An Aid to the Improvement of Filter Approximation

A simple visual aid to the correction of filter responses is presented. The type of correction considered is the addition of one pole and one zero in the second quadrant of the complex plane, and at their respective conjugate locations. The incremental phase and logarithmic loss produced by such a pole-zero pair, called for brevity, a pair-shaped correction, is easy to evaluate at any frequency by means of the overlay contours described and illustrated. The equal phase contours and the equal logarithmic loss contours are families of mutually orthogonal circles. Variation of the position and orientation of these contours over the complex frequency plane, suitably scaled, facilitate rapid attainment of a nearly optimal pair-shaped correction.