Research Letters in Communications

Volume 2008, Article ID 860368, 5 pages

http://dx.doi.org/10.1155/2008/860368

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

^{1}Laboratoire d’Electronique et des Technologies de l’Information (LETI), ENIS, Route Sokra km 3.5, Sfax 3038, Tunisia^{2}ETIS, UMR 8051, CNRS, ENSEA, University Cergy-Pontoise, F-95000 Cergy, France^{3}Institut 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 [2–4].

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 where 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 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 .
In this letter, we consider the BPSK modulation (). The symbols are then transmitted over a
multipath channel. We assume that transmissions are organized into bursts of symbols. The channel is assumed to be
invariant during the transmission. The received baseband signal sampled at the
symbol rate at time *k* iswhere is the channel memory and are the transmitted symbols. In this expression, are modeled as independent and identically
distributed (iid) samples from a random variable with normal probability
density function (pdf) where denotes a Gaussian distribution with mean and variance .
The term is the th tap gain of the channel.

The initial channel estimate is provided to the receiver by a least square estimator using training symbols with [1], where 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 pilot symbols and estimates of the coded symbols coming from the
decoder. Let be the sequence containing the training symbols and the data symbols .
The output of the channel corresponding to the vector is given bywhere is the vector of channel taps, is the Hankel matrix having the first column and the last row and is the corresponding noise vector.

In order to estimate the channel, the observation vector is approximated as follows:where is the estimated version of the matrix 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 corresponds to the estimated symbols at the last iteration.

The least square estimate of is given by [1]In general, is chosen to give a good complexity/performance trade-off. We suppose in the following that is fixed such that . 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]where is the mathematical expectation, is the number of erroneous hard decisions on the coded symbols at the output of the decoder, used by the channel estimator.

Let and . Hence, (5) can be rewritten as

#### 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 [8–10]. 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 bywhere and is the channel estimation MSE given in (6).
The quantities and are defined aswhere is the set of all nonzero error events
[11], is the number of decision errors in and is the convolution of 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 .
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 () 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 pilot symbols and estimates of the data symbols. The 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 [8–10]. We consider two reliability levels of *a
posteriori* information: and and two reliability levels of *a priori* information: and .
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.

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 asOur aim is to find the TS length maximizing this quantity. Hence, we define the following optimization problemLet , andwhere

Thus, .

When , can be approximated bywhich is a decreasing function.

When the decisions on the data symbols added to the training sequence are reliable, . Thus, the optimal length of the training sequence solution of (10) isWhen the hard decisions used by the channel estimator are not reliable, the approximation 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 pilot symbols and 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 .
The equalizer is also fed with artificial *a priori* LLRs modeled as iid
samples from a random variable with pdf .
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 and ,
for dB, and .
We consider two values of : and and two values of : and .
We note that the training sequence length maximizing is equal to when the decisions on the data symbols added
to the TS become reliable ( corresponding to ). We also note that the optimal choice of the
training sequence length can significantly improve the effective SNR.

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 .
The information data are encoded using the rate 1/2 convolutional code with generator polynomials 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 ,
where SNR is the signal-to-noise ratio at the input of
the MAP equalizer, for , and for different values of the length of the training sequence. The 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 .

#### 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 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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