Simplified Random-Walk-Model-Based Kalman Filter for Slow to Moderate Fading Channel Estimation in OFDM Systems

Abstract

Abstract-This study deals with multi-path channel estimation for orthogonal frequency division multiplexing systems under slow to moderate fading conditions. Advanced algorithms exploit the channel time-domain correlation by using Kalman Filters (KFs) based on an approximation of the time-varying channel. Recently, it was shown that under slow to moderate fading, near optimal channel multi-path complex amplitude estimation can be obtained by using the integrated Random Walk (RW) model as the channel approximation. To reduce the complexity of the high-dimensional RW-KF for joint estimation of the multi-path complex amplitudes, we propose using a lower dimensional RW-KF that estimates the complex amplitude of each path separately. We demonstrate that this amounts to a simplification of the joint multi-path Kalman gain formulation through the Woodbury's identities. Hence, this new algorithm consists of a superposition of independent single-path single-carrier KFs, which were optimized in our previous studies. This observation allows us to adapt the optimization to the actual multi-path multi-carrier scenario, to provide analytic formulae for the mean-square error performance and the optimal tuning of the proposed estimator directly as a function of the physical parameters of the channel (Doppler frequency, Signal-to-Noise-Ratio, Power Delay Profile). These analytic formulae are given for the first-, second-, and thirdorder RW models used in the KF. The proposed per-path KF is shown to be as efficient as the exact KF (i.e., the joint multipath KF), and outperforms the autoregressive-model-based KFs proposed in the literature.

On the use of tracking loops for low-complexity multi-path channel estimation in OFDM systems

“…Each single-path KF is optimized by the channel state noise variance. It has been shown that the per-path processing KF can have almost the same performance as the joint multipath KF for the AR1 model [29], and this approximation has been analytically proved for RW models [28]. The per-path and joint multipath KF are identical when the discrete-time equivalent channel is uncorrelated, i.e., F H p F p = N p I L , or, equivalently, the multipath delays are multiples of the sample time, i.e., τ (l) are integers ( [28], appendix A).…”

“…In general, λ > 1, while in the special case that F H p F p is a diagonal matrix equal to N p I L , one has λ = 1. This occurs only when the path delays of channel τ (l) T s are multiples of the sampling time T s , i.e., if the τ (l) are integers (see [28], appendix A for more details).…”

“…However, in the multi-path multicarrier scenario, the equivalence between the RWr-LS-CATL and the RWr-KF is not aways available: the former processes the paths separately, and optimized by the average path power, whereas the latter processes the multi-path channels jointly. However, a link between the RWr-LS-CATL and a per-path processing RWr-KF [28] can be found.…”

“…We have thus extended these previous results to the fourth-and sixth-order moments, which can be exploited to calculate the AMSE of the RW2-and RW3-LS-CATL. A last contribution of this paper is a performance comparison between the AMSE of the tracking loops and that of the Kalman-based channel estimator for OFDM recently proposed in the literature [28] [29].…”

Second-order autoregressive model-based Kalman filter for the estimation of a slow fading channel described by the Clarke model: Optimal tuning and interpretation

“…This prompts the need for a comprehensive study of 2 Model (1) assumes that symbols are normalized and known (or decided), in addition to the flat fading assumption. Although this model is admittedly simplistic, it can be applied to different (more involved) contexts, such as pilot-aided multi-carrier systems in frequency-selective wireless channels [42]. channel estimation for f d T ≤ 10 −2 1 .…”