Frequency Domain Convolution of Rational Transfer Functions

Abstract

The convolution of two rational transfer functions is also rational, but a formula for the convolution has never been derived. This paper introduces a formula for the convolution of two rational functions in the frequency domain by two new methods. The first method involves a partial fraction expansion of the rational transfer functions where the problem gets reduced to the sum of the convolution of the partial fractions of the two functions, each of which can be solved by a new formula. Since the calculation of the roots of a high-order polynomial can be very time-consuming, we also demonstrate new methods for performing the convolution without calculating these roots or undergoing partial fraction expansion. The convolution of two rational Laplace transform denominators can be calculated using their resultant, while that of the two rational Z-transform transfer functions can be found using Newton’s identities. The numerator can be easily found by multiplying the numerator with the initial values of the power series of the result.