Coefficient Accuracy and Digital Filter Response

The frequency response of a digital filter realized by a finite word-length machine deviates from that which would have been obtained with an infinite word-length machine. An "ideal" or "errorless" filter is defined as a realization of the required pulse transfer function by an infinite word-length machine. This paper shows that quantization of a digital filter's coefficients in an actual realization can be represented by a "stray" transfer function in parallel with the corresponding ideal filter. Also, by making certain statistical assumptions, the statistically expected mean-square difference between the real frequency responses of the actual and ideal filters can be readily evaluated by one short computer program for all widths of quantization. Furthermore, the same computations may be used to evaluate the rms value of output noise due to data quantization and multiplicative rounding errors. Experimental measurements verify the analysis in a practical case. The application of the results to the design of the digital filters is also considered.