Improved performance of hybrid error control techniques for real-time digital communications over noisy channels

IM P R O V E D P E R F O R M A N C E OF H Y B R ID

E R R O R C O N T R O L T E C H N IQ U E S

F O R R E A L -T IM E DIG ITAL C O M M U N IC A T IO N S

O V E R N O IS Y C H A N N E L S

by C h a r l i e Q i n g Y a n g

R.S.H.E, 'University of Science and Technology of China, 1982

n t ' t V I '

T 5. I J '

s ,.V( M.A.Sc., University of Victoria, 1990

A Dissertation S ubm itted in P artial Fulfillment of the

L '

l/

* ' . u y y Y R equirem ents for th e Degree of

) %

D O C T O R , O F P H I L O S O P H Y

in the D e p a rtm e n t of Electrical and C o m p u te r Engineering

We accept this dissertation as conforming to th e requ ire d s ta n d a rd

Dr. Vi jay K 11 ha 1'gava, Supervisor (D e p a rtm e n t of EC E )

Dr. Qiang Wang, D epartm ental M em ber (D e p a rtm e n t of ECE)

Dr. P a u a jo tis Agatlioklis, D epartm ental M em ber (D e p a rtm e n t of E C E )

Dr, H ans A/iYf-rrl-kua O utside M em ber (D e p a rtm e n t of C o m p u te r Science)

Dr, S1 op 1 ■ioTTIl r, External E xam iner (Georgia I n s titu te of Tcclmology)

© C H A R L I E QING YANG, 1993 University of V ictoria

A ll riyh ls I'c sc rm t, D isse rta tio n m a y n o t be reproduced in w hole or in part,

by p h o to co p yin g or o th e r m e a n s

,

w ithout the p e rm issio n o f the. a u th o r.

ii

fhtpervisor; I)r. V ijay K. Bhargava

A B S T R A C T

H ybrid e rro r control techniques to im prove d a t a com m unication perform ance for noisy channels h av e been extensively studied. However, a growing concern h com m unication system design is th e im pact of delay d ue to retransm issions

and / o r delay-prone technologies on system performance, Previous analyses have not considered various delay aspects of a hybrid error control system. Efficient error control techniques which are aide to provide im proved coding gain a n d th ro u g h p u t by p ro m p tly m atching the error correction coding capability w ith th e changing channel conditions Imve yet to be developed and investigated,

In this thesis, delay-related perform ance characteristics are investigated for asynchronous time division m u ltiplexing links. Tw o different m e th o d s based on an im bedded Markov chain model are developed and applied to th e system with a noisy feedback channel, yielding analytical expressions for th e buffer occupancy and the block delay. A recursive expression for packet loss p ro b ab ility for system s with a finite tr a n s m itte r buffer is obtained.

T h e concept of delay-lim ited error control coding is in troduced for real-tim e com m unications. P erform ance im provem ent by tru ncation of a ty pe-II hybrid- AH Q protocol with one retransmission is investigated in detail. It is shown th a t the tru n c a te d protocol has a bounded delay and bounded cjucue length u n d e r typical com m unication traffic conditions. T h e error perform ance of th e tru n ca ted protocol is further analysed for various mobile fading channels.

M a tch e d -rate hybrid error control coding for both ad a p tiv e and non-adaptive cases is also studied. A new adaptive error control protocol using Reed-Solomon codes is proposed. T h e protocol uses novel feedback transm issions to achieve faster estim ation of channel states. N um erical optim iz atio n is carried o u t by

introduc-ili

ing overall th ro u g h p u t and modified th ro u g h p u t as oOieioucy criteria. Based on channel b it error r a te m e asurem ent, o p tim u m overall th ro u g h p u t is obtained with m in im u m im p lem entation complexity,

O u r general conclusions are: (1) B oth delay and packet, loss can be greatly re ­ duced. by incorporating a Keed-Solomon code into the data-link protocol for noisy channels. (2) T h e tru n ca ted hybrid error control protocol can provide coding gain

im provem ent and reduced delay over the conventional (u n tru n c a te d ) protocol, (!{)•

T h ro u g h p u t efficiency of a type-1 or type-II hybrid-A R Q protocol can be signifi­ can tly im proved by using th e proposed m atch ed -rate error control techuhpies,

E xam iners:

Dr. Vijay K, Bhargava, Supervisor• (D e p a rtm e n t of EGE)

Dr; Qiang Wang, D e p a rtm e n ta l Mem her (D e p a rtm e n t of BCE)

Dr, P a n a jo tis/A g ath o k lis, D e p artm en tal M em ber (D e p a rtm e n t of EGE)

Dr. H ans A. Muller, Ot^sidiTivfyjmber (D e p a rtm e n t of (lam p liter .Sen n< i )

i v

C on ten ts

T i t l e P a g e i A b s t r a c t ♦«n T a b l e o f C o n t e n t s i v L i s t o f T a b l e s v ii L i s t o f F i g u r e s v iii A c k n o w 1 e d g m e n t s x ii i 1. I n t r o d u c t i o n 1

1.1 M otivation for Research . ... . . . . '1

1.2 C ontributions of 1,lie Dissertation . . . , ... ... . . . 4

1.3 Organization of the D i s s e r t a t i o n ... . . . . G 2 F u n d a m e n t a l s o f H y b r i d E r r o r C o n t r o l 8 2.1 T h e Classic Coding Problem ... . . . . 8

2.1.1 Forward Error Correction C o d i n g ... ... . . . . 8

2,1.2 Basie Coding P r o c e s s ... , , , . 10

2.1.3 Coding C la in ... ... . . . . . . 11

2.2 From A,HQ to H ybrid Error C o n t r o l ... ... . . . 13

2.2.1 T h e Idea of A R Q and H y b r i d - A f t Q ... . . . . 13

2.2.2 R ate-A daptive Hybrid Error C o n t r o l ... . . . . 15 2.3 OanacHv of C om m u n ica tio n Channels with Feedback . , . .

. . . . i e

C O N T E N T S

2.3.1 M emoryless Channel M o d e l s ...

2.3.2 Feedback for Memoryless Channels . . .

2.3.3 Feedback For Channels with M emory . . . . 2.4 M ethods for Channel Error R a te Estim ation , , , , , , , , ,

2.5 P erform ance Evaluation M ethods ...

2.5.1 Reliability of Coding with Retransm issions 2.5.2 T h r o u g h p u t E f f i c i e n c y ...

2.5.3 Analysis of Coding Cain , , « . ... ... 2.6 S u m m a ry ... , . , ... ...

3 I m b e d d e d M a r k o v C h a i n A n a l y s i s o f D e l a y a n d P a c k e t L o s s 3.1 Introduction . ... ... ...

3.2 T h e System and Its Model . , , ... 3.3 Delay with an Infinite B u f f e r ... . , 3.4 P ack et Loss with a F inite b u f f e r ... ... 3.5 R e s u l t s ... ... ... ... ...

3.6 S u m m a ry . , , ,

4 R e d u c i n g T i m e D e l a y b y P r o t o c o l T r u n c a t i o n

4.1 I n t r o d u c t i o n ... ... 4.2 D elay-lim ited A dap tiv e Coding , ...

4.3 Description of the T ru ncated Protocol ... . 4.4 Delay Analysis , , ... , 4.4.1 'Time Delay and Queueing Models . . , , ... 4.4.2 Q ueueing A n a l y s i s ... . . . . 4.4.3 Transmission D e l a y ... , ... 4.5 N umerical Results and D i s c u s s i o n ... 4.6 S u m m a ry ... 5 C o d i n g G a i n I m p r o v e m e n t b y P r o t o c o l T r u n c a t i o n 5.1 Introduction . . , v 16 IS

20

23 25

25

31 31 37 38 38

30

•14 51 5!) Of)

70

70 72 76 77 77 78 81 •VI 84 80

80

C O N T E N T S

.5.2 Description of Channel M o d e ls ... 87

5.2.1 T h e Nakagami-m Fading Model... ... . . , , 87

5.2.2 Mobile Satellite Channel M o d e l s , , ... 89

5,8 Coding (lain A n a ly s is ... 91

5.3.1 Generalized Coding Gain . 91

5.3.2 Transmission Efficiency... . , , . 93

5.3.3 Protocol 'Error P robability < ... 95

5.4 Numerical Results and D i s c u s s i o n ... , 97

5.4.1 Coding Cain Comparison 97 5.4.2 O ptim um Error D e t e c t i o n ... 105

5.4.3 Queueing Results on Nakagarni Fading C h a n n e l ... 106

5.5 S u m m a ry . 107 0 M a t c l i c d - r a t e A d a p t i v e C o d i n g w i t h F e e d b a c k T r a n s m i s s i o n s 110 6.1 introduction ... 110

6.2 'Description of the Proposed Protocol ...112

6.3 T h ro u g h p u t A n a l y s i s ... 115

6.4 T h e M alohed-rato A daptive A lgorithm . . . 121

6.5 N umerical Results and D i s c u s s i o n ... . . . . 127

6.6 Sum m a ty ... 132

7 N u m e r i c a l O p t i m i z a t i o n o f C o d i n g P a r a m e t e r s 133 7.1 I n t r o d u c t i o n ... 133

7.2 C hannel Statistics and Error MeasurcmeM... 134

7.3 P erform ance C rite ria for O p t i m i z a t i o n ...138

7.4 O p tim u m Overall T h r o u g h p u t . . . 139

7.4.1 Design of Hybrid Error Control with BC1I Codes . . . 139

7.4.2 Design of Hybrid Error Control with RS Codes ... 144

7.4.3 Modified T h ro u g h p u t and More N um erical R e s u l t s 147 7.5 S u m m a ry ... 152

C O N T E N T S

8 Conclusions and Further Research

155

3.1 S u m m a ry of the D i s s e r t a t i o n ... 155

8.2 Suggestions for F uture W o r k ... 15(5

8.3 Concluding Rem arks . ... 157

A List

of Sym bols

158

B List

of Abbreviations

100

List o f Tables

.*5.1 O p tim u m error d e t e c t i o n . . . 107

7.1 Compulation, results of overall th r o u g h p u t for C B N w ith B C E codes 142 7.2 C om p u tatio n results of overall th ro u g h p u t for S-Ii w ith B C H codes 142 7.3 C om putation results of overall th ro u g h p u t for S -11 w ith RS codes . 147 7.4 C o m p u latio n results of modified th ro u g h p u t for S-R, w ith RS codes 150

List o f Figures

2.1 Digital com m u n ica tio n using forward e rro r correction coding , . , 2/2 P erform ance of uncoded P S K over AYV( 1N chan n e l' ... 2.3 A hybrid-A H Q error control coding s y s t e m ... 2.4 C apacity of the binary erasure channel ... ... , 2.5 Decoder undefo',lcd error probability of US c o d e s ... ... 2.6 T h r o u g h p u t perform ance of various error control schemes , . . , , 2.7 Coding gain analysis of various coding s c h e m e s . . , . . ...

3.1 ATDM system with hybrid-A R Q error control using a Heed Solomon code . . ... ... ... 3.2 D a ta transm ission based on SAW hybrid-A HQ error control . . ,

3.3 D a ta transm ission ba ied on C B N hybrid* A HQ error control , , , 3.4 T h e JVl/(t/l Markov chain m o d e l ... ..

3.5 P robability of u n d e te c te d error vs. channel bit error probability, where a block length of 2000 bits is assumed for ( IRC-16 and C H C 32 c o d e s ... . . ... ... ... ...

3.6 Average buffer occupancy for SAW protocol with; (a) C R CdO code

given

pi,

as I0“ !H (b) 011(1*16 code given-

pt,

ss lO"4; (c) (255,223) RS code given

pi,

~ l.0“ a } (d) (31,15) HS code given

pi,

~ J.(ra . . 3.7 Average buffer occupancy for CU N protocol with: (a) O R C I 6 code

given

pi,

s 10“ ;,i (b) CRC-10 code given

pi,

ss I0 " 4; (c.) (255,223) RS code given

pi,

= t0~3; (cl) (31,15) RS c ale given

pi, ^

10 '! . ,

M T O F F K I U R F S

x

3.8 Average block delay for SAW protocol with: (a) CRO-16 code; (b)

(2-15,223) ItS code; (e) (31,15) RS code, given A = 0.2 blocks/sec

(solid lines) and A = 1.0 blocks/sec (dashed l i n e s ) ... 06

3.9 Average block delay for C B N protocol with; (a) CRO-16 code; (b)

(255,223)

lt,S

code; (c) (31,15) RS code, given A = 1,0 blocks/sec (solid lines) and A ~ 2.5 blocks/sec (dashed line

■)

. . . . 66 3.10 P robability of packet loss for SAW protocol with: (a) CRO-1,6 code;

(b) (255,223) IRS code; (<*) (31,1.5) RS code, given A = 1.0 blocks/sec

(solid lines); A * 2,0 blocks/sec. (dash-dot lines); A ~ 3.0 blocks/sec

(dashed l i n e s ) ... 67

3.11 P robability of packet lose for C B N protocol with; (a) ORC-16 code;

(b) (255,223) RS code, (c) (31,15) RS code, given A — 1.0 blocks/sec

(solid lines); A - 2.0 blocks/sec (dash-dot lines); A = 3,0 blocks/sec

(dashed lines) ... 67

3.12 P robability of packet loss for SAW protocol with (255,223) RS code

(dashed lines) and (31,15) RS code (solid lines), given (a) A « 1.0

blocks/sec; (b) A “ 1,5 blocks/sec; (c) A — 2.0 b l o c k s / s e c ... 68

3.13 P ro b ab ility of packet loss for CIBN protocol with (255,223) RS code

(dashed lines) and (31,15) RS code (solid lines), given (a) A » 1.0

blocks/sec; (b) A » 3.0 blocks/sec; (c) A s= 5.0 b l o c k s / s e c ... 68

4,1 Digital com m unication using delay-limited ad a p tiv e error control

coding ... 74

4.2 T h ro u g h p u t of various error control t e c h n i q u e s ... 74 4.3 Q ueue length of the tru n ca ted protocol for arrival rates 0.5, 1.0,2.0,

2 , 5 ... 82

4.4 Q ueueing delay of the tru n c a te d protocol for arrival rates 0.1, 0.5,

L I S T O F F I O V

k

F S

XI

4.5 Transmission delay comparison of various coding p r o t o c o l s

83-5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.S 5.9 5 10 5.11 5.12 5.13 5.14 5.15 5.16 5.17

,'oe Nnkagumi-m distribution Coding gain analysis

- ( m =

I, Rayleigh fading chaunot)

T h r o u g h p u t comparison of ”arious coding protocols, w hen' si mu la tion result:’ shown count only error-free blocks . . . .

Coding gain analysis

(m

= 0,5) . ... C oding gain analysis ( r u

=

2) ... Coding gain analysis (m - 5) ...

Coding gain analysis (m a= 10) ,

C oding gain analysis

(m

« 2 0 ) ... Coding gain analysis on a Rayleigh fading channel . . C oding gain analysis on a Rician fading channel , . , C oding gain analysis on an AWGN channel

Im p a c t of e rro r detection capability on perform ance (m « 1, n

k

18

) .

I m p a c t of error detection capability on performance' (m ro 1,

n - k

- . 071

itl J « » «< » # t 4 I > • ^ » < » • « « * 4 « • t < * 4 l » • » » • < « • Q ueu e length of the tru ncated protocol over Nakagami fading chan nel (mssI.) for arrival rates 0.5, 1,0, 2,0, 2,5 . ... .

Q ueu e length of the truncated protocol over Nakagami fading chan­

nel (mss 10) for arrival rates 0,5, 1.0, 2,0, 2.5 , . . ...

Q ueueing delay of the tru ncated protocol over Nakagami fading channel (ms=l) for arrival rates 0.1, 0.5, 1.0, 2.0, 2.5

Q ueueing delay of the tru n c a te d protocol over Nakagami fading channel (msslO) for arrival rates 0.1, 0,5, 1.0., 2.0, 2.5 . ...

88 98 99 101 101 102

102

103 103 101 101 105 100 108 108 109 109

6.1 lly b rid -A R Q error control using a rate-adaptive Reed-Solomon ro d e 112 6.2 T h e relati ve values of

P at)

and

P ta n

... . , 118

U S T O F F l G U I U t t x li

6.3 T h ro u g h p u t of a m is-m atched protocol and an im proved type-I hybrid-A R Q protocol, where C R C codeword length is 260 symbols and RS codeword length is 256 s y m b o l s ...124 6.4 P robability distribution of symbol errors in an RS codeword , . . , 125 6.5 T h e envelopes of symbol error probability d i s t r i b u t i o n s ... 125 6.6 Ideal m atch ed -rate th ro u g h p u t observed from th e m a x im u m thro u g h ­

p ut envelope • • 127

6.7 T h ro u g h p u t perform ance of type-I and type-II hybrid-A R Q p ro to ­ cols, where C R C codeword length is 260 symbols and RS codeword

length is 256 symbols . . ... 128

6.8 Tlirouglipul, of the type-II hybrid-A R Q protocol com pared with the ideal m atch ed -rate th ro u g h p u t ... 128 6.9 P erform ance of th e proposed protocol for different time-varying chan­

nel conditions

u

— 1 (case 1); // = 0,9 (case 2);

u =

0 (case 3) . . 131 6.10 Performance' of the m is-m atched protocol for different tim e-varying

channel conditions -

u =

1 (case 1);

v —

0.9 (case 2);

v

= 0 (case 3) 131

7.1 T h ro u g h p u t perform ance of h y b rid error control protocols, where CllCJ codeword length is 260 symbols and RS codeword length is

256 s y m b o l s ... 134 7.2 Block diagram of bit error rate m e a s u r e m e n t ... 136 7.3 T h e results of a typical bit error ra te m e a su re m e n t . . . 137 7.4 C o n to u r plot of overall th ro u g h p u t for C B N with BCH Codes . . . 143 7.5 C on to u r plot of overall ti n o u g h p u t for S-R with BCH Codes . . . . 143 7.6 T h e results of a. BKR m e asure m ent for a noisy c h a n n e l ... 145 7.7 Cont our plot of overall, th ro u g h p u t (G llC-16 and 16 symbols overhead) 148

7.8 Illustration of 3-dimensional overall th ro u g h p u t s u r f a c e ... 14S

7.9 C on to u r plot, of modified th ro u g h p u t (ORC-16 a n d 16 symbols over­

L I S T O F F I G U R E S

xiii

7.10 Illustration of ’Hmensional modified th ro u g h p u t surface1 . . . 151 7.11 C ontour plot of modified th ro u g h p u t ((111(1*10 and 8 symbols over­

head) . ... 1 5 2 7.12 C ontour plot of modified th ro u g h p u t (ORO-IG and 32 .symbols over­

head) ... 103

7.13 C ontour plot of modified th ro u g h p u t (OR 0-3 2 and 16 symbols over­

head) . . ...133

7.14 C on to u r plot of modified th ro u g h p u t (ORC-32 and 8 symbols over­

head.) ... ...I hi

7.15 C on to u r plot of modified th ro u g h p u t (ORC-32 and 32 symbols over­

A c k n o w le d g m e n ts

I feci deeply indebted to my supervisor, Dr. Vi ja y K. Bhargava, for h av­ ing offered me th e o p p o rtu n ity to pursue my g ra d u a te studies in V ictoria and introducing me to the sub ject of error control coding. W ith o u t his s u p p o rt and e ncouragem ent th ro u g h o u t my research, this work would never h av e been possible.

I would like to thank l)rs. Q. Wang, P. Agathoklis, and II. A. Muller for their services as my supervisory com m ittee m embers, and Dr. S. B. W icker for his valu­ able com m ents on the work I have done.

M y th a n k s are: also extended to all the faculty m em bers and staff in t h e E C E d e p a rtm e n t. T h e help I,hey offered m e during my P h.D . p rogram is deeply appre­ ciated.

I thank my colleagues a t the Digital C om m unication G roup, which is under th e guidance of Dr. V. K, Bhargava and Dr. Q. Wang, for creating th e friendly atm o s p h e re in which I have had the pleasure to work. In p a rticu la r, I wish to th a n k Mr. Gordon W ebster and Mr. Dave Peterson for their helpful suggestions which have improved the quality of this thesis.

T his research is s u p p o rte d by th e C anadian In stitu te for Telecom m unications

Research, a Fedora! Network, of Centers of Excellence, a p o s tg rad u ate scholarship from the N atural Science and Engineering Research Council of C anada, a Univer­ sity of V ictoria Fellowship, and a P resid en t’s Research Scholarship.

Finally, 1 wish to dedicate this dissertation to my parents, Dr. J u n h u a Yang and Dr. C hong Zhao, m y wife Ju lie Qing Zhu and my daug h ter Yffei Yang. T h eir love and care mack' all my work worthwhile.

Without delay constraint,

electrical communication makes no sense

,

I'o achieve

real-time communication

over a noisy channel

remains

the m o s t challenging problem

to researchers

C hapter 1

In trod u ction

1/1

M otivation for R esearch

T h e stu d y of error control coding began in 19d8 with th e publication of Claude Shan non *s famous p aper [98], Shannon dem o n strated the existence of codes achiev­ ing reliable com m unication whenever th e

dala m lv.

is sm aller th a n a threshold

0

called the

ch a n n el rapacity,

hot* the additive, white G aussian noise (AW GN) chan­ nel, the channel capacity is given by

c = m o g <2 l - b “ 1 , _{+ /VJ »} s l . l )_{A 'G}

where

13

is the channel bandw idth and

S / N

is the ratio of signal to noise power falling within the bandw idth. This rem arkable result indicates t h a t th e u ltim a te p erfo rm a n ce limit caused by channel noise is not reliability, as generally believed before S h a n n o n ’s work, but the rate a t which d a t a can be reliably tra n s m itte d .

T h e concept of channel capacity is fundam ental to co m m u n ica tio n theory and is surprisingly powerful and general. It can be applied to a large class of channel models, w hether memoryless or not, discrete or nondiscrete. However, S h a n n o n ’s celebrated coding theorem s are only existence theorems; they do n o t show how promising coding schemes can be constructed. Since the. publication of S h a n n o n ’s paper, a considerable am o u n t of research lias addressed th e design and analysis

C H A P T E R I. I N T R O D U C T I O N

2

of practical coding and decoding techniques p erm ittin g reliable com m unication at

th e d a t a rates promised by the theory [9] [11] [1.7],

Today, error control coding is an area of increasing im p o rta n c e in digital com ­ m unications. System planners and designers m ust use sophisticated techniques to cope with the ever-increasing d em and for digital com m unication services. Perhaps th e m o s t im p o rta n t advantage digital com m unications has over its analog coun­ te r p a r t is th e ease with which coding can be im plem ented to significantly improve com m unication performance.

T h e efficiency and reliability of a d a t a com m unication system depend heavily on the chosen data-link* protocol, T h e basic function of a data-link protocol is to provide an ap p arently error-free link between com m unication nodes in a network, M ost existing error control protocols, such as high d a t a link control (Ill)b ( l),

em ploy cyclic redundancy check (0.110) bits to detect errors [97], This type of error

control, which is called. A u to m a tic R e p e a t re.Quest ( AHQ), involves segm enting the bit stream into blocks and adding error detection redundancy b its 'to each block. W henever errors are detected, th e blocks d e te c te d in error are re tra n sm itte d ,

A m a jo r concern in mobile d a t a com m unications is the control of tra n s m is ­ sion errors when channel conditions are poor due to m ultip a th fading, Doppler effects and oth e r interference. U n d e r poor conditions in mobile d a t a links, simple AR.Q-ba.sed protocols often provide very low channel efficiency, In recent yearn,

m a n y hybrid error control protocols1 have been proposed to im prove th e channel efficiency [I] [123]. Those more sophisticated protocols make use of b o th error

dem otion and error correction coding in order to achieve high th ro u g h p u ts and low u n d etecte d error probabilities for two-way mobile data, com m unications,

Most of us have a reasonably good idea of the role, of feedback in control system s, where it is easier to make recursive corrections than open loop corrections. Feedback also plays an im p o r ta n t role in com m unications by simplifying th e design of error control coding. For exam ple, ft DUG with a suitably large buffer lias been

C H A P T E R , I, I N T R O D U C T I O N

3

widely used for d a t a transmission [10], Moreover, a com m unications system using a 'hybrid protocol m ay be m ore reliable than a system using only p u re forward e rror correction, and have a higher th ro u g h p u t than a system w ith retransm ission only [08]. Since the error statistics on m ost real channels vary w ith tim e, ad aptive

error control coding has recently become an active research topic [61] [62] [73] [100]

[109] [110]. Obviously, ad ap tiv e error control m ust utilize some ty p e of feedback

protocol,

In the com m unications context, a feedback channel is often available, b u t both transmission tim e and system complexity may be increased if feedback is utilized. P roper use of feedback can greatly affect the trade-offs in a co m m unication system design, However, to achieve real-time, com m unication over a noisy channel rem ains the m ost challenging problem to researchers in th e a re a of hybrid error control tech­ niques, For example., the next-generation of cellular and microcellular system s are expected to,provide efficient wireless transmission of speech, d a ta , image, facsimile and video signals between portable term inals and the wireline network w ith in real­

tim e requirem ents, Here we use the word “real-tim e” 2 to m ean t h a t inform ation is delivered to the end user within the desired tim e period, As will b e shown in this work, hybrid error control techniques can offer promising solutions to various real-tim e com mmiieatioris cllallcnges.

Most p erform ance analyses for error control coding reported in th e literatu re have been carried o u t by out' of two research groups. One group has focused on various ARQ and hybrid-A RQ protocols. For perform ance evaluation of these feed­ back protocols, th ro u g h p u t and undetected error probability a,re norm ally used, 'Phe other group has only been concerned with forward error correction, using cod­ ing gain as a m easure of the system performance. Com m on perform ance m easures are lacking, due to the fact t h a t in th e early days th e applications of

FISC were m ainly motivated by satellite com m unications [101], while A RQ was

* According to- Webster's Ninth New C ollegiate Dictionary, real-time is the actual tim e during which s o m eth in g takes place,

C H A P T E R I I N T R O D U C T I O N

very p opular in c o m p u ter d a t a com m unications [107]. Consequently, It has been

difficult to com pare the performance of various coding schemes with and without feedback.

W e believe th a t much m o re research needs to be done in applying common s ta n d a rd s to effectively evaluate and com pare the perform ance of different coding schemes. M ore promising techniques can then be identified and developed to im ­ prove system perform ance in a cost-effective m anner, To pursue this u lt im a te goal, we a re going to s tu d y various perform ance aspects of hybrid error control coding techniques for digital com m unications in this work.

1.2

C on tributions o f th e D issertation

H ybrid error control techniques to improve d a t a com m unications perform ance for noisy channels have been extensively studied. However, a g ro w in g concern in com m unication system design is th e im pact of delay due to retransmissions and/or.

delay-prone technologies on s y ste m perform ance, Previous analyses have not con ­ sidered various delay aspects of a hybrid error control system. Rflieieut error control techniques which are able to provide improved coding gain and th ro u g h p u t by p ro m p tly m a tch in g the error correction coding capability with the changing channel conditions have yet to be developed and investigated.

I n this work, the following three m a jo r contributions concerning improved p e r­ formance of hybrid error control techniques are reported,'

1. Delay-related perfo rm a n ce characteristics are investigated for a,synchronous

tim e division m ultiplexing (ATOM) links with a hybrid-A RQ error control

protocol. T h e protocol employs a Reed-,Solomon code for b o th error detection and e rro r correction, T h e system is modeled as an im bedded Markov chain with e ith e r finite or infinite tra n s m itte r buffer, T w o different m ethods based on th e im bedded M a rk o , chain model are applied to the system with a noisy feedback channel, yielding analytical expressions for the buffer occupancy

o f i A p m i i, i N T H O D v c r m

5

and th e block delay, A recursive expression for packet loss probability for system s with a finite tra n s m itte r buffer is also obtained.

2. T h e concept of delay-limited error control coding is introduced. In particu la r, performance, im provem ent by truncation of a type-II hybrid-A R Q protocol with one retransmission is investigated in detail. It is show n th a t th e tr u n ­ cated type*l i hybrid protocol has a bounded delay and bou n d ed queue length under typical com m unication traffic conditions. T h e error perform ance of the tru n ca ted protocol is further analysed for mobile channels w ith

R a y le ig h

,

M ae, hog-N orm als

and normalized

N a ka g a m > m

fading distrib u tio n s. M ean­ while, an analytical approach is developed to com pare t h e power savings of various error control protocols, including the tru n ca ted protocol and th e

conventional (u n tru n c a te d ) protocols.

3. A new adaptive error control protocol using Reed-Solomon codes is proposed and its th ro u g h p u t perform ance for a b inary sy m m e tric channel (BSC) with

a tim e-varying channel error probability is analyzed. T h e novel protocol makes use of received blocks sent back to th e tr a n s m itte r to achieve faster estim ation of channel states in a two-way d a t a com m unication link. T h e tlirouglipul, capacity of hybrid error control protocols is identified and an em pirical adaptive algorithm is proposed. M a tch e d -rate e rro r control coding lor nou-aclaptive cases is also studied. A novel th r o u g h p u t criterion is p ro ­ posed to evaluate perform ance based on channel b it error r a t e m easurem ent, and numerical optim ization of th e perform ance is u n d ertaken.

T h e perform ance analysis results repo rte d in this work have shown t h a t the p ro ­ posed hybrid error control techniques offer improved p erform ance in th e following th ree aspects,

1. Both delay and packet, loss can be greatly reduced by incorporating a Reed- Solomon code into the data-link protocol, especially for m obile d a t a links,

C H A P T E R L I N T R O D U C T I O N

(i

where severe fading may result in very poor performance when a pure ARQ protocol is used.

2. T h e tru n c a te d protocol provides coding gain im provem ent over the u n tru n ­ cated ty p e -11 hybrid-A R Q protocol. T h e tru ncated protocol is therefore best suited to power- and delay-lim ited mobile applications.

3. T h r o u g h p u t efficiency of a type-I or type-II hybrid-A R Q protocol can be significantly im proved by using th e matched-rate. error control coding tech­ niques.

1.3

O rganization of th e D isserta tio n

A n outline of the retmiinder of this dissertation is as follows!

•

C h a p te r

5 describes the fundam ental concepts in hybrid error control research and introduces import,ant m e th o d s for perform ance evaluation in later chap­ ters.

•

C h a p te r 3

investigates t he im pact of a noisy feedback channel on transm is­ sion delay and packet loss for AT DM links with a hybrid-A RQ error control protocol and a finite tr a n s m itte r buffer.

•

C h a p te r J,

studies the efficient use of a small n u m b e r of feedback signals (w ith or without; retransmissions) to achieve real-tim e com m unications. In

particular, tim e delay of a truncated type-II hybrid-A R Q protocol with one retransm ission is analyzed by using the m ethod developed in C h a p te r 3.

•

C h a p te r 5

em ploys a unified analytical approach to evaluate the coding gain of tru n ca ted protocols on various mobile fading channels. The. results are

C H A P T E R , h I N T R O D U C T I O N

7

•

C h a p ter 6

proposes a new ad aptive error control protocol w ith feedback tran s­ missions, and analyses its th ro u g h p u t perform ance for a BSC w ith a time- varying channel error probability. T h e m atch ed -rate th r o u g h p u t capacity of hybrid error control protocols is also discussed.

•

C lm p ie r

7 develops a pragm atic m ethod lor num erical optim iz atio n of overall throughput. By numerical optim ization, the o p tim u m system p a ra m e te rs are found for type-I hybrid-ARQ w ith BCH codes and RS codes.

•

C h a p te r

<S’ concludes tills dissertation and provides suggestions for fu tu re work which will extend the results of our Investigation.

C hap ter 2

F u n d am en tals o f H ybrid Error

C ontrol

In this c h ap ter we give a historic overview of error control for digital com muuica- tions and exam ine tin role of feedback in com m unication. T h e fundam ental issues presented in this c h a p te r form the basis of th e whole thesis and most of them are used in later chapters.

2.1

T he C lassic C oding P roblem

2.1 .1

Forward Error C orrection C oding

E rr o r control coding is concerned with m e thods of delivering inform ation from a source to a destination with a m inim um n u m b e r of errors. There, are three basic techniques for error control coding which are used in com m unication systems:

forward error correction (E E C ), ARQ and a proper com bination of E E C with

A R Q (hybrid) [LI] [68]. All three techniques add redundancy to data, prior to transm ission in o rder to reduce the effect of errors a t the receiver. EEC does n o t need to have a feedback channel, while ARQ requires a feedback channel, for necessary retransm issions.

T h e com m unication system depicted in Figure 2.1 employs EEC). The. source g en era tes d a t a bits or messages th a t m ust be tra n s m itte d to a distant user over a noisy channel. Generally speaking, a specific signal is assigned to each of

M

C H A P T E R 2, F U N D A M E N T A L S

9

Discrete _ Duta Channel

Figure 2.1: Digital com m unication using forward error correction coding

possible messages th a t can be e m itte d by th e source. 'The selection rule th a t assigns a tr a n s m itte d signal to each message is the

code.

'The

encoder

im plem ents the selection rule, while the

decoder

perform s th e corresponding inverse m apping, Because of channel noise, th e tra n s m itte d signals may not arrive at t h e receiver exactly as tra n s m itte d , causing errors to occur a t the decoder input. A natural design objective is to select, a code th a t will p erm it most of the errors to b e

corrected

by the d e c o d e r , thereby providing an acceptable level of reliability.

Coding is a design technique which can fundam e n tally change the trade-offs in a digital com m unication system. T h e m ost trivial ex am p le of coding is t h e repetition of th e sam e message on the transm ission channel. Here it is clear th a t redundancy, and therefore reliability, is obtained a t the expense of transm ission efficiency, or bandw idth utilisation, In general, error control coding can increase signal quality from problem atic to acceptable levels. If the a t te n d a n t increases in com plexity a t the tra n s m itte r and receiver is economically justifiable, a n d b a n d w id th utilization is not unduly com prom ised, useful perform ance im provements may result. For exam ple, with coding, less power m ay be required to com m unicate between a s atellite and a mobile term inal. F urtherm ore, coding m ay result in an increase in the m a x im u m n u m b e r of mobile term inals per satellite.

C H A P T E R 2. F U N D A M E N T A L S

It)

2,1.2

B asic C od ing P rocess

F E C coding system s have traditionally been divided into

block

and

convolulionnl

error-correction techniques.

In an (n,

k )

linear block code, a sequence of

k

inform ation bits is used to obtain a set of

n

—

k

p a rity bits, yielding an encoded, block of « bits, Usually modulo-2 a rith m e tic 3s used to c o m p u te the parity bits. Modulo-2 arith m e tic is particularly su ite d to digital logic; addition corresponds to the K X d M lS IV H -O R operation, while m u ltiplication can be realized as an AND operation. T h e code rate

r

is defined as

r

=

k / n

and

n

is called th e block length. Id near codes form a linear vector space; two code words can be added (m odulo 2) to produce a th ird cotie

word,

T h e H a m m in g weight of a code word e is delined to be the num ber of nonzero com ponents of c, For exam ple, the code word

c

~ ( 1 10101) has a H am m in g weight of 4. T h e H a m m in g distance between two code words ci and o>, denoted d(ci,r*a),

is th e n um ber of positions in which they differ. For exam ple if c t •= (IIUJ.01) and

c%

= (111000) then f/(c(,

c-p) —

3, T h e

m in im u m , d ista n ce d

of a linear block code is equal to the m inim um weight of its nonzero code words, A code can correct all p a tte rn s of

I

or fewer random errors and d etect all p a ttern s having no more than

»

errors, provided t h a t

s

+

21

+ 1 <

d.

If the code is used for error correction alone, any p a t te r n of

I

or fewer ran d o m errors can be corrected, provided th a t 214 - 1

£ d,

A convolutional code of rate

\ / v

may be generated by a.

K

stage shift register and

v

modulo-2 a d d e r s . For each in p u t information bit, the o u tp u t of the modulo 2 adders provides

v

channel bits. T h e

co n stra in t lenglh.

of th e code is defined as th e n u m b e r of shifts over which a single Information bit can influence the encoder o u tp u t, For the simple binary convolutional code, the co n strain t Jengi.h is equal

to

K i

the length of the shift register,

W h e th e r block coding or convolutional coding is used, the encoded sequence e. m a p p e d to suitable waveforms by th e m odu lato r and tra n s m itte d over the noisy

C H A P T E R 2. F U N D A M E N T A L S

11

channel. T h e physical (or waveform) channel consists of all th e h ard w are (for exam ­

ple, filtering and amplification devices) and th e physical m e d ia t h a t th e waveform passes through, from the o u tp u t of the m o d u la to r to th e in p u t of t h e dem odulator.

T h e d em odulator estim ates which of th e possible symbols was tra n s m itte d based upon an observation of th e received signal. Finally, the decoder estim ates the tra n s m itte d information sequence from the d em odulator o u tp u t. T he decoder makes use of the fact th a t t h e tra n s m itte d sequence is com posed of codewords. Transmission errors are likely t o result in th e reception of a noncode sequence.

2.1.3

C oding Gain

It is often useful to express coding perform ance not in term s of th e error ra te

reduction for a given sigual-to-nowe ratio (SNR), b u t as th e S N R difference at a fixed hit error rate. Consider an AvVGN channel with one-sided noise spectral density

Nq

and no resl notion

o n

bandw idth. Let

E/,

denote the received energy per bit. Jt can bo shown [11] t h a t if th e SN R

J'A/Nq

exceeds —1.6

d B t

there exists a coding scheme which allows error-free com m unication, while reliable com m uni­ cation is n ot generally possible at lower signal-to-noise ratios. O n the other hand, it. is well known th a t e n c o d e d phase1 shift keying ( P S K ) m odulation over th e same

channel requires a b o u t 9.6

did

to achieve a b it error rate of 10~5. T hus, as shown in Figure 2.2, a potential coding gain of 11.2

d B

is theoretically possible.

C oding g a in

(in decibels, or

d B )

is defined as the difference in values of E b / N o

required to a tta in a particular error r a t e w ith and w ithout coding [12]. Notice

t h a t coding gain is oft en o bta ined a t th e expense of transm ission b a n d w id th 1. T he b a n d w id th expansion is the reciprocal of the code rate. Coding schemes delivering 2 to 8

d.B

coding gain are widely used in m odern digital co m m unication systems. This is because of th e phenom enal decrease in the cost of digital h ardw are and the much less significant decrease in th e cost of analog com ponents such as power

B it E rr o r R a te

C H A P T E R 2, F U N D A M E N T A L S

B in ary P S K (no cod in g)

P er fo r m a n ce lim it (h ard q u an tization )

P e r fo r m a n ce lim it (soft q u a n tiza tio n )

R eg io n o f p o ten tial cod in g gain

-2,00 0.00 2.00 4,00

6.00

8.00

10.00

E b / N o

C H A P T E R 2, F U N D A M E N T A L S

13

arnpli fiors, an tern me and so on.

P ractical com m unication systems rarely provide the ability to make full use of the actu al analog voltages of the received signal. T h e norm al practice is to quan tise these voltages, if binary quantization is used, we say t h a t a

hard decision

is m ade a t th e receiver as to which level was actually sent. For exam ple, in coherent P S K with equally likely tran s m itte d symbols, the optim u m threshold is zero [12]. T h e dem o d u lato r o u tp u t is a one or a zero depending on w hether the voltage is above or below the threshold. With coding, it is desirable to m aintain an indication of th e reliability of this decision. A

soft,-decision

dem od u lato r first decides w h eth er the voltage is above'or below the decision threshold, and then com putes a “confidence” n u m b e r which specifies how far from the decision threshold tire d e m o d u la to r o u tp u t is. This num ber in theory could be an analog quantity, but in m ost practical

applications a three -b it (eight-level) q uantization is used. It is known t h a t soft decision decoding is a b o u t 3

did

more efficient than hard decision decoding a t very high

E b / N o ,

A tigure of 2

d l l

is more likely a t realistic values of

E b / N o .

2.2

From ARQ to H ybrid Error Control

2.2.1 T h e Id ea o f A R Q and H ybrid-A R Q

In an

a u to m a ti c repeat request

(ARQ) system, whenever th e receiver detects an er­ ror in the tra n s m itte d message, it sends a retransmission request to the tr a n s m itte r over a feedback channel. T h ese requests are repeated until the message is received correctly. T h ree basic types of ARQ protocols are com m only used: stop-and-w ait, go-back-N, and selective-repeat [26] [68] [97].

Because of its simplicity, ARQ is used in m any d a ta com m unications systems. However, the technique lias a m a jo r shortcoming: the th ro u g h p u t efficiency may be highly depen d e n t on channel conditions. At low signal-to-noise ratios, th e n um ber of retransm issions required before a message is received correctly m a y be very large, and hence involve a long tim e delay, T his delay is unacceptable in some

C H A P T E R

2.

F U N D A M E N T A L S

D ata Out D ata In ARQ FEC ENCODER F E C DECODER FORWARD CHANNEL FEEDBACK CHANNEL

Uinary Forw ard C hannel

Figure 2.3: A hybrid-A RQ error eon I, ml coding system

applications. One approach to reducing the tim e delay is the

hybrid-A R Q

protocol of Figure 2.3. Here an A RQ protocol is used to o b ta in it. desired error rate. FEC coding is used to correct low-weight error p attern s in each message* reducing the n u m b e r of retransm ission requests.

In a

ty p c - I hybrid,-A HQ

system, the message and error detecting parity bits generated by th e ARQ protocol are fu rth er encoded with an F E C code [91]. At

th e receiver, t h e error correction p a rity bits are used to correct channel errors. TJie F E C decoder produce's an e s tim a te of the message ami the e rro r detection parity bits. Tire result is then tested by th e error detection system to d eterm ine if the message should be accepted as error free, or rejected as containing errors. If the channel signal strength is poor (high, bit e rro r rate), or if the. message* is long* the

probability of error-free transmission may approach zero. Under these conditions, th e efficiency may be improved by using a type-1 hybrid protocol rathe r than a sim ple A R Q protocol. However, if signal strength is adequate,, the type-1 hybrid protocol involves a waste of bandw idth, because the error correction parity bits are unnecessary.

In a

ty p c-II hybrid-A H Q

system [69], t h e first transm ission of a message is coded with error detection parity bits alone, as in a stan d ard ARQ protocol. If th e receiver detects errors in the received block, it saves the, erroneous block in a buffer and requests a retransm ission. T h e re tra n sm itte d information is not the*

OH A P T FAi 2, P U N D A M E N T A L S

15

original coded message, but a block of parity bits derived by applying an F E C

code to the message. The. receiver uses these parity bits (which m a y themselves bo in error) to correct the block stored in the receiver buffer. If error correction

is riot successful, subsequent retransm issions are. requested', which m ay consist of th e original codeword or ano th er block of error correction p a rity bits. The retransm ission form at depends ori the strategy and on th e error correction code used. The intention of type-II hybrid protocols is to provide th e efficiency of s ta n d a rd A R Q under good channel conditions while achieving th e perform ance im provem ent of ty p e d hybrid protocols u n d er poor channel conditions.

2.2.2

R ate-A clap tive H ybrid Error C ontrol

In o rder to cope with diverse system requirem ents and service options,

rate-adaptive

h yb r id error control

protocols, in which the code rate is a d a p te d to suit prevail­ ing transm ission conditions [75] [115] [123], are very a ttractiv e. Such protocols are applicable to satellite or mobile com m unication systems, where a d a p tiv e sig­ nal com pensation m ay provide ad e q u a te performance, when channel conditions are

poor (due to rainfall or fading, etc.) w ithout a p e rm a n e n t b a n d w id th reduction

caused by excessive coding overhead. In a rate-adaptive coding scheme, a high rate code is used under good channel conditions to m aintain a high in fo rm ation rate, b ut as the channel condition worsens, th e information rate is reduced by applying lower rate codes to m aintain th e desired reliability.

T raditional convolutional codes are not well suited to ra te -ad ap tiv e coding tech­ niques, because only low rate codes have been used, However, as d a t a transm ission rates increase, the need increases for good high rate convolutional codes w ith prac­ tical encoding and decoding techniques,

High rate pu n ctu re d convolutional codes

are a significant recent developm ent in which the. difficulties usually associated with decoding high rate codes are significantly reduced [19] ['hi], V ite rb i or sequential

C H A P T E R , 2. F U N D A M E N T A L S

Ki

decoding of r a te

l / v

codes, In fact, th e p u n ctu re d code is obtained in a straightfor­ w ard m a n n er from a “p a re n t” rate

l / v

code, while a Viterbi or sequential decoder for th e la tte r code can be easily modified to decode the p u n ctu re d code. 'Phis pro p e rty makes punctu re d convolutional codes a ttra c tiv e for rate -adaptive appli­ cations - a single decoder can acco m m o d a te the entire family of p u n ctu re d codes arising from a single p a re n t code.

A m ong th e class of block codes,

m a x i m u m distance separable

(MDS) codes are particu la rly well suited to rate-ad ap tiv e coding techniques. An (n,

k)

cock1 for which the m in im u m distance' equals

n

- 1 is said to be m ax im u m distance separable, MDS codes are optim al in the sense th a t no (•//, /:) code has m inim um H a m m in g distance exceeding

i i . - k -

1-1, T h e MDS codes most often encountered in practice are Reed-Solomon (US) codes. IIS codes are a ttractiv e not only because of

th e ease with which they can be decoded using Berlekam p’s decoding algorithm , b u t also because they feature a wide variety of possible code rates and block lengths. RS codes are suitable for rate -a d a p tiv e coding schemes because a low r a te paren t RS code can be broken down into several high rate subcodes, in much th e sam e way th a t a paren t convolutional code yields high ra te punctu re d codes. T h e subcodes are themselves linear block codes having the MDS property, With ap p ro p ria te modifications, the decoder for the parent code can also be used to decode the subcodes. T h u s a single decoder may be used in a rate adapti ve system w here the tr a n s m itt e r employs th e subcode most suitable for th e current channel s tate.

2.3

C apacity o f C om m unication C hannels w ith

Feedback

2.3.1 M em oryless C hannel M od els

A

m o d e l

of a com m unication channel is a m a th em a tic al representation defined to ap p ro x im ate ly describe the channel characteristics, ft is conventional to define a

C I I A P T M l 2, F U N D A M E N T A L S

17

channel model to include the m odulator, the d em odulator, and all th e intervening

transmission eq uipm ent and m edia [81]. A

discrete m e m o r y l e s s c h a n n el

(DM C) is defined by an M -ary set of input symbols { « f,

(i -

0, l , . „ , A f — 1)}, a Q-ary set of o u tp u t symbols

{ijj,

(j t=

0,

Q -

1)}, an d a set of conditional probabilities, called

transition probabilities

, which we can w rite as

P ( y

s

y j \ x

= a->) =

P ( v j \ m ) .

(2.1)

T h e

b in a r y s y m m e t r i c elu mn vl

(BSC) is an im p o r ta n t DMC m odel having a binary input, a 1 ' y o u tp u t, and transition probabilities given by

P ( y

~ 11a: ~ 0) =

P ( y

— 0 1;r — 1) — pt

p ( y

a 0|,r = 0) a

P { y

= 1|® = 1) = 1 - p6, (2.2)

where

pt,

is called the

bit error probability

and 0 <

pi,

< I.

A nother important. I) M( J model is th e

additive white G a u ss ia n noise

(AWGN) channel, which can be described by

y

- » +

n o ,

(2.3)

where

n a

is a '/.ero-mean Gaussian random variable with variance

<r2,

a nd th e in p u t ,r can have any one of M discrete values. T h a t is, th e conditional probability density function of the o u tp u t ;ty, with an in p u t ;r, is given, by

p {y\* = x t ) =

.

(2.4)

For the AYVCNf channel model, th e probability of a received bit being in error is not depen d e n t on w hether one or m ore preceding bits were in error. As the

probability of error on each received bit is the sam e, the perform ance of this m odel can be com pletely described by a single num ber, which is denoted by pt as in th e

BSC model. 91

C H A P T E R , 2, F U N D A M E N T A L S

1.8

2.3.2

Feedback for M em oryless C hannels

T h e idea of hybrid error control brings up the fundam ental question of the role of feedback in com m unication,

A m easure of inform ation for general sources was provided by ,Shannon [98] w ith the application of the concept of

entropy

to an inform ation source. Consider a source which produces any one of A/ symbols a:i , a g , ...»^*a/» where the probabilities of occurrence are

P ( x

i),

P[x-i)r

P (

x m

)\

die entropy of th e source is defined by

M

II ^ - E P ^ i ) l o y P ( x i ) ,

(2.5)

iss'l

I I

represents th e uncertainty a b o u t th e information before it is received from th e source. In the case of error-free transmission* the user receives uncorrupted source ay" bols, and the uncertainty a b o u t the source inform ation vanishes com­ p letely as symbols a re received, lii this ideal case, the channel transfers information from the source to th e user a t an average rate of

II

bits per symbol.

In all realistic cases, of course, transmission through the channel is not error- free, and th u s after reception the user is left with some residual uncertainty, M in ­ im izing this residual u n certainty while m aking efficient use of signal energy is the essence of t h e com m unication system design problem.

Consider a DM C model with a set of o u tp u ts ;f/i,v/2, T h e

m u t u a l infor­

m a t i o n

associated with the transmission of at,- and reception of

yj

is defined by

= (2,0)

T h e overall rate of transfer of information through the channel can then be

calculated by simply averaging /(»,•; &/j) over all possible values of

Xi

and

y 3i

giving us th e

average m u t u a l i n f o r m a t i o n

, defined by

O U A P m i 2. F U N D A M E N T A L S

19

(2.7)

which lias u n its of hits per symbol when the logarithm base is 2, T h e

capacity

of a discrete memoryless channel is defined as the m a x im u m value of

I { X \ Y )

with respect to all input distributions, th a t is,

shows th a t it is possible (b u t not how!) to tra n s m it d a t a with an a rb itrarily small

error rate, as long as the d a t a rate

lid

<

C .

As an example,, it can be easily shown th a t the capacity of the BSC is given by

If th e bit error probability

p ~

0,

O s s a

— J> he., we can tra n s m it 1 bit per s y m b o l On th e other hand, if

pi,

— 0.5,

C a s e —

0, and the channel fails to tra n s m it any information a t all. Therefore, in using the B S C model, we need only consider 0 <

pb

< 0.5.

A rem arkable conclusion ab o u t feedback, which is also d ue to S hannon [99],

is th a t

feedback docs n o l i m p r o v e the capacity o f m e m o r y l e s s channels.

However, m ost real communication' channels have been found to exhibit memory. Even when considering th e pure memoryless case, there, are numerous results t h a t indicate th a t feedback may reduce th e coding complexity. T h e sim plest ex am ple of this m ig h t be the binary erasure channel, as shown in Figure 2.4, in which one sends

X

g {0,1} and receives I ’ £ {0,1, e}, where

c

denotes erasure. Suppose

Y

—

X

with probability I - />, and V' — c with probability

S.

It is easy to show

C

= m a x

I ( X ] Y )

(2.8)

T h e m ost remarkable' of S h an n o n ’s results is th e

cha n n el coding theorem

, which

C H A P T E R 2, F U N D A M E N T A L S

20

o P ( y = o ) = i / 2 ( i - o c )

P(x=0)=1/2

e P(y=e)= ol

O

P(y=1)=1/2(1- oO

1

Pl[x=1)=1/2

1

₁

_-

₀

₍

Figure. 2,4: C apacity of the binary erasure channel

t h a t th e capacity of th e channel is

C —

1 —

8,

but is difficult to achieve this capa city w ithout feedback. W hen feedback is used, however, th e intended symbol can be re tr a n s m itte d whenever an erasure is received. T h e expected, n um ber of transm issions required to reveal a given symbol

X

is 1/(1' — A), T h u s (1 -

8)n

t r a n s m itt e d bits require

n

transm issions and th e resulting capacity is

(■

I -

8.

2.3.3

Feedback for C hannels w ith M em ory

I t lias been recognized [*17] t h a t when

real s y s t e m s

are measured

u n d e r real op ­

erating c o n d itio n s

in th e field, the errors are found to arrive mostly in bursts, a. b ehavior which is n o t in accord with th e AWC.1N model, and which cannot be well described by a bit error probability. Such channels are said, to exhibit

m e m o r y

, i.e., statistic al dependence in th e occurrence of errors.

To control th e errors effectively through some coding technique, it is necessary

to stu d y the statistical dependence of the errors. T h e fact t h a t the errors tend to occur in bursts, i.e., som ew hat “predictably,” should prove to be advantageous, In inform ation theory parlance, this means channel m em ory increases capacity. B u t how can m e m o ry be effectively exploited to realize the additional capacity? T his basic question has provided motivation for the developm ent of channel models

C H A P T E R

2,

F U N D A M E N T A L S

21

which reflect the statistical dependence of error occurrences [52], Often, channels with m em ory are modeled by using a Markov chain consisting of a finite n u m b e r of states with defined transition probabilities. Such models a t t e m p t to sim ulate the transitions in real channel behavior from good to bad, and vice versa.

As was pointed o u t in [52], one underlying reason th a t led to th e developm ent of a variety of channel models is the quest to obtain a model th a t represents real channels “b e tte r,” in the sense th a t a model should c ap tu re essential behavior of a real channel. W ith successive refinements, it is of course n a tu ra l t h a t these models tend to become increasingly com plicated, som etim es becoming so u n m a n ­ ageable as to bo of little practical use for the original intended purpose of being an. in term ed iate step.

It seems doubtful th a t any detailed models which could be c o n s tru cted would have general validity, even if enough d a t a crndd be collected to constru ct th e m [47], Our purpose in using a channel model is n o t to a p p ro x im a te a real error

source as closely as possible, but; ra th e r , w ith m inim al complexity, to o b ta in th e m a jo r channel characteristics with th e statistics necessary for evaluation of error control schemes, In this work, therefore, we will locus on perform ance evaluation by using only the IhSO and some basic fading channel models, For real channels with memory, ideal interleaving will be assumed. Nevertheless, we would still like to have a look at the role of feedback for channels with memory.

Consider the capacity of tim e-varying additive Gaussian noise channels with feedback, It has been shown [25] [31] th a t th e feedback capacity

G

f b and the nonfeedback capacity

C

satisfy th e inequalities

C m

< 2(7 (2.10)

and

Cm < C + i

C H A P T E R 2. F U N D A M E N T A L S

22

in bits p er symbol. We can see th a t feedback' can a t m ost double the capacity, and a t m ost one half bit per symbol can be added, for an additive G aussian noise' channel, Therefore,

feedback could i n c re a

" ;

the capacity o f channels with m em o ry,

I t has recently been shown [63] th a t th e channel capacity in a. Rayleigh fading

e n v ironm ent [57] is always lower th a n in a Gaussian noise environm ent. However, th e calculation was carried out in an average sense which did not take into account th e channel memory.

W e wish to clarify the point t h a t

m e m o r y increases channel capacity f o r binary

discrete channels.

It has been shown [116] th a t the discrete memoryless process has m a x im u m entropy am ong the class of binary discrete stochastic processes with error r a te

p

and arb itra ry m em ory length. In other words, the entropy for the binary channels with m em ory has an upper bound, i.e.,

I I < U m

(2,12)

w here

Ho

is the entropy for a DMO with error rate

pb

given by

M) =

- pbloypb -

(1 -

Pb)loy{

1 -

pb).

(2,13)

A process with a given error rate

p

can be said to have memory if its

I f

is less t h a n

Ho.

T h e m easure for memory, denoted as

0,

can be defined by

0 = —

(2.M)

I I

0

Thus a memoryless

(0

= Q) process has m axim um entropy

H a,

while a process w ith 0 2*1 has aero entropy, i.e., there is no uncertainty a b o u t the relative error

locations, such as in a cyclic process.

In the case of’ a binary com m unication channel with m em ory and arb itrary error rate

p,

th e channel capacity is given by

C K A P T P I l 2. F V h i ) A M E N T A L S

23

C

= 1 - / / bit/sym bol, (2.15)

Let

C

q denote the capacity for th e BBC with error rate p, then

Co

= 1 - //o, (2.16)

where

U

q is given by (2.13).

From (2.14), (2.15), and (2.16), we have

C

= Co +

OH

q

.

(2.17)

T h e q u a n tity

Oil

o, therefore, characterizes th e additional capacity of the c h an ­ nel with m em ory com pared to th e BSC. An AR.Q or hybrid-A RQ protocol is m ore efficient in a burst channel than a memoryless random channel. W hen a burst of very unreliable d a t a is received, retransm issions could provide a m ore reliable group of symbols. Moreover, retransmissions can provide several copies of the m essage which could be used to obtain a soft-decision sequence for th e decoder by

using the code com bining technique [21]. As a result, the additional capacity in a

channel with m em ory could be realized by using feedback.

2 .4

M eth od s for C hannel Error R ate E stim a ­

tion

E rror detection with retransm issions is in fact

adaptive,

because transm ission of re d u n d an t inform ation is increased when errors occur. Again, feedback plays an im p o rta n t role during th e coding adaptation. A plain ARQ protocol ad ap ts slowly, and the quest for more rapid a d a p ta tio n leads to the concept of using variable redundancy codes and forms th e basis of ty p e -tl hybrid-A RQ protocols and other ad ap tiv e coding schemes.