Variable leaky steepest descent algorithm for autoregressive fading estimation in OFDM systems

Abstract

This paper presents an adaptive algorithm for unbiased estimation of autoregressive Rayleigh flat fading parameters in OFDM systems. To this end, a variable leaky steepest descent algorithm is proposed. The authors use a variable leak parameter in the algorithm to remove the noise induced bias from the autoregressive estimates. Theoretical performance analyses of the proposed method are conducted. Moreover, computer simulations are presented to evaluate the performance of the proposed algorithm and to compare it with other existing methods.

Introduction

Channel estimation is a crucial task in signal processing and in the new generation of wireless communications (see [1], [2], [3], [4], [5]). Inter-symbol interference (ISI) is avoided by using orthogonal frequency division multiplexing (OFDM). The frequency selective multipath channel can be converted to a set of parallel flat fading channels by using OFDM systems [5]. In orthogonal frequency division multiplexing (OFDM) systems carrier frequency offset (CFO) is a major problem in achieving orthogonality between subcarriers and introduces inter-carrier interference (ICI) [6], [7], [8] between subcarriers. CFO estimation scheme targeted for OFDM systems, using the subspace method, has been proposed in [9]. Many papers considered channel estimation in the presence of the CFO, that's why in this paper, we focused on autoregressive channel parameters estimation by an adaptive algorithm. For m-th subcarrier in OFDM system, the received signal, using a training sequence, can be expressed as

$$y_m(n)=s_m(n)⁎h_m(n)+b_m(n)$$

where $m\text{=0,1,}...\text{, }M\text{-1}$ while M is total number of subcarriers, $b(n)$ is an additive zero-mean white noise process, and $h(n)$ is a Rayleigh flat fading which can be modeled as an autoregressive (AR) process [10], [11] given by

$$h(n)=a_1^{⁎}h(n-1)+a_2^{⁎}h(n-2)+...+a_p^{⁎}h(n-p)+u(n)$$

The driving noise, $u(n)$, is a zero-mean white noise process, and ${\mathbf{w}}_o={\left[\begin{array}{cccc}\hfill a_1\hfill & \hfill a_2\hfill & \hfill \dots \hfill & \hfill a_p\hfill \end{array}\right]}^H$ is the vector of AR parameters. The order of AR model, p, is assumed to be known. The noise processes $u(n)$ and $b(n)$ are assumed to be orthogonal, i.e. ($E\{u(n)b{(n)}^H\}=0$), where E is the expectation operator and ${(.)}^H$ denotes conjugate transpose (${(.)}^{T⁎}$). The variances of driving noise and observation noise are ${\sigma }_u^2$, and ${\sigma }_b^2$ respectively. The autocorrelation function (ACF) for the fading process is given by [12]

$$r_h(q)=E\{h(n)h^{⁎}(n-q)\}={\sigma }_h^2{J}_0(2\pi f_{d}T\left|q\right|)$$

where $q\text{=0,1,}...\text{, }Q\text{-1}$ while Q is number of autocorrelation lags, $J_0$ is the zero order Bessel function of the first kind, $f_{d}T$ is the normalized Doppler spread, and ${\sigma }_h^2$ is the variance of fading process. By taking the Fourier transform of $r_h(q)$, the power spectrum can be obtained

$$S(f)=\frac{{\sigma }_h^2}{\pi f_{d}T\sqrt{1-{(\frac{f}{f_{d}T})}^2}}\text{ ; }\left|f\right|\le f_{d}T$$

The AR modeling for the fading process can be accomplished by using the well-known Yule-Walker equations. We can select a proper AR order for fading channel by using AR order selection methods [13]. Then the AR model coefficients can be estimated by Yule-Walker equations.

In this paper, it is aimed to estimate the channel parameters ${\mathbf{w}}_o$, ${\sigma }_u^2$, and ${\sigma }_b^2$ from the received signal samples i.e. ($y(n)\text{ ; }n=1,...,N$) where N is the number of data samples.

From the above discussions, we see that the fading estimation in OFDM systems can be considered as a problem of AR estimation in noise. Many researchers have dealt with this problem (see [14], [15], [16], [17], [18], [19], [20], [21]). In the improved least squares (ILS)-based methods presented in [14], [15], [16], the inverse of the covariance matrix of the observations must be computed, which is a critical drawback for implementing these algorithms in terms of computational complexity and numerical instability. In [17] a computationally simple least mean squares (LMS) algorithm is devised for unbiased AR modeling in the presence of white noise, assuming that the observation noise is known; however, in fading estimation, the observation noise variance is unknown and we must estimate the observation noise variance. The method presented in [18] is based on the eigenanalysis of the covariance matrix of observations, which is difficult for online implementation. In [20], an errors-in-variables (EIV)-based method is presented. The method includes searching the noise variances whose noise compensated covariance matrix of observations is positive semi-definite. In spite of a good accuracy, the method has a high computational complexity. In [21], authors proposed vector process disturbed by an additive white noise. When this process, which can be corresponding to a mobile fading channel, is modeled by a multivariate autoregressive (M-AR) process, optimal filters such as Kalman filter can be used for prediction or estimation from noisy observations. Investigating the relevance of a structure based on two cross-coupled H∞ filters for the joint estimation of time-varying frequency-flat Rayleigh fading channel and its AR parameters proposed in [22]. In [23] authors addressed the problem of estimating a channel supposed to follow a Rayleigh model with Jakes' Doppler spectrum using a second order autoregressive model with Kalman filter and using a Minimization of Asymptotic Variance (MAV) criterion to improve the estimation performance.

In this paper, we present an autoregressive channel estimation scheme based on a variable leaky steepest descent (VLSD) algorithm for Rayleigh flat fading in OFDM systems. We show that the leak parameter in the VLSD should be considered equal to the observation noise variance. A computationally simple method for estimating the observation noise variance or the leak parameter is presented. The convergence analysis of the proposed adaptive algorithm is also performed and a stability criterion is derived.

Finally, given the AR parameters estimates, the fading process is estimated using Kalman filter and evaluated the Bit Error Rate (BER) performance in AWGN, Rayleigh fading channel before and after elimination of fading effect. The results are derived for BPSK modulation. In the simulation study, the performance of the proposed method is compared with the other existing methods.

This paper is organized as follows: Section 2 introduces the OFDM system and the AR modeling for fading processes. In Section 3, we derive the proposed VLSD algorithm for AR estimation in white noise. Section 4 presents the simulations to evaluate and compare the performance of the proposed method with the other existing methods. Finally, some conclusions are given in Section 5.