Subsampling to estimate delay with application to echo cancelling

A much-studied problem is the estimation of the delay Δ between a known signal x(k) and a delayed version x(k - Δ) observed in additive white noise ε(k). In this paper, we analyze for delay a generalized version of this problem in which the observed signal is not simply x(k - Δ) + ε(k), but rather h(k) * x(k) + v(k) where * denotes convolution and h(k) is an unknown system impulse response. We suggest a computationally attractive structure for obtaining a system delay estimate and discuss the application of this structure to echo cancelling.