Iterative decoding with imperfect channel estimation for wireless systems.

University of Windsor

University of Windsor

Scholarship at UWindsor

Scholarship at UWindsor

Electronic Theses and Dissertations

Theses, Dissertations, and Major Papers

1-1-2006

Iterative decoding with imperfect channel estimation for wireless

Iterative decoding with imperfect channel estimation for wireless

systems.

systems.

Yibo Zhou

University of Windsor

Follow this and additional works at: https://scholar.uwindsor.ca/etd

Recommended Citation

Recommended Citation

Zhou, Yibo, "Iterative decoding with imperfect channel estimation for wireless systems." (2006). Electronic Theses and Dissertations. 7146.

https://scholar.uwindsor.ca/etd/7146

Iterative D ecoding w ith Im perfect Channel

E stim ation for W ireless System s

by

Y i b o Z h o u

A Thesis

Subm itted to the Faculty of G raduate Studies and Research through Electrical Engineering

in P artial Fulfillment of the Requirements for the Degree of M aster of Applied Science at the

University of W indsor

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A b s tr a c t

In th is thesis, we propose a new iterative algorithm for tu rb o decoding w ith integrated

channel estim ation for Rayleigh flat-fading channels. T h e design of th e proposed algorithm

is based on a new tu rb o decoding m etric for binary phase-shift keying (B PSK ) signaling

on fast fading Rayleigh channels w ith noisy channel estim ates. T h e algorithm consists of a

channel estim ator th a t can reduce th e error variance of th e channel estim ate iteratively by

using th e soft extrinsic inform ation o u tp u t from th e tu rb o decoder. T he extrinsic inform a­

tion generated from th e tu rb o decoder has some a priori inform ation of th e tran sm itte d d a ta

sym bols which can be used to refine th e channel estim ate after each iteration of decoding.

T he refined channel estim ate is th e n fed back to th e tu rb o decoder for th e next iteration

of decoding. T he resulting iteration betw een th e channel estim ato r and th e tu rb o decoder

using th e new m etric, w ith interm ediate exchange of soft channel-sym bol inform ation, yields

very impressive results. Sim ulation results verify th a t th e proposed algorithm can o u tp e r­

form conventional algorithm s significantly and even its perform ance can approach th a t of a

tu rb o decoding algorithm w ith perfect knowledge of th e channel.

In m ost existing tu rb o decoding algorithm s for fading channels, th e received d a ta sym ­

bols are m ultiplied by th e complex conjugate of an estim ate of th e channel gain before

tu rb o decoding w ithout any m athem atical justification. However, in th is thesis, we prove

th a t th e result of th is kind of preprocessing (or filtering) o p eration on th e received signal

A B S T R A C T

m inim um -m ean-square-error (MMSE) estim ates of th e tra n sm itte d d a ta symbols.

We also stu d y th e effect of signal-to-noise ratio (SNR) m ism atch on th e bit-error ra te

(B ER ) perform ance of tu rb o decoding algorithm s. According to th e previous results ob­

tained by o ther researchers, turbo-decoding w ith th e M ax-Log-M AP decoder is independent

of SN R for Rayleigh fading channels when th e channel is assum ed to be perfectly known to

th e receiver. Using such an assum ption, th e Rayleigh fading channel may th en be viewed

as an additive w hite G aussian noise (AWGN) channel conditioned on th e fading coefficient.

Since in practical com m unication system s, th e channel inform ation is not available at th e

receiver, th e channel m ust be estim ated a t th e receiver. Because of th e low signal to noise

ratios typical of tu rb o coded system s, it is difficult to obtain perfect estim ates of th e fading

coefficients. Thus, we show th a t th e M ax-Log-M AP decoder is sensitive to SNR m ism atch

over fading channels w ith noisy channel estim ates. Also, it is shown th a t th e Max-Log-MAP

decoder is less sensitive to SNR m ism atch th a n th e M AP or Log-M AP decoder.

Finally, we propose a new SNR m ism atch m odel to exam ine th e sensitivity of tu rb o

decoding algorithm s to SNR m ism atch for system s w ith an SN R estim ator. T his m odel is

m ore practical th a n th e conventional one. U nder th is model, we still can verify th a t th e

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

I would like to th a n k all who m ade th is work possible: my advisor Dr. B ehnam Shahrrava,

for thousands of suggestions and guidelines to m aintain overall high stan d ard s; my internal

reader Dr. Kemal Tepe and my external reader Dr. Scott Goodwin. I would also like to

th a n k my parents who have always supp o rted me w ith my g ra d u ate study. Lastly, I th a n k

my dearest girl friend Yanjie Mao, who is always by my side and has helped me to prepare

C o n ten ts

A b s tr a c t iv

A c k n o w le d g e m e n t v i

L ist o f F ig u r e s x

1 I n to d u c tio n 1

1.1 T urbo (iterative) decoding algorithm s ... 1

1.2 A new algorithm for tu rb o decoding w ith channel e s t i m a t e s ... 3

1.3 P reprocessing (fading com pensation) for tu rb o d e c o d in g ... 4

1.4 Effect of SN R m ism atch on B E R p e rfo rm a n c e ... 4

1.5 A New SNR m ism atch m o d e l ... 5

1.6 O rganization of th e t h e s i s ... 6

2 C h a n n e l M o d e ls 7 2.1 AWGN channels ... 7

2.2 Fading c h a n n e ls ... 8

2.2.1 T ype of sm all-scale f a d i n g ... 9

2.2.2 Fading channel m odel in th is t h e s i s ... 10

3 T u r b o C o d e s w ith I te r a tiv e D e c o d in g A lg o r ith m s 12 3.1 C hannel c o d i n g ... 12

C O N TE N T S

3.3 T urbo e n c o d e r s ... 13

3.3.1 C oncatenation of RSC codes e n c o d e r s ... 15

3.4 T urbo decoding algorithm s over AWGN c h a n n e l s ... 15

3.4.1 M AP algorithm over AWGN c h a n n e l s ... 16

3.4.2 T he modified M AP algorithm for tu rb o d e c o d in g ... 19

3.4.3 T urbo d e c o d e r ... 21

3.4.4 Log-MAP algorithm over AWGN c h a n n e ls ... 22

3.4.5 M ax-Log-M AP algorithm over AWGN c h a n n e ls ... 23

3.5 T urbo decoding algorithm s over fading c h a n n e ls... 24

4 T u rb o D e c o d in g w ith I n te g r a te d C h a n n e l E s tim a tio n 27 4.1 E stim ation th eo ry ... 28

4.2 A new a l g o r i t h m ... 29

4.2.1 T he first iteration of d e c o d in g ... 29

4.2.2 T he next iterations of d e c o d in g ... 30

5 F a d in g C o m p e n s a tio n 3 4 5.1 Linear M M SE estim ator of th e tra n sm itte d b i t s ... 35

5.2 T h e preprocessing before tu rb o d e c o d in g ... 35

6 E ffe ct o f S N R M is m a tc h on t h e P e r fo r m a n c e o f T u r b o D e c o d in g A lg o ­ r ith m s 3 7 6.1 Effect of an SNR m ism atch (AWGN C h a n n e l s ) ... 38

6.1.1 T h e M ax-Log-M AP a lg o rith m ... 38

6.1.2 T he Log-MAP algorithm ... 40

6.2 Effect of an SNR m ism atch (Fading C hannels) ... 41

6.2.1 T h e M ax-Log-M AP a lg o rith m ... 41

6.2.2 T h e Log-MAP algorithm ... 43

C O N TE N TS

7.2 B E R perform ance of tu rb o decoding algorithm s:

Effect of SN R m i s m a tc h ... 49

8 C o n c lu sio n a n d F u tu r e R e se a r c h D ir e c tio n s 54

8.1 Overview ... 54

8.2 Suggestions for future s t u d i e s ... 56

R e fe r e n c e s 57

List o f F igu res

2.1 AWGN channel m o d e l ... 8

2.2 Fading channel m o d e l ... 10

3.1 RSC codes e n c o d e r ... 14

3.2 T urbo codes e n c o d e r ... 14

3.3 System m odel over AWGN c h a n n e l s ... 15

3.4 T urbo codes d e c o d e r ... 20

3.5 System m odel over fading c h a n n e l s ... 24

4.1 T urbo decoding integrated w ith channel e s tim a tio n ... 29

7.1 Perform ance of C hannel estim ator 2 ... 46

7.2 Perform ance of C hannel estim ator 2 ... 47

7.3 P erform ance of C hannel estim ator 2 ... 47

7.4 B E R perform ance of th e proposed a lg o rith m ... 48

7.5 Effect of SNR m ism atch (fading c h a n n e l s ) ... 49

7.6 Effect of SNR m ism atch (fading c h a n n e l s ) ... 50

7.7 Effect of SNR m ism atch (fading c h a n n e l s ) ... 50

7.8 Effect of SN R m ism atch w ith new m o d e l... 52

7.9 Effect of SNR m ism atch w ith new m o d e l... 52

C h a p ter 1

I n to d u c tio n

1.1

Turbo (iterative) d ecod in g algorithm s

T urbo codes w ith iterative decoding algorithm (M AP algorithm ) were introduced by B errou

and Glavieux in 1993 [3]. A fter introduction of tu rb o codes, a lot of research has been

conducted on th e s tru c tu re of th e tu rb o encoder and tu rb o decoder.

In [6], th e trellis term in atio n of tu rb o encoder and extrinsic inform ation generated by

th e tu rb o decoder have been described thoroughly by R obertson. T h e sam e au th o r in [7]

introduced two kinds of th e m axim um a posteriori (M AP) algorithm s, th e Log-MAP and

M ax-Log-M AP decoding algorithm s. T he perform ance of th e M AP decoding algorithm has

been proved to be rem arkably well over additive w hite G aussian noise (AWGN) channels [3].

Since th e M AP algorithm is very complex, th e Log-MAP and M ax-Log-M AP decoding

algorithm s have been introduced to solve th is problem. T h e Log-M AP algorithm operates

in th e logarithm ic dom ain and reduces th e num ber of additions and m ultiplications in th e

decoding process while its perform ance is th e sam e as th a t of th e M AP decoder. T he Max-

Log-M AP algorithm fu rth er reduces th e com putational com plexity by approxim ating all th e

non-linear functions in th e Log-MAP algorithm w ith th e linear functions. T h e perform ance

1. IN TO D U C TIO N

In [5], Sklar sum m arized th e ideas behind tu rb o decoding. He discussed th e com ponent

codes and th e interleaver in th e tu rb o encoder. He also derived th e channel reliability

factor L c for tu rb o decoding algorithm s over AWGN channels. It was shown th a t th e

channel reliability factor depends on SNR in th is case.

T h e perform ance of tu rb o decoding algorithm s over fading channels has also been studied

since tu rb o decoding algorithm s have been introduced. In [16], B arbulescu assum ed the

fading am plitude and phase are known at th e receiver and th e channel reliability factor

L c for th is case depends on th e SNR and th e fading am plitude. In [18], Hall and W ilson

assum ed only th e phase of th e fading is known at th e receiver and studied th e decoding

process. In [19], th e sam e auth o rs studied th e decoding process w hen th e fading am plitude

is assum ed to be known at th e receiver. In [12], Frenger derived a new tu rb o decoding

m etric over fading channels by considering th e uncertainty of b o th fading am plitude and

phase. Frenger also indicated th a t if th e tra n sm itte d bits experience fading distortion w ith

im perfect channel estim ation, th e channel reliability factor L c depends on b o th SNR and th e

error variance of th e channel estim ate. In [12], using com puter sim ulations, Frenger proved

th a t tu rb o decoding based on his new m etric outperform s th e m etric w ithout considering

th e error variance of th e channel estim ate. He also showed th a t Log-M AP decoder can

achieve b e tte r b it error ra te (BER) perform ance w ith sm aller error variance of channel

estim ate.

T urbo decoding over AWGN channels needs SNR [5], and tu rb o decoding over fading

channels needs b o th SNR and channel fading factor [16], Practically, neither SN R or fading

factor are known at th e receiver. A lot of research th u s has been done on SN R and channel

estim ation schemes for tu rb o decoding. In [28] and [35], SNR and channel(fading) estim ation

schemes for tu rb o decoding have been discussed. All these estim ation schemes can be

classified into th ree categories: pilot-assisted channel estim ation, blind channel estim ation,

and hybrid channel estim ation. T he pilot-assisted and blind channel estim ation m ethods

have been discussed in [36] and [37], respectively. T he hybrid channel estim ation scheme

com bines two schemes. Most of th e proposed m ethods perform channel estim ation w ithout

1. IN TO D U C TIO N

scheme is introduced which uses th e extrinsic inform ation generated from th e tu rb o decoder.

L ater, m ore work was done on th e iterative channel estim ation in [24].

1.2

A new algorithm for turbo d ecod in g w ith channel esti­

m ates

In th is thesis, we propose a new tu rb o decoding algorithm for B PSK signaling on Rayleigh

flat-fading channels w ith noisy channel estim ates.

Conventional channel estim ators for tu rb o decoding have no feedback from th e tu rb o

decoder and ignore th e extrinsic inform ation generated from th e tu rb o decoder. W hile, th e

extrinsic inform ation can be fed back to th e channel estim ator to refine its previous estim ate.

In [23], Valenti proposed a tu rb o decoding algorithm w ith integrated pilot-assisted channel

e stim ato r w ithout using th e tu rb o decoding m etric proposed by Frenger in [11]. In [23],

Valenti used th e channel reliability factor L c derived by B arbulescu in [16] and tried to

re-estim ate th e fading factor and th u s derived a new L c. However, the reliability factor

derived by B arbulescu was based on th e assum ption th a t b o th th e am plitude and phase

of fading are perfectly known a t th e receiver. Since in practical com m unication system s,

th e channel inform ation is not available a t th e receiver, th e channel m ust be estim ated at

th e receiver. Because of th e low signal to noise ratios typical of tu rb o coded system s, it is

difficult to obtain perfect estim ates of th e fading coefficients. Hence, for fading channels

w ith noisy channel estim ate, we will use th e channel reliability factor L c derived by Frenger

in [11],

T hen, using a modified version of th e channel reliability factor L c derived by Frenger,

we propose a new iterative algorithm for tu rb o decoding w ith integrated channel estim ation

for Rayleigh flat-fading channels. F urther, we use th e hybrid channel estim ation scheme

in th e proposed algorithm . For th e first iteration, a pilot-assisted channel estim ator is

assum ed to provide us w ith th e initial channel estim ate and its error variance. A fter each

iteratio n of decoding, a blind linear M M SE (m inim um m ean square error) channel estim ator

1. IN TO D U C TIO N

back from th e tu rb o decoder. T he refined channel estim ate is th en sent back to th e Log-

M AP tu rb o decoder for th e next iteratio n of decoding. T his way, in th e blind channel

estim ator, we m ay reduce th e error variance of th e channel estim ate iteratively by using

th e a priori inform ation of th e d a ta bits buried in th e extrinsic inform ation. Finally, th e

resulting iteration between th e channel estim ator and th e tu rb o decoder, w ith interm ediate

exchange of soft channel-sym bol inform ation, can improve th e perform ance of b o th th e

channel estim ator and th e channel decoder.

1.3

P rep rocessin g (fading com p en sation) for tu rb o d ecod in g

In practical system s, in order to undo th e effect of fading, th e received d a ta symbols are

m ultiplied by th e com plex conjugate of an estim ate of th e channel gain before tu rb o de­

coding, [44], Since th ere has been no m athem atical justification in th e lite ratu re for this

kind of fading com pensation for system s w ithout perfect knowledge of th e channel, we try

to come up w ith some justification. F irst, we derive a linear M M SE estim ato r to o btain

optim al estim ates of th e tra n sm itte d d a ta symbols w ith noisy channel estim ates. Then, it is

shown th e linear M M SE estim ator can b e viewed as a norm alized version of th e preprocess­

ing (fading com pensation) operation used in m ost existing tu rb o decoding algorithm s, [11].

Also, we show th a t th e norm alization has no effect on th e B E R perform ance of th e tu rb o

decoder.

1.4

Effect o f S N R m ism atch on B E R perform ance

One requirem ent of th e Log-M AP tu rb o decoder over fading channels is th e knowledge

of SN R and th e error variance of channel estim ate. Since th e proposed tu rb o decoding

algorithm composed of th e Log-MAP decoder, b o th th e channel SNR and th e channel

fading m ust be estim ated. Adding any estim ator to th is algorithm increases th e com plexity

of th e algorithm . Therefore, any decoding algorithm th a t does not need perfect knowledge

of SN R or fading factor is m ore desirable. T he effect of SNR m ism atch on th e perform ance

1. IN TO D U C TIO N

and Nichols. In [30] and [28], th e effect of SNR m ism atch over th e fully interleaved channels

has been discussed. Valenti and W oerner fu rth er discussed th e effect of SNR m ism atch over

correlated fading channels in [27]. In [9], W orm indicated th a t M ax-Log-M AP tu rb o decoder

is insensitive to SNR m ism atch over AWGN channels and fading channels. However, this

result is valid for th e M ax-Log-M AP tu rb o decoding over AWGN channels. W hereas, for

th e M ax-Log-M AP tu rb o decoding over fading channels, it was assum ed th e fading factors

are known at th e receiver. Using such an assum ption, th e Rayleigh fading channel may

th en be viewed as an AWGN channel conditioned on th e fading coefficient. However, for

fading channels w ith im perfect channel estim ation, th e error variance of channel estim ate

m ust be included in th e channel reliability factor L c, [12].

In th is thesis, we stu d y th e sensitivity of th e Log-MAP and M ax-Log-M AP algorithm s

to SNR m ism atch over fading channels by using th e channel reliability factor L c derived

in [12]. We show th a t b o th th e Log-MAP and M ax-Log-M AP decoder are sensitive to

SNR m ism atch over fading channels when th e channel is not perfectly known. However, we

find th e M ax-Log-M AP decoder is less sensitive to th e SNR m ism atch th a n th e Log-MAP

decoder.

1.5

A N ew S N R m ism atch m odel

At last, we also propose a new SNR m ism atch model. T he new SNR m ism atch model is

m ore practical th a n th e conventional m odel used in th e literatures. T h e conventional model

considers all th e d a ta fram es experience th e sam e SNR m ism atch offset at a given SNR. In

th e new model, we consider th e SNR m ism atch is a random variable and different frames

experience different SNR m ism atch. U nder th e new model, we can still verify th a t b o th

Log-MAP and M ax-Log-M AP decoder are sensitive to SNR m ism atch over fading channels

1. IN T O D U C TIO N

1.6

O rganization of th e th esis

T he organization of th is thesis is as follows. In C h ap ter 2, th e channel models are introduced.

In C h ap ter 3, th e tu rb o encoder and tu rb o decoding algorithm s are described. In C hapter

4, we propose a new Log-MAP tu rb o decoding w ith integrated w ith channel estim ation

over fading channels. In C h ap ter 5, th e preprocessing or fading com pensation is discussed

from th e estim ation th eo ry point of view. In C h ap ter 6 , th e effect of SNR m ism atch on th e

perform ance of tu rb o decoding algorithm s is investigated. In C h ap ter 7, sim ulation results

are presented and also we introduce a new model to stu d y th e effect of SN R m ism atch

on th e B E R perform ance of tu rb o decoding algorithm s. Finally, conclusions and futu re

C h a p ter 2

Channel M o d els

To design th e channel estim ator and analyze th e effect of SN R m ism atch on th e perform ance

of tu rb o decoding algorithm s, we have to u nderstand th e channel th a t th e tra n sm itte d

d a ta b its experiences. In this thesis, we need two kinds of channel models to discuss our

contributions. In th is chapter, we first introduce additive w hite G aussian noise (AWGN)

channel model, and th e n fading channel model.

2.1

A W G N channels

T he additive w hite G aussian noise (AWGN) channel model to g eth er w ith bin ary phase shift

keying (B PSK ) m od u lato r is depicted as in figure 2.1. W here dk G (0,1) are th e tra n sm itte d

d a ta bits, x k G { — \ f E l , \ f E s ) are B PSK sym bols, n k are th e additive w hite G aussian noise

and y k are received symbols. In AWGN channel models, a AWGN noise n k is added to th e

tra n sm itte d symbols xk. Here, n k are comp lex-valued and G aussian d istrib u te d random

variables. T he m ean and variance of n k are

E [n k} = 0,

2. CHANNEL M ODELS

BPSK

A W G N n k

x±—

y t = x * + * *

Figure 2.1: AWGN channel model

T h e signal to noise ratio (SNR) for this case is

Eb E s

(2.2)

N 0 R x 2 a2 ’

where R is th e coding ra te and E s = Eb x R . E s is th e tra n sm itte d sym bol energy and Eb

is th e source bit energy. T he received symbols y k are

y k = x k + n k. (2.3)

For com m unication system s over AWGN channels, at th e receiver, we need to know th e

signal to noise ratio. Normally, th e symbol energy E s is set to 1 and th e coding ra te R is

known. T he noise variance No is th e only value we need to estim ate to get th e SNR.

2.2

Fading channels

For wireless com m unications, th e tra n sm itte d d a ta bits will experience a much more compli­

cated channel th a n AWGN channel. Due to th e com plex environm ents between th e wireless

com m unication term inals (buildings, forests and vehicles) and movem ent of th e term inals,

th e tra n sm itte d d a ta bits will be affected by unpredictable errors. N orm ally these errors are

called channel fading. And th e wireless channels are usually m odeled as fading channels.

C hannel fading can be classified into two categories:

• Large-scale fading

2. CHANNEL MODELS

In [40], it has been m entioned “Large-scale fading is caused by reflection, diffraction and

scatting o f the signals when they are obstructed by buildings, forests and vehicles”. This

ty p e of fading usually reflects a long period of channel characteristics.

Small-scale fading, or sim ply fading, on th e o th er hand reflects a short period of channel

characteristics. Small-scale fading is caused by th e m u ltip ath interference waves and th e

movem ent of th e wireless term inals. T he m u ltip ath interference waves are th e different

versions of th e tra n sm itte d d a ta bits arriving a t th e receiver a t different tim es. T he sum of

th e m u ltip ath waves at th e receiver will result in a received signal which can varies widely

in am plitude and phase. T he movement of th e term inals will also introduce channel errors.

In th is thesis, we only use th e small-scale fading channel model. Below, we will only talk

ab o u t th e sm all-scale fading.

2.2.1

T ype o f sm all-scale fading

T he ty p e of sm all-scale fading depends on th e two im p o rtan t factors in th e wireless com m u­

nication channels: m u ltip ath delay and D oppler spread. M ultip ath delay is introduced by

m u ltip ath interference waves. Doppler spread is introduced by th e movement of th e wireless

term inals. In this thesis, we only briefly introduce th e fading types, for detail, see th e book

by S tiiber, [40].

• Fading types due to m u ltip ath delay

M u ltip a th delay causes th e tra n sm itte d signal to experience eith er flat or frequency

selective fading.

If th e tran sm itte d d a ta bits period is greater th a n th e m u ltip ath delay, th e tra n sm itte d

d a ta bits will experience a flat fading. On th e o th er hand, if th e d a ta bits period is shorter

th a n th e m ultip ath delay, th e d a ta bits will experience a frequency selective fading.

• Fading types due to Doppler spread

D oppler spread causes th e tra n sm itte d signal to experience eith er slow or fast fading.

If th e signal b an d w id th is much greater th a n th e D oppler spread, th e tra n sm itte d bits

will experience a slow fading. If th e signal band w id th is sm aller th a n or close to th e D oppler

2. CHANNEL MODELS

a k AWGN nk

BPSK

y* = «»■ *»+ »*

Figure 2.2: Fading channel model

Two kinds of fading classification are independent.

2.2.2

Fading channel m odel in th is th esis

T he fading channel model used in th is thesis is a sm all-scale flat fading channel. T he

Rayleigh fading model is th e most common channel model used to describe th e flat fading

channel. T he fading channel model to g eth er w ith th e B PSK m odulator is depicted as

Fig 2.2. C om pared w ith th e AWGN channel model, th e m ultiplicative fading factors a*

are added. T he fading factors are complex-valued and G aussian d istrib u te d random

variables. T he m ean and variance of a*, are

We assum e a*. and n*, are independent and th e th e variance of th e received bits y& are:

A nd th a t is why th is fading channel model is called Rayleigh fading channel model.

For com m unication system s over Rayleigh fading channels, a t th e receiver, we need

b o th signal to noise ra tio and th e fading factors. Fading channels are m ore complex to E[ak] = 0,

E[\ak |2] = 2 a \. (2.4)

(2.5)

T he am plitude \a^\ has a Rayleigh d istrib u tio n w ith p d f

2. CHANNEL MODELS

analysis th a n AWGN channels because we need to estim ate th e fading factors. T here are

m any channel estim ation schemes. T hey can be classified into th ree categories: pilot-

assisted channel estim ation, blind channel estim ation and hybrid channel estim ation. In

C h a p ter 3

Turbo Codes with I te r a tiv e Decoding

A lg o r ith m s

3.1

C hannel coding

T he fading and AWGN noise induced by channels will affect th e tra n sm itte d d a ta bits a

lot. At th e receiver, it is very hard to recover w hat have been originally sent out. Channel

coding is an effective m ethod for deducing th e effects introduced by channels.

T here are m any different types of error control codes, b u t th ey can be classified into

block codes and convolutional codes. In [40], it has been m entioned th a t “B oth block codes

and convolutional codes are used in the mobile radio system s. Som e second generation digital

cellular standards (e.g., GSM, IS-54) use convolutional codes, while others (e.g., P D C ) use

block codes”.

Block codes can detect and correct a lim ited num ber of errors. T h e detecting and

correcting error ability of th e block codes depends on th e code free distance. T raditional

block codes include H am m ing codes, BCH codes, Reed-Solomon codes etc. H ard decision

block decoders are easy to im plem ent.

3. TU RBO CODES W IT H IT E R A T IV E DECODING A LG O R ITH M S

into blocks and th e n encoding, convolutional codes use linear finite shift registers (LFSR)

to encode th e d a ta bits sequence. Convolutional codes outperform block codes. T he most

com m on decoding algorithm for convolutional codes is V iterbi algorithm .

T urbo codes, introduced by B errou and Glavieux in 1993 [3], also known as parallel

concatenated recursive system atic convolutional codes, can outperform m ost codes.

3.2

Turbo codes

T urbo codes w ith iterative decoding algorithm (M AP algorithm ) were introduced by B errou

and Glavieux in 1993 [3]. T he tu rb o encoder in [3] consists of two RSC encoders, a random

interleaver and a m ultiplexer. T h e m ultiplexer concatenates th e o u tp u ts from th e two

RSC encoders and sends th em out serially. W h a t makes tu rb o code different from other

channel codes is its special decoding algorithm s. T he tu rb o decoder in [3] consists two

M AP decoders. Two M AP decoders generate and exchange the extrinsic inform ation. T he

decoding process runs several tim es to obtain a b e tte r B ER perform ance.

3.3

Turbo encoders

T he tu rb o encoder in [3] consists of two recursive system atic convolutional (RSC) codes

encoders. A sim ple binary code ra te 1/2 RSC codes encoder w ith code m em ory 4 is depicted

in Fig. 3.1, where th e bits in th e m em ory u k is recursively calculated as

3

^ k ~ dk -f ^ (^*1)

i=0

T he corresponding codeword is th e b it pair (d|,d]()

d/; = dfc ,

3

dpk = dk + ^ g u i i k - i , m o d i 52; = 0,1 (3.2) i=0

where g n = (11111) and c/2i = (10001) are th e code generators. T hey can also be expressed

3. TURBO CODES W IT H IT E R A T IV E DECODING A LG O R ITH M S

A

U /

Figure 3.1: RSC codes encoder

Data <4 d *

Multiplexer

RSC Encoder 2

RSC Encoder 1

Random Interleaving

3. T U RBO CODES W IT H IT E R A T IV E DECODING ALG O RITH M S

AWGN

BPSK

T urbo e n c o d e r T u rb o d e c o d e r

y t = xt + nk

Xk ^ V \ }

Figure 3.3: System model over AWGN channels

3.3.1 C oncatenation o f R SC codes encoders

T h e tu rb o encoder in [3] is shown in Fig. 3.2. T he source d a ta bits are sent to th e first RSC

codes encoder. T he interleaved version of d a ta bits will be sent to th e second RSC codes

encoder. T h e codewords are dsk , dpk l and dpk 2, where dsk are source d a ta b its and dFk l ,cPk 2 are

th e p arity bits from th e first and second RSC encoder respectively. T h e code ra te for this

tu rb o encoder is R = 1/3, th e o u tp u t sequence from th e m ultiplexer is {dsk ,cPk v <Pk 2 • • • }.

For higher code rate, we can add a switch to p u n ctu re th e codeword. For example, for

R = 1/2, th e o u tp u t sequence from th e m ultiplexer is {dsk, cPk 1; dsk+1, dpk+l 2 • • • }.

However, this is no t th e only construction of th e tu rb o encoder. In [4], th e tu rb o encoder

can consist of N RSC encoders, th e coding ra te for th a t system will be 1 / N . Also in [14]

and [15], it has been indicated th a t th e com ponent codes of tu rb o encoder can b e any block

codes and convolutional codes, not only RSC codes. Different design of tu rb o codes encoder

will result in different codes perform ance and decoding complexity.

3.4

Turbo d ecod in g algorithm s over A W G N channels

T h e tu rb o codes system over AWGN channels is depicted in Fig. 3.3. M axim um a poste­

riori probability (M AP) decoding algorithm s is th e optim al algorithm for tu rb o decoding.

Max-3. TU RBO CODES W IT H IT E R A T IV E DECODING ALG O R ITH M S

Log-MAP decoding algorithm s have been introduced to make th e im plem entation practical.

Below, we discuss th e detail of these decoding algorithm s over AWGN channels and fading

channels.

3.4.1

M A P algorithm over A W G N channels

T he optim al sequence decoding algorithm for convolutional codes is V iterbi algorithm .

V iterbi algorithm is a very sim ple decoding algorithm and its decoding process depends

on th e stan d ard of m axim izing a posteriori probability of a sequence received bits. M AP

algorithm , on th e o ther hand, tries to decode th e received d a ta bits depending on m axim iz­

ing a posteriori probability of each received bit. V iterbi algorithm is a sequence decoding

algorithm and M AP algorithm is a bit-by-bit decoding algorithm . T h e M AP algorithm is

discussed thoroughly in [1] and [3], bellow we tr y to explain th e fundam entals.

In th e M AP decoder, we need to calculate th e a posteriori probability (A P P ) of each

received bit. For binary phase shift keying (B PSK ), th e tra n sm itte d bits can only be 0 or 1,

so we need to calculate two A P P P r { d k = i / observat ions}, i = 0,1. T he decoder decision

depends on:

dk = 1 i f Pr { dk = 1 /observations} > Pr {dk = 0 /observations},

dk = 0 i f Pr { dk = 1 /observations} < Pr {dk = 0/observations}. (3.3)

T h e observations are th e received bits. We assum e th e frame size is N and th e observations

are th e received bits from 1 to N, noted as R ^ , Rk = {VkiVk}- "Phe APP* can be expressed

as

Pr { dk = i / R \ } * = 0,1 (3.4)

T he A P P can be derived from th e joint probability:

P r { d k = i / R i } = ^ T P r { d k = i , S k = m / R ? } , m

-

EE

where S k is th e encoder trellis sta te at tim e k and Sk~i is th e encoder trellis s ta te a t tim e

3. TU RBO CODES W IT H IT E R A T IV E DECODING A LG O R ITH M S

as

P r { d k = i / R % } = — 1

EE

*- 1 -* m rri

F urther, if th e trellis s ta te S k is known, th e trellis s ta te and received bits after tim e k are

not affected by observation R \ and b it dk. Using th e Bayes’ rule, th e A P P can be expressed

as

P r { d k = i / R i } = ~ j ^ Y l l E P r U k = h s k = 'rri,Sk_ l = m /, R k1~ l , R k, R $ r+1}

f 1 ■* m m l

= P r { R N \ ^ ^ P r { R L i / S k = m }

t 1 J

rn m!

x P r { d k = i, S k = m , S k^ i = to ', P^- 1 , R k }

1 E E P r { 4 A = ™} P r { P f }

x P r { d k = i , S k = m, R k/ S k„i = m ' } P r { S k_i = to ', P * - 1 } (3.7)

m m '

J ryk—l i

In [1], th ree probability functions have been defined to help th e calculation of th e APP:

a k {m) = P r { S k = m , R \ } ,

(3k (m) = P r { R ^ +1/ S k = m},

7i ( R k, m ' , m ) = P r { d k = i, S k = to, R k/ S k^ i = m '}. (3.8)

where a k ( m), 0 k(m), and 7;(P&, m ', to) are called forward probability, backw ard probability,

and tran sitio n probability, respectively. By taking (3.8) into (3.7), we obtain

P r { d k = i / R = 'Yi(Rk , m ' , m ) a k_i(m')(3k (m) (3.9) 1 ' m m '

3. TU RBO CODES W IT H IT E R A T IV E DECODING A LG O R ITH M S

follows:

a k {m) = P r { S k = m , R 1} l

-

EE

m ' i= 0

1

= ' Y L ' Y l P r { dk = S k = m , R k, R \ ~ 1}

m ! i= 0

1

= X X P r i d k = i , s k = rn, R k/ S k^ i = m ' } P r { S k^ i = t o ', P j - 1 } m'

1

= EE

m l 2 = 0

Pk(m) = = to}

i

- E E

m' 2=0 1

-

EE

m ' 2 = 0

1

= X X

^ + 1 = TO'> -Rfe+l/^ = ™}-P»'{^f+2/S'fc+l =

m ! i= 0

1

=

EE

m l 2 = 0

T h e tran sitio n probability can be fu rth e r separated into th ree term s:

7i ( R k , m ' , m ) = P r { R k/ d k = i, S k = to, S k- i = m ' } P r { d k = i, S k = m / S k- i = to'}

= P r { R k/ d k = i , S k = to, S k^ i = m ' } P r { d k = i / S k = to, S k_i = to'}

x P rlP fe = m / S k - i = to'}. (3.12)

3. TU RBO CODES W IT H IT E R A T IV E DECODING ALG O R ITH M S

d a ta bits y sk are independent of th e trellis states:

P r { R k/ d k = i , S k = m , S k„ x = m ' } = P r { y k/ d k = i, S k = m , S k- \ = m ' }

y~Pr{i/k/ d k = i , S k = m , S k_ x = m ' }

= P r { y k/ dk = i}

x P r { y pk/ d k = i , S k = to, S k^ x = to'} (3.13)

P r { y k/ dk = i}

i (yj-xi) ‘2 ■ e 2<dl rr Or,

1 (Vfc x)c)

P r { y pk/ d k = i , S k = m , S k- X = to'} = - = e 2<n> (3.14) V 2 r r a n

T he second term of (3.12), P r { d k = i / S k = m , S k^ x — to '} , is 0 or 1 depending on th e

trellis. T he th ird te rm of (3.12), P r { S k = m / S k^ x = to'}, is 1/2 depending on trellis.

3.4.2 T he m odified M A P algorithm for turbo decoding

In [3], in order to use th e M AP algorithm in tu rb o decoding, B errou modified th e original

M AP algorithm . Since B PSK signal only have two possible values, for simplicity, instead

of calculating each a posteriori probability, we calculate th e log-likelihood ra tio (LLR) of

th e a posteriori probabilities. T he LLR can be expressed as

P r { d k = I/ ob s er va ti o n s }

m ) = log > U 4 = 0 M » en m t W T (3-15)

At th e decoder, after iteratively decoding, we can make a decision dk by com paring A(dk)

to a threshold equal to zero

dk = 1 i f A(<4) > 0,

dk = 0 i f A ( d k) < 0. (3.16)

T h e LLR A(dk) can be expressed as th e function of th ree probabilities:

A W = log E " ^

3. TURBO CODES W IT H IT E R A T IV E DECODING ALG O R ITH M S

MAP

DEC1

MAP

DEC2

DEMUX

Inter­

leaving

D

einter-leaving

D

einter-leaving

Inter­

leaving

D eco d er decision

Figure 3.4: T urbo codes decoder

As indicated in [6], th e LLR A (dk) can be fu rth er separated into th re e term s

Em Em' 7 i ( l / k , m \ m ) a k _ 1( j v / y h ( m )

+ log D M E i z A + log

P r l v l / d ,

P r { i k

L e{dk) + L c ■ y \ + L a{dk). (3.18)

w here L e{dk) is th e extrinsic inform ation generated from th e decoder, L c ■ y sk represents th e

reliability value of th e channel, L c is called th e channel reliability factor and L a{dk) is th e

a priori inform ation of th e tra n sm itte d bits. From equation (3.14), th e channel reliability

factor L c for discrete memoryless G aussian channels [5] is

4 \XE7 L r =

N 0 (3.19)

T he extrinsic inform ation L e(dk) is independent of th e channel reliability value L c ■ y k and

3. TU RBO CODES W IT H IT E R A T IV E DECODING ALG O R ITH M S

3.4.3

Turbo decoder

T h e tu rb o decoder in [3] is depicted in Fig. 3.4. In th e decoder, a de-m ultiplexer first

accepts th e serial received bits and parallel sends th em out. T h e received d a ta bits y k

and p arity bits y k 2 are sent to th e first M AP decoder. T he received p arity bits ypk 2 and

interleaved version of th e received d a ta bits y k are sent to th e second M AP decoder.

In th e first M AP decoder, for th e first iteration of decoding, th e a priori probability

L a,i{dk) is set to 0. T he extrinsic inform ation is derived as:

l £ } ( d k) = A {^ \ d k ) — L c - y k — L a,i(dk). (3.20)

T he subscript 1 in L ^ \ ( d k) and A^ \ d k) represents th e first decoder and th e superscript

(* = 1) represents th e first iteration of decoding. T he subscript 1 in L ap{dk) represents

th e a priori probability of th e first tu rb o decoder. T he channel reliability factor L c is

independent of th e num ber of iterations and decoder. T his extrinsic inform ation L ^ \ ( d k),

after interleaved, is sent to th e second M AP decoder. T he second M AP decoder will tak e th e

extrinsic inform ation L ^ \ [ d k) as th e a priori inform ation of th e tra n sm itte d d a ta bits. In

th e second M AP decoder, after decoding, a new extrinsic inform ation is L ^\ {d k) generated:

L ^ ( d k) = h ^ \ d k) - L c - y sk - L a a {dk),

= k f i d ^ - L ^ - y i - L ^ d k ) (3.21)

T he L ^ \ { d k), after de-interleaved, is fed back to th e first M AP decoder.

For th e next iterations, th e first M AP decoder will take th e de-interleaved version of

Tg*2 l \ d k) as the a priori inform ation of th e tra n sm itte d d a ta bits; th e second M AP decoder

will take th e interleaved version of l\ d k) as th e a priori inform ation of th e tra n sm itte d

d a ta bits. T he general formula for generating extrinsic inform ation w hen th e num ber of

iterations i > 1 are:

L {:\ ( d k) = h . f { d k) - L c - y { - L ^ l\ d k),

L % ( d k) = A ^ ( d J - L c - y i - L ^ i d k ) . (3.22)

3. TU RBO CODES W IT H IT E R A T IV E DECODING ALG O R IT H M S

3.4.4

Log-M A P algorithm over A W G N channels

Log-MAP decoder does not change th e way to o b tain th e extrinsic inform ation. Log-

M AP decoder changes th e way to calculate th re e probability functions a k ( m ) , /3k ( m)

and j i ( R k , m ' , m ) . Instead of calculating th ree probabilities directly, Log-MAP decoder

calculates th em in th e logarithm ic dom ain.

7 il(yk>yk)’m , ’ m } - los(7i ( R k , m ' , m ) ) ,

a k( m) A log(a!jfe(m)),

/? k(m) = log(Pk (m)). (3.23)

Jacobian logarithm is used in [7] to derive th e Log-MAP algorithm . T h e Jacobian logarithm

for two param eters is expressed as

ln(e'Sl + e&2) = m a x {5i ,5 2) = m a x ( 5i , 8 2) + ln (l + e_ l'52_'5ll). (3-24)

T he Jacobian logarithm for more th a n two param eters can be derived recursively from

(3.24):

ln(ec?1 + • • • + eSn) = m a x ( S i, • • • , Sn )

= ln(A + eSn) w i th A = eSl -) -f e15" -1 = e&

= max(5, Sn ) + ln (l + e~ ^n_<^) (3.25)

T h e S can derived from 5' = l n ^ 1 + • • • + eSn~2) and so on.

T h e way to derive th e probabilities in logarithm ic dom ain is out of range of th is thesis.

For detail, please see in [7]. Here, we use th e sam e notatio n as in [7] and sim ply present th e

way to calculate th e th ree probabilities:

7 i[{ySk , y Pk ) , m ' , m ] = ^ L a{dk) x sk(i) + ^ L cy skx sk{i) + ^ L cy pkx pk{i)

a k(m) = m a x { 7 i M , ^ ) , m ', m ] + 5 ^ ( 171')}

(m' ,i)

Pk(m ) = m ax{7i[(2/A +i>^+i)>m >m /l _(m'.t) +Pk+i{™')}

3. TU RBO CODES W IT H IT E R A T IV E DECODING ALG O R IT H M S

We see 7i{(ysk, y ^ ) , m ' , m \ depends on th e channel reliability factor L c, th e a priori in­

form ation L a(dk) and received bits. a k ( m) and /3k (m) can be calculated recursively by

7i [(ysk , y pk) , m ' , m \ w ith non-linear m ax functions. These results are very im p o rtan t for th e

discussion of th e contributions later.

T h e way to derive LLR A(dk ) [7] is:

A ( d k) = Tnax -f a n ( m ' )

(771,771' )

+0*(™ )} - m S r { 7 0[(ysk , y pk) , m ' , m ]

(771,771' )

+ a k^ l ( m /) + P k(m) } (3.27)

T he calculation of LLR also needs th e non-linear functions max.

3.4.5

M ax-L og-M A P algorithm over A W G N channels

T h e Log-MAP algorithm reduces com putational complexity, bu t it still has m any nonlinear

functions max. T he M ax-Log-M AP algorithm solves th is problem by using approxim ations.

T he M ax-Log-M AP decoder is deduced from th e Log-MAP decoder by su b stitu tin g th e non­

linear functions m ax w ith th e linear functions ma x . T he LLR A ( d k) and probabilities can

be expressed as

A(dk) = m ax {7 1[ ( ^ , ^ ) , m / ,m] + a k_ i ( m ' )

(771,771' )

+P k ( m) } - m ax {7 0 [(y*,*/£),to ',m ]

(771,771' )

+ a k^ i ( m ' ) +(3k{m)}, (3.28)

a k (m) = m ax{7 i [ ( ^ , ^ ) , m /,m ] + a k_1(m')}, (Til' ,l)

Pk(m) = p a x { 7 i[(y * +1,yfc+1) , m , m ' } + P k+1{m')}.

(Til' ,l)

(3.29)

3. TU RBO CODES W ITH ITERATIVE DECODING A LG O R ITH M S

Decoder decision Preprocessor I Encoder

; & BPSK

Turbo Decoder Channel Estimator 1

F igure 3.5: System model over fading channels

3.5

Turbo d ecod in g algorithm s over fading channels

T h e tu rb o decoding algorithm s over fading channels are sim ilar w ith those over AWGN

channels. In [12], it has been indicated th a t “when S N R is known, the decoding algorithms

that are used f or A W G N channels remain unchanged and only the channel reliability factor

L c needs to be redefined”. By including th e tu rb o decoder, fading channel m odel and tu rb o

decoder, th e system model over fading channels is depicted in Fig. 3.5.

In th is thesis, we assum e th a t an initial channel estim ation is available from a

pilot-assisted channel estim ator (channel estim ato r 1). T he initial channel estim ate hk is m odeled

as in [12]

h k = ak + m k , (3.30)

where ak is the actual channel fading factor, m* is th e channel estim ation error. T he m* is

a complex-valued G aussian d istrib u te d random variable w ith m ean and variance

E[rnk] = 0,

3. TU RBO CODES W IT H IT E R A T IV E DECODING A LG O R ITH M S

We assum e ak and m k are independent, th e m ean and variance of channel estim ate h k are

E[ hk\ = 0,

E[\hk\2] = 2 a l = 2 ( a l + a l ) . (3.32)

By com paring th e system m odel over fading channels w ith th a t over AWGN channels,

besides th e fading factor a k and pilot-assisted channel estim ator, a new preprocessor is also

added. T his processor will be discussed in detail in chapter 5. In th e preprocessor, w ith th e

received bits yk and initial channel estim ate h k available, we calculate th e decision variables

Zk ~ Uk^k ~ z k^r j z k,ii (3.33)

w here j is th e im aginary unit and z k r and z k % are th e real and im aginary p a rt of z k. In [12],

th e cross correlation coefficient of y k and h k is defined as:

E[ Vkh*k ]

x k

= \y\e~jsk. (3.34)

F urther, Frenger in [12] derived th e probability of decision variable conditioned on th e

tra n sm itte d bits as follows:

P r { z k/ d k } = - — ---- — exp 2irazha ^ ( l - \ y \ ^

^■[zkRk]

* • 1 d ^ 1 ’ ( 3 ' 3 5 )

° v°>»(i - H 2).

w here K q { x ) is th e zeroth-order Hankel function of x, and 5R[ar] is th e real com ponent of x.

Over AWGN channels, when we derive th e channel reliability factor L c, we use th e

probability of received d a ta bits on th e tra n sm itte d bits:

1

P r { y i / d k} = — - (3.36) v27rcrn

and th e channel reliability factor L c is derived as:

j P r i V k / d k = 1} = L s g P r { y sJ d k = 0} cVk’

3. TU RBO CODES W IT H IT E R A T IV E DECODING A LG O R ITH M S

Over fading channels, based on th e analysis of conditional probability of decision variable

Zk, the channel reliability factor for tu rb o decoding algorithm s over fading channels can be

redefined as follows:

As seen, by assum ing th a t th e variance of fading factor a % is known, th e channel reliability

factor L c over fading channels depends on b o th SNR and th e error variance of channel

estim ate a

T h e way of th e Log-MAP and M ax-Log-M AP decoding algorithm s over fading channels

will stay th e sam e as over AWGN channels. T here are only two m odifications. T h e original

received bits yk is replaced by th e decision variables Zk and th e channel reliability factor L c

is redefined as in (3.39).

P r j z k / d k = 1}

(3.38)

where

C h a p ter 4

Turbo Decoding w ith In tegrated Channel

E s t im a t io n

Log-M AP tu rb o decoding algorithm over fading channels needs th e inform ation of SNR and

th e error variance of channel estim ate to obtain th e channel reliability factor L c. It has

been proved in [12] th a t sm aller error variance results in b e tte r B E R perform ance. M any

estim ation schemes have been introduced to reduce the erro r variance of channel estim ate

and th u s improve th e B E R perform ance. Most of th e schemes are processed before th e

s ta rt of th e tu rb o decoding. However, th e extrinsic inform ation generated and exchanged

betw een th e tu rb o decoders has some a priori inform ation of th e tra n sm itte d d a ta bits.

T his inform ation can help us to refine th e channel estim ate and reduce th e cr^ after each

iteration of decoding.

In th is chapter, we propose a new algorithm of tu rb o decoding integrated w ith channel

estim ation over fading channels. We use th e Log-MAP tu rb o decoder in our new algo­

rithm . A nd th e hybrid channel estim ation scheme is used to refine th e channel estim ation.

We first review th e estim ation theory in th e m ean-square-error sense. T hen th e working

4. TU RBO DECODING W IT H IN T E G R A T E D CH ANN EL E STIM A T IO N

4.1

E stim ation th eory

Using th e theory of estim ation, we can derive an estim ator to estim ate an unknown p ara­

m eter from our observations. Usually, in th e estim ation process, we tr y to minimize a cost

function. In th e m ean-square-error sense, th e cost function is

C[x, x\ = E[(x — x ) 2\, (4.1)

w here x is th e unknow n signal and x is th e estim ate. Given a collection of observations y,

th e m inim um -m ean-square-error (MMSE) estim ato r of x given y is given by, [42],

x = E \ x \ y } = ( x f x\y{x\y), (4.2) J s x

w here S x is th e dom ain of x. T he m ean value of th e estim ate and th e resulting m inim um

cost is given by

E[x\ = x,

E [ { x - x ) 2} = E [ x2] - E [ x 2}. (4.3)

However, it is not always easy to obtain a closed form expression for th e M M SE estim ator.

T his difficulty limits m any engineers to th e linear estim ators. T he linear M M SE estim ator

and its m inim um cost function or error variance can be expressed as

x = x + K 0(y - y ) ,

m in(£'[(x - x )2]) = R x - R xyR ~ [R yx, (4.4)

where Ko is any solutions to th e equation KqRv = R xy, and

R x = E[(x - x ) ( x - x)*},

R xy = E[{x - x ) ( y - y ) * ] ,

R yx = E [ { y - y ) { x - x ) * ) ,

4. TU RBO DECODING W IT H IN T E G R A T E D CH ANN EL E STIM A TIO N

Fading channel

DEMUX

Delay

Channel

Estimator 2

Preprocessor

Channel

Estimator 1

Turbo Decoder

z « = * / r

Figure 4.1: T urbo decoding integrated w ith channel estim ation

4.2

A new algorithm

In th is thesis, we propose a new tu rb o decoding algorithm w ith integrated channel esti­

m ation over Rayleigh fading channels as depicted in Fig. 4.1. T he tu rb o decoder used in

th is algorithm is a Log-MAP decoder. C om paring w ith th e original Log-M AP decoder over

fading channels, we have added th e following blocks: a blind channel estim ator (channel

estim ator 2), a de-m ultiplexer, and a delay module.

4.2.1 T he first iteration of decoding

For th e first iteration, we assum e th e re is a pilot-assisted channel estim ato r (channel esti­

m ator 1) which provides us w ith th e initial channel estim ate hk and th e ir error variance

<7^. T his initial channel estim ate hk go th ro u g h th e channel estim ato r 2 which is a blind

linear M M SE channel estim ator w ithout any processing. A fter th a t, th ey are sent to b o th

4. TU RBO DECODING W IT H IN T E G R A TE D CH ANN EL E ST IM A T IO N

estim ato r 2 and is sent to b o th th e tu rb o decoder and th e delay m odule. T he delay m odule

reserves hk and cr^ for th e next iteration of channel estim ation. In th e tu rb o decoder, after

th e first iteration of decoding, we have th e first version of th e extrinsic inform ation L ^ \ ( d k )

of th e tra n sm itte d d a ta bits. T h e de-interleaved version of th e extrinsic inform ation L^X\(dk)

from th e second Log-MAP decoder, instead of being fed back to th e first Log-MAP decoder,

is sent to th e channel estim ator 2.

4.2.2

T he n ext iterations o f decoding

For th e next iterations, a de-m ultiplexer takes out th e received d a ta bits y sk and sends th em

to th e channel estim ato r 2, th e proposed estim ator. D epending on th e reserved channel

(i — 1)

estim ate h k and its error variance, th e received d a ta bits y k and extrinsic inform ation

Lg*2 X\ d k ) , we refine th e channel estim ate for d a ta bits h!~k and its erro r variance in th e

linear M M SE estim ator (channel estim ato r 2). For simplicity, we use th e following notation:

A 4 a 2m . (4.6)

T h e refined error variance for iteration i and bit dk is T he refined channel estim ate h^k

and th e ir error variance A ^ are again saved in th e delay m odule for th e next iteration of

channel estim ation. T hey are also sent to th e tu rb o decoder for th is iteratio n of decoding.

A fter decoding, we have an u p d ated version of th e extrinsic inform ation L^\ {dk). This

inform ation will be fed back to th e channel estim ator 2, th e proposed estim ator, for th e

next iteratio n of channel estim ation. T he whole process will ru n iteratively. In following,

we go into detail to explain th e process of exchanging inform ation betw een th e channel

estim ato r 2 and tu rb o decoding.

T he channel estim ator 2, depending on th e received d a ta bits y k , th e reserved channel

estim ate for d a ta bits together w ith its error variance A^-1 ^ and th e de-interleaved

extrinsic inform ation L^ 2 l\ d k ) , derives th e new channel estim ate of ak and its error

variance

h ^ = K0 y \

4. TU RBO DECODING W IT H IN T E G R A TE D CH ANN EL E STIM A T IO N

Here we assign a* as m atrix 1 and 4 - » as m atrix 2, th e n Kq can be expressed as

(4.8) Vk

K0 = R u R ^ 1,

where subscripts 1 and 2 are for m atrix 1 and 2, respectively. T h e error variance is

E [ \ mk |2] = 2 \ f = R i - R n R'2l R2i.

T he correlation functions are

Rx = E[aka*k] — 2a \

(4.9)

R1 2

i?21

R2

E I ak

E

-2 <(*%)* [2°'a 2 a l ( x skr

E -■A-1) Vi 1 7 i

' 1 2( al + \ % - ])) 2 a l ( x skr

y k J 2a l x k 2(a l \ x k\2 + <7n) _ (4.10)

T h e new channel estim ate and its variance can th en b e expressed as

A )

l + A k 1 +

+

( i - l )

1,(0

N n ~x k Vk

0\ (l—1)

1 -L ,r s*

1 + TVo k

(4.11)

(4.12) 1 -I- x(*^0/2_Bg i l l

L + A k (77o' + ^ ) >

In (4.11) and (4.12), we see x k is in th e equation. However, th ey are th e unknown tra n s ­

m itted d a ta bits. W h a t we have is th e extrinsic inform ation L ^2l\ d k) for th e d a ta bits

from th e last iteration of tu rb o decoding. We assign P r ( d k = 1) = P r ( x sk = + 1) = p,

P r ( d k = 0) = P r { x| = —1) = 1 — p, th e extrinsic inform ation from th e last

iteratio n of decoding will be used as th e a priori inform ation

4. TU RBO DECODING W IT H IN T E G R A T E D CH ANN EL E STIM A T IO N

so we have

exP [L e.2 1](dk)]

1 + e x p [ L ^ {dk )}

T h e p d f of th e tra n sm itte d d a ta bits x sk can be expressed as

f x (x) = p S ( x sk - 1) + ( 1 ~ p ) S ( x sk + 1).

T he m ean value of th e tra n sm itte d d a ta bits x sk can be derived as

f L {e2l\ d k)

E [ xsk] = / x f x {x) dx = 2p — 1 = t a n h( — ---).

(4.14)

(4.15)

(4.16)

We try to approxim ate th e new channel estim ate and its error variance by taking th e

expected value to (4.11) and (4.12) on x sk as follows:

h(i) E xs [h_{k i' '} (Oi _{k J}

1 4. \

1 + Ak { N o

{ < - »,

+

N0

■ta n h (-

_)yt

and

/

x f = E x l [ \(? } = 2 e ,_k (Oi L' k

2\(i-i) r ( i - D

1

-1 + ^ - t a n h C ^ V ^ )

(4.17)

(4.18)

V

Now, we save b o th inform ation in th e delay m odule for th e next iteratio n of channel esti­

m ation. At th e sam e tim e, th e refined th e channel estim ate h^k (only th e channel estim ate

for d a ta bits have been refined) are sent to th e preprocessor to derive th e new decision

variables z(■i) .

,(0

°k ~yk{h^k )*. (4.19)

T h e refined error variance A ^ are sent to th e tu rb o decoder to u p d a te th e channel reliability

factor:

4 ° ( 4 ) 4 sfWs

No -a „ A ? ( ~ a 2a + 1 ) + a \ ( 2 E s

(4.20) k \ N o

For th e sake of com parison, we also show th e channel reliability factor L c derived by Frenger,

[1 1]:

4i/ E l

No C o L (\ N +0 1 ) + of