On the Laguerre series expansion and transform.

Abstract

The Laguerre transform is defined and motivated based on the Laguerre series expansion and Laguerre filter. The Laguerre transform is represented by a unitary transformation matrix of infinite size and is controlled by the Laguerre parameter. When realised as a finite size matrix, the case where the Laguerre parameter approaches zero results in the identity matrix. Otherwise the transform is less and less unitary and the inverse transform is not complete. However, we can choose the size of the matrix such that orthogonality is preserved and use the matrix transform for signal compression. We show that modeling an acoustic duct impulse response using the Laguerre transform has less mean-squared error than the truncated DCT transform for high compression ratios. We also show results using least-square solutions and adaptive filtering where the Laguerre filter gave improved results over classical methods.