Turbo Equalization Of Nonlinear ISI-channels Using High Rate FEC Codes

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Advances in Radio Science, 3, 259–263, 2005 SRef-ID: 1684-9973/ars/2005-3-259

Advances in

Radio Science

Turbo Equalization Of Nonlinear ISI-channels Using High Rate

FEC Codes

M. Siegrist, T. Schorr, A. Dittrich, W. Sauer-Greff, and R. Urbansky

University of Kaiserslautern, Communications Engineering, Erwin-Schr¨odinger-Strae, 67653 Kaiserslautern, Germany

Abstract. Turbo equalization is a widely known method

to cope with low signal to noise ratio (SNR) channels cor-rupted by linear intersymbol interference (ISI) (Berrou and Galvieux, 1993; Hagenauer et al., 1997). Recently in this workshop it was reported that also for nonlinear channels a remarkable turbo decoding gain can be achieved (Siegrist et al., 2001). However, the classical turbo equalization relies on code rates at 1/3 up to 1/2 which makes it quite unattrac-tive for high rate data transmission. Considering the potential of iterative equalization and decoding, we obtain a consider-able turbo decoding gain also for high rate codes of less than 7% redundancy by using punctured convolutional codes and block codes.

1 Introduction

As linear ISI can be regarded as a special convolutional code, it may be considered as part of a serially concatenated cod-ing scheme. Therefore the concept of iterative equalization and decoding, which is denoted as “turbo equalization”, can be applied. Due to the iterative decoding process, the degra-dation caused by linear ISI can be reduced remarkably or even compensated almost completely (Berrou and Galvieux, 1993; Hagenauer et al., 1997). Furthermore, the performance of the FEC encoded system can be enhanced without an in-crease in complexity by using an additional ISI channel pre-coder (Koetter et al., 2004).

Many communication systems are corrupted by nonlinear ISI like e.g. high rate optical data transmission. Non-linearity and ISI are due to propagation effects along the fiber de-scribed by the vectorial nonlinear Schroedinger equation and due to the use of photo diodes as signal detectors with square law characteristic.

In general, nonlinear ISI can be described by a finite state nonlinear channel model approximating the received signal after the electrical detection where state tables with con-ditional probabilities are used, enabling the application of turbo equalization to nonlinear ISI (Siegrist et al., 2001; Sauer-Greff et al., 2001).

Correspondence to: M. Siegrist ([email protected])

In this paper, we refer to the transmission of digital data over nonlinear channels with constraint length L. Besides the specification of the turbo equalizer, we focus on high data and code rate transmission. In order to increase the code rate with respect to the classical iterative equalization and decod-ing approach, we first apply punctured convolutional codes with a code rate of 0.933. The possibility of changing the code rate simply by modifying the puncturing scheme could be seen as an advantage of punctured convolutional codes. However, as it is expected from theory, suitable block codes of the same code rate require less SNR for the same bit error ratio (BER). Therefore we also consider a turbo scheme in-cluding a block code. Whereas for punctured convolutional codes equalizer and decoder make use of the same maximum a posteriori probability (APP) symbol-by-symbol detecting algorithm, the APP decoder for block codes requires a differ-ent approach.

In the following sections we give a short description of the nonlinear channel model based on noiseless, state depen-dent signal chips as introduced in Siegrist et al. (2001) and present the investigated systems for punctured convolutional and block codes. Simulation results outline the achievable decoding gain by using appropriate turbo equalization.

2 Model of nonlinear channels

For nonlinear channels and additive Gaussian noise the ana-log front-end that utilizes the nonlinear channel model con-sists of a bank of filters matched to theqL+1chips (Benedetto et al., 1987; Sauer-Greff et al., 2001), which can be reduced to a minimum effort front-end including a noise limiting low-pass filter with “square root Nyquist” characteristic and a sampler at the symbol rate. However, this leads to a non-recoverable degradation which in return is small compared to the ideal case (Haunstein et al., 2004).

Assuming stationary or slowly time varying channels, also a probability based nonlinear channel model can be derived, which in contrast is not restricted to signal independent addi-tive Gaussian noise (Siegrist et al., 2001).

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260 M. Siegrist et al.: Turbo equalization of nonlinear ISI-channels

TURBO EQUALIZATION OF NONLINEAR

ISI-CHANNELS USING HIGH RATE FEC CODES

M. Siegrist

+

, T. Schorr

+

, A. Dittrich

+

, W. Sauer-Greff

+

, R. Urbansky

+

University of Kaiserslautern, Communications Engineering, 67653 Kaiserslautern, Germany

email:

[email protected]

Abstract: Turbo equalization is a widely known method to

cope with low signal to noise ratio (SNR) channels cor-rupted by linear intersymbol interference (ISI) [1,2]. Recently in this workshop it was reported that also for nonlinear channels a remarkable turbo decoding gain can be achieved [3]. However, the classical turbo equali-zation relies on code rates at 1/3 up to 1/2 which makes it quite unattractive for high rate data transmission. Con-sidering the potential of iterative equalization and decod-ing, we obtain a considerable turbo decoding gain also for high rate codes of less than 7% redundancy by using punctured convolutional codes and block codes.

1. INTRODUCTION

As linear ISI can be regarded as a special

convo-lutional code, it may be considered as part of a

seri-ally concatenated coding scheme. Therefore the

concept of iterative equalization and decoding,

which is denoted as “turbo equalization”, can be

applied. Due to the iterative decoding process, the

degradation caused by linear ISI can be reduced

remarkably or even compensated almost completely

[1, 2]. Furthermore, the performance of the FEC

encoded system can be enhanced without an

in-crease in complexity by using an additional ISI

channel precoder [3].

Many communication systems are corrupted by

nonlinear ISI like e.g. high rate optical data

trans-mission. Non-linearity and ISI are due to

propaga-tion effects along the fiber described by the vectorial

nonlinear Schroedinger equation and due to the use

of photo diodes as signal detectors with square law

characteristic.

In general, nonlinear ISI can be described by a

finite state nonlinear channel model approximating

the received signal after the electrical detection

where state tables with conditional probabilities are

used, enabling the application of turbo equalization

to nonlinear ISI [3, 5].

In this paper, we refer to the transmission of

digi-tal data over nonlinear channels with constraint

length

L

. Besides the specification of the turbo

equalizer, we focus on high data and code rate

transmission. In order to increase the code rate with

respect to the classical iterative equalization and

decoding approach, we first apply punctured

convo-lutional codes with a code rate of 0.933. The

possi-bility of changing the code rate simply by modifying

the puncturing scheme could be seen as an

advan-tage of punctured convolutional codes. However, as

it is expected from theory, suitable block codes of

the same code rate require less SNR for the same bit

error ratio (BER). Therefore we also consider a

turbo scheme including a block code. Whereas for

punctured convolutional codes equalizer and

de-coder make use of the same maximum a posteriori

probability (APP) symbol-by-symbol detecting

algo-rithm, the APP decoder for block codes requires a

different approach.

In the following sections we give a short

de-scription of the nonlinear channel model based on

noiseless, state dependent signal chips as introduced

in [3] and present the investigated systems for

punc-tured convolutional and block codes. Simulation

results outline the achievable decoding gain by

us-ing appropriate turbo equalization.

2. MODEL OF NONLINEAR CHANNELS

For nonlinear channels and additive Gaussian

noise the analog front-end that utilizes the nonlinear

channel model consists of a bank of filters matched

to the

L1

q+

chips [5, 6], which can be reduced to a

minimum effort front-end including a noise limiting

lowpass filter with “square root Nyquist”

character-istic and a sampler at the symbol rate. However, this

leads to a non-recoverable degradation which in

return is small compared to the ideal case [7].

0 5 1 0 1 5 2 0 2 5 3 0 3 5

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2

N o n l i n e a r C h a n n e l : P M D 8

S a m p l i n g P h a s e φ

Amplitude

Fig. 1. Eye diagram of nonlinear channel

Assuming stationary or slowly time varying

channels, also a probability based nonlinear channel

model can be derived, which in contrast is not

re-Fig. 1. Eye diagram of nonlinear channel.

stricted to signal independent additive Gaussian noise [3].

The eye diagram of the nonlinear channel shows that the eye is closed at all times, Fig. 1. Without loss of generality the simulations are based on the channel values at sampling phase φ=10 / 32.

3. CONVOLUTIONAL CODE TURBO EQUALIZATION

The performance of the serially concatenated punctured convolutional code (SCPCC) turbo scheme in the presence of nonlinear ISI channels is investigated. The channel effects are modeled as mentioned in section 2. An arbitrary bit sequence is encoded by a recursive systematic convolutional (RSC) encoder with code rate 1

2, punctured and

transmitted over the nonlinear channel.

The equalizer (Max-Log APP) using a Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm then deter-mines the state transition probabilities referring to the channel output _r( )i

µ

% by using the conditional probabilities

(

( )i ,

)

P rµ =r% eµ sµ [3, 5, 8]. The derived extrinsic soft information is handed over to the code decoder that consists of a puncturing/ depuncturing and a Max-Log APP decoder part in case of convo-lutional code decoding, Fig 2.

Equalizer BCJR Deinterleaver 1 − ∏ Decoder BCJR Interleaver 1 ∏ Turbo-Decoding ( ) ( ) a

L e ,

ˆ ( )

D i ext

L e LDext( )eˆ

( )

* ˆ E ext L e r % _Depunct. Puncturing 1..0..1 1000001 ( ) ( ) a L e ˆ ( ) D L a Code-Decoder

Fig. 2. SCPCC Turbo Decoder. Channel equalizer and decoder are fed with log-likelihood

ratios Lext as extrinsic information.

Equalizer and decoder refer to the received val-ues and additional soft valval-ues as in conventional SCC systems. The decoder assumes a log likelihood ratio (LLR) of zero for all punctured bits and refers to the received extrinsic information for all other bits. Since the RSC code with G₁=23₈ and G₂=35₈ is systematic, puncturing can be done just by cancel-ing an appropriate number of code bits. Assumcancel-ing uniform puncturing and a code rate of 0.933 only the first code word consisting of one information and one code bit is transmitted completely. The

succeed-ing 13 code words are transmitted only by half, i.e. only the information bits are sent [9].

A nonlinear optical channel is examined by transmitting 65025 coded bits with code rate of 0.933 over the channel with memory M=5. Having the technical implementation in mind, pseudo ran-dom as well as block interleavers with the size of 65025 bits have been used.

The simulation results of the nonlinear channel with a fixed sampling phase and additive Gaussian noise are presented in Fig. 3. It can be seen that the interleaver structure has a remarkable influence on the results. The degradation between an AWGN channel and the nonlinear channel with a pseudo random interleaver is only 0.5dB after the sixth it-eration at a BER of 5

10− . Consequently, even in nonlinear systems SCPCC turbo equalization deliv-ers a remarkable decoding gain.

0 1 2 3 4 5 6 7 8 9

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

Channel: PMD8, φ =10, R=0.933

E

b/N0/dB

BER

w/o ISI, BlockInt 0. It. 1. It. 6. It. 0. It., RandInt

4. BLOCK CODE SOFT DECODING

Puncturing of convolutional codes to achieve the desired code rates higher than 0.9 results in codes with low performance. A better alternative for these code rates are block codes, e.g. Bose-Chaudhuri-Hocquenghem (BCH) codes, which show a good performance at high code rates using hard decision decoding, like Bounded-Minimum-Distance (BMD) decoding. One drawback of BMD decoding is the restriction to correct less errors than half the designed distance of the code. The designed distance of a BCH-code often remains below the minimum distance, especially at lower code rates.

Another drawback that has to be circumvented is the lack of soft-out information of a BMD de-coder, because it is essential in a turbo decoding scheme. Therefore, another decoder has to be found. It is known, that, like convolutional codes, block codes can be represented by a trellis diagram [10]. Using this representation, it is possible to apply the

Fig. 3. SCPCC-Turbo-Equalization of a nonlinear optical channel.

Fig. 2. SCPCC Turbo Decoder. Channel equalizer and decoder are

fed with log-likelihood ratiosLextas extrinsic information.

3 Convolutional code turbo equalization

The performance of the serially concatenated punctured con-volutional code (SCPCC) turbo scheme in the presence of nonlinear ISI channels is investigated. The channel effects are modeled as mentioned in Sect. 2. An arbitrary bit se-quence is encoded by a recursive systematic convolutional (RSC) encoder with code rate1_/₂_{, punctured and transmitted} over the nonlinear channel.

The equalizer (Max-Log APP) using a Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm then determines the state transition probabilities referring to the channel output r˜µ(i) by using the conditional probabilities P rµ= ˜r(i)

eµ,sµ (Siegrist et al., 2001; Sauer-Greff et al., 2001; Bahl et al., 1972). The derived extrinsic soft information is handed over to the code decoder that consists of a puncturing/ depunctur-ing and a Max-Log APP decoder part in case of convolutional code decoding, Fig. 2.

Equalizer and decoder refer to the received values and ad-ditional soft values as in conventional SCC systems. The decoder assumes a log likelihood ratio (LLR) of zero for all punctured bits and refers to the received extrinsic informa-tion for all other bits. Since the RSC code with G1=238 andG2=358 is systematic, puncturing can be done just by canceling an appropriate number of code bits. Assuming uniform puncturing and a code rate of 0.933 only the first code word consisting of one information and one code bit is transmitted completely. The succeeding 13 code words are

stricted to signal independent additive Gaussian

noise [3].

The eye diagram of the nonlinear channel

shows that the eye is closed at all times, Fig. 1.

Without loss of generality the simulations are based

on the channel values at sampling phase

φ =10 / 32

.

3. CONVOLUTIONAL CODE TURBO

EQUALIZATION

The performance of the serially concatenated

punctured convolutional code (SCPCC) turbo

scheme in the presence of nonlinear ISI channels is

investigated. The channel effects are modeled as

mentioned in section 2. An arbitrary bit sequence is

encoded by a recursive systematic convolutional

(RSC) encoder with code rate

1

2

, punctured and

transmitted over the nonlinear channel.

The equalizer (Max-Log APP) using a

Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm then

deter-mines the state transition probabilities referring to

the channel output

( )i

r%µ

by using the conditional

probabilities

(

( )i ,

)

P rµ =r% eµ sµ

[3, 5, 8]. The derived

extrinsic soft information is handed over to the code

decoder that consists of a puncturing/ depuncturing

and a Max-Log APP decoder part in case of

convo-lutional code decoding, Fig 2.

Equalizer BCJR Deinterleaver 1 − ∏ Decoder BCJR Interleaver 1 ∏ Turbo-Decoding ( ) ( ) a

L e , ˆ ( )

D i ext

L e LDext( )eˆ

( )

* ˆ E ext L e r % _Depunct. Puncturing 1..0..1 1000001 ( ) ( ) a L e ˆ ( ) D L a Code-Decoder

Fig. 2. SCPCC Turbo Decoder. Channel equalizer

and decoder are fed with log-likelihood

ratios

Lext

as extrinsic information.

Equalizer and decoder refer to the received

val-ues and additional soft valval-ues as in conventional

SCC systems. The decoder assumes a log likelihood

ratio (LLR) of zero for all punctured bits and refers

to the received extrinsic information for all other

bits. Since the RSC code with

G₁=23₈

and

G₂=35₈

is systematic, puncturing can be done just by

cancel-ing an appropriate number of code bits. Assumcancel-ing

uniform puncturing and a code rate of 0.933 only the

first code word consisting of one information and

one code bit is transmitted completely. The

succeed-ing 13 code words are transmitted only by half, i.e.

only the information bits are sent [9].

A nonlinear optical channel is examined by

transmitting 65025 coded bits with code rate of

0.933 over the channel with memory

M

=5. Having

the technical implementation in mind, pseudo

ran-dom as well as block interleavers with the size of

65025 bits have been used.

The simulation results of the nonlinear channel

with a fixed sampling phase and additive Gaussian

noise are presented in Fig. 3. It can be seen that the

interleaver structure has a remarkable influence on

the results. The degradation between an AWGN

channel and the nonlinear channel with a pseudo

random interleaver is only 0.5dB after the sixth

it-eration at a BER of

₁₀−5

_{. Consequently, even in}

nonlinear systems SCPCC turbo equalization

deliv-ers a remarkable decoding gain.

0 1 2 3 4 5 6 7 8 9

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

Channel: PMD8, φ =10, R=0.933

E_b/N₀/dB

BER

w/o ISI, BlockInt 0. It. 1. It. 6. It. 0. It., RandInt

4. BLOCK CODE SOFT DECODING

Puncturing of convolutional codes to achieve

the desired code rates higher than 0.9 results in

codes with low performance. A better alternative for

these code rates are block codes, e.g.

Bose-Chaudhuri-Hocquenghem (BCH) codes, which show

a good performance at high code rates using hard

decision decoding, like

Bounded-Minimum-Distance (BMD) decoding. One drawback of BMD

decoding is the restriction to correct less errors than

half the designed distance of the code. The designed

distance of a BCH-code often remains below the

minimum distance, especially at lower code rates.

Another drawback that has to be circumvented

is the lack of soft-out information of a BMD

de-coder, because it is essential in a turbo decoding

scheme. Therefore, another decoder has to be found.

It is known, that, like convolutional codes, block

codes can be represented by a trellis diagram [10].

Using this representation, it is possible to apply the

Fig. 3. SCPCC-Turbo-Equalization of a

nonlinear optical channel

.

Fig. 3. SCPCC-Turbo-Equalization of a nonlinear optical channel.

transmitted only by half, i.e. only the information bits are sent (Haccoun and B´egin, 1989).

A nonlinear optical channel is examined by transmitting 65 025 coded bits with code rate of 0.933 over the channel with memoryM=5. Having the technical implementation in mind, pseudo random as well as block interleavers with the size of 65 025 bits have been used.

The simulation results of the nonlinear channel with a fixed sampling phase and additive Gaussian noise are pre-sented in Fig. 3. It can be seen that the interleaver structure has a remarkable influence on the results. The degradation between an AWGN channel and the nonlinear channel with a pseudo random interleaver is only 0.5 dB after the sixth it-eration at a BER of 10−5. Consequently, even in nonlinear systems SCPCC turbo equalization delivers a remarkable de-coding gain.

4 Block code soft decoding

Puncturing of convolutional codes to achieve the desired code rates higher than 0.9 results in codes with low perfor-mance. A better alternative for these code rates are block codes, e.g. Bose-Chaudhuri-Hocquenghem (BCH) codes, which show a good performance at high code rates using hard decision decoding, like Bounded-Minimum-Distance (BMD) decoding. One drawback of BMD decoding is the re-striction to correct less errors than half the designed distance of the code. The designed distance of a BCH-code often re-mains below the minimum distance, especially at lower code rates.

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M. Siegrist et al.: Turbo equalization of nonlinear ISI-channels 261 soft-in information and to deliver soft-out information.

Un-fortunately, this approach demands a prohibitive high com-plexity decoder, because, e.g., a BCH(255,239) code has a trellis with 216=65 536 states. Another proposal is to use the dual code for soft-out decoding (Hagenauer et al., 1996), which leads to less, but still quite high effort.

A reduced complexity approach is the modified Chase de-coder (Chase, 1972), also known from the decoding of block turbo codes (Pyndiah et al., 1994), as a sub-optimum soft-out decoding method. The Chase algorithm is a list decoding algorithm operating on a code word list. Consider a linear block code with parameters (n, k,δ)and a transmission of bi-nary symbols –1, +1 over a AWGN channel with standard de-viationσ. For a transmitted code worde=(e0, ..., el, ..., en) the received sequencer=(r0, ..., rl, ..., rn)is given by

r =e+w, (1)

where w=(w0, ..., wl, ..., wn) are uncorrelated Gaussian noise samples. The optimum decision considering maximum likelihood (ML) decoding is the code word that fulfills the condition

d=ciwith r−c

i 2

≤ r−c

l 2

,∀l∈h1,2ki, (2) withci=(c₀i, ..., ci_l, ..., ci_n)being a code word of the consid-ered block code. At high SNR, the ML code wordd is lo-cated in a sphere of radius(δ−1)around the received hard decision wordy=(y0, ..., yl, ..., yn), with yl=sgn(rl). I.e., only the code words within this sphere have to be consid-ered in order to find the ML code word. To further reduce the complexity, we take into account only the most probable code words, which are assumed to be given by the positions of the p= bδ/2c least reliable symbols inr. Test patterns

tq _{are created by toggling the bits of an}_n_-dimensional all-one vector to “–1” at the least reliable positions, starting at the least reliable position with an increasing number of “–1”. Feeding all test sequenceszq, achieved by component-wise multiplication ofy andtq, into an algebraic BMD decoder using e.g. the Berlekamp-Massey-Algorithm (BMA), a set of valid codewordscqis delivered. The optimum decisiondcan be found by applying Eq. (2) to all code wordscq. The relia-bility measure is given by log-likelihood ratio (LLR) (Berrou et al., 1993)

L (yi)=ln

Prej = +1|rj Prej = −1|rj

! =

₂

σ2

rj, (3)

where the least reliable positionsj are the positions with smallestrj

. For each bit of the hard decision decoder out-putd, a soft decision in terms of an LLR value has to be found. The LLR value ofdj is given by

L (di)=ln

Prej = +1|r Prej = −1|r

!

. (4)

Introducing the setsS_j+1 as the set of all code wordscq

equal to +1 at positionj and theS_j−1as the set of all code

decoder decision for the hard decision bit j. It re-sults in a valid LLR value with SNR dependent magnitude, which might be less accurate than that delivered by (5).

5. RESULTS OF BLOCK TURBO EQUALIZATION

Figure 4 shows simulation results using the same channel as investigated with SCPCC Turbo Equalization. A BCH(255,239)-code with a code rate r=0.937 was applied to a block of length N=65025. It compares block turbo equalization with block interleaver, random interleaver and ISI-free transmission.

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

10-10 10-8 10-6 10-4 10-2 100

SNR / dB

Output BER 0. Iter. block

1. Iter. block 4. Iter. block 9. Iter. block 0. Iter. random 1. Iter. random 4. Iter. random 9. Iter. random w/o ISI

As already mentioned the ISI-free transmission using block codes shows a performance gain versus convolutional codes of 1.5dB at BER=10-6. Obvi-ously, the usage of simple block interleaving does not lead to high iteration gains. It remains a gap of 1.6dB between the 9. iteration and the ISI free transmission at BER=10-5. As assumed, a random interleaver gives a performance improvement in terms of higher iteration gains. It is possible to ap-proach the ISI - free bound up to 0.25dB at BER = 10-6.

6. PRECODING

The overall properties of the concatenated scheme are strongly dependent on the properties of the component codes. So it might be reasonable that an IIR system as inner code could be advantageous compared to the FIR structure in the case of an ISI channel. As a given channel can not be changed, one way to generate an IIR structure is to apply a recur-sive precoder in the binary domain. In conjunction with the FIR channel this results in an IIR system (cf. Fig. 5).

Encoder ∏1

T

0

( )

z

h_µ ϕ Precoder

FIR channel

Encoder ∏1

_T

0

( )

z

h_µ ϕ IIR channel a)

b)

Fig. 5: Usage of a precoder with approaches for a) implementation and b) modeling.

Figure 5 shows how this scheme a) would be im-plemented and b) has to be interpreted. The APP detector at the receiver side only has to consider the change of the channel model. It can be easily seen that there is no increase in complexity, since the number of states remains unchanged compared to the system without precoder.

Figure 6 illustrates the performance results of Block Turbo Equalization with precoder compared with the results without precoder. Again, a block length of N=32640 with the same outer BCH code is used.

Fig. 6: Block Turbo Equalization with and without precoder.

Obviously at high SNR the system with precoder delivers an additional gain, which can be explained by an increased minimum distance of the concate-nated scheme due to the inner IIR code. It is possible to cut the no ISI bound at 6.75dB. At low SNR the performance of the precoder system is inferior. Here, the effect has to be considered that single channel errors lead to double bit errors. Note that the region where the precoder shows superior results is at low output BERs.

The application of the precoder in the case of convo-lutional coding leads to similar results, except that Fig. 4: Block Turbo Equalization of a nonlinear

optical channel.

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 10-10

10-8 10-6 10-4 10-2 100

SNR / dB

Output BER

0. Iter. w/o prec. 1. Iter. w/o prec. 4. Iter. w/o prec. 9. Iter. w/o prec. 0. Iter. precoder 1. Iter. precoder 4. Iter. precoder 9. Iter. precoder w/o ISI

Fig. 4. Block Turbo Equalization of a nonlinear optical channel.

wordscqequal to –1 at positionj, numerator and denomina-tor of Eq. (4) can be rewritten as sums, resulting in

L (di)=ln 

  

P

cq_∈_S+1

j

p{r|e=cq|}

P

cq_∈_S−1

j

p{r|e=cq_|} 

  

. (5)

A further simplification of Eq. (5) is to consider only the code word in S_j+1and S−_j1, respectively, closest to the re-ceived sequencer. This eliminates the sums in the numer-ator and denominnumer-ator of Eq. (5), avoiding the summation of probabilities given in the logarithmic domain. Obviously (5) delivers LLR values only for positionsj for which bothS_j+1

andS_j−1are non-empty sets. This is not the case for all posi-tionsj, if only a finite sphere around the received sequence is considered. One method to overcome this problem is to in-crease the number of test patternstqby increasing the num-berpof observed least reliable symbols. Hence, more valid code words around the received sequence are obtained, how-ever, requiring a higher decoding effort. Because of this high complexity another solution is chosen. For allj with empty

S_j+1orS_j−1a fixed reliability is given with an absolute value

β, and we obtain:

L (di)=β·dj. (6)

An equation to calculate a nearly optimumβ is given by

β ≈ln Pr

dj =ej

Pr

dj 6=ej !

, (7)

where Pr

dj=ej is the probability of a correct decoder de-cision for the hard dede-cision bitj. It results in a valid LLR value with SNR dependent magnitude, which might be less accurate than that delivered by Eq. (5).

5 Results of block turbo equalization

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262 M. Siegrist et al.: Turbo equalization of nonlinear ISI-channels decoder decision for the hard decision bit j. It

re-sults in a valid LLR value with SNR dependent magnitude, which might be less accurate than that delivered by (5).

5. RESULTS OF BLOCK TURBO EQUALIZATION

Figure 4 shows simulation results using the same channel as investigated with SCPCC Turbo Equalization. A BCH(255,239)-code with a code rate r=0.937 was applied to a block of length N=65025. It compares block turbo equalization with block interleaver, random interleaver and ISI-free transmission.

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

10-10 10-8 10-6 10-4 10-2 100

SNR / dB

As already mentioned the ISI-free transmission using block codes shows a performance gain versus convolutional codes of 1.5dB at BER=10-6. Obvi-ously, the usage of simple block interleaving does not lead to high iteration gains. It remains a gap of 1.6dB between the 9. iteration and the ISI free transmission at BER=10-5. As assumed, a random interleaver gives a performance improvement in terms of higher iteration gains. It is possible to ap-proach the ISI - free bound up to 0.25dB at BER = 10-6.

6. PRECODING

The overall properties of the concatenated scheme are strongly dependent on the properties of the component codes. So it might be reasonable that an IIR system as inner code could be advantageous compared to the FIR structure in the case of an ISI channel. As a given channel can not be changed, one way to generate an IIR structure is to apply a recur-sive precoder in the binary domain. In conjunction with the FIR channel this results in an IIR system (cf. Fig. 5).

Encoder ∏1

T

0

( )

z

h_µϕ Precoder

FIR channel

Encoder ∏1

T

0

( )

z

h_µ ϕ IIR channel a)

b)

Fig. 5: Usage of a precoder with approaches for a) implementation and b) modeling.

Figure 5 shows how this scheme a) would be im-plemented and b) has to be interpreted. The APP detector at the receiver side only has to consider the change of the channel model. It can be easily seen that there is no increase in complexity, since the number of states remains unchanged compared to the system without precoder.

Figure 6 illustrates the performance results of Block Turbo Equalization with precoder compared with the results without precoder. Again, a block length of N=32640 with the same outer BCH code is used.

Fig. 6: Block Turbo Equalization with and without precoder.

Obviously at high SNR the system with precoder delivers an additional gain, which can be explained by an increased minimum distance of the concate-nated scheme due to the inner IIR code. It is possible to cut the no ISI bound at 6.75dB. At low SNR the performance of the precoder system is inferior. Here, the effect has to be considered that single channel errors lead to double bit errors. Note that the region where the precoder shows superior results is at low output BERs.

The application of the precoder in the case of convo-lutional coding leads to similar results, except that Fig. 4: Block Turbo Equalization of a nonlinear

optical channel.

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 10-10 10-8 10-6 10-4 10-2 100

SNR / dB

Output BER

Fig. 5. Usage of a precoder with approaches for (a) implementation and (b) modeling.

decoder decision for the hard decision bit

j

. It

re-sults in a valid LLR value with SNR dependent

magnitude, which might be less accurate than that

delivered by (5).

5. RESULTS OF BLOCK TURBO

EQUALIZATION

Figure 4 shows simulation results using the

same channel as investigated with SCPCC Turbo

Equalization. A BCH(255,239)-code with a code

rate

r

=0.937 was applied to a block of length

N

=65025. It compares block turbo equalization with

block interleaver, random interleaver and ISI-free

transmission.

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 10-10 10-8 10-6 10-4 10-2 100

SNR / dB

As already mentioned the ISI-free transmission

using block codes shows a performance gain versus

convolutional codes of 1.5dB at BER=10

-6

.

Obvi-ously, the usage of simple block interleaving does

not lead to high iteration gains. It remains a gap of

1.6dB between the 9. iteration and the ISI free

transmission at BER=10

-5

. As assumed, a random

interleaver gives a performance improvement in

terms of higher iteration gains. It is possible to

ap-proach the ISI - free bound up to 0.25dB at BER =

10

-6

.

6. PRECODING

The overall properties of the concatenated

scheme are strongly dependent on the properties of

the component codes. So it might be reasonable that

an IIR system as inner code could be advantageous

compared to the FIR structure in the case of an ISI

channel. As a given channel can not be changed, one

way to generate an IIR structure is to apply a

recur-sive precoder in the binary domain. In conjunction

with the FIR channel this results in an IIR system

(cf. Fig. 5).

Encoder

1

∏

T

0

( )

z

h

_µ

ϕ

Precoder

FIR channel

Encoder ∏1

_T

0

( )

z

h

µ

ϕ

IIR channel a) b)

Fig. 5: Usage of a precoder with approaches for

a) implementation and b) modeling.

Figure 5 shows how this scheme a) would be

im-plemented and b) has to be interpreted. The APP

detector at the receiver side only has to consider the

change of the channel model. It can be easily seen

that there is no increase in complexity, since the

number of states remains unchanged compared to

the system without precoder.

Figure 6 illustrates the performance results of Block

Turbo Equalization with precoder compared with

the results without precoder. Again, a block length

of

N

=32640 with the same outer BCH code is used.

Fig. 6: Block Turbo Equalization with and

without precoder.

Obviously at high SNR the system with precoder

delivers an additional gain, which can be explained

by an increased minimum distance of the

concate-nated scheme due to the inner IIR code. It is possible

to cut the no ISI bound at 6.75dB. At low SNR the

performance of the precoder system is inferior.

Here, the effect has to be considered that single

channel errors lead to double bit errors. Note that

the region where the precoder shows superior results

is at low output BERs.

The application of the precoder in the case of

convo-lutional coding leads to similar results, except that

Fig. 4: Block Turbo Equalization of a nonlinear

optical channel.

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

10-10 10-8 10-6 10-4 10-2 100

SNR / dB

Output BER

Fig. 6. Block Turbo Equalization with and without precoder.

BCH(255,239)-code with a code rater=0.937 was applied to a block of lengthN=65 025. It compares block turbo equal-ization with block interleaver, random interleaver and ISI-free transmission.

As already mentioned the ISI-free transmission using block codes shows a performance gain versus convolutional codes of 1.5 dB at BER=10−6. Obviously, the usage of sim-ple block interleaving does not lead to high iteration gains. It remains a gap of 1.6 dB between the 9. iteration and the ISI free transmission at BER=10−5_{. As assumed, a} ran-dom interleaver gives a performance improvement in terms of higher iteration gains. It is possible to approach the ISI-free bound up to 0.25 dB at BER=10−6.

6 Precoding

The overall properties of the concatenated scheme are strongly dependent on the properties of the component codes. So it might be reasonable that an IIR system as inner code could be advantageous compared to the FIR structure in the case of an ISI channel. As a given channel can not be

the loss at low SNR is smaller. Figure 7 illustrates

the performance results of SCPCC Turbo

Equaliza-tion of the punctured RSC code with and without

precoder.

0 1 2 3 4 5 6 7 8 9 10

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

E_b/N₀/dB

Output BER

w/o ISI, R=0.933, 65025 0. It.

1. It. 6. It.

R=0.933, 65025 precoder

Fig. 7: SCPCC Turbo Equalization with

and without precoder

7. CONCLUSIONS

We have shown that „Turbo Equalization”,

which is well investigated for linear channels and

for low code rates

≤

0.5, can also achieve high

itera-tion gains using high rate outer codes on nonlinear

channels. This allows for applications in fiber

opti-cal transmission, where high code rates are required

due to the challenges in high-speed systems.

The performance simulations of punctured

con-volutional codes and block codes with a code rate of

0.933 for a nonlinear ISI channel have shown that

even for strongly punctured convolutional codes a

remarkable decoding gain can be achieved. A

feasi-ble concept has been presented for iterative

equali-zation of suitable block codes, which have superior

performance compared to punctured convolutional

codes. Considering the interleaver design, a

trade-off between feasibility and performance has to be

found, regarding the large gains achieved by the

usage of random interleavers.

Finally, without increasing the receiver

com-plexity, a precoding scheme in conjunction with

iterative equalization has been presented, which

allows for additional performance gain.

REFERENCES

[1] C. Berrou, A. Galvieux, P. Thitimajshima:

“Near Shannon limit error-correcting coding

and decoding: Turbo Codes (1)”,

Proc. IEEE

Int. Conf. On Communications

, Geneva,

Swit-zerland, pp. 1064-1070, 1993.

[2] J. Hagenauer, G. Bauch, H. Khorram: “Iterative

Equalization and Decoding in Mobile

Commu-nications Systems”,

Proc. EPMC ’97

, pp.

307-312, 1997.

[3] M. Siegrist, A. Dittrich, W. Sauer-Greff, R.

Urbansky: “Iterative Equalization for Nonlinear

Channels with Intersymbol Interference

”, Proc.

IEEE Signal Processing ‘2001

, Poznan, Poland,

pp. 95-98, 2001.

[4] R. Koetter, A. C. Singer, M. Tüchler: “Turbo

Equalization; An iterative equalization and

de-coding technique for coded data transmission”,

IEEE Signal Processing Magazine

, pp. 67-80,

Jan. 2004.

[5] W. Sauer-Greff, A. Dittrich, H. Haunstein, M.

Lorang, K. Sticht, R. Urbansky: “Adaptive

Viterbi Equalizers for Nonlinear Channels”,

Proc. German U.R.S.I. Conference

Kleinheubach, Germany, pp.215-222, 2001.

[6] S. Benedetto, E. Biglieri, V. Castellani:

Digital

Transmission Theory

, Prentice Hall, 1987.

[7] H. F. Haunstein, W. Sauer-Greff, A. Dittrich,

K. Sticht, R. Urbansky: “Principles for

Elec-tronic Equalization of Polarization-Mode

Dis-persion”,

IEEE Journal of Lightwave

Technol-ogy

, vol. JLT-22, pp 1169-1182, 2004

[8] L. R. Bahl, J. Cocke, F. Jelinek, J. Raviv:

“Op-timal decoding of linear codes for minimizing

symbol error rate”,

Abstracts, Int. Symp.

In-form. Theory,

Asilomar, USA, p. 90, 1972.

[9] D. Haccoun, G. Bégin: “High-Rate Punctured

Convolutional Codes for Viterbi and Sequential

Decoding”,

IEEE Trans. on Comm.

, vol.

COM-37, pp. 1113-1125, 1989.

[10] J.K. Wolf: “Efficient maximum likelihood

de-coding of linear block codes using a trellis”,

IEEE Trans. Inform. Theory

, vol. IT-24, pp.

76-80, Jan. 1978.

[11] J. Hagenauer, P. Höher: “A Viterbi algorithm

with soft-decision outputs and its applications“,

Proc. IEEE Globecom,

Dallas, USA, pp.

1680-1686, 1989.

[12] J. Hagenauer, E. Offer, L. Papke: “Iterative

decoding of binary block and convolutional

codes“,

IEEE Trans. Inform. Theory

, vol.

IT-42, pp. 429-445, 1996.

[13] D. Chase: “A class of algorithms for decoding

block codes with channel measurement

infor-mation“,

IEEE Trans. Inform. Theory

, vol.

IT-18, pp. 170-182, 1972.

[14] R. Pyndiah, A. Glavieux, A. Picart, S. Jacq:

“Near optimum decoding of product codes”,

Proc. IEEE Globecom

, San Francisco, USA,

pp. 339-343, 1994.

Fig. 7. SCPCC Turbo Equalization with and without precoder.

changed, one way to generate an IIR structure is to apply a recursive precoder in the binary domain. In conjunction with the FIR channel this results in an IIR system (cf. Fig. 5).

Figure 5 shows how this scheme (a) would be imple-mented and (b) has to be interpreted. The APP detector at the receiver side only has to consider the change of the chan-nel model. It can be easily seen that there is no increase in complexity, since the number of states remains unchanged compared to the system without precoder.

Figure 6 illustrates the performance results of Block Turbo Equalization with precoder compared with the results with-out precoder. Again, a block length ofN=32 640 with the same outer BCH code is used.

Obviously at high SNR the system with precoder delivers an additional gain, which can be explained by an increased minimum distance of the concatenated scheme due to the in-ner IIR code. It is possible to cut the no ISI bound at 6.75 dB. At low SNR the performance of the precoder system is infe-rior. Here, the effect has to be considered that single channel errors lead to double bit errors. Note that the region where the precoder shows superior results is at low output BERs.

The application of the precoder in the case of convolu-tional coding leads to similar results, except that the loss at low SNR is smaller. Figure 7 illustrates the performance re-sults of SCPCC Turbo Equalization of the punctured RSC code with and without precoder.

7 Conclusions

We have shown that “Turbo Equalization”, which is well in-vestigated for linear channels and for low code rates≤0.5, can also achieve high iteration gains using high rate outer codes on nonlinear channels. This allows for applications in fiber optical transmission, where high code rates are required due to the challenges in high-speed systems.

(5)

M. Siegrist et al.: Turbo equalization of nonlinear ISI-channels 263 be achieved. A feasible concept has been presented for

iterative equalization of suitable block codes, which have superior performance compared to punctured convolutional codes. Considering the interleaver design, a trade-off be-tween feasibility and performance has to be found, regarding the large gains achieved by the usage of random interleavers. Finally, without increasing the receiver complexity, a pre-coding scheme in conjunction with iterative equalization has been presented, which allows for additional performance gain.

References

Bahl, L. R., Cocke, J., Jelinek, F., and Raviv, J.: Optimal decoding of linear codes for minimizing symbol error rate, Abstracts, Int. Symp. Inform. Theory, Asilomar, USA, 90, 1972.

Benedetto, S., Biglieri, E., and Castellani, V.: Digital Transmission Theory, Prentice Hall, 1987.

Berrou, C., Galvieux, A., and Thitimajshima, P.: Near Shannon limit error-correcting coding and decoding: Turbo Codes (1), Proc. IEEE Int. Conf. On Communications, Geneva, Switzer-land, 1064–1070, 1993.

Chase, D.: A class of algorithms for decoding block codes with channel measurement information, IEEE Trans. Inform. Theory, vol. IT-18, 170–182, 1972.

Haccoun, D. and B´egin, G.: High-Rate Punctured Convolutional Codes for Viterbi and Sequential Decoding, IEEE Trans. on Comm., vol. COM-37, 1113–1125, 1989.

Hagenauer, J. and H¨oher, P. : A Viterbi algorithm with soft-decision outputs and its applications, Proc. IEEE Globecom, Dallas, USA, 1680–1686, 1989.

Hagenauer, J., Offer, E., and Papke, L.: Iterative decoding of binary block and convolutional codes, IEEE Trans. Inform. Theory, vol. IT-42, 429–445, 1996.

Hagenauer, J., Bauch, G., and Khorram, H.: Iterative Equalization and Decoding in Mobile Communications Systems, Proc. EPMC ’97, 307–312, 1997.

Haunstein, H. F., Sauer-Greff, W., Dittrich, A., Sticht, K., and Urbansky, R.: Principles for Electronic Equalization of Polarization-Mode Dispersion, IEEE Journal of Lightwave Tech-nology, vol. JLT-22, 1169–1182, 2004.

Koetter, R., Singer, A. C., and T¨uchler, M.: Turbo Equalization: An iterative equalization and decoding technique for coded data transmission, IEEE Signal Processing Magazine, 67–80, Jan. 2004.

Pyndiah, R., Glavieux, A., Picart, A., and Jacq, S.: Near optimum decoding of product codes, Proc. IEEE Globecom, San Fran-cisco, USA, 339–343, 1994.

Sauer-Greff, W., Dittrich, A., Haunstein, H., Lorang, M., Sticht, K.,and Urbansky, R.: Adaptive Viterbi Equalizers for Nonlin-ear Channels, Proc. German U.R.S.I. Conference Kleinheubach, Germany, 215–222, 2001.

Siegrist, M., Dittrich, A., Sauer-Greff, W., and Urbansky, R.: Iter-ative Equalization for Nonlinear Channels with Intersymbol In-terference, Proc. IEEE Signal Processing ‘2001, Poznan, Poland, 95–98, 2001.