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IEEE TRANSACTIONS ON COMMUNICATIONS,VOL.53,NO.2,FEBRUARY2005263

Redundancy Allocation in Turbo-Equalizer Design Matthew J.M.Peacock,Student Member,IEEE,and Iain B.Collings,Senior Member,IEEE

Abstract—This paper considers properties of the extrinsic infor-mation transfer(EXIT)functions of turbo equalized intersymbol interference channels andﬁnite-impulse response precoders.An analytic expression is derived for the maximum value of the EXIT function of the http://www.wendangku.net/doc/0b01f68171fe910ef12df868.htmling this parameter,a design strategy is proposed for allocating redundancy between the equalizer and the decoder.The key quantities are pilot-symbol rate and code punc-turing,forﬁxed overall data and symbol rates.

Index Terms—Extrinsic information transfer(EXIT)charts, turbo equalization.

I.I NTRODUCTION

T URBO equalization[1],[2]has been shown to yield remarkable bit-error rate(BER)performance over fre-quency-selective channels[3].Similar to closely related turbo-coding techniques,the approach is to iterate between equalization and decoding stages in order to converge to a vastly improved data estimate[4].Unfortunately,turbo-equalization schemes are notoriously difﬁcult to analyze,and have tended to be investigated using Monte Carlo BER simulations.Recently, in theﬁeld of turbo coding,extrinsic information transfer (EXIT)functions have been proposed[5]as a new analysis tool.EXIT functions have been used to determine a stopping criterion for iterative decoding[6],and have also been applied to turbo-equalization schemes overﬁxed multipath channels [7],[8].

It has been observed that the maximum value of the EXIT function for the equalizer stage in a turbo equalizer is less than one,for generalﬁnite-impulse response(FIR)channels and signal-to-noise ratios(SNRs)of interest.In some cases, this value is signiﬁcantly less than one.This implies that for the turbo equalizer to have a low BER,the outer code must be strong enough such that its EXIT function intersects the equalizer EXIT function on the right-hand axis of the EXIT chart.Clearly,this leads to an interesting design problem.

In this paper,we derive an analytic expression for the max-imum value of the equalizer EXIT function forﬁxed known intersymbol interference(ISI)channels.This value is related to the matched-ﬁlter bound[9],and as such,is a function of only the channel SNR(i.e.,independent of the particular values of the ISI taps).We show that this value is independent of the pilot-symbol rate,and that the expression is also applicable to

Paper approved by K.Chugg,the Editor for Signal Processing and Itera-tive Design of the IEEE Communications Society.Manuscript received August 29,2002;revised March1,2004.This work was supported in part by CSIRO Telecommunications and Industrial Physics,Australia.

The authors are with the Telecommunications Laboratory,School of Elec-trical and Information Engineering,University of Sydney,NSW2006,Australia (e-mail:mpeac@http://www.wendangku.net/doc/0b01f68171fe910ef12df868.html.au;i.collings@http://www.wendangku.net/doc/0b01f68171fe910ef12df868.html.au).

Digital Object Identiﬁer10.1109/TCOMM.2004.841984FIR precoders.It is also directly extendable to fading channels by averaging over the channel SNR distribution.

An important design issue is the choice of appropriate code-puncturing and pilot-symbol rates.Pilot bits are added to aid equalizer performance,while code-parity bits are added to aid decoder performance.For practical systems withﬁxed data-and modulation-symbol rates(i.e.,ﬁxed data-rate services em-ploying hardware with aﬁxed transmit bandwidth and modula-tion format),for each SNR,a design tradeoff exists between the pilot-symbol rate and the code(puncturing)rate.In this paper, we develop an analytically-based design strategy for optimizing the choice of pilot rate and code puncturing,in order to pro-vide the best initial iteration for the turbo equalizer,while still meeting a BER requirement.The design is based on our ana-lytic expression for the maximum value of the equalizer EXIT function,coupled with our observation that adding pilot sym-bols increases the area under the EXIT function.The analysis also allows us to determine the SNR range for which the equal-izer will converge to a target BER for particular code-puncturing rates.

II.T RANSMISSION S YSTEM M ODEL

In this paper,we assume a standard model for data commu-nications when using a turbo equalizer,as shown in Fig.1(e.g., as used in[9]and others).Binary data is encoded with a binary convolutional encoder of

rate to obtain a sequence of coded bits

(indicates the rate of the unpunctured outer code).Op-tionally,the coded sequence is punctured at a

rate,that is,

every th bit from the coded sequence is removed.This sequence is then interleaved and interspersed with pilot bits with

frequency,that is,a pilot symbol is inserted after

every data symbols.We will call this

sequence,which,for sim-plicity,we assume is modulated using binary phase-shift keying (BPSK)to obtain the

sequence,

where

and

.The data is processed in blocks of

length ,

i.e.,.

Since we are interested in optimizing the allocation of redun-dancy(code puncturing and pilot symbols)in the system,all analysis and simulations in this paper are for aﬁxed data rate and aﬁxed channel-symbol rate.In other words,we are opti-mizing the choice

of

and for aﬁxed energy per bit.There-fore,the channel-symbol SNR is alsoﬁxed.Note that if this ap-proach is not taken,then the channel-symbol SNR would need to be scaled in order to account for the waste of energy resulting from using pilots,when comparing different pilot frequencies. Assume the channel symbols are transmitted with symbol

period,and denote the discrete-time equivalent impulse re-sponse of the combined channel and transmit/receiveﬁlters by

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