Stability considerations on filters realized with delta modulation

The designer of recursive filters realized that in digital or in hybrid form using delta modulation may encounter certain stability problems. In the present paper, these filters are shown to have two kinds of limit cycles which are called "slope overload cycles" and "granular cycles," referring to their origin. Filters showing slope overload cycles can normally not be used because their amplitude is mostly prohibitively large. Amplitudes covering one half of the signal range are not infrequent. Therefore, it is very important to find methods to avoid these oscillations. At first, a model is presented which permits the study of slope overload cycles, while neglecting the less disturbing granular ones. Then, applying a theorem of Popov and the describing-function concept, two stability conditions for the filter coefficients are obtained. Finally, using the stability criteria found, a method is outlined which permits the calculation of filters free of slope overload cycles. The proposed procedure is based on numerical optimization under constraints.