e transmitter , w e hav e a sequenc e o f bit s t o b e sen t t o th e receiver . T h e bits-to-symbol mapping bloc k take s severa l bit s o f inpu t an d map s th e bits t o a rea l o r comple x modulation symbol s(n). Som e processin g ma y b e applied t o thes e symbol s an d th e discrete-tim e outpu t x(n) i s the n converte d to a continuous-tim e signa l x a(t). T h e transmitte d signa l x a(t) propagate s through th e channel . A t th e receiver , th e receive d signa l r a(t) i s converte d t o a discrete-tim e signa l r(n). Usuall y som e signa l processin g i s applie d t o r(n) before th e receive r make s a decisio n o n th e transmitte d symbol s an d obtain s s~(n) (symbol detection). T h e symbol-to-bits mapping bloc k map s th e symbol s s~(n) back t o bi t stream . T h e reade r ca n fin d relevan t backgroun d materia l i n [50, 67 , 120 , 137] . W h e n a signa l propagate s throug h th e channel , distortio n i s invariabl y 1

1. Introductio n transmitter 01001100... bit strea m

bits-tosymbol mapping

transmitter! signal processing

a?o(0

D/C

Pi(«)

channel

r(n)

»■-(*)

H Pa W

C/D

receiver signal processing receiver

s(n)

i(n)

symbol detection!

symbolto-bits mapping

bit stream

Figure 1.2 . Simpl e bloc k diagra m fo r a digita l communicatio n system .

introduced t o th e t r a n s m i t t e d signal . I n additio n t o channe l noise , ther e is als o interferenc e fro m othe r symbols . A t tim e n th e receive d signa l r(n) depends no t onl y o n s(n) , bu t als o o n pas t t r a n s m i t t e d symbol s s(n — 1), s(n — 2 ) , . . . Thi s dependenc y i s terme d inter symbol interference (ISI) . T h e processing applie d t o r(n) a t th e receive r i s carrie d ou t t o obtai n estimate s of th e t r a n s m i t t e d symbol s befor e symbo l detection . Th e proces s i s generall y known a s equalizatio n an d th e signa l processin g bloc k i s calle d a n equalizer . W h e n th e receive r ca n perfectl y regenerat e th e t r a n s m i t t e d symbol s s(n) i n the absenc e o f channe l noise , w e sa y th e equalizatio n i s zero-forcing. I n man y applications, th e transmitte r als o help s wit h equalization . I n thi s cas e som e signal processin g i s applie d t o th e symbol s s(n) , an d th e resultin g outpu t x(n) is t r a n s m i t t e d a s show n i n Fig . 1.2 . One wa y t h a t th e transmitte r ca n greatl y eas e th e tas k o f equalizatio n a t the receive r i s t o divid e th e t r a n s m i t t e d signa l int o block s an d ad d redundan t samples, als o calle d a guar d interval , t o eac h block . Figur e 1. 3 show s tw o examples o f guar d interval s calle d zero padding an d cyclic prefix. I n th e zero-padding scheme , th e guar d interval s consis t o f "zeros. " W i t h cycli c prefix , the las t fe w sample s o f eac h bloc k ar e copie d an d inserte d a t th e beginnin g o f the bloc k a s show n i n th e figure. Th e guar d interva l act s a s a buffe r betwee n consecutive blocks . I f th e guar d interva l i s sufficientl y long , th e interbloc k interference (IBI ) ca n b e avoide d o r ca n b e late r remove d a t th e receive r b y discarding th e redundan t samples . W h e n ther e i s n o IBI , interferenc e come s only fro m th e sam e block . I n thi s case , intrabloc k interferenc e ca n b e cancele d easily usin g matri x operations . T h e mos t notabl e exampl e o f system s t h a t us e cycli c prefi x a s a guar d interval i s th e D F T (Discret e Fourie r Transform)-base d transceive r show n i n

3 block #1 bloc

(a)

s(n)

•" lllllHIIIII

M

k #2

l

2M

zeros padding

(b)

x(n)

zeros padding

illlllll Ihlllll. copy cyclic prefix

(c)

x(n)

ll l copy

cyclic prefix

M

II■ I

illl.l..ll

F i g u r e 1 . 3 . T w o example s o f guar d intervals , (a ) A signa l s(n) w i t h sample s divide d i n t o blocks ; ( b ) t h e sequenc e x{n) afte r zero s ar e padde d t o eac h bloc k o f s ( n ) ; (c ) t h e sequence x(n) afte

r a cycli c prefi x i s inserte d i n eac h bloc k o f s(n).

Fig. 1.4 . Th e signa l processin g a t th e transmitte r applie s IDF T (Invers e Dis crete Fourie r Transform ) t o th e inpu t bloc k o f modulatio n symbol s an d add s a cycli c prefi x t o th e IDF T outputs . Th e receive r discard s th e prefi x an d performs a DF T o f eac h block . Du e t o th e combinatio n o f cycli c prefi x an d IDFT/DFT operations , zero-forcin g equalizatio n ca n b e achieve d b y onl y a set o f simpl e scalar s calle d frequenc y domai n equalizer s (FEQs) . Whe n IS I is canceled , th e overal l syste m fro m th e transmitte r input s t o th e receive r outputs i s equivalen t t o a se t o f paralle l subchannel s a s show n i n Fig . 1.5 . In genera l th e subchannel s hav e differen t nois e variances . I f th e informatio n of the subchanne l nois e variance s i s availabl e t o th e transmitter , th e symbol s Si(n) ca n b e furthe r designe d t o bette r th e performance . Fo r example , th e symbols o f different subchannel s ca n carr y differen t number s o f bits (bi t load ing) [23] , and th e powe r o f the symbol s ca n als o be differen t (powe r loading) . The transmitte r ca n optimiz e bi t loadin g an d powe r loadin g t o maximiz e th e transmission rat e [24] . The cyclic-prefixe d DFT-base d syste m i s widel y use d i n bot h wire d an d wireless communicatio n systems . I t i s generall y calle d a n OFD M (orthogo nal frequenc y divisio n multiplexing ) syste m [27 ] in wireles s transmissio n an d a DM T (discret e multitone ) syste m [24 ] i n wire d DS L (digita l subscribe r lines) transmission . Fo r wireles s applications , th e channe l stat e informatio n is usuall y no t availabl e t o th e transmitter . Th e transmitte r i s typicall y in -

1. Introductio

n

modulation symbols

s(n)

IDFT

•• •

•• •

parallel to serial conversion

s0(n)

->

cyclic prefix

x(n)

W") transmitter signa l processing

FEQ

r(n)

discard prefix

DFT

§1

receiver signa l processing

Figure 1.4 . DFT-base d transceive r wit h cycli c prefi x adde d a s a guard interval .

dependent o f th e channe l an d ther e i s n o bi t o r powe r loading . Havin g a channel-independent transmitte r i s als o a ver y usefu l featur e fo r broadcastin g systems. Fo r broadcas t applications , ther e i s onl y on e transmitte r an d ther e are man y receivers , eac h wit h a differen t transmissio n p a t h . I t i s impossi ble fo r th e transmitte r t o optimiz e fo r differen t channel s simultaneously . I n O F D M system s fo r wireles s applications , usuall y withou t bi t an d powe r al location, th e transmitter s hav e th e desirabl e channel-independenc e property . T h e channel-dependen t par t o f the transceive r i s the se t o f F E Q coefficient s a t the receiver . I n D M T system s fo r wire d DS L applications , signal s ar e trans mitted ove r coppe r lines . Th e channe l doe s no t var y rapidly . Thi s give s th e receiver tim e t o sen d bac k th e channe l stat e informatio n t o th e transmitter . T h e transmitte r ca n the n allocat e bit s an d powe r t o th e subchannel s t o max imize th e transmissio n rate . Mor e detail s o n DS L transmissio n ca n b e foun d in [14 , 122 , 144 , 145] . Both th e O F D M an d D M T system s hav e bee n show n t o b e ver y usefu l transmission systems . Th e D M T syste m wa s adopte d i n standard s fo r ADS L (asymmetric digita l subscribe r lines ) [7 ] an d VDS L (very-high-spee d digita l subscriber lines ) [8 ] transmission . Th e O F D M system s hav e bee n adopte d in standard s fo r digita l audi o broadcastin g [39] , digita l vide o broadcastin g

^0

S]

'■

Figure 1.5 . Equivalen t paralle l subchannels .

[40], wireles s loca l are a network s [54] , an d broadban d wireles s acces s [55] . A variation o f th e cyclic-prefixe d DFT-base d transceive r i s th e so-calle d cyclic prefixed single-carrie r (SC-CP ) syste m [129] . T h e modulatio n symbol s ar e directly sen t ou t afte r a cycli c prefi x i s added . A s i n th e O F D M system , th e redundant cycli c prefi x greatl y facilitate s equalizatio n a t th e receiver . T h e SC-CP syste m i s par t o f th e broadban d wireles s acces s s t a n d a r d [55] . transmitter receive

\{n) _^ | ftf

wZ

sfo) -► ! fN

W T?

w*)-Httf

7 f~\

Hb o W | H

wZ

r

t

•• •

7 (~\

H *M-\V-) \ transmitting filters

(~\

H M o\z) \

J

U

* jv w

f

^

1

-►| H x{z) \-+\ ±N

(~\

i W|

wU

w

k

i

1

\

•• •• ••

H^M-IOOM^ receiving filters

Figure 1.6 . Filte r ban k transceiver , i n whic h onl y th e transmitte r signa l processin g and receive r signa l processin g part s ar e shown .

T h e insertio n an d remova l o f redundan t sample s ca n b e represente d us ing multirat e buildin g blocks . (Definition s o f multirat e buildin g block s wil l be give n i n Chapte r 4. ) Base d o n th e multirat e formulatio n th e DFT-base d system ca n als o b e viewe d a s a discrete-tim e filter bank transceiver (Fig . 1.6) , or a transmultiplexer. T h e transmitte r an d receive r eac h consist s o f a ban k o f discrete-time filters. Suc h a formulatio n lend s itsel f t o th e frequenc y domai n analysis o f th e transceiver . Fo r example , fo r th e transmitte r sid e i t offer s additional insigh t o n th e effec t o f individua l transmittin g filters o n th e trans mitted spectrum . Fo r th e receive r side , w e ca n analyz e th e subchanne l nois e

6

1. Introductio n

using a frequenc y domai n approach . Thes e observation s ar e ver y usefu l fo r designing th e transceive r fo r differen t criteria . I n DS L applications , goo d fre quency separatio n amon g th e transmittin g filters i s important fo r reducin g th e so-called spectral leakage, whic h i s a n undesire d spectra l componen t outsid e the transmissio n band . W h e n th e transmittin g filters hav e highe r stopban d attenuation, th e t r a n s m i t t e d spectru m ha s a faste r spectra l rollof f an d les s spectral leakage . Fo r th e receivin g filters, frequenc y separatio n i s als o impor t a n t fo r th e suppressio n o f interferenc e fro m radi o frequenc y signal s whic h share th e sam e spectru m wit h DS L signals . T h e filter ban k framewor k i s als o usefu l fo r designin g transceiver s wit h better spectra l efficiency . I n th e DFT-base d transceiver , a lon g guar d interva l is required i f the channe l impuls e respons e i s long. Th e us e o f a lon g redundan t guard interva l decrease s th e spectra l efficiency , s o w e woul d lik e th e guar d interval t o b e a s shor t a s possible . O n th e othe r hand , i t i s desirabl e t h a t th e guard interva l b e lon g enoug h s o t h a t F I R equalizatio n i s possible . Th e filter bank transceive r ca n b e use d t o introduc e guar d interval s o f a ver y genera l form. I n mos t cases , zero-forcin g equalizatio n ca n b e achieve d usin g a guar d interval muc h shorte r t h a n wha t i s neede d i n th e DFT-base d transceiver .

Outline Chapter 2 give s a n overvie w o f a digita l communicatio n system . Fro m a continuous-time channe l impuls e respons e an d channe l noise , th e equivalen t discrete-time channe l an d channe l nois e wil l b e derived . Th e equivalen t discrete-time channe l mode l i s ver y usefu l i n th e analysi s an d desig n o f digita l communication systems . W i t h suc h a model , ther e i s n o nee d t o rever t t o the continuous-tim e channe l an d noise . Terminolog y an d fundamental s suc h as modulatio n symbols , equalization , an d transmissio n ove r paralle l channel s are als o reviewed . Chapter 3 i s a stud y o f channe l equalization . W e wil l discus s th e desig n of F I R equalization , i n whic h th e receive r contain s onl y F I R filters. A ver y powerful too l calle d th e orthogonalit y principl e wil l b e introduced . Th e prin ciple i s o f vita l importanc e i n th e desig n o f MMS E (minimu m mea n squar e error) receivers . I t ca n b e use d fo r th e equalizatio n o f scala r channel s a s wel l as paralle l channels . Chapter 4 give s th e basic s o f multirat e signa l processing . Multirat e build ing block s ar e introduced . Th e operation s o f blockin g an d unblockin g t h a t appear frequentl y i n digita l transmissio n ar e describe d usin g multirat e build ing blocks . I n addition , polyphas e decompositio n o f filters i s reviewed . Base d on th e decomposition , th e polyphas e representatio n o f filter bank s ca n b e de rived an d efficien t polyphas e implementatio n ca n b e obtained . Reader s wh o are familia r wit h multirat e system s an d filter bank s ca n ski p thi s chapter . Chapter 5 formulate s som e moder n digita l communicatio n system s usin g multirate buildin g blocks . Th e filter ban k transceive r i s introduce d an d con ditions o n th e transmitte r an d receive r fo r zer o IS I ar e derived . Usin g th e multirate formulation , redundan t sample s ca n b e inserte d i n th e t r a n s m i t t e d signal. Tw o type s o f redundan t sample s ar e discusse d i n detail : cycli c prefi x and zer o padding . Th e matri x for m representation s o f thes e system s ar e use d frequently i n th e discussion s o f application s i n late r chapters .

7

1.1. Notation s

Chapter 6 i s devote d t o th e stud y o f som e usefu l DFT-base d transceivers . T h e O F D M , D M T , an d SC-C P system s wil l b e presente d an d th e performanc e will b e analyzed . T h e correspondin g filte r ban k representatio n wil l als o b e derived. Thes e transceiver s hav e foun d man y practica l application s du e t o the fac t t h a t the y ca n b e implemente d efficientl y usin g fas t algorithms . Chapter 7 deal s wit h th e desig n o f optima l transceiver s whe n th e trans mitter doe s no t hav e th e channe l stat e information , whic h i s usuall y th e cas e for wireles s applications . A s th e transmitte r doe s no t hav e th e channe l knowl edge, ther e i s n o bit/powe r allocation . W e conside r th e desig n o f minimu m bi t error rat e (BER ) transceiver s b y addin g a unitar y precode r a t th e transmitte r and a post-code r a t th e receive r o f th e O F D M system . W e wil l se e t h a t th e derivation o f th e minimu m B E R transceive r nicel y tie s th e O F D M an d th e SC-CP system s together . Chapter 8 deal s wit h th e desig n o f optima l transceiver s whe n th e channe l state informatio n i s availabl e t o th e transmitter . I n additio n t o bi t an d powe r allocation, th e transmitte r an d receive r ca n b e jointl y optimized . Fo r a give n error rat e an d targe t transmissio n rate , th e transceive r wil l b e designe d t o minimize th e transmissio n power . Chapter 9 describe s a metho d t o improv e th e frequenc y separatio n amon g the subchannel s fo r th e DFT-base d transceivers . Som e shor t F I R filter s calle d subfilters ar e introduce d i n th e subchannel s t o enhanc e th e stopban d atten uation o f th e transmittin g an d receivin g filters . B y usin g a slightl y longe r guard interval , w e ca n includ e th e subfilter s withou t changin g th e ISI-fre e property. W h e n subfilter s ar e adde d t o th e receiver , th e transmissio n rat e can b e increase d considerabl y i n th e presenc e o f narrowban d R F I (radi o fre quency interference) . Fo r th e transmitte r side , th e subfilter s ca n improv e th e spectral rollof f o f th e transmitte d spectru m whil e havin g littl e effec t o n th e transmission rate . Chapter 1 0 i s a stud y o f minimu m redundanc y fo r F I R equalization . Fo r a given channel , w e consider th e minimu m redundanc y t h a t i s required t o ensur e the existenc e o f F I R equalizers . W e wil l se e t h a t th e answe r i s directl y tie d t o what w e cal l th e congruou s zero s o f th e channel . T h e minimu m redundanc y can b e determine d b y inspectio n onc e th e zero s o f th e channe l ar e known .

1.1 Notation

s

• Boldface d lowe r cas e letter s represen t vector s an d boldface d uppe r cas e letters ar e reserve d fo r matrices . T h e notatio n A T denote s th e transpos e of A , an d A ^ denote s th e transpose-conjugat e o f A . • T h e functio n E [y] denotes th e expecte d valu e o f a rando m variabl e y. • T h e notatio n 1M i s use d t o represen t th e M x M identit y matri x an d 0 m n denote s a n m x n matri x whos e entrie s ar e al l equa l t o zero . T h e subscript i s omitted whe n th e siz e of the matri x i s clear fro m th e context . • T h e determinan t o f a squar e matri x A i s denote d a s d e t ( A ) . T h e nota tion diag[A o A i . . . A M - I ] denote s a n M x M diagona l matri x wit h the fcth diagona l elemen t equa l t o Afe .

8

1. Introductio n

• Th e notatio n W i s use d t o represen t th e M x M unitar y DF T matrix , given b y [Wlfcn = —Le-'M* ™ for 0 VM

< k,n < M - 1 .

• Fo r a discrete-tim e sequenc e c(n) , th e z-transfor m i s denote d a s C(z) and th e Fourie r transfor m a s C(e^). Fo r a continuous-tim e functio n xa(t), th e Fourie r transfor m i s denote d a s X a(jQ,).

2 Preliminaries o f digita l communications In thi s chapter , w e shal l revie w som e introductor y material s t h a t ar e usefu l for ou r discussio n i n subsequen t chapters . Fo r convenience , w e reproduc e i n Fig. 2. 1 th e bloc k diagra m fo r digita l communicatio n system s introduce d i n Chapter 1 . transmitter 01001100.. bit strea m

bits-tosymbol mappingl

s(n)

transmitter! signal processing

x(n)

xa(t) D/C

PiW

channel

r«(t)

r

r(n) P2(t)

C/D

receiver signal processing

u(n)

s(n)

symbol detection

receiver

symbolto-bits mapping

bit stream

Figure 2 . 1 . Simpl e bloc k diagra m fo r a digita l communicatio n system .

2.1 Discrete-tim

e channe l model s

In th e stud y o f communicatio n systems , th e transmissio n channe l i s ofte n modeled a s a continuous-tim e linea r tim e invarian t (LTI ) syste m wit h impuls e 9

10

2. Preliminarie s o f digita l communication s Qa(t)

xa(t)

ra(t)

channel

xa{t) Ca(t)

Figure 2 . 2 . LT I channe l model .

qa(f) x(n)

D/C

T

xa(t)

Px(t)

j; r

Ca(t)

a(t)

J

p2(t)

^

C/D

r(»)

T

T

(a)

x(n)

> c(»

-► r(n)

)

(b) Figure 2.3 . (a ) Th e syste m fro m x(n) t o r(n). (b model.

) Equivalen t discrete-tim e channe l

response c a(t) an d additiv e nois e q a(t). Thi s LT I channe l mode l i s show n i n Fig. 2.2 . Give n th e inpu t x a(t), th e channe l produce s th e o u t p u t CO

/

-co

ca(r)xa(t -

r)dr + q

a(t).

Letting th e symbo l V denot e convolution , w e ca n writ e ra{t) =

(x a *c a)(t) +

q

a(t).

Though th e channe l i s a continuous-tim e system , i t i s ofte n mor e convenien t to wor k directl y o n a n equivalen t discrete-tim e system . I n man y aspect s o f digital communications , a discrete-tim e channe l mode l i s ofte n adequat e an d much easie r t o wor k with . I n thi s section , w e shal l sho w t h a t th e syste m fro m x(n) a t th e transmitte r t o r(n) a t th e receive r (Fig . 2.1 ) i s equivalen t t o a discrete-time LT I system . T h e syste m fro m x(n) t o r(n) i s show n separatel y i n Fig . 2.3(a) . Suppos e t h a t th e sample s x{n) ar e space d apar t b y T seconds . Th e D / C converte r take s

11

2.1. Discrete-tim e channe l model s

the discrete-tim e sequenc e x(n) an d produce s a continuous-time impuls e trai n spaced apar t b y T: ^x(n)6a(t-nT), n

where S a(t) i s th e continuous-tim e Dira c delt a function . Afte r th e impuls e train passe s throug h th e transmittin g puls e pi(t) , w e ge t a continuous-tim e signal x a(t): oo

xa{t)= J2 x{k)pi(t-kT). /c= —o o

The signa l x a(t) i s transmitte d throug h th e channel . A t th e receivin g end , the receive d signa l i s oo

ra{t) =

(x a * ca){t) + q a(t) =

Y, x(k)(pi*c

a){t-kT)

+

q a(t).

/ c = — oo

The receive d signa l r a(t) i s first passe d throug h a receiving pulse P2(t), whic h produces oo

Wa{t) = (r a*P2)(t)= Y,

x(k)(

Pi*ca*P2)(t-kT)

+

(q a*P2)(t). (2.1

)

k= — oo

Then w a(t) i s uniformly sample d ever y T second s to produce the discrete-tim e output r(n) = w a(nT). Thi s unifor m samplin g operatio n i s denote d b y th e box labele d C/D . Defin e th e effectiv e continuous-tim e channe l an d effectiv e noise, respectively , a s follows : ce(t) =

(pi * ca *p 2 )(*) an d q e(t) = (q a *P2)(*)-

Then th e receive d discrete-tim e signa l i s oo

r(n) = Y^ x(k)c

e(nT

-

kT) + q e{nT).

A;= —oo

The abov e expressio n ca n b e rewritte n a s oo

r(n) =

y , x(k)c(n — k) + q(n), k= — oo

where c(n) an d q(n) are , respectively , th e discrete-tim e equivalen t channe l and nois e give n b y c(n) = (p i * c a *p2)(t) q{n) = (q

a*P2)(t)

t=nT (2.2

)

t=nT

Thus, th e syste m show n i n Fig . 2.3(a ) ca n b e represente d a s i n Fig . 2.3(b) , which contain s onl y discrete-tim e signal s an d systems . Th e transfe r functio n of th e discrete-tim e channe l i s give n b y oo

C(z)= Y


~n-

12

2. Preliminarie s o f digita l communication s

Observe t h a t c(n) i s th e sample d versio n o f th e cascad e o f th e transmittin g pulse pi (t) , th e channe l c a (t), an d th e receivin g puls e pi (t) . Choosin g differen t transmitting an d receivin g pulse s wil l affec t th e discrete-tim e channel . I n practice, th e channe l i s ofte n modele d a s a finit e impuls e respons e (FIR ) filter. Fro m (2.2) , w e ca n se e t h a t th e channe l lengt h i s inversel y proportiona l to th e samplin g perio d T . Reducin g T b y one-hal f wil l doubl e th e lengt h o f c(n). W h e n th e channe l C(z) ha s mor e t h a n on e nonzer o t a p , sa y c(0 ) an d c ( l ) , i t wil l introduc e interference betwee n th e receive d symbols . T o se e this , suppose t h a t ther e ar e n o "signa l processing " block s a t th e transmitte r an d receiver i n Fig . 2.1 , the n th e t r a n s m i t t e d signa l x(n) = s(n). Th e receive d signal wil l b e r(n) =

c(0)s(n) +

c(l)s(n —

1 ) + q(n);

the curren t symbo l s(n) i s contaminate d b y th e pas t symbo l s(n — 1) . Thi s phenomenon i s know n a s intersymbo l interferenc e (ISI) . Th e tas k o f symbo l recovery i s complicate d b y b o t h th e additiv e nois e q(n) an d ISI . Example 2. 1 Conside r a transmissio n syste m wit h effectiv e continuous-tim e channel c e(t) = (p i * c a *P2)(t) give n i n th e Fig . 2.4 . Vv (Pi*c ^- -

0

a*p2)(t)

^►

1

2t

Figure 2.4 . A n exampl e o f (p± * c a *P2)(t).

Suppose w e sen d on e sampl e o f x(n) pe r second , i.e . th e samplin g perio d T = 1 . The n th e discrete-tim e equivalen t channe l i s c(n) = 5{n — 1), a delay . T h e channe l doe s no t introduc e ISI . W h e n w e increas e ou r transmissio n rat e to tw o sample s pe r second , the n th e samplin g perio d become s T = 0. 5 an d the discrete-tim e equivalen t channe l i s C(z) = O.bz -1 + z~ 2 + 0.5z~ 3. T h e channel become s a three-ta p F I R channel . W e se e t h a t th e faste r w e sen d th e samples x(n), th e longe r th e F I R channe l c{n). ■ Note t h a t ther e i s n o carrie r modulatio n i n th e syste m show n i n Fig . 2.1 . This i s know n a s baseban d communication . I n wireles s communications , th e signal x a(t) i s modulate d t o a carrie r frequenc y f c fo r transmission , a s show n in Fig . 2.5 . Thi s i s know n a s passban d transmission . Fo r passban d commu nications, afte r carrie r modulatio n th e signa l t h a t i s t r a n s m i t t e d throug h th e channel c a(t) i s give n b y

va(t) =

2Re{x

a(t)e^^},

where i?e{« } denote s th e rea l part . A t th e receiver , th e receive d signa l i n thi s case become s y a(t) = (v a * c a)(t) + q a(f). Carrie r demodulatio n i s performe d to obtai n th e baseban d signal 1 ra{t) =

y

a(t)e-^^.

13

2 . 1 . Discrete-tim e channe l model s Va(t])

Xa(t)

x(n) D/C

t

-+

P,(t)

carrier mod.

ya(t)

channel

ra{t)

carrier w demod.

P2(t)

—> C/D

t

T

T

Figure 2.5 . Passban d communicatio n channel .

By followin g a simila r procedure , on e ca n sho w (Proble m 2.3 ) t h a t th e syste m sandwiched betwee n th e C / D an d D / C converter s i n a passban d communica tion syste m i s als o equivalen t t o a discrete-tim e LT I system . I n thi s case , th e equivalent discrete-tim e channe l an d nois e are , respectively , give n b y c(n)

(pi *c a *P2)(t)

q(n)

t=nT

(2.3)

where c a(t) = c a(t)e~j27r^ct an d q a(t) = q a(t)e~j27r^ct. Fro m thi s relation , on e can clearl y se e th e effec t o f carrie r modulation : wha t th e transmitte d signa l xa(t) see s i s a frequency-shifte d versio n o f th e origina l channe l c a(t) an d nois e qa(t). Not e t h a t b o t h th e channe l impuls e respons e c(n) an d th e channe l noise q(n) ar e comple x fo r passban d transmissio n du e t o th e t e r m e _ j 2 7 r ^ c t . For baseban d transmission , thes e quantitie s ar e real . Channel nois e Throughou t thi s book , w e wil l assum e t h a t th e channe l noise q(n) i s a zero-mean wide sense stationary (WSS) Gaussian rando m process. Fo r baseban d transmission , q(n) i s rea l an d th e probabilit y densit y function (pdf ) o f a zero-mea n Gaussia n nois e q(n) i s give n b y

fM

I

V/2A/" 0

(2.4)

where A/ o i s th e nois e variance . Figur e 2. 6 show s th e well-know n bell-shape d Gaussian pd f fo r differen t value s o f A/o, mor e widesprea d fo r a large r variance . For passban d transmission , th e channe l nois e q(n) i s i n genera l modele d a s a zero-mea n circularly symmetric complex Gaussian rando m variabl e whos e pdf i s give n b y

fq(q) = - U - ( « ) M / - o, (2.5 7TA/0

)

where q = qo + jqi an d A/ o i s th e varianc e o f q. T h e rea l an d imaginar y part s are b o t h zero-mea n Gaussia n an d the y hav e equa l variance : E[q^] = E[q\] = AA0/2. Thi s pd f i s show n i n Fig . 2. 7 fo r Af 0 = 1 . In th e following , w e will describ e som e commonl y use d model s o f equivalen t discrete-time channels . Thes e models , thoug h simplified , ar e usefu l fo r th e analysis o f digita l communicatio n systems . The y ar e als o frequentl y employe d in numerica l simulation s t o evaluat e th e syste m performance . 1 T h e high-frequenc y componen t centere d aroun d 2f c i s i n genera l eliminate d b y a low pass filte r i n th e proces s o f carrie r demodulation .

14

2. Preliminarie s o f digita l communication s

0.5 0.4

0.3 0.2 0.1 0 -

4

-

2

0

2

4

q Figure 2.6 . Th e pd f o f a zero-mean rea l Gaussia n rando m variable .

AGN an d AWG N channel s A channe l i s calle d a n AG N (additiv e Gaus sian noise ) channe l whe n th e channe l satisfie s th e followin g tw o properties . (1) It s channe l impuls e respons e i s

c(n) =| v J v= S(n) J ^

1

'U

=

°'

0 , otherwise

.

(2) Th e channe l nois e q(n) i s a Gaussia n rando m process . If i n additio n t o bein g a Gaussia n rando m process , q(n) i s als o white , t h a t is , E{q(n)q*(m)} = 0 wheneve r m ^ n , the n th e channe l i s a n AWG N (additiv e white Gaussia n noise ) channel . W h e n th e channe l i s a n AG N o r AWG N channel, ther e i s n o IS I an d th e transmissio n erro r come s fro m th e channe l noise only . FIR channel s I n man y applications , th e channe l no t onl y introduce s ad ditive nois e q(n), bu t als o distort s th e t r a n s m i t t e d signal . Th e channe l c(n) i s no longe r a n impulse , an d i n genera l i t ha s a causa l infinit e impuls e respons e (IIR). Fo r th e purpos e o f analysis , th e channe l i s ofte n modele d a s a finit e impulse respons e (FIR ) filter, t h a t i s L

C(z) =

J2c(n)z- n. (2.6 n=0

)

2.1. Discrete-tim e channe l model s

15

-5 -

5

Figure 2.7 . Th e pd f o f a circularl y symmetri c comple x Gaussia n rando m variable .

T h e impuls e respons e i s nonzer o onl y fo r a finit e numbe r o f coefficient s (o r taps). T h e integer s L an d L + 1 are , respectively , th e channe l orde r an d channel length . B y makin g L larg e enough , a causa l II R filte r ca n b e wel l approximated b y a n F I R filter . I n th e frequenc y domain , th e magnitud e re sponse o f th e F I R channe l | C ( e j a ; ) | i s no t fla t unles s c{n) ha s onl y on e nonzer o t a p . T h e channe l ha s differen t gains fo r differen t frequenc y components . T h u s such channel s ar e als o know n a s frequency-selectiv e channels . W h e n c{n) has onl y on e nonzer o t a p , i t i s calle d frequency-nonselective . Random channel s wit h uncorrelate d tap s I n man y situations , th e exac t channel impuls e respons e ma y no t b e available , an d onl y th e statistic s o f th e channel i s known . On e o f th e widel y adopte d channe l model s assume s t h a t the tap s ar e zero-mea n uncorrelate d rando m variable s wit h know n variances . In thi s case , c(n) satisfie s th e followin g conditions : (1) E[c(n)} = 0, (2) E[c{n)c*{n-k)] = a

2

J{k).

(2.7)

T h e se t o f quantitie s {cr^} i s calle d th e powe r dela y profile . W e sa y t h a t th e channel ha s a n exponentia l powe r dela y profil e whe n a^ decay s exponentiall y

16

2. Preliminarie s o f digita l communication s

with respec t t o n (se e Fig. 2.8) . I f th e channe l impuls e response s c(n) ar e in dependent identica l rando m variables , i t i s often calle d a n i.i.d . (independen t identically distributed ) channel . Thes e channe l model s ar e ofte n employe d i n numerical simulation s whe n w e want t o lear n th e syste m performanc e "aver aged ove r al l channels. " 1

1/2

2 n

1/4 , 1/ 8 1/ 16 || , 1/3 2 01

2

3

4

5

►

Figure 2.8 . Exponentia l powe r delay profil e with x(n)

Figure 2.9 . LT I equalizer .

where th e syste m dela y n o = 0 i n thi s case . T h e n th e erro r e(n ) = (q * a)(n) consists onl y o f noise . I n practice , th e II R zero-forcin g equalize r 1/C(z) i s not frequentl y use d becaus e th e equalize r 1/C(z) wil l b e unstabl e whe n C(z) has zero s outsid e th e uni t circle . T o avoi d thi s problem , w e ca n us e a n F I R equalizer a(n). T h e outpu t o f th e equalize r i s x(n) =

(a * c* x)(n) + ( a * q)(n).

There ar e tw o commo n way s o f designin g a(n). On e i s t o choos e a(n) suc h t h a t th e IS I i s smal l i n som e sense . T h a t is , w e woul d lik e th e convolutio n (c * a)(n) t o b e a s clos e t o a dela y 5{n — no) a s possible . Anothe r wa y o f designing a(n) i s t o includ e th e effec t o f b o t h channe l nois e an d ISI . Not e t h a t whe n C(z) ha s a zer o zo ^ 0 , th e produc t A(z)C(z) canno t b e a dela y z~n° fo r an y F I R equalize r A(z) becaus e A(z)C(z) wil l hav e a zer o a t ZQ. In particular , whe n a n F I R channe l ha s mor e t h a n on e nonzer o t a p , a n F I R equalizer ca n neve r b e zero-forcing ; th e outpu t erro r wil l contai n b o t h th e channel nois e an d ISI . W e wil l stud y thes e solution s o f F I R equalizer s i n mor e detail i n Chapte r 3 . Signal t o nois e rati o (SNR ) I n a digita l communicatio n system , w e ofte n measure th e performanc e b y evaluatin g th e rati o o f signa l powe r £ x t o th e mean square d erro r a 2e = E[\x(n) — x(n — no)|2 ]. Thi s rati o i s know n a s th e signal t o nois e rati o (SNR ) an d i t i s give n b y

W h e n th e equalize r i s no t zero-forcing , th e erro r e(n ) = x(n) — x(n — no) contains no t onl y noise , bu t als o IS I terms . I n thi s case , /3 is als o know n a s the signa l t o nois e interferenc e rati o (SINR) , bu t w e shal l refe r t o i t simpl y as SNR .

2.3 Digita

l modulatio n

In digita l communicatio n systems , th e transmitte d bi t strea m consistin g o f "zeros" an d "ones, " i s ofte n partitione d int o segment s o f length , say , b. Eac h segment (codeword ) i s the n mappe d t o on e membe r i n a se t o f 2 b rea l o r complex numbers . Thi s proces s i s know n a s digita l modulation . T h e rea l or comple x number s representin g th e codeword s ar e know n a s modulatio n

18

2. Preliminarie s o f digita l communication s

symbols. A t th e receiver , base d o n th e receive d informatio n a decisio n wil l be mad e o n th e symbo l transmitted . Thi s proces s i s calle d symbo l detection . T h e resultin g symbo l i s the n decode d t o a 6-bi t codewor d (symbol-to-bit s mapping). Man y type s o f digita l modulation s hav e bee n developed . I n th e following, w e wil l describ e tw o widel y use d digita l modulatio n scheme s know n as th e puls e amplitud e modulatio n (PAM ) an d q u a d r a t u r e amplitud e mod ulation (QAM) . W e wil l analyz e thei r performanc e fo r transmissio n ove r a n AWGN channel . Not e t h a t i n a baseban d communicatio n system , wher e th e channel i s real , PA M i s ofte n employed , wherea s i n a passban d syste m th e channel i s complex , an d QA M i s usuall y employed .

2.3.1 Puls

e amplitud e modulatio n (PAM )

In 6-bi t puls e amplitud e modulatio n (PAM) , a codewor d o f b bits i s m a p p e d to a rea l numbe r G { 0 , 1 , . . ., 2 b~1 - 1} . (2.8

ek

s = ±(2f c + l ) A , wher

)

Figure 2.1 0 show s th e possibl e value s o f a PA M symbo l fo r b = 2 an d 6 = 3 , respectively [120] . Th e Gra y cod e indicate d i n th e figure wil l b e explaine d later. Suc h figures ar e calle d signa l constellations . Th e minimu m distanc e between tw o constellatio n point s i s 2A . Assum e t h a t al l constellatio n point s are equiprobable . The n th e signa l powe r o f a 6-bi t PA M constellatio n i s give n by gs = E[s 2] =

^-(2 2b-l). (2.9

)

T h e signa l powe r i s proportiona l t o th e squar e o f th e minimu m distanc e 2A . W h e n th e minimu m distanc e i s fixed, th e signa l powe r wil l increas e b y roughl y 6 d B fo r ever y additiona l bit .

(a)

(b)

-3A -

AA

-7A -5

3

A -3 A -

A

AA

3

A5

A7

A

Figure 2 . 1 0 . PA M constellations : (a ) 2-bi t PAM ; (b ) 3-bi t PAM .

Suppose t h a t a 6-bi t PA M symbo l s o f th e for m i n (2.8 ) i s t r a n s m i t t e d through a rea l zero-mea n AWG N channe l wit h nois e varianc e A/o . Th e re ceived signa l i s r = s + q. Assum e t h a t s an d q ar e independent . T h e conditional pd f o f th e receive d signa l r give n t h a t s i s t r a n s m i t t e d i s fr\s(r\s)=

f r\3(s +

q\s) = f

q(q),

2.3. Digita

l modulatio n

19

where f q(q) i s th e Gaussia n pd f give n i n (2.4) . Fro m th e receive d signa l r , w e make a decisio n o n th e transmitte d symbol . T h e commonl y adopte d decisio n rule i s th e neares t neighbo r decisio n rul e ( N N D R ) . I n a n NNDR , w e mak e a decisio n s = (2 z + l ) A , i

f \r- (2 z + 1)A | < \r - (2 j + 1)A | fo r al l j . (2.10

)

T h e decisio n J i s th e constellatio n poin t t h a t i s closes t t o th e receive d sig nal r . Fo r al l th e interio r constellatio n points , th e symbo l wil l b e detecte d erroneously i f th e channe l nois e ha s \q\ > A . Fo r th e tw o exterio r constella tion point s s = (2 b — 1)A an d s = (—2 6 + l ) A , w e wil l mak e a n erro r whe n q < — A an d q > A , respectively . Therefor e th e probabilit y t h a t th e detectio n is erroneou s i s give n b y P(s + s\s) =

( P(q > A ) , fo r s = ( - 2 6 + 1)A ; I P(q< - A ) , fo r s = (2 6 - 1)A ; [ P(\q\ > A ) , otherwise .

Using th e formul a fo r th e Gaussia n pd f give n i n (2.4) , th e conditiona l proba bility o f symbo l erro r i s give n b y

P(s^s\s)

\ Q ^W^W^ for* =

± (2>-l)A;

1 2Q i)l {2»-W 0)>0thCTWiSe '

where th e functio n Q(x) i s th e are a unde r a Gaussia n tail , define d a s

i r°°

Q{x)=

^mL e

~r2/2dT- (2

-n)

For equiprobabl e PA M symbols , th e symbo l erro r rat e (SER ) i s give n b y

SERpam(b) =

\ h E P&*

s

\g) = 2(1 " 2- b )Q(J ( 2 2 b 3 _ g s 1 ) A / - o ). (2.12

)

As th e functio n Q(x) decay s rapidl y wit h respec t t o x , fo r a moderat e SN R value mos t symbo l error s happe n betwee n adjacen t constellatio n points . W e can us e a mappin g i n whic h th e codeword s o f adjacen t constellatio n point s differ b y onl y on e bit . A n erro r betwee n adjacen t constellatio n point s result s only i n on e bi t error . T h e widel y use d Gra y cod e i s a mappin g schem e t h a t possesses thi s property . Figure s 2.11(a ) an d (b ) sho w a Gra y cod e mappin g for 2-bi t PA M an d 3-bi t PAM , respectively . W h e n th e Gra y cod e i s employe d in a 6-bi t PA M modulation , th e bi t erro r rat e (BER ) an d SE R ar e relate d approximately b y [120 ] BERpam(b) «

-SER

pam(b).

(2.13

)

From th e abov e formulas , w e se e t h a t th e bi t erro r rat e depend s o n th e signa l power S s an d th e nois e powe r A/o - T h e B E R curve s ar e ofte n plotte d agains t the SNR . Fo r AWG N channels , th e SN R i s simpl y th e rati o £ 8/N0. I n orde r

2. Preliminarie s o f digital communication s

20

to evaluat e th e accurac y o f th e BE R formul a derive d above , w e comput e th e BER curve s throug h Mont e Carl o simulatio n i n Fig. 2.12 . In the Monte Carlo simulation , sufficientl y man y round s o f simulation ar e carrie d ou t an d the result s ar e average d t o give a n accurate estimat e o f the actua l BER . As a rul e o f thumb, fo r a BER o f 10—z, w e nee d t o generate a t least 10 0 * 10z bits i n th e simulation s to obtain a n accurat e estimate . I n Fig . 2.12 , th e BE R approximations obtaine d fro m (2.13 ) ar e plotte d i n the dotte d curve s an d th e BER curve s obtained fro m th e Mont e Carl o simulation ar e plotted i n the soli d curves (a s the dotte d curve s almos t overla p with th e soli d curves , we see onl y the soli d curve s i n the figure) . Sinc e th e tw o curve s matc h almos t perfectly , the formul a i n (2.13 ) give s a very good approximatio n o f the tru e BE R values . Comparing th e PA M o f different constellatio n sizes , w e se e tha t fo r a BE R of 10~ 4 , w e nee d a n SNR o f around 11. 7 dB , 18. 2 dB , an d 24. 2 d B fo r 1-bit, 2-bit, an d 3-bi t PAM , respectively . T o achiev e th e sam e BER , th e require d SNR increase s roughl y b y 6 dB fo r ever y additiona l bit . (a) "

[V)

00

01

11

10

-3A

-A

A

3A

000

001

011

010

110

-7A -5 A -3 A - A A

3

111

101

A5

100

A7 A

Figure 2 . 1 1 . Gra y cod e mapping : (a ) 2-bit PAM ; (b ) 3-bit PAM .

Equation (2.12 ) relate s th e erro r rat e t o the SN R £ s/Afo. I t can b e used to obtai n th e numbe r o f bits tha t ca n b e transmitted fo r a given SN R an d target erro r rate . B y rearrangin g th e term s i n (2.12), w e ge t >=ilo&(l+

2v ^\

where

r para

L

r

^, ( 2 . 4

)

para

^ J-J -ftpam

2(1-2-^ If on e compare s th e formul a fo r b with th e channe l capacity , whic h i s given by 0.5log2 ( 1 + Ss/Afo) (bit s pe r use) , the quantit y T pam represents th e differenc e i n the require d SN R betwee n th e PAM schem e an d th e channe l capacity . Therefor e T parn is also know n a s th e SNR gap . Fo r a moderate erro r rate , th e inverse Q function i s relatively flat. Therefor e th e SN R ga p is well approximate d b y Tpam « \ [Q- 1 (SER pam/2)]2 .

(2.15

)

The SN R ga p i s a quantity tha t depend s onl y o n th e erro r rate . I n Table 2.1, we list th e value s of Tpam for som e typica l SER parn.

21

2.3. Digita l modulatio n

10 — B — 2-bi t PA M 0 3-bi t PA M 10"

10"' DC LU CO

10"'

10

10

10 1

52 SNR(dB)

0

25

30

Figure 2 . 1 2 . BE R performanc e o f PA M i n a real zero-mea n AWG N channel . Th e soli d curves ar e th e experimenta l value s obtaine d fro m th e Mont e Carl o simulatio n an d th e dotted curve s (almos t indistinguishabl e fro m th e soli d curves ) ar e th e theoretica l value s obtained fro m th e formul a i n (2.13) .

O Hi -LLpam

r L

para

Fpam i n d B

10"

2

2.21

3.44

10"

3

3.61

5.57

10" 4

5.05

7.03

icr icr 6

6.50

8.13

7.98

9.02

7

9.46

9.76

5

lO"

Table 2.1 . Th e SN R ga p T parn in (2.15 )

Binary phas e shif t keyin g ( B P S K ) modulatio n Fo r th e specia l cas e o f PAM wit h 6 = 1 , ther e ar e onl y tw o constellatio n points , + A an d —A , an d thi s is mor e commonl y know n a s binar y phas e shif t keyin g (BPSK ) modulation . For B P S K symbols , th e bi t erro r rat e an d symbo l erro r rat e ar e th e same .

22

2. Preliminarie s o f digita l communication s

T h e formul a (2.12 ) reduce s t o BERbpSk=

SERbp Sk=

Q

Unlike (2.13) , th e abov e B E R formul a fo r B P S K i s exact ; n o approximatio n is made . Example 2. 2 Suppos e th e transmitte r i s t o sen d th e followin g bi t stream : 000 01 0 11 1 11 0 01 0 10 0 10 1 110 . Assume t h a t th e modulatio n schem e i s a 3-bi t PA M wit h Gra y cod e mappin g as i n Fig . 2.10 . Th e "bits-to-symbo l mapping " bloc k take s ever y thre e inpu t bits an d map s t h e m t o a 3-bi t PA M symbol . Th e first thre e bit s ar e "000 " an d thus fro m Fig . 2.1 0 w e kno w t h a t th e first PA M symbo l i s — 7A. The n th e next thre e bit s ar e "010 " an d fro m th e figure w e hav e th e nex t PA M symbo l as —A . Continuin g thi s process , w e find t h a t th e abov e sequenc e o f 2 4 bit s i s m a p p e d t o th e followin g PA M symbols : s(n) :

- 7 A , - A , 3A , A , - A , 7A , 5A , A .

Now suppos e t h a t th e abov e PA M symbol s ar e t r a n s m i t t e d ove r a n AWG N channel an d du e t o channe l nois e th e receive d sequenc e i s r(n) :

- 9 A , - 0 . 7 A , 1.9A , 1.1A , - 1 . 3 A , 6.6A , 4.8A , 2.1A .

Assume t h a t a t th e receive r ther e i s n o additiona l signa l processin g an d t h a t N N D R i s applie d directl y t o r(n). Th e outpu t o f th e symbo l detecto r wil l b e s(n) :

- 7 A , - A , A , A , - A , 7A , 5A , 3 A .

Comparing s~(n) wit h s(n) , w e find t h a t w e hav e mad e tw o symbo l error s out o f eigh t t r a n s m i t t e d symbol s (indicate d b y boldface d symbols) . I n thi s experiment, th e symbo l erro r rat e i s given b y SER = 0.25 . Afte r th e "symbol to-bits mapping " bloc k usin g Gra y code , w e obtai n th e followin g sequence : 000 01 0 11 0 11 0 01 0 10 0 10 1 111 . Comparing th e decode d sequenc e wit h th e t r a n s m i t t e d sequence , tw o bit s (indicated b y boldface d numbers ) ar e receive d erroneously . Th e bi t erro r rat e is BER = 1/12 , whic h i s equa l t o SER/3 i n thi s exampl e becaus e th e tw o erroneously detecte d symbol s ar e adjacen t t o th e actua l t r a n s m i t t e d symbols .

2.3.2 Quadratur

e amplitud e modulatio n ( Q A M )

Unlike PA M symbols , QA M symbol s ar e comple x numbers . Fo r 26-bi t QAM , a codewor d o f 2b bit s i s m a p p e d t o a symbo l o f th e form 2 s = ±(2f c + l ) A ± j ( 2 Z + l ) A , w

h e r e k,

I G { 0 , 1 , . . ., 2 6 _ 1 - 1} . (2.16 )

2 Unless mentione d otherwise , th e QA M symbol s i n this boo k hav e a square constellation . Hence eac h QA M symbo l carrie s a n eve n numbe r o f bits .

23

2.3. Digita l modulatio n

Figure 2.13(a ) an d (b ) show , respectively , th e signa l constellation s fo r 2-bi t and 4-bi t QA M wit h th e correspondin g Gra y codes . T h e specia l cas e o f 2 bit QA M i s als o know n a s q u a d r a t u r e phas e shif t keyin g ( Q P S K ) . Al l th e four constellatio n point s i n Q P S K hav e th e sam e magnitude . Fro m (2.8 ) an d (2.16), w e se e t h a t th e rea l an d imaginar y part s o f a 26-bi t QA M symbo l ca n be viewe d a s tw o 6-bi t PA M symbols . Usin g thi s relation , man y result s fo r the QA M symbo l ca n b e obtaine d b y modifyin g thos e o f th e PA M symbols . For example , th e signa l powe r o f th e 26-bi t QA M symbo l i s

£s=E[\Sf]=

2

-^(2*b-l).

It i s twic e t h a t o f a 6-bi t PA M symbol .

•

01

-A 00

•

• A

• -A

A •

11

10

•

•

0110

1110

0011

•

• A 0111

1111

1011

0001

• -A 0101

1101

1001

•

.-3A

^3A •

0000

(a) (b

#3A_

0010

• I

^A

A •

•

0100

1100

1010

•

3A •

•

1000

)

Figure 2 . 1 3 . QA M constellatio n an d it s Gra y cod e mapping : (a ) 2-bi t QA M (als o known a s QPSK) ; (b ) 4-bi t QAM .

Suppose t h a t a 26-bi t QA M symbo l wit h powe r S s i s transmitted throug h a zero-mean comple x AWG N channe l wit h nois e varianc e A/o - Suppos e t h a t th e noise i s circularl y symmetri c s o t h a t it s pd f i s a s give n i n (2.5) . T h e n th e rea l part an d th e imaginar y par t ar e b o t h Gaussia n wit h varianc e A/o/2. Therefor e the transmissio n o f a 26-bi t QA M symbo l throug h a comple x AWG N channe l with nois e varianc e A/ o ca n b e viewe d a s th e transmissio n o f tw o 6-bi t PA M symbols, eac h wit h powe r £ s / 2 , throug h tw o rea l AWG N channels , eac h wit h noise varianc e A/o/2 . Fro m earlie r discussions , w e kno w t h a t whe n a 6-bi t PAM symbo l wit h powe r £ s/2 i s transmitte d throug h a rea l AWG N channe l

24

2. Preliminarie s of digital communication s

with nois e varianc e A/o/2, th e SE R i s given b y

5£fip„.„(» = 2(l-l)0( v / ( 2 2 t 3 _^ / 2 ). A QA M symbo l i s correctly decode d whe n bot h th e rea l an d imaginar y part s are correctl y decoded . Th e probabilit y fo r this i s (1 — S E Rparn{b))2. Thu s the SE R o f a 26-bit QA M symbo l i s given b y SERqam(2b) =

2SERpam(b) -

2

SER

pam(b).

When th e erro r rat e i s small, w e ca n ignor e th e second-orde r ter m an d th e SER i s well approximate d b y

SERqam{2b) « 2SERpam(b) = 4(l - ^ ) Q ( J^[^J- (

2 17

- )

Similar t o th e PA M case , if we use Gra y code s to ma p th e rea l an d imaginar y parts, respectively , a s shown i n Fig. 2.13 , the n an y QA M symbo l an d its nearest neighbo r wil l diffe r onl y b y on e bit . I n thi s case , th e BE R o f a 26-bit QAM ca n b e approximate d b y 1 -SER qam(2b). (2.18 26~

BERqam(2b) «

)

Figure 2.1 4 show s th e BE R performanc e o f 26-bit QA M fo r differen t b using Monte Carl o simulatio n an d th e formul a i n (2.18). Agai n w e se e tha t the experimental and theoretical BER curves match almos t perfectly . Th e formul a in (2.18 ) give s a very goo d approximatio n o f th e actua l BER . Comparison o f 6-bit PA M an d 26-bi t QA M Usin g the formulas i n (2.12) , (2.13), (2.17) , an d (2.18) , fo r th e sam e SN R £ 8/Af0 we hav e BERqarn(2b) «

BER

parn(b);

the BER s o f a 26-bi t QA M an d a 6-bi t PA M ar e approximatel y th e same , but th e bi t rat e o f QA M i s twice tha t o f PAM. However , w e should not e tha t the compariso n i s based o n differen t channe l settings . Fo r PAM , th e symbol s are rea l an d th e channe l nois e i s also rea l wit h varianc e A/o . Fo r QAM , th e symbols ar e comple x an d th e channe l nois e is complex wit h th e variance s o f both th e rea l and imaginar y part s equa l to A/o/2. I n other words , in passban d communication i f we choose a QAM ove r a PAM o f the sam e bi t rate , w e will have a gain i n SNR. Fo r example , fo r a BER o f 1 0- 4 , w e se e fro m Fig . 2.1 2 and Fig . 2.1 4 tha t a 2-bit QA M need s a n SN R o f aroun d 11. 7 d B wherea s a 2-bit PA M need s a n SNR o f around 18. 2 dB ; w e hav e a saving of 6.5 d B b y using QAM . For QA M symbol s w e can als o expres s 2 6 in term s o f the SNR , £ s/A/o, as we di d fo r PA M symbol s in (2.14). B y rearrangin g (2.17) , w e obtai n 2b = log2 (l + \-

£

-^) ,

L qam J

(2.19

)

25

2.3. Digita l modulatio n

* — 2-bi t QA M e — 4-bi t QA M 3 — 6-bi t QA M

10 1

52 SNR(dB)

0

Figure 2.14 . BE R performanc e o f QA M i n zero-mea n AWG N channels . Th e soli d curves ar e th e experimenta l value s obtaine d fro m th e Mont e Carl o simulatio n an d th e dotted curve s (almos t indistinguishabl e fro m th e soli d curves ) ar e th e theoretica l value s obtained fro m th e formul a i n (2.18) .

where T qarn i s the SN R ga p give n b y SERa 4(1 -2~

b

)

(2.20)

Again fo r moderat e erro r rates , th e followin g expressio n give s a very accurat e approximation o f th e SN R gap : ^qam ~ ~ \Q~ (S E Rqarn / 4)]

(2.21)

In Tabl e 2.2 , w e list th e value s o f Tqarn fo r som e typica l SER qarn. Althoug h the formul a i n (2.19 ) i s derive d fo r even-bi t QA M symbols , th e right-han d side i s als o use d fo r estimatin g th e numbe r o f bit s tha t ca n b e transmitte d when ther e i s n o even-bi t constrain t [38] . Quadrature phase shift keyin g (QPSK ) modulatio n Whe n a QAM sym bol ha s onl y tw o bits , i t i s als o commonl y know n a s a QPS K symbol . Th e constellation an d a Gra y cod e mappin g fo r QPS K ar e show n i n Fig . 2.13(a) . Suppose a QPS K symbo l wit h signa l powe r £ s i s transmitte d throug h a n AWGN channe l wit h nois e varianc e A/o - Fo r equiprobabl e QPS K symbols , we ca n comput e th e BE R b y computin g th e BE R fo r an y constellatio n poin t because o f th e symmetry . Suppos e tha t "11 " is transmitted . Le t q r an d qi

26 2

. Preliminarie s o f digita l communication s 3 -t^ -t^qam 2

ioicr 3 io- 4 io~5 io~6 io- 7

-1- qam

2.63

4.04 5.48

J- qam m

4.19

6.06 7.39

6.95

8.42

9.91

9.96

8.42

9.25

Table 2.2 . Th e SNR gap Tqarn i n (2.21)

be, respectively , th e rea l an d imaginar y part s o f nois e q. The n th e firs t bi t i s in erro r whe n q r < — A an d th e secon d bi t i s i n erro r whe n qi < —A . Thu s the BE R o f QPS K i s given b y BERqpsk=

0.5P(q r < - A ) + 0.5P(f t < - A ) = Q

The abov e BER formula fo r QPS K i s exact an d i t i s identical to that o f BPSK . Other modulatio n scheme s PA M an d QA M ar e th e mos t commonl y used modulatio n scheme s du e t o thei r simplicity . Ther e ar e man y othe r mod ulation schemes , for exampl e phas e shif t keyin g (PSK) , frequenc y shif t keyin g (FSK), differentia l phas e shif t keyin g (DPSK) , minimu m shif t keyin g (MSK) , and s o forth. I n practice , w e may choos e on e modulatio n schem e ove r others , depending o n th e application . Fo r example , i n som e application s i t migh t be desirabl e t o hav e modulatio n symbol s wit h a constan t magnitude , tha t is \sk\ = £ s for al l k. I n thi s case , w e ca n us e PS K modulatio n (show n i n Fig. 2.15) . I n a PS K modulatio n scheme , al l th e constellatio n point s ar e uniformly distribute d o n a circl e an d th e radiu s o f th e circl e determine s th e symbol power . Fo r a mor e detaile d an d complet e coverag e o f variou s digita l modulations, th e reader s ar e referre d t o [120 ] an d [137] . Example 2. 3 Suppos e w e wan t t o sen d th e 24-bi t sequenc e i n Exampl e 2. 2 using 4-bi t QA M symbols . Le t th e constellatio n an d Gra y cod e mappin g b e as show n i n Fig . 2.13(b) . A s eac h symbo l carrie s fou r bits , w e grou p th e 2 4 bits int o codeword s o f fou r bits : 0000 101 1 111 0 010 1 0010 1110 . From Fig . 2.13(b) , w e fin d th e correspondin g si x QA M symbol s s(n). Thes e QAM symbol s ar e sen t ove r a n AWG N channel . Suppos e th e receive d signal s r(n) ar e a s give n i n Tabl e 2.3 . Th e erro r probabilitie s i n thi s exampl e ar e fo r the sak e o f demonstration . Th e actua l error s ar e usuall y muc h smaller , suc h as 1 0 - 2 , 1 0 - 4 , etc . Assum e tha t a t th e receive r ther e i s n o additiona l signa l

2.3. Digita l modulatio n

27

(a)

(b)

Figure 2 . 1 5 . Phas e shif t keyin g modulation : (a ) 8-PSK ; (b ) 16-PSK .

n

s{n)

0 -3A - 3Aj 3A + Aj 1 2 A + 3Aj -A - Aj 3 4 -3A + 3Aj A + 3Aj 5

r(n)

-4.lA-2.6Aj 3.7A + 2.1AJ 0.9A + 2.8Aj -1.1A-Aj -4A + 1.7AJ 2.1A + 1.1AJ

s(n)

-3A - 3Aj 3A + 3Aj A + 3Aj -A - Aj -3A + Aj 3A + Aj

Table 2.3 . Transmitte d QA M symbol s s(n), receive d signal s r(n), an d detecte d sym bols 7i{n).

processing an d th e NND R i s applie d directl y t o r(n). Afte r symbo l detectio n we ge t s"(n) . Comparin g s(n ) wit h s(n) , w e find tha t w e hav e mad e thre e symbol error s (s"(l) , s"(4) , s"(5) ) ou t o f si x transmitte d symbols . Th e symbo l error rat e i s SER = 1 / 2 . Afte r symbol-to-bit s mappin g usin g th e Gra y cod e provided i n Fig . 2.13(b) , w e obtain th e followin g sequence : 0000 101 0 111 0 0101 0011 1011. Comparing th e decode d sequenc e wit h th e transmitte d sequence , fou r bit s (indicated b y boldface d numbers ) ar e receive d erroneously . Th e bi t erro r rat e is BER = 4/2 4 = 1/6 , whic h i s large r tha n SER/A = 1/8 . Thi s i s becaus e s(5) i s not adjacen t t o s(5) , whic h cause s a n erro r o f two bit s rathe r tha n on e

bit. ■

2. Preliminarie s o f digita l communication s

28

+s o

so

+s i

> -s ,

equalizer

channel

Figure 2 . 1 6 . A se t of M paralle l channel s and the corresponding zero-forcing equalizer .

2.4 Paralle

l subchannel s

In man y wideban d communicatio n systems , a wideban d channe l i s divide d into a se t o f subchannels , eac h wit h a smalle r bandwidth . Example s includ e the widel y use d O F D M an d D M T systems , whic h wil l b e studie d i n detai l in Chapte r 6 . I n thes e systems , a n F I R channe l i s converte d t o a se t o f parallel ISI-fre e channel s a s show n i n Fig . 2.16 . Th e receive d signa l o f th e i t h subchannel i s where Si i s th e symbo l t r a n s m i t t e d ove r th e i t h subchannel . Th e quantitie s di an d qi are , respectively , th e i t h subchanne l gai n an d noise . Becaus e eac h subchannel ha s onl y a singl e t a p , zero-forcin g equalizatio n ca n b e don e b y using simpl e scala r multiplier s 1/a ^ (i f ai ^ 0 ) a s indicate d i n th e figure . Le t s an d s " be, respectively , th e inpu t an d outpu t vectors . Defin e th e outpu t erro r vector a s e = s — s. T h e n w e ca n redra w th e paralle l channel s a s i n Fig . 2.17 . I t i s clea r t h a t th e output erro r o f th e i t h subchanne l i s e ^ = qi/ai. Th e erro r varianc e fo r th e i t h subchanne l i s

2_ M

where Mi i s th e varianc e o f q^. Fo r man y applications , th e subchannel s hav e the sam e nois e variance s Mi = A/o , bu t th e subchanne l gain s ca n b e ver y different. Thu s th e erro r variance s a^ ca n b e ver y differen t fo r differen t sub channels. Signal s t r a n s m i t t e d ove r differen t subchannel s encounte r differen t levels o f distortion . Fo r subchannel s wit h larg e erro r variances , th e bi t erro r rate wil l b e hig h an d th e overal l performanc e o f th e paralle l channel s wil l be limite d b y thes e bad subchannels . T o se e thi s effect , le t u s conside r th e following exampl e wit h onl y tw o subchannels .

29

2.4. Paralle l subchannel s

so

S\

£.

-► * o

->A l

S

M-\

Figure 2 . 1 7 . Equivalen t paralle l channel s o f Fig . 2.16 .

Example 2. 4 Suppos e t h a t ther e ar e onl y tw o subchannel s an d th e gain s ar e ao = 1 and a\ = 0.1 , respectively. T h e transmissio n powe r i s fixed a t 10 . T h e subchannel noise s qi ar e AWG N wit h varianc e J\fi = 1 . A quic k calculatio n shows t h a t th e outpu t erro r variance s ar e

Suppose t h a t th e maximu m signa l powe r allowe d o n eac h subchanne l i s £ m a x = 10. Le t th e transmitte d signal s Si b e B P S K symbol s wit h si = ± \ / l 0 s o t h a t the powe r i s Ei = 10 . Fo r B P S K modulation , B E R i s equa l t o SER . Fro m (2.12), th e B E R o f th e zerot h subchanne l i s Q ( \ / l 0 ) = 7.8 3 x 10~ 4 , wherea s the B E R o f th e first subchanne l i s Q ( \ / o T ) = 0.38 . T h e averag e B E R o f th e parallel channel s i s approximatel y 0.5*Q(VoT) = 0.19 . T h e performanc e o f th e syste m i s severel y limite d b y th e first subchannel . Instead o f transmittin g B P S K symbol s o n b o t h th e subchannels , suppos e t h a t we no w transmi t a 2-bi t PA M symbo l o n th e zerot h subchanne l an d th e first subchannel i s no t utilize d fo r transmission ; th e zerot h an d first subchannel s are allocate d tw o bit s an d zer o bits , respectively , s o t h a t th e tota l numbe r o f bits transmitte d i s stil l two . Usin g th e B E R formul a fo r 2-bi t PAM , w e find t h a t BER « 3/ 4 * Q{y/lQ/5)= 0.059 , whic h i s muc h smalle r t h a n 0.19 . ■ From th e abov e example , w e se e t h a t b y properl y loadin g th e bit s amon g the subchannels , w e ca n significantl y improv e th e B E R performanc e fo r th e same transmissio n rate . I n wha t follows , w e wil l sho w ho w t o achiev e this . Bit loadin g W h e n th e power s o n th e subchannel s ar e fixed a t £ m a x an d the subchanne l erro r variance s o 2e. ar e differen t fo r differen t z , the bit s assigne d to th e subchannel s bi can b e adjuste d t o improv e th e erro r rate . Thi s i s calle d bit loadin g (als o know n a s bi t allocation) . Below , w e wil l first conside r bi t loading fo r th e PA M cas e unde r th e pea k powe r constraint . Suppos e t h a t the erro r rate s o f al l th e subchannel s ar e th e sam e an d thei r SER s ar e equa l

2. Preliminarie s o f digital communication s

30

to SERQ. Fo r P A M symbols, w e kno w t h a t t h e numbe r o f bit s i s relate d t o the SN R by (2.14) . Therefor e t h e numbe r o f bit s t h a t ca n be t r a n s m i t t e d o n the i t h subchanne l i s

fc = i l o g 2 ( l +

% ^Y (2.22

)

where o\. i s t h e nois e powe r o f t h e i t h subchanne l output . T h e averag e bi t rate i s give n b y M-l

M^ T h e bit s compute d i n (2.22 ) ar e not intege r i n general . W h e n t h e constrain t of intege r bi t i s applied , t h e averag e bi t rat e i s b = ( 1 / M ) ^2i=^ \pi\, w n e r e [x\ denote s t h e larges t intege r les s t h a n o r equa l t o x. For Q A M symbols, suppos e t h e i th subchannel carrie s 2b i bits . Then , fro m (2.19), w e get t he number o f bits t h a t ca n be sent throug h t h e i th subchanne l as 2bi = log 2 1

\-

+—

%

L qam J

-.

T h e averag e bi t rat e i s give n b y M-l i=0

For t h e cas e o f intege r bi t loading , t h e averag e bi t rat e i s 1 / M ^ - = Q 2 [ ^ J . Example 2. 5 Conside r t h e paralle l channel s i n Exampl e 2.4 . Suppos e t h a t PAM symbol s ar e sent , t h e desire d S E R is SERQ= 1 0 - 7 , and t he m a x i m u m transmission powe r allowe d i s Smax = 1000 . Using Tabl e 2.1 , we have T pam = 9.46. T h e subchanne l SNR s ar e S W ? o = 1000 , SNRi =

10.

T h e maximu m achievabl e bit s fo r t h e two subchannels ar e b0 = 3.37 , 6

i = 0.52 .

T h u s t h e overal l maximu m achievabl e bi t rat e i s b = 1.95 . I f a n intege r bit allocatio n i s desired , thi s valu e become s b = 1.5 . Not e t h a t althoug h b\ = 0.52 , we roun d i t dow n t o zer o s o t h a t t h e desire d quality-of-servic e o f SER0= 1 0 - 7 is not violated . ■ In t h e abov e discussions , w e assum e t h a t t h e pea k signa l powe r o f eac h subchannel i s limite d b y S max. I n som e applications , w e ma y b e concerne d about t h e averag e signa l powe r rathe r t h a n t h e peak signa l power . T h e prob lem o f bit loadin g fo r thi s cas e wil l b e studie d i n Chapte r 8 .

2.5. Furthe r readin g

2.5 Furthe

31

r readin g

Some basi c concept s fo r digita l communicatio n system s wer e briefl y reviewe d in thi s chapter . Ther e ar e man y textbook s t h a t provid e a mor e detaile d an d comprehensive t r e a t m e n t o f thes e topics . T h e intereste d reader s ar e referre d to [50 , 67 , 120] , t o nam e jus t a few .

2.6 Problem

s

2.1 Suppos e w e hav e a communicatio n syste m wit h th e wavefor m (jpi * ca * p2)(t) give n b y ( \u\ f ( P i * ^ **»)(' ) =

-0.5|t-2 |+ l , for0