A Bayesian Performance Bound for Time-Delay of Arrival based Acoustic Source Tracking in a Reverberant Environment

Abstract—Acoustic source tracking in a room environment based on a number of distributed microphone pairs has been widely studied in the past. Based on the received microphone pair signals, the time-delay of arrival (TDOA) measurement is easily accessible. Bayesian tracking approaches such as extended Kalman filter (EKF) and particle filtering (PF) are subsequently applied to estimate the source position. In this paper, the Bayesian performance bound, namely posterior Cramér-Rao bound (PCRB) is derived for such a tracking scheme. Since the position estimation is indirectly related to the received signal, a two-stage approach is developed to formulate the Fisher information matrix (FIM). First, the Cramér-Rao bound (CRB) of the TDOA measurement in the noisy and reverberant environment is calculated. The CRB is then regarded as the variance of the TDOAs in the measurement function to obtain the PCRB. Also, two different TDOA measurement models are considered: 1) single TDOA corresponding to the largest peak of the generalized cross-correlation (GCC) function; and 2) multiple TDOAs from several peaks in GCC function. The later measurement model implies a higher probability of detection and heavier false alarms. The PCRB for both measurement models are derived. Simulations under different noisy and reverberant environments are organized to validate the proposed PCRB. I