Cramér–Rao Lower Bound of adaptive filtering algorithms for acoustic echo cancellation

Abstract

Numerous adaptive filtering algorithms have been proposed for acoustic echo cancellation. However, whether the performance of the algorithms approaches the optimal performance or if there has been intentionally overstated remains challenging to evaluate. Fortunately, the Cramér–Rao Lower Bound (CRLB) provides a theoretical minimum variance for any unbiased estimator under given observational data and statistical models. This paper derives the CRLB of adaptive filtering algorithms for acoustic echo cancellation (AEC), in which the generalized Gaussian distribution (GGD) is utilized to model the Gaussian/non-Gaussian background noises. To accelerate the CRLB calculation process, the recursive resolution of the CRLB is presented by using the matrix inversion lemma, and the computational complexity is also analyzed. The derivation results indicate that CRLB for AEC model depends on the acoustic input (i.e., speaker’s voice) and the statistical properties of GGD noise but is unaffected by the channel sparsity. The CRLB derived in this paper can serve as a benchmark to evaluate whether the performance of the adaptive filtering algorithms is optimal and to exclude some adaptive filtering algorithms that deliberately exaggerate their performance.