Recent insights in the Bayesian and deterministic CRB for blind SIMO channel estimation

Abstract

The performance of channel estimation is often assessed by deriving the proper Cramér-Rao Bound (CRB). Depending on how to treat the symbols and the channel, we have previously derived different versions of CRB. Specifically, we have dealt with the cases where the symbols and/or the channel are assumed to be either deterministic unknowns or random. Moreover, the symbols have been considered to be either jointly estimated with the channel or marginalized. All in all, we have derived six different versions of Bayesian and deterministic CRBs. However, we have shown that many of these CRBs are too optimistic in the sense that they are not strict enough to be attained by any deterministic or Bayesian estimator. In this paper we propose modified versions of those loose CRBs in the context of SIMO FIR system that are valid at least in the moderate and high SNR regimes. The analytical formulas for the lower bounds introduced are validated by some Monte-Carlo simulations.

Cramer-Rao bounds in the estimation of time of arrival in fading channels

“…Remembering that the correlation matrix R φ described in (27) is related to (26), the expected spatial signature in (28) results. (25) and from the fact that temporal variation due to Doppler may again be modeled as a first-order AR Markov process (AS 7), as the temporal vector, α t (α), is a function of α, and it takes the form…”

“…The angular spread corresponds to the standard deviation of the direction of arrivals from multipath components at the receiver when a normalized version of the PAS is used as the weighting function. Some works report that Laplacian distribution can provide a good match to this angular distribution [15,53,61]; but when it is used in (27) instead of a Gaussian distribution, meaningless variations are achieved.…”

“…However, it has been reported that in certain cases, results predicted by CRBs or BCRBs are too optimistic and some modifications to the classical CRBs have been proposed lately. This requires the postponement of the application of an expectation operator required for Fisher information matrix (FIM) computation, in a way that matrix inversion is performed first and then as a second step, an expectation operator is applied to compute the modified CRB (MCRB) [27]. These latter variations of the CRBs are out of the scope of this paper.…”

“…where dependency with the source mean bearing and its angular spread meets through the correlation vector ρ, as will be explained later in (27) T(α) is modeled as a first-order AR Markov process,…”