Abstract-In problems of enhancing a desired signal in the presence of noise, multiple sensor measurements will typically have components from both the signal and the noise sources. When the systems that couple the signal and the noise to the sensors are unknown, the problem becomes one of joint signal estimation and system identification. In this paper, we specifically consider the two-sensor signal enhancement problem in which the desired signal is modeled as a Gaussian autoregressive (AR) process, the noise is modeled as a white Gaussian process, and the coupling systems are modeled as linear time-invariant finite impulse response (FIR) filters. Our primary approach consists of modeling the observed signals as outputs of a stochastic dynamic linear system, and we apply the Estimate-Maximize (EM) algorithm for jointly estimating the desired signal, the coupling systems, and the unknown signal and noise spectral parameters. The resulting algorithm can be viewed as the time-domain version of our previously suggested frequency-domain approach [4] , where instead of the noncausal frequency-domain Wiener filter, we use the Kalman smoother. This time-domain approach leads naturally to a SequentWadaptive algorithm by replacing the Kalman smoother with the Kalman filter, and in place of successive iterations on each data block, the algorithm proceeds sequentially through the data with exponential weighting applied to allow adaption to nonstationary changes in the structure of the data. A computationally efficient implementation of the algorithm is developed by exploiting the structure of the Kalman filtering equations. An expression for the log-likelihood gradient based on the Kalman smootherhlter output is also developed and used to incorporate efficient gradient-based algorithms in the estimation process. are unknown, one must deal with the more difficult problem of joint signal estimation and system identification. An approach to the two-sensor signal enhancement problem is presented in [4] . In that approach, the desired (speech) signal is modeled as an autoregressive (AR) Gaussian process, the noise is modeled as a white Gaussian process, and the coupling systems are modeled as linear time-invariant finite impulse response (FIR) filters. The problem is formulated as a maximum likelihood (ML) estimation problem, and the iterative estimate-maximize (EM) algorithm is applied for its solution. The resulting algorithm consists of iteratively applying the Wiener filter to the two-sensor data to estimate the signal and the noise and using these estimates to identify the coupling systems and the unknown spectral parameters of the signal and the noise. When applied to speech signals in room acoustic environment, this algorithm has shown encouraging results that improve upon Widrow's LMS noise cancellation method [21]. In order to deal with the nonstationarity of the signal, the noise, and the coupling systems, it is suggested in [4] that the algorithm be applied on consecutive time frames using a sliding window. This approach involves two contradicting requirements: The window should be short enough so that the algorithm will respond to nonstationary changes in the signal and noise statistics. However, the window should be long in order to improve the statistical stability of the resulting signal and parameter estimates and in order to obtain a computationally tractable algorithm in which noncausal frequency-domain Wiener filtering can be applied. In this paper, we present a time-domain approach to the twosensor signal enhancement problem based on the development in [20]. In this approach, we model the observed signals as outputs of an unknown stochastic dynamic linear system and apply the EM algorithm for jointly estimating the signal, the noise, the coupling systems, and the unknown signal and noise spectral parameters. The proposed algorithm is similar in structure to that in [4] . However, in place of the noncausal Wiener filter, we employ a Kalman smoother. In this way, many of the computational and conceptual difficulties associated with the prior frequency-domain approach in [4] are avoided. Furthermore, the time-domain formulation leads naturally to a sequential/adaptive algorithm by replacing the Kalman smoother with the Kalman filter, and in place of successive iterations on each data block, the algorithm proceeds sequentially through the data with exponential weighting applied to allow adaptation to nonstationary changes in the 1053-587X/94$04.00 0 1994 IEEE
In REF , authors consider two-sensor signal enhancement problem in a noisy environment.
Iterative and sequential algorithms for multisensor signal enhancement
IEEE Trans. Signal Process.
IEEE Trans. Signal Process.
Computer Science Mathematics