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

Authors: Imed Hadj Kacem; Noura Sellami; Inbar Fijalkow; Aline Roumy
Journal: Research Letters in Communications
 (2008)

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## Abstract

We consider the problem of optimization of the training sequence length when
a turbo-detector composed of a maximuma 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.


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## Body

## 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 thea 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 2L−1, 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
Figure1, 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 is(1)yk=∑l=0L−1hlxk−l+nk,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 hl is the lth tap gain of the channel.Figure 1
Transmitter structure.The initial channel estimate is provided to
the receiver by a least square estimator usingTp training symbols with 2L−1≤Tp≤T [1], where Tp is the parameter to be optimized.


## 3. Decision-Directed Channel Estimation

As shown in
Figure2, 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+δT−1,…,x0)T be the sequence containing the Tp training symbols (xTp−1,…,x0)T and the δT data symbols (xTp+δT−1,…,xTp)T.
The output of the channel corresponding to the vector x¯ is given by(2)y¯=Xh¯+n¯,where h¯=(h0,…,hL−1)T is the vector of channel taps, X is the (Tp−L+1+δT)×L Hankel matrix having the first column (xTp+δT−1,…,xL−1)T and the last row (xL−1,…,x0); and n¯ is the corresponding noise vector.Figure 2
Receiver structure.In order to estimate the channel, the observation
vectory¯ is approximated as follows:(3)y¯≈X^h¯+n¯,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 estimateh¯^=(h^0,…,h^L−1)T of h¯ is given by [1](4)h¯^=(X^TX^)−1X^Ty¯.In general, δT is chosen to give a good
complexity/performance trade-off. We suppose in the following that δT is fixed such that δT≫L.
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](5)E(∥δh¯∥2)=σ2LTp−L+1+δT+4E(n2)+(L−1)E(n)(Tp−L+1+δT)2,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.Letβ¯=E(n/δT) and β2¯=E(n2/δT2).
Hence, (5) can be rewritten as(6)E(∥δh¯∥2)=σ2LTp−L+1+δT+4β2¯δT2+(L−1)β¯δT(Tp−L+1+δT)2.


## 4. Performance Analysis of the MAP Equalizer

We want now to
study the impact of thea 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
thea 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/σa2,4/σa2) [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 by(7)SNReq,DD=(d′2+4m′μ24σ2)(1+1σ2E(∥δh¯∥2)1+4m′μ2/d′2)−1,where μ=σ/σa and E(∥δh¯∥2) is the channel estimation MSE given in (6).
The quantities m′ and d′ are defined as(8)(m',d')=argminm(e¯),∥d¯(e¯)∥∥d¯(e¯)∥2+4m(e¯)μ22σ×(1+E(∥δh¯∥2)1+4m(e¯)μ2/∥d¯(e¯)∥2)−1/2,where e¯∈E,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/σx2,4/σx2) [8–10]. We consider two reliability levels of a
posteriori information: σx2=0.5 and σx2=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.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.


## 4.1. Equivalent SNR at the Output of the Equalizer

We assume that
thea 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/σa2,4/σa2) [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 by(7)SNReq,DD=(d′2+4m′μ24σ2)(1+1σ2E(∥δh¯∥2)1+4m′μ2/d′2)−1,where μ=σ/σa and E(∥δh¯∥2) is the channel estimation MSE given in (6).
The quantities m′ and d′ are defined as(8)(m',d')=argminm(e¯),∥d¯(e¯)∥∥d¯(e¯)∥2+4m(e¯)μ22σ×(1+E(∥δh¯∥2)1+4m(e¯)μ2/∥d¯(e¯)∥2)−1/2,where e¯∈E,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/σx2,4/σx2) [8–10]. We consider two reliability levels of a
posteriori information: σx2=0.5 and σx2=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.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 as(9)SNReff,eq,DD=T−TpTSNReq,DD=T−TpT(d′2+4m′μ24σ2)×(1+1σ2E(∥δh¯∥2)1+4m′μ2/d′2)−1.Our aim is to find the TS length
maximizing this quantity. Hence, we define the following optimization
problem(10)Tp*=argmax2L−1≤Tp≤T−δTSNReff,eq,DD.Let x∈R+*, x≥2L−1 and(11)f1(x)=T−xT(d′2+4m′μ24σ2)(1+g(x))−1,where g(x)=(1/σ2(1+4m′μ2/d′2))(σ2L/(x−L+1+δT)+4((β2¯δT2+(L−1)β¯δT)/(x−L+1+δT)2)).Thus,SNReff,eq,DD=f1(Tp).Wheng(x)≪1, f1(x) can be approximated by(12)f1(x)≈T−xT(d′2+4m′μ24σ2)(1−g(x))which is a decreasing function.When theδT 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)
is(13)Tp*=2L−1.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 usingTp 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/σx2,4/σx2).
The equalizer is also fed with artificial a priori LLRs modeled as iid
samples from a random variable with pdf 𝒩(±2/σa2,4/σa2).
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 σx2: σx2=0.2 and σx2=0.4; and two values of σa2: σa2=0.5 and σa2=0.65.
We note that the training sequence length maximizing SNReff,eq,DD is equal to 2L−1 when the decisions on the data symbols added
to the TS become reliable (σx2=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.Figure 4
SNReff,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 useChannel 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 SNReff=((T−Tp)/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=2L−1.Figure 5
MAP equalizer BER performance versusSNReff 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 value2L−1 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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*Source: 860368-2008-06-12.xml*