Serial Concatenated Single Di�erential Space-TimeCoded OFDM System

Yi Yao and M. M. K. HowladerWireless Communications Research Group, ECE Department

The University of Tennessee, Knoxville, TN 37996Email: {yyao1, howlader}@utk.edu

Abstract–In this article, we propose serial concatenatedsingle di�erential space-time (SDST) coded OFDM systems,in which blind SDST coding is serially concatenated with con-volutional coding. The OFDM is incorporated for maintainingsatisfactory performance in both time and frequency selectivefading channels at low SNR. The encoding and iterativedecoding process are thoroughly studied and the improvedperformance is demonstrated via simulations.

I. Introduction

Recently, there exits interest in concatenation of space-time coding with convolutional coding to achieve improvedperformance at relatively low SNR. Among several pro-posed structures, the serial concatenation of di�erentialspace-time block coding (DST) and convolutional codinghas been proposed in [1]. DST is promising because it canbe decoded without knowing channel state information(CSI), and expressed in trellis form, which enables theapplication of iterative decoding techniques.

Based on the implementation of [1], we explore furtherto improve its performance in time varying and frequencyselective fading channels. First, we observe that as thetotal number of iterations grows, noncoherent detectioncannot o�er as conspicuous improvement as coherentdetection can provide, which will be illustrated in thefollowing simulations. Thus, we propose to combine non-coherent and coherent detection so as to fully explorethe advantages provided by iterative decoding. Second,we note that there is a trade-o� between the frame lengthof DST and the frame length of convolutional coding. DSTneeds small frame length to prevent error from expanding,while convolutional coding and interleaver perform betterwith large frame length. In our schemes, di�erent framelengths are chosen for DST and convolutional coding toeliminate the afore-mentioned contradiction and, further-more, to improve system performance in time varyingfading channels. Finally, we incorporate OFDM into ourproposed schemes for the robustness in frequency selectivefading channels.

The rest of this article is organized as follows. InSection II, we discuss our system model and the encodingprocess. Section III studies the iterative decoding process.The performance of our proposed schemes are demon-

strated and compared in Section IV. Section V concludesthis article.

II. Encoding Process

A. System Model

We consider a system with Nt transmitters and Nrreceivers. At time t, the coded symbol is denoted as Ct,a matrix of dimension N ×Nt, which is expressed as,

Ct =

�����

ct1,1 ct1,2 ... ct1,Nt

ct2,1 ct2,2 ... ct2,Nt

......

. . ....

ctN,1 ctN,2 ... ctN,Nt

����� , (1)

where N is the length of space-time symbol. In SDST, thecurrent space-time coded symbol is related to the previouscoded symbol by,

Ct = GtCt�1 �t � 1, (2)

where Gt, an N ×N unitary matrix, is a unique mappingof input message symbol, st.

The trellis diagram of SDST is shown in Fig. 1,where the trellis states depend on output coded sym-bols, i.e., state 1, 2, 3, and 4 correspond to output

coded symbol

µ1 �11 1

¶,

µ1 1

�1 1

¶,

µ�1 1�1 �1

¶,

and

µ�1 �11 �1

¶, respectively.

For a flat fading channel, disregarding the ISI, thereceived signal at time t can be expressed as yt = Ctht,

Fig. 1. Trellis diagram of single di�erential space-time coding.

31500-7803-7802-4/03/$17.00 © 2003 IEEE

where yt is the received signal matrix of dimension N×Nrat time t, Nr is the number of receiver antennas, and

ht =

�����

ht1,1 ht1,2 ... ht1,Nr

ht2,1 ht2,2 ... ht2,Nr

......

. . ....

htNt,1htNt,2

... htNt,Nr

����� , (3)

an Nt × Nr matrix, represents the channel fading coe�-cients at time t.

However, in a frequency selective fading channel, theimpulse response from the ith transmitter to the jth

receiver at time t is modeled byPL�1

l=0 hi,j(t, l)�(t � l),where L = �/T denotes the number of symbol period Tspanned by the maximum channel excess delay � . Thenthe received signal at the jth receiver at time t can beexpressed as yjt =

PNt

i=1

PL�1l=0 hi,j(t � l)c

t�li,j + �

jt , where

�jt is Gaussian noise at the jth receiver at time t with zeromean and variance 1/2 per dimension. If OFDM is appliedto the system, the frequency response at the kth tone of thetth block corresponding to the ith transmit antenna can beexpressed as Hi,j(t, k) =

PL�1l=0 hi,j(t, l)W

klL . The signal

from each receive antenna can be expressed as yjt (k) =PNt

i=1 Hi,j(t, k)cit(k) + w

jt (k), where Hij(t, k) denotes the

channel frequency response at the kth tone of the tth

OFDM block, corresponding to the ithtransmit and the jth

receive antenna, and wjt (k) denotes the additive complexGaussian noise on the jth receive antenna and is assumedto be zero mean with variance 1/2 on each dimension.

B. Encoding Process

The block diagram of serial concatenated SDST codedOFDM is shown in Fig. 2. The message bits are convo-lutionally encoded and interleaved, first. Di�erent fromthe schemes proposed in [2], in which serial coded bitsare fed into di�erential space-time decoder directly, theserial coded bits in our scheme are divided into parallelstreams with the total number of sub-streams denotedas K, so that di�erential space-time coding is carried outwithin each sub-stream. The above arrangement, essentialto the application of OFDM, is similar to the rearrangingoperation in [3]. The IFFT block modulates parallel SDSTcoded symbols onto K sub-carriers. As the last step,cyclic prefix is added to OFDM frames and signals aretransmitted through Nt transmit antennas.

Since convolutionally coded bits are fed into di�erentsub-streams and di�erential space-time coding is imple-

Fig. 2. Block diagram of encoder.

mented independently within each sub-stream, the framelengths of convolutional encoder, Nc, and SDST, Nst, aredi�erent and related by Nc = Nst×K. In each di�erentialspace-time coded frame, some extra symbols are added,for example, one reference symbol at the beginning of eachframe and tail symbols, which are used to terminate trellisfor the decoding process. These extra symbols decreasesthe system rate. For clarity, let us take the SDST examplefrom our simulations. The number of sub-streams K = 32,Nst = 100, and Nc = 3200. The total symbols perconvolutional code frame is (100 + 3)× 32 = 3296 for ourscheme, while 3200 + 3 = 3203 for the case where SDSTand convolutional encoder have the same frame length [2].The system rate drops slightly by 1� 3203/3296 = 2.8%.

The phenomenon of di�erent frame lengths for con-volutional encoder and di�erential space-time encoder ismore advantageous compared with its impairment onsystem rate. In di�erential coding, current symbol isencoded and decoded based on several previous symbols.Incorrect detection of previous symbols will a�ect thedetection of following symbols. Therefore, once a symbolis mis-detected, the error will propagate. For larger framelength, the symbols at the back are more vulnerable andthe detection error is more prone to expand, which resultsin burst error in one frame. Comparatively, in the case withsmaller frame length, the probability of burst error in oneframe is much lower. So, we are particularly interestedin di�erential coding with smaller frame length. On theother hand, longer frame length leads to better perfor-mance of convolution coding and higher interleaving gain.Therefore, we propose to choose di�erent frame lengths forconvolutional encoder and di�erential space-time encoderso that we can not only prevent burst error in di�erentialcoding but also maintain the performance of convolutioncoding and the interleaving gain. Furthermore, in OFDMcoded system, the message bits must be rearranged inorder that di�erential space-time coding is performedacross a OFDM frame. By dividing the output of theouter encoder intoK sub-streams and applying di�erentialcoding within sub-streams, we can achieve both di�erentframe length for outer and inner encoders and di�erentialcoding across OFDM frames.

III. Iterative Decoding Process

A. Non-OFDM System

We will briefly describe the iterative decoding processof schemes without OFDM, proposed in [2]. The blockdigram of conventional SISO decoder is depicted in Fig.3 by solid lines. For brevity, the definition of someexpressions and the detailed explanation of the structureare referred to [2], [4] and [5]. In [4], the output extrinsiclog likelihood ratio (LLR) at time t is computed as:

�t(c;O)= maxe:c(e)=c{�t�1[s

S(e)] + �t[u(e); I] + �t[sE(e)]},

(4)

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and

�t(u;O)= maxe:u(e)=u{�t�1[s

S(e)] + �t[c(e); I] + �t[sE(e)]}.

(5)In SDST, the transition metric can be computed by thefollowing equation employing di�erential decoding withoutCSI [6].

pXi=1

ut;i(e)�t[ut;i(e); I] +ReTr(Gyt�1yHt )

2�2, (6)

where �2 is noise variance.If the transition metric in (6) is used, the improvement

due to the iterative decoding process is not as conspicuousas the case in which coherent iterative decoding is applied.The expressions in (6) illustrates that for each endingstate, upon a state transition, the metric only di�ers ina priori probability. The extrinsic information exchangedbetween constituent decoders is calculated by subtractinga priori probability from the output LLR. As a result,the extrinsic information remains almost constant fromiteration to iteration. However, it is the extrinsic infor-mation that helps to enhance the performance. Iterativeoperation will not yield much benefit if the extrinsic infor-mation remains pseudo-constant. Powerful convolutionalencoder with longer constraint length can ameliorate theimprovement from iteration to iteration, while introducesextra complexity at the same time. Therefore, in orderto fulfill the advantage of iterative decoding, we resortto coherent decoding, in which the transition metric iscalculated by the following equation,

pXi=1

ut;i(e)�t[ut;i(e); I]�kyt � Cthtk2

2�2. (7)

Fig. 3 shows the block diagram of our proposedscheme, where the decoding process is divided into twostages. The first stage includes the first iteration, shown indashed lines, in which the received signal is decoded di�er-entially without CSI. Then the message bits, reconstructedfrom the first iteration, are used to estimate the channelfading coe�cients, which means the recovered message bitsof the first iteration work as pilot bits for the successiveiterations. In the second stage, based on the estimated

Fig. 3. Block diagram of serial concatenated decoder.

channel fading coe�cients, the coherent iterative decodingis carried out by calculating the transition metric in (7).After each iteration, the latest reconstructed messagebits are updated to refine the estimated channel fadingcoe�cients.

B. OFDM Systems

Fig. 4 illustrates the block diagram of decoding processof a serial concatenated OFDM system. In this figure,the bold lines indicate parallel data stream. The receivedsignals is transferred from serial stream into K parallelstreams. Then FFT, the counterpart of IFFT, decouplesthe parallel symbol streams from K sub-carriers. In thefirst iteration, which comprises of the first decoding stage,the parallel decoupled symbol streams pass the di�erentialspace-time decoder, yielding a set of parallel extrinsicinformation �(ui;O). After being transferred into serialsequence, this extrinsic information is de-interleaved andfed into convolutional decoder, which generates anotherset of extrinsic information, �(co;O), for the next iteration.Accordingly, before it can be used by space-time decoder,�(co;O) should be interleaved, required by serial structure,and transferred into parallel streams again, required byspace-time decoder.

The channel estimator used in OFDM schemes isdi�erent from that used in non-OFDM schemes, in whichper-survivor and linear prediction are used to estimatethe channel fading coe�cients. In OFDM system, therecovered message bits are considered in frequency domainwhile channel fading coe�cients are considered in time do-main, which introduces di�culties in channel estimation.In this paper, we follow the estimation scheme proposedin [7], where the detailed explanation is provided. Channelestimator takes the uncouple received signals, which are intime domain, and the recovered message bits reconstructedin each iteration, which are in frequency domain as twoinputs and provides coherent SDST decoder with theestimated channel fading coe�cients.

IV. Simulation Results

In our simulations, the entire channel bandwidth issupposed to be 800 kHz and the total number of sub-

Fig. 4. Block diagram of serial concatenated decoder for OFDMsystems. x̂t represents recovered message bits

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Fig. 5. Performance of serial concatenated SDST coded OFDM.

carriers is 32. Thus, symbol duration T is 1.25 µs. Two-ray Rayleigh fading channels with maximum excess delay� as T = 1.25 µs, 2T = 2.5 µs, 3T = 3.75 µs, and4T = 5 µs are used to simulate wireless communica-tion environments. The resultant rms delay spread �

�

is approximately 625 ns, 1.25 µs, 1.875 µs, and 2.5 µs,respectively. We also consider time varying fading channelswith fd = 200 Hz, fd = 400 Hz, and fd = 800 Hz.

We choose to present two sets of performance mea-surement, bit error rate (BER) vs. SNR and OFDMframe error rate (FER) vs. SNR, to provide more detailedevaluation of our proposed schemes. Fig. 5 shows theperformance of serial concatenated SDST coded OFDMin frequency selective channel with � = 2T = 2.5 µs andfd = 800 Hz. From the plots, the turbo cli� occurs atSNR = �1 dB and the performance is enhanced with theincrease of the iteration times N . Moreover, the improve-ment introduced by iteration operation becomes moreand more obvious with respect to the increasing SNR.However, further increase of N will provide diminishingimprovement, while introduce much more computationalcomplexity. Therefore, we choose N = 5 in our followingsimulations.

The performance of our system depends partly on theaccuracy of channel estimation. The turbo cli� usuallyoccurs when the BER of the first iteration is roughly0.05. If the first iteration cannot generate the recoveredmessage bits, which are accurate enough to guaranteesatisfactory channel estimation, the following coherent it-eration operation will not provide expected improvement.As Fig. 5 shows, the performance of each iteration isalmost the same when SNR is less than �1 dB. The systemperformance dependency on the first iteration will decreaseif a better channel estimator is used.

For comparison, we present the performance of non-coherent decoding scheme in Fig. 6 under the same fadingconditions. The improvement of noncoherent detectionresulting from interaction operation is not as conspicuous

Fig. 6. Performance of serial concatenated SDST coded OFDMwith noncoherent detection.

as that of coherent detection. At SNR = 1 dB, the BERof noncoherent decoding at N = 5 approaches 10�3, whilethe BER of our scheme achieves the value of 10�5. TheFER of noncoherent scheme approaches 10�2, while theFER of our scheme is improved from 10�1 to 10�4 atSNR = 1 dB, which clearly verifies the superiority of ourscheme using reasonably more computation due to chan-nel estimation. In noncoherent scheme, the improvementbetween iterations is primarily due to the improved perfor-mance of outer (convolutional) decoder. If the performanceof noncoherent scheme is to be improved to the samemagnitude of our scheme, convolutional coding with largerconstrain length must be utilized, which brings much morecomputational complexity. Therefore, considering bothperformance and computational complexity, our schemeoutperforms noncoherent schemes.

Figs. 7, 8, and 9 depict the BER and FER perfor-mance of our schemes under di�erent fading channelswith fd = 800 Hz, fd = 400 Hz, and fd = 200 Hz,respectively. As expected, channels with smaller channelexcess delay present superior performance compared withchannels with larger � . However, we observe only anaverage 2 dB performance degradation with � varyingfrom � = T = 1.25 µs to � = 4T = 5 µs at BER = 10�5

and FER = 10�4, which illustrates the e�ciency of ourschemes in frequency selective channels, from suburbanarea, whose averaged typical �� is 200 - 310 ns, to urbanarea, whose averaged typical �� is 3.5 µs [8]. Moreover, inour simulations, only 32 sub-channels are applied. In orderto maintain system performance, we can further increasethe number of sub-channels, K.

Finally, we compare the performance of our schemeswith di�erent fd. For brevity, we only demonstrate thecomparison with � = 4T in Fig. 10, since we observethat the comparison under other conditions with � = T ,� = 2T , and � = 3T present the same trend. With theincreased fd, system performance drops accordingly. FromFig. 10, we can find that the degradation is only roughly

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Fig. 7. Performance comparison at fd = 800 Hz.

Fig. 8. Performance comparison at fd = 400 Hz.

Fig. 9. Performance comparison at fd = 200 Hz.

Fig. 10. Performance comparison at � = 4T .

0.5 dB at BER = 10�4 and FER = 10�3. Our systemo�ers consistent performance in time varying channel,although SDST requires the applicable channels to bequasi-static. This is because we apply di�erent length forSDST and convolutional coding, which e�ectively precludeburst error in time varying channels.

V. Conclusions

In this article, we implemented serial concatenatedSDST coded OFDM and thoroughly discussed the encod-ing and decoding process. Simulation results demonstratedthe e�ciency of our proposed schemes in frequency selec-tive and time varying fading channels. Future work willfocus on the implementation of serial concatenated doubledi�erential space-time coding (DDST) coded OFDM toinvestigate whether system performance could be furtherimproved.

References

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[4] S. Benedetto, D. Divsalar, G. Montorsi and F. Pollara,“Serialconcatenation of interleaved codes: performance analysis, de-sign, and iterative decoding”, IEEE Trans. on InformationTheory, pp. 909-26, vol. 44, no. 3, May 1998.

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