Performance Study of Least-Squares Channel Estimation Based on Hard Decisions Samson Lasaulce 1 , Noura Sellami 2 , Yi Yuan 1 and Ahmed Saadani 1 1 France Télécom R&D, 38-40 rue du Général Leclerc, 92794 Issy-les-Moulineaux, France, samson.lasaulce@francetelecom.com 2 ENSEA Laboratoire ETIS, 6 avenue du Ponceau, 95014 Cergy-Pontoise, France, sellami@ensea.fr Abstract ─ The main purpose of this paper is to study the performance of the bootstrap channel estimation scheme, which consists in this paper in using the hard-decided outputs of the channel decoder/equalizer in order to extend the training sequence. Through a simple asymptotic analysis, we derive the expression of the channel estimation MSE in order to evaluate the impact of using wrong decisions on the Least-Squares channel estimation performance. In particular, it is showed that it depends on the first and second moments of the number of errors per block of symbols. Interestingly, it is also proven that the bootstrap estimator does not always improve channel estimation accuracy. The impact of using bootstrap channel estimation on the overall receiver performance is studied through an SNR analysis. I. INTRODUCTION The most conventional way of estimating the propagation conditions in mobile radio systems is to model the channel by a Finite Impulse Response filter and to transmit a data sequence known to the receiver. If the channel noise is additive, white and Gaussian, the Maximum Likelihood (ML) estimator based on the observations generated by the training sequence (TS) consists in minimizing the mean square error between the signal received during the emission of the TS and its noiseless counterpart (the filtered version of the TS), which is a Least-Squares (LS) estimator. It is well known that the performance (in terms of error variance) of the Least-Squares channel estimator is roughly inversely proportional to the training sequence length (see [1] for example). Therefore, a simple idea to increase the number of training symbols is to use the hard decisions associated with the outputs of the symbol detector or channel decoder (e.g. [2]). Of course, it is also possible to exploit the soft information associated with the latter if it is available. In this paper, we will only consider the case where the channel estimator is fed with hard decisions. As the decoder output is generally more reliable than the symbol detector output, it is more efficient to use the decoded bits. The latter are then re-encoded and used to extend the initial training sequence. Generally, several iterations are performed in order to improve the overall receiver performance. The corresponding estimation strategy is often referred to as bootstrap estimation or iterative estimation based on hard decisions. In contrast with the conventional estimation procedure, in which the training symbols are perfectly known, the bootstrap estimator does not coincide with the ML estimator because of the presence of errors in the extended training sequence. Such an introduction of "noise" into the training data affects the performance of the bootstrap channel estimator. The corresponding performance degradation has been evaluated in [2] and [3] based on the Cramer-Rao Bound (CRB). Performance analyses respectively rely on an asymptotic analysis (large extended TS length) in [2] and a deterministic/Gaussian symbol model in [3]. One of the disadvantages of considering lower performance bounds is that the behavior of the real estimation performance is partially hidden. In particular, the fact that the CRB decreases with the "extended TS" length does not imply that the bootstrap estimator variance is also a decreasing function of this parameter. A relevant analysis of iterative channel estimation performance in turbo equalization has been done in [4] but only the soft information case is considered. It turns out that under the assumptions of [3], it is possible to obtain the estimation variance of the bootstrap estimator itself. This is precisely the main purpose of this paper. Furthermore, papers dealing with iterative channel estimation generally do not consider the receiver performance through a Block Error Rate (BLER) analysis but a Bit Error Rate (BER) analysis whereas the "irregular behavior" of the number of errors per block plays an important role in the design of the bootstrap estimator. This paper is organized as follows. The signal model and description of the bootstrap estimation are provided in section II. The main part of this paper is section III where it is showed how to derive the estimation error variance expression. The corresponding expression is discussed in section IV. In section V, the overall receiver performance is analysed. Section VI reviews the main results of this paper. General notations. In this paper, notations , v s and M stand for scalar, vector and matrix respectively. Notations T (.) and H (.) stand for transpose and Hermitian operators. II. PROBLEM STATEMENT A. Framework In this paper we consider a simple scenario where there is one transmitter (figure 1) and one receiver (figure 2). A single antenna is assumed at both sides of the transmission. The transmitter sends coded and interleaved BPSK symbols { } { } Z t t x ∈ + − ∈ , 1 , 1 ) ( to the receiver over a multipath channel (figure 3) that includes the