Cramér-Rao bounds for rate-constrained distributed time-delay estimation

Abstract

The discussion in this paper is limited to the case of two microphones. The following signal model holds for the TD estimation problem:(2) (1) XI(t) = s(t) + nl(t) X2(t) = s(t -T) + n2(t),R(>') = 4~i: max ( 0, log2 ipx~(w)) dMJ, 1r -00The distortion here is the mean-squared error (MSE) between X2 (t) and its reconstruction. Equation (2) provides the relation where Xl (t) and X2 (t) are the received microphone signals, s(t) is the source signal, T is the time delay, and nl (t) and n2(t) are the noise processes at the two sensors. The signals s(t), nl(t), and n2(t) are assumed to be zero-mean stationary Gaussian random processes with power spectral densities~s(w),~nl (w), and e., (w), respectively.It is assumed that the first node is the fusion center and that the second node transmits its signal X2 (t) at a rate R to the first node, where the TD estimate is obtained from the two signals. When the signal X2 (t) is quantized at a rate R, the relation between R and the resulting quantization error can be expressed by the following parametric rate-distortion relation [11]:the communication bit-rate and the resulting accuracy of TD estimation by analyzing the Cramer-Rao lower bound (CRLB) on the variance of the estimation error as a function of the bitrate.For a given observation duration, the CRLB is tight, i.e., the bound is attainable, only if the signal-to-noise ratio (SNR) exceeds a certain threshold [4], [5]. In the rate-constrained case, it is intuitive that the CRLB is attained only beyond certain bit-rates. For a given observation time and SNR, an expression that establishes a condition on the minimum bit-rate required to ensure attainability of the CRLB is also derived in this paper.The remainder of the paper is organized as follows. Section II introduces the signal model employed, and the relevant ratedistortion relations used in the analysis. The CRLB in the rate-constrained case is obtained in Section III. The minimum bit-rate that renders the CRLB tight is derived in Section IV. Conclusions are summarized in Section V.Abstract-This paper investigates time-delay estimation for acoustic source localization in a distributed microphone array. The microphones are assumed to be part of a wireless sensor network, with a constraint on the number of bits that may be exchanged between the sensors. Consequently, at the fusion center, time-delay estimation needs to be performed using quantized signals. In this paper, the relation between the communication bitrate and the Cramer-Rae lower bound (CRLB) on the variance of the time-delay estimation error is explored. The minimum bit-rate required to ensure that the CRLB is attained is also derived.