Frequency estimation in the presence of Doppler spread: performance analysis

We are concerned with the estimation of the frequency of a complex sinusoid that has been corrupted by complex-valued multiplicative and additive noise. This problem is important in many applications including array processing in the case of spatially distributed sources and synchronization in the context of time-selective channels. The multiplicative noise smears the spectral line due to the sinusoid. This smearing, which is often called Doppler spreading, may significantly degrade the estimation accuracy. The goal of this paper is to analytically assess this degradation. The finite-sample Cramer-Rao bounds (CRBs) are derived, and closed-form expressions are given for the large-sample CRB. The latter gives insights into the effective coherent and noncoherent SNRs for frequency estimation. We then analyze the accuracy of frequency estimators that are based on the angles of the sample covariances. Simulations results are presented to illustrate the theoretical results.

I. Introduction

Frequency estimation in the presence of multiplicative noise is important for synchronization of communications signals in the context of time-selective fading channels. The fading-induced multiplicative noise is often referred to as short-scale fading [14]. In mobile communications, this fading is due to local scatterers around the mobile unit [15]. The bandwidth of the fading process (the Doppler spread) increases with the speed of the mobile. Multiplicative noise is also encountered in coherent radar processing when the dynamics of the changes in the radar cross section are significant. The constant amplitude model is inadequate for pulse-to-pulse fluctuating targets [20]; these fluctuations contribute to a spectral spread of the received signal. This implies that the received pulses in a batch have neither the same amplitude nor the same initial phase. However, the phases of pulses are not uncorrelated, as in noncoherent radar processing. They are (highly) correlated, and the strength of this correlation is inversely proportional to the rate of the fluctuations. It is this correlation spread, which is sometimes identified as the scatterer “dwell” time, that makes the phase information useful. Multiplicative noise models are also appropriate in array processing when the source signal is spatially distributed [19], as well as for backscattered acoustic signals [1] and SAR imagery systems.