Analysis of rational sample rate conversion using image response combining

Using discrete-time response modelling, we analyse the complicated frequency domain input-output relationship of sample rate conversion with a rational, non-integer factor and address the ambiguity of the concept of frequency response in fractional multirate systems. The dependence of a system response on the phase of a real signal is shown. We develop a method to translate a magnitude response from a higher sample rate to a lower one to allow better comparison between filter properties and system specifications when rational sample rate conversion is used. The method is based on the idea of combining the responses of all (aliased) images of an input frequency into a scalar quantity. The method is hence referred to as image response combining (IRC) and the resulting response function as image-combined magnitude response. Two different IRC schemes are proposed. To overcome the sometimes very high computational complexity of frequency domain IRC, we derive a drastically faster time domain IRC algorithm referred to as sampled autocorrelation of impulse response (SACIR). The paper concentrates on decimation, but the analysis and proposed methods can be easily applied to interpolation, as well.