Elsevier

Signal Processing

Volume 85, Issue 12, December 2005, Pages 2394-2411
Signal Processing

Efficient semi-blind estimation of multipath channel parameters via a delay decoupling optimization approach

https://doi.org/10.1016/j.sigpro.2005.02.018Get rights and content

Abstract

In this paper a parametric method for estimating the unknown multipath channel impulse response (CIR) in a semi-blind manner is proposed. The main trait of this method is that instead of seeking the whole CIR sequence, only the unknown time delays and attenuation factors of the physical channel multipath components are estimated. The technique is based on a suitable application of the sub-channel response matching (SRM) criterion. The resulting cost function is separable with respect to the two sets of unknown parameters, i.e., time delays and attenuation factors, and thus a two step optimization procedure can be applied. Concerning the first step, which is the most difficult one, it is proven that the resulting non-linear cost function can be decoupled in terms of the respective time delay parameters. Thus, only a small number of simple linear searches needs to be executed in order to estimate the time delays of the multipath channel. The new method offers significant computational savings and a lower mean square estimation error as compared to existing semi-blind channel estimation methods. It performs well even for closely-spaced delays and is quite robust to channel overmodeling.

Introduction

Multipath channel estimation is an issue of major importance in reliable wireless communications system design. Due to the multipath phenomenon in wireless applications, the introduced InterSymbol Interference (ISI) may cause a severe degradation in a system's performance. The problem is much more severe in high-rate applications, since, as the symbol rate becomes higher, the number of symbols spanned by a channel impulse response (CIR) with a certain delay spread is increased. Thus, an efficient and accurate estimation of the CIR is highly desirable, in order to mitigate interference and achieve reliable data detection.

In many high-speed wireless applications, the propagation channel can be modeled as a specular channel consisting of a relatively small number of dominant rays [1], with each one being characterized by its time delay and attenuation factor. Then, provided that the transmitter and receiver filters are known [2], the channel estimation task is reduced to that of estimating the parameters of the multipath channel components, i.e., the time delays and the attenuation factors. Two are the main advantages of using such a parametric approach. First, the number of required data is reduced, resulting in a significant saving in complexity, processing delay and, possibly, in bandwidth. Second, the mean square channel estimation error is expected to be lower since a more parsimonious parameterization of the channel is adopted, compared to conventional CIR estimation.

In a wireless communication channel, bandwidth is a precious resource, therefore the need for training sequence reduction is imperative. Thus, conventional training based channel estimation methods are inappropriate, especially when the channel span is large as in high rate applications. On the other hand, blind channel estimation methods require a high number of data and have very often limited performance. In such cases semi-blind estimation techniques may be employed [3], [4], [5], [6]. The main trait of a semi-blind scheme is that a purely blind criterion is suitably modified so as to incorporate information from a short training sequence [3]. The performance of a semi-blind method, if properly designed, can be significantly improved against the corresponding blind method.

The estimation of a multipath channel using a parametric modeling has gained considerable attention in recent years. Blind [7], [8], [9] as well as training based [10], [11], [12] channel parametric estimation methods have been recently proposed. These methods either rely on preliminary estimates of the CIR [8], [10], [12] or directly estimate the parameters of the multipath channel components [7], [9]. Recently, parametric channel modeling has also been applied to channel estimation in multicarrier (OFDM) [14] and multiuser (CDMA) [15], [16] system design. In a blind context, an important implication of parametric channel modeling is that the channel overmodeling problem, which is inherent in second-order statistics blind identification algorithms, can be overcome [9]. That is, channel parameterization with respect to delays and attenuation factors, leads to consistent channel estimates even if the channel order is overestimated. However, such an improvement is traded off with a need to perform a computationally costly multidimensional search, the order of which equals the number of multipath components.

In this paper, we propose a novel semi-blind technique, which exploits the specular channel structure by incorporating the knowledge of the pulse shaping filter. By applying the well-known sub-channel response matching (SRM) criterion [18] to the problem at hand and after incorporating limited input information, we end up with a least squares (LS) problem, which is separable with respect to the unknown parameter sets, i.e. time delays and attenuation factors. Specifically, it is shown that the optimization problem can be separated to two different subproblems. One subproblem which is non-linear with respect to the time delays and another subproblem which is linear with respect to the attenuation parameters. After revealing the special structure of the non-linear problem, a computationally efficient linear search method for the estimation of the unknown time delays is developed. In the sequel, the Gauss–Newton (G–N) algorithm may be applied in order to further improve the accuracy of the estimated values. Finally, the attenuation parameters are estimated by solving a low-order linear LS problem. The new method is very simple to implement and achieves a lower channel estimation error compared to other related methods, e.g. [3], at a reasonable computational cost. A basic feature of the proposed technique is that, due to the special form of the non-linear cost function, a highly inefficient multidimensional search is avoided and a small number of simple linear searches is executed instead. Moreover, the new method offers the possibility of trading off performance to complexity in an easy manner and yields good estimates even in cases of closely spaced time delays. The performance of the new method has been justified theoretically and tested through extensive simulations. Some preliminary results of this work have been presented in [17].

The paper is organized as follows. In Section 2 the multipath channel model is defined and the problem is formulated. In Section 3 the new method is derived and in Section 4 an efficient version of the proposed method is described. Performance issues of the new semi-blind technique are briefly discussed in Section 5 and simulation results are provided in Section 6. Concluding remarks are presented in Section 7.

Section snippets

Channel model

In general, the CIR encountered in wireless communication systems has a form which varies significantly depending on several factors, such as physical environment, transmission rate, mobility etc. However a common trait in most cases, particularly in high-speed applications, is that the multipath channel tends to be of a discrete form (i.e., it consists of a number of dominant multipath components). More specifically, if the CIR is assumed to be time invariant within a small-scale time

Derivation of the algorithm

In this section an efficient method for estimating the multipath channel parameters is described, by exploiting the special form of the non-linear cost function given in (12). In particular, the cost function is shown to possess the following two properties. First, the optimization problem can be split up into two subproblems in terms of the delay and attenuation parameters, respectively. Second, the delays are shown to be decoupled from each other, allowing for their efficient estimation.

An alternative version of the DECP method

In this section it is shown that by further exploiting the form of F(τ), an alternative equivalent procedure can be developed for estimating in an even more efficient manner the delay parameters. Let us first define the following quantities:φi=YSg(τi),Φi:j=[φiφi+1φj],i,j=0,1,,p-1,ji.That is, φi is the (i+1)th column of matrix Φ(τ) and Φi:j stands for its sub-matrix formed by columns i+1 through j+1.

Based on these definitions, matrix C(τ) given in (19) can be written asC(τ)=Φ0:p-2HΦ0:p-2Φ0:p-2

Computational complexity

It can be easily shown that the two versions of the new method presented in Tables 1 and 2 (i.e., DECP and DECPa, respectively) are completely equivalent, in terms of performance, by assuming a perfect decoupling of the delay parameters. However, even in practical cases, where decoupling is not perfect, the two algorithms perform in a very similar manner. This has been verified by extensive simulations, some of which are presented in Section 6.

To compare the two algorithms in terms of

Simulation results

In this section, we investigate the performance of the new parametric estimation method under various conditions. The two versions of the new method (i.e., DECP and DECPa) are compared with the semi-blind SRM-based method presented in [3], in which the CIR is estimated in a non-parametric manner, using a linear combination of a blind and a non-blind cost function. As a comparison measure we have used the root-mean-square-error between actual and estimated CIRs, i.e. RMSE=(1/P)k=1Pi=12L(hact(i)

Conclusion

In this paper, we have proposed a new semi-blind estimation algorithm that exploits the specular channel structure and a-priori knowledge of the pulse shaping filter. The main contribution of this work is the observation that the optimized cost function is almost decoupled with respect to the time delays, a fact which allows for efficient estimation of the delay parameters. The new method is very simple to implement, it works well even for closely-spaced multipath components and is quite robust

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