Additional correction to the impulse invariance method for the design of IIR digital filters

Abstract

Almost four decades after the impulse invariance (i.i.) method has been in vogue, a correct way of its application has been emphasized independently, but at the same time, by Jackson and Mecklenbrauker. When there is a discontinuity at t=0 in the analog impulse response, the correct way requires that one should take half of its value. In spite of this correction, due to under-sampling there persists aliasing error in the frequency domain. In this paper the effect of aliasing is viewed as a 3 dB frequency warping and a pre-warping step prior to its application is proposed to reduce it, thereby making the technique appropriate from the viewpoint of frequency domain also. Simulation examples comparing this approach with the traditional method reveal significant improvement in the frequency response. Further, according to Jackson and Mecklenbrauker and also as written in many a standard textbook available today, i.i. technique cannot be used when the degrees of the numerator and denominator polynomials of the parent analog transfer function are equal. An example under this category is the Pre-emphasis circuit or a simple high pass filter. We shall show how the technique can still be used even for a Pre-emphasis circuit. Antoniou proposed (an intelligently) modified impulse invariance technique for certain types of filters by which, aliasing error is nearly eliminated. These cases are neither admitted by the traditional i.i. method nor by that of Jackson and Mecklenbrauker at all. Antoniou's modified method results in a digital filter of order almost twice as great with an additional task of stabilization. The method in this paper retains the stability and order but gives satisfactory response. In the standard textbooks, the i.i. technique is explained with the help of a first-order low-pass filter and for higher orders, partial fraction expansion is used. For the situation of repeated poles, application of i.i. method calls for extreme care and special handling. This paper alternatively employs the use of state variables to make i.i. technique more elegant.