OVERFLOW OSCILLATIONS IN DIGITAL FILTERS

The cascade and parallel realizations of an arbitrary digital filter are both formed using second order sections as building blocks. This simple recursive filter is commonly implemented using 2's complement arithmetic for the addition operation. Overflow can then occur at the adder and the resulting nonlinearity causes self‐oscillations in the filter. The character of the resulting oscillations for the second order section are here analyzed in some detail. A simple necessary and sufficient condition on the feedback tap gains to insure stability, even with the presence of the nonlinearity, is given although for many desired designs this will be too restrictive. A second question studied is the effect of modifying the “arithmetic” in order to quench the oscillations. In particular it is proven that if the 2's complement adder is modified so that it “saturates” when overflow occurs, then no self‐oscillations will be present. © 1969 The Bell System Technical Journal