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Adaptive Impulse Response Estimation Algorithms

We often encounter a situation that the impulse of a black box system has to be estimated. In echo and feedback cancellation applications, we first estimate the impulse responses of the echo or feedback paths before we apply cancellation techniques. For dereverberation, the reverberation impulse response can be first estimated before the reverberation is equalized.

System impulse response estimation problem can be described by the simple diagram below.

Diagram showing acoustic path and system impulse response estimation

The acoustic path appear to be a black box. But we can apply an input signal x(n) and we observe an output signal y(n). For this generic model, we will describe a unified solution which encompasses the most commonly used adaptive estimation algorithms.

If we group the input-output pairs into the following sequence, with block length K,

y(n) = x(n) * h(n) + noise(n)

y(n – 1) = x(n – 1) * h(n) + noise(n – 1)

y(n – K – 1) = x(n – K – 1) * h(n) + noise(n – K – 1)

With the assumption on the ambient noise as uncorrelated white Gaussian noises, we obtain the following mathematically optimal estimation from Penrose inversion.

where X(n) is the blocked input signal matrix and y(n) is the blocked output vector. With this blocked expression, we can then roughly classify the solutions into the following three practical categories.

  • K = 1, Normalized Least Mean Squared (NLMS)
  • K = +∞, Recursive Least Squred (RLS) without a time decay
  • K  between 1 and +∞, Affine Projection Algorithm (APA)

According to each individual use case, VOCAL Technologies chooses and optimizes the algorithms for performance and resource utilization.

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