Evaluation of a linear system impulse response and its Fourier transform†

Abstract

Impulse response of a linear time invariant system is partitioned by dividing the time axis into equal intervals of time. Then the impulse response is expressed as a sum of these partitioned portions. Each individual portion is approximated by a finite sum of orthogonalized sinusoids satisfying integral squared error criteria. Four different sets are given for this purpose. If the time reversed functions from these sets are applied to the system then the sampled values of the system response at the partitioning instants directly yield the system coefficients as required for the least integral squared error. Knowing these coefficients the best approximation to the impulse response can be constructed as illustrated by the examples considered. Sampled values of the Fourier transform of system impulse response are obtained as a by-product.