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Research Letters in Communications
Volume 2008 (2008), Article ID 860368, 5 pages
http://dx.doi.org/10.1155/2008/860368
Research Letter

Training Sequence Length Optimization for a Turbo-Detector Using Decision-Directed Channel Estimation

1Laboratoire d’Electronique et des Technologies de l’Information (LETI), ENIS, Route Sokra km 3.5, Sfax 3038, Tunisia
2ETIS, UMR 8051, CNRS, ENSEA, University Cergy-Pontoise, F-95000 Cergy, France
3Institut National de Recherche en Informatique et Automatique (INRIA), Campus de Beaulieu, 35042 Rennes Cedex, France

Received 1 April 2008; Accepted 27 May 2008

Academic Editor: Guosen Yue

Copyright © 2008 Imed Hadj Kacem et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the problem of optimization of the training sequence length when a turbo-detector composed of a maximum a posteriori (MAP) equalizer and a MAP decoder is used. At each iteration of the receiver, the channel is estimated using the hard decisions on the transmitted symbols at the output of the decoder. The optimal length of the training sequence is found by maximizing an effective signal-to-noise ratio (SNR) taking into account the data throughput loss due to the use of pilot symbols.

1. Introduction

To combat the effects of intersymbol interference (ISI) due to the frequency selectivity of mobile radio channels, an equalizer has to be used. In order to efficiently detect the transmitted symbols, the equalizer needs a good estimate of the channel. The channel is classically estimated by using a training sequence (TS) known at the receiver [1]. When the length of the TS increases, the channel estimate becomes more reliable. However, this leads to a loss in terms of data throughput. Thus, instead of using the training sequence only, the information carried by the observations corresponding to the data symbols can also be used to improve iteratively the channel estimation. At each iteration, the channel estimator refines its estimation by using the hard or soft decisions on the data symbols at the output of the data detector or the channel decoder [24].

A question that one can ask concerns the length of the TS to choose in order to obtain a satisfactory initial channel estimate without decreasing significantly the data throughput. Several methods have been proposed to answer this question. In [5], a solution based on the maximization of a lower bound of the capacity of the training-based scheme was proposed for a transmission over a frequency selective channel. In [6], we considered the case where a maximum a posteriori (MAP) equalizer is used. We proposed to maximize an effective signal-to-noise ratio (SNR) taking into account the loss in terms of data throughput due to the use of the pilot symbols. These studies have been performed for a noniterative receiver. In [2], an iterative data detection and channel estimation scheme was considered for a transmission over a flat fading channel. The TS length optimization was performed by minimizing the ratio of the channel estimation mean square error (MSE) to the data throughput.

In this letter, we consider a coded transmission over a frequency selective channel. At the receiver, a turbo-detector composed of a MAP equalizer and a MAP decoder is used. The channel is iteratively estimated by using hard decisions on the coded bits at the output of the decoder. This estimation strategy is usually referred to as decision-directed or bootstrap estimation. We derive the expression of the equivalent SNR at the output of the MAP equalizer fed with the a priori information (from the decoder) and the channel estimate. We define, based on this expression, an effective SNR taking into account the loss in terms of data throughput due to the use of the TS. We propose to find the length of the TS maximizing this expression. We show that when the decisions provided by the decoder are enough reliable, the optimal TS length is equal to its minimum value 2𝐿1, where 𝐿 is the channel memory. Notice that a similar result was found in [5] by maximizing a lower bound on the training-based channel capacity when the training and data powers are allowed to vary.

Throughout this letter, scalars and matrices are lower and upper case, respectively, and vectors are underlined lower case. The operator ()𝑇 denotes the transposition.

2. Transmission System Model

As shown in Figure 1, the input information bit sequence is encoded with a convolutional code, interleaved and mapped to the symbol alphabet 𝒜. In this letter, we consider the BPSK modulation (𝒜={1,1}). The symbols are then transmitted over a multipath channel. We assume that transmissions are organized into bursts of 𝑇 symbols. The channel is assumed to be invariant during the transmission. The received baseband signal sampled at the symbol rate at time k is𝑦𝑘=𝐿1𝑙=0𝑙𝑥𝑘𝑙+𝑛𝑘,(1)where 𝐿 is the channel memory and 𝑥𝑘 are the transmitted symbols. In this expression, 𝑛𝑘 are modeled as independent and identically distributed (iid) samples from a random variable with normal probability density function (pdf) 𝒩(0,𝜎2), where 𝒩(𝛼,𝜎2) denotes a Gaussian distribution with mean 𝛼 and variance 𝜎2. The term 𝑙 is the 𝑙th tap gain of the channel.

860368.fig.001
Figure 1: Transmitter structure.

The initial channel estimate is provided to the receiver by a least square estimator using 𝑇𝑝 training symbols with 2𝐿1𝑇𝑝𝑇 [1], where 𝑇𝑝 is the parameter to be optimized.

3. Decision-Directed Channel Estimation

As shown in Figure 2, we consider a turbo-detector composed of a MAP equalizer and a MAP decoder. At each iteration, the equalizer and the decoder compute a posteriori probabilities (APPs) and extrinsic probabilities on the coded bits [7]. They exchange the extrinsic probabilities which will be used as a priori probabilities, to improve iteratively their performance. In order to refine the channel estimate, the channel estimator uses the hard decisions on the transmitted coded symbols based on the APPs at the output of the decoder. Indeed, the channel estimator is fed with 𝑇𝑝 pilot symbols and 𝛿𝑇 estimates of the coded symbols coming from the decoder. Let 𝑥=(𝑥𝑇𝑝+𝛿𝑇1,,𝑥0)𝑇 be the sequence containing the 𝑇𝑝 training symbols (𝑥𝑇𝑝1,,𝑥0)𝑇 and the 𝛿𝑇 data symbols (𝑥𝑇𝑝+𝛿𝑇1,,𝑥𝑇𝑝)𝑇. The output of the channel corresponding to the vector 𝑥 is given by𝑦=𝑋+𝑛,(2)where =(0,,𝐿1)𝑇 is the vector of channel taps, 𝑋 is the (𝑇𝑝𝐿+1+𝛿𝑇)×𝐿 Hankel matrix having the first column (𝑥𝑇𝑝+𝛿𝑇1,,𝑥𝐿1)𝑇 and the last row (𝑥𝐿1,,𝑥0); and 𝑛 is the corresponding noise vector.

860368.fig.002
Figure 2: Receiver structure.

In order to estimate the channel, the observation vector 𝑦 is approximated as follows:𝑦𝑋+𝑛,(3)where 𝑋 is the estimated version of the matrix 𝑋 containing the hard decisions on the coded symbols at the output of the decoder. The iteration process can be repeated several times and here the matrix 𝑋 corresponds to the estimated symbols at the last iteration.

The least square estimate =(0,,𝐿1)𝑇 of is given by [1]=𝑋𝑇𝑋1𝑋𝑇𝑦.(4)In general, 𝛿𝑇 is chosen to give a good complexity/performance trade-off. We suppose in the following that 𝛿𝑇 is fixed such that 𝛿𝑇𝐿. We also assume that the vector of errors on the coded symbols at the output of the decoder is independent of the noise vector. In average, the errors are assumed to be uniformly distributed over the burst. The channel estimation mean square error (MSE) is given by [3]𝐸𝛿2=𝜎2𝐿𝑇𝑝𝐸𝑛𝐿+1+𝛿𝑇+42+(𝐿1)𝐸(𝑛)(𝑇𝑝𝐿+1+𝛿𝑇)2,(5)where 𝐸() is the mathematical expectation, 𝑛 is the number of erroneous hard decisions on the coded symbols at the output of the decoder, used by the channel estimator.

Let 𝛽=𝐸(𝑛/𝛿𝑇) and 𝛽2=𝐸(𝑛2/𝛿𝑇2). Hence, (5) can be rewritten as𝐸𝛿2=𝜎2𝐿𝑇𝑝𝐿+1+𝛿𝑇+4𝛽2𝛿𝑇2+(𝐿1)𝛽𝛿𝑇(𝑇𝑝𝐿+1+𝛿𝑇)2.(6)

4. Performance Analysis of the MAP Equalizer

We want now to study the impact of the a priori information and the channel estimation errors on the MAP equalizer performance.

4.1. Equivalent SNR at the Output of the Equalizer

We assume that the a priori (extrinsic) log likelihood ratios (LLRs) at the input of the equalizer, fed back from the decoder, are iid samples from a random variable with the conditional pdf 𝒩(±2/𝜎2𝑎,4/𝜎2𝑎) [810]. The equivalent signal-to-noise ratio at the output of the MAP equalizer fed with the a priori LLRs from the decoder and the channel estimate (from the decision-directed channel estimator) can be approximated at high SNR bySNReq,DD=𝑑2+4𝑚𝜇24𝜎211+𝜎2𝐸𝛿21+4𝑚𝜇2/𝑑21,(7)where 𝜇=𝜎/𝜎𝑎 and 𝐸(𝛿2) is the channel estimation MSE given in (6). The quantities 𝑚 and 𝑑 are defined as𝑚,𝑑=argmin𝑚(𝑒),𝑑(𝑒)𝑑(𝑒)2+4𝑚(𝑒)𝜇2×𝐸2𝜎1+𝛿21+4𝑚(𝑒)𝜇2/𝑑(𝑒)21/2,(8)where 𝑒𝐸,𝐸 is the set of all nonzero error events [11], 𝑚(𝑒) is the number of decision errors in 𝑒, and 𝑑(𝑒) is the convolution of 𝑒 with the channel.

Remark 1. We prove the result given in (7) similarly to [12, Proposition 1]. However, in [12], we assumed that the channel was estimated by using a perfect training sequence and the covariance matrix of the channel estimation error was then diagonal. When a decision-directed channel estimator is used, as it is the case in this work, this covariance matrix is not diagonal which makes the proof more complicated. We omit here the proof for the sake of space.

4.2. Simulation Results

In our simulations, we consider the channel with impulse response (0.37;0.6;0.6;0.37). The transmissions are organized into bursts of 256 symbols. The modulation used is the BPSK. We do not consider the channel coding and the turbo-detector yet. Figure 3 shows the bit error rate (𝐵𝐸𝑅) curves with respect to the SNR at the input of the MAP equalizer when the channel is estimated by the decision-directed channel estimator using 𝑇𝑝=15 pilot symbols and 𝛿𝑇=30 estimates of the data symbols. The 𝛿𝑇 estimates of the data information symbols at the input of the channel estimator are generated by making hard decisions on artificial LLRs modeled as iid samples from a random variable with pdf 𝒩(±2/𝜎2𝑥,4/𝜎2𝑥) [810]. We consider two reliability levels of a posteriori information: 𝜎2𝑥=0.5 and 𝜎2𝑥=0.2 and two reliability levels of a priori information: 𝜇=0.1 and 𝜇=0.5. The solid lines indicate the theoretical MAP equalizer performance. The dotted ones are obtained by simulations. We note that the theoretical curves approximate well the curves obtained by simulations.

860368.fig.003
Figure 3: Comparison of the equalizer performance (dotted curves) and the theoretical performance (solid curves) when the channel estimate is provided by the decision-directed channel estimator.

In the following, we propose to optimize the training sequence length by maximizing an effective signal-to-noise ratio that we will define taking into account the data throughput loss due to the use of the pilot symbols.

5. Optimization of the Training Sequence Length

Increasing the training sequence length leads to an improvement of the channel estimate quality but also to a loss in terms of data throughput. Thus, in order to take into account this loss, we define, based on (7), an effective SNR at the output of the MAP equalizer asSNRe,eq,DD=𝑇𝑇𝑝𝑇SNReq,DD=𝑇𝑇𝑝𝑇𝑑2+4𝑚𝜇24𝜎2×11+𝜎2𝐸𝛿21+4𝑚𝜇2/𝑑21.(9)Our aim is to find the TS length maximizing this quantity. Hence, we define the following optimization problem𝑇𝑝=argmax2𝐿1𝑇𝑝𝑇𝛿𝑇SNRe,eq,DD.(10)Let 𝑥𝐑+, 𝑥2𝐿1 and𝑓1(𝑥)=𝑇𝑥𝑇𝑑2+4𝑚𝜇24𝜎21+𝑔(𝑥)1,(11)where 𝑔(𝑥)=(1/𝜎2(1+4𝑚𝜇2/𝑑2))(𝜎2𝐿/(𝑥𝐿+1+𝛿𝑇)+4((𝛽2𝛿𝑇2+(𝐿1)𝛽𝛿𝑇)/(𝑥𝐿+1+𝛿𝑇)2)).

Thus, SNRe,eq,DD=𝑓1(𝑇𝑝).

When 𝑔(𝑥)1, 𝑓1(𝑥) can be approximated by𝑓1(𝑥)𝑇𝑥𝑇𝑑2+4𝑚𝜇24𝜎21𝑔(𝑥)(12)which is a decreasing function.

When the 𝛿𝑇 decisions on the data symbols added to the training sequence are reliable, 𝑔(𝑇𝑝)1. Thus, the optimal length of the training sequence solution of (10) is𝑇𝑝=2𝐿1.(13)When the hard decisions used by the channel estimator are not reliable, the approximation 𝑔(𝑇𝑝)1 becomes inaccurate and the optimization problem cannot be solved analytically.

6. Simulation Results

We first consider a MAP equalizer fed with the channel estimate calculated by the decision-directed channel estimator using 𝑇𝑝 pilot symbols and 𝛿𝑇 estimates of the data symbols. These estimates are obtained by making hard decisions on artificial a posteriori LLRs modeled as iid samples from a random variable with pdf 𝒩(±2/𝜎2𝑥,4/𝜎2𝑥). The equalizer is also fed with artificial a priori LLRs modeled as iid samples from a random variable with pdf 𝒩(±2/𝜎2𝑎,4/𝜎2𝑎). Figure 4 shows the effective SNR given in (9) as a function of the training sequence length for Channel 3 and Channel 4 with respective impulse responses (0.5;0.71;0.5) and (0.37;0.6;0.37;0.6), for SNR=9 dB, 𝑇=512, and 𝛿𝑇=100. We consider two values of 𝜎2𝑥: 𝜎2𝑥=0.2 and 𝜎2𝑥=0.4; and two values of 𝜎2𝑎: 𝜎2𝑎=0.5 and 𝜎2𝑎=0.65. We note that the training sequence length maximizing SNRe,eq,DD is equal to 2𝐿1 when the decisions on the data symbols added to the TS become reliable (𝜎2𝑥=0.2 corresponding to 𝛽=0.013). We also note that the optimal choice of the training sequence length can significantly improve the effective SNR.

860368.fig.004
Figure 4: SNRe,eq,DD versus 𝑇𝑝 for Channel 3 and Channel 4, SNR=9 dB, 𝑇=512, and 𝛿𝑇=100.

Now, we consider the whole system with the channel coding at the transmitter and the iterative receiver composed of a MAP equalizer and a MAP decoder. We use Channel 4 with impulse response (0.37;0.6;0.6;0.37). The information data are encoded using the rate 1/2 convolutional code with generator polynomials (7,5) in octal. At the first iteration, the channel is estimated by using the TS [1]. At the next iterations, it is estimated by using the decision-directed technique. Figure 5 shows the BER performance (on the coded bits) at the output of the MAP equalizer after two iterations of the iterative receiver and at the convergence (after three iterations) with respect to SNRe=((𝑇𝑇𝑝)/𝑇)SNR, where SNR is the signal-to-noise ratio at the input of the MAP equalizer, for 𝑇=512, 𝛿𝑇=100 and for different values of the length 𝑇𝑝 of the training sequence. The 𝛿𝑇 estimates of the coded symbols at the input of the channel estimator are obtained by making hard decisions on the a posteriori LLRs at the output of the MAP decoder. Simulations confirm that the MAP equalizer best performance is achieved, at high SNR, when 𝑇𝑝=2𝐿1.

860368.fig.005
Figure 5: MAP equalizer BER performance versus SNRe for different values of the length of the training sequence after two and three iterations of the iterative receiver for Channel 4, 𝑇=512 and 𝛿𝑇=100.

7. Conclusion

In this letter, we considered the problem of optimization of the training sequence length for the iterative receiver composed of a MAP equalizer, a MAP decoder and a decision-directed channel estimator. We proved that the training sequence length maximizing the effective SNR at the output of the MAP equalizer is equal to its minimum value 2𝐿1 when the decisions provided by the decoder are reliable. A future work will consider the case where the channel estimator uses soft decisions provided by the decoder on the coded symbols.

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