Optimization of the second order autoregressive model AR(2) for Rayleigh-Jakes flat fading channel estimation with Kalman filter

Abstract

Abstract-This paper deals with the estimation of the flat fading Rayleigh channel with Jakes' Doppler spectrum (model due to R.H. Clarke in 1968) and slow fading variations. A common method in literature consists in approximating the variations of the channel using an auto-regressive model of order p (AR(p)), whose parameters are adjusted according to the "correlation matching" (CM) criterion and then estimated by a Kalman filter (KF). Recent studies based on first order AR (1) showed that the performance is far from the Bayesian CramerRao bound for slow to moderate channel variations, which is the case for many applications. The same studies on first order model have shown the importance of replacing the CM criterion with a MAV criterion (Minimization of Asymptotic Variance). Moreover, it has been shown in literature that increasing the order of the model by passing from AR(1) to AR(2) did not improve the performance when the CM criterion is considered. In order to obtain an improvement in performance, it is essential to consider the MAV criterion with second order autoregressive model AR(2), as shown in this article. By imposing a linear relation between one of the parameters and the Doppler frequency, we derive analytic formulas for suboptimal adjustment of the parameters of AR(2) as a function of the noise level and the Doppler frequency of the channel. Theoretical assumptions are validated via simulation.

Kalman Filter-Based Channel Estimation for Mobile-to-Mobile and Relay Networks

“…In this section, the approach of [12] that was restricted to the Jakes' Doppler spectrum corresponding to the fix-to-mobile channel is extended to a more general band-limited Doppler spectrum Γ α (f ), possibly including relays. This has been done by rewriting the results as a function of the fourth moment µ (4) .…”

“…from which the KF equations can be established as in [12] with the two-dimension Kalman gain vector K (k) = K 1(k) K 2(k) to produce the estimate vectorα (k|k) = α (k|k) ,α (k−1|k)…”

“…The CM criterion is simple to implement and adequate for fast fading channels. However, for a slow fading scenario (normalized Doppler frequency less than 10 −2 ), which is valid for usual vehicular communications, using the CM criterion gives poor results compared to the minimum asymptotic variance (MAV) criterion, as seen in [9]- [11] for AR with order p = 1, and in [12] for Fix-to-Mobile channel. The main contribution of this paper is to provide analytic expressions for the parameters of a KF based on secondorder AR model (AR(2)) tuned under MAV criterion, together with the corresponding steady-state MSE performance, for M-M and mobile relays channels.…”

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

“…Part of this work has been presented in [1], but without rigorous justifications. In [1], a sub-optimal adjustment of {a 1 , a 2 } was proposed by giving an analytical expression for a 1 in terms of a 2 , where a 2 is given by imposing a linear constraint obtained through experimentation, with respect to the Doppler frequency. In the following, the optimal tuning of r and ζ AR(2) is given in terms of the Doppler frequency and the SNR.…”

“…Note that part of the results of the present paper was presented in the conference paper [1]. However, [1] followed a sub-optimal approach that was slightly different: the adjustment of the parameter a 1 imposed a linear constraint with the Doppler frequency for the parameter a 2 deduced from grid search. Furthermore, no complete proof, interpretation, or extensive comparison was provided.…”