How do multirate systems introduce 'efficiency' in computation?

Question

I've started reading more into multirate systems recently and am somewhat confused. The concepts introduced seem to make sense, that is, how the decimation and interpolation can change certain characteristics of signals (e.g. aliasing and imaging in the frequency domain, etc).

However, I keep hearing the term 'computational efficiency' when subject of multirate systems is introduced. What confuses me is how exactly do we gain this efficiency? What are some applications where implementing multirate introduces efficiency? From a quick glance, the need to downsample and upsample requires more computations in the form of anti-aliasing and anti-imaging. Does the ability to operate at lower sampling rates always counteract this added cost?

Say for example I have a signal that was sampled at 44.1kHz. Let's also say that signal's spectral content is primarily based from 20-2000Hz and there is a disturbance noise at 3000Hz that we would like to remove. Now a single rate system would essentially just apply a digital filter that is capable of removing that 3000Hz disturbance. Let's assume we are using an FIR filter. This filter would have to operate at a 44.1kHz update rate which is pretty fast. In order to create a filter that have an appropriate cutoff frequency of say 2500Hz, the filter length would also have to be pretty long due to the fast update rate.

An alternative method would be to downsample the signal to get to rate where the frequency spectrum ranges from 0 - 3000 Hz (instead of 0 - 22.05kHz). This frequency ranges seems more appropriate for the operations we're trying to do on our signal. If we were to slow down to that rate, the FIR filter length would be smaller(could I get a confirmation on this statement?) and the filter would operate at slower rate. We would essentially downsample, modify the signal, and then upsample.

However, in order to not pollute our lower sample rate signal, it's necessary to introduce anti-aliasing and anti-imaging for the downsample and upsample operations respectively. I realize there are efficient methods for doing integrating the filters with the downsampler/upsampler (i.e. polyphase realization), but how do I know whether these added computations are worth the cost?

TLDR: How does a multirate system introduce computation efficiencies when it's necessary to add more computations when upsampling and downsampling?

Answer 1

I will try to explain from my understanding.

For interpolation, the whole process can be viewed as two parts: up-samping with padding zeros, and the low-pass filtering. The zero padding will increase the length and thus the computational costs. For the low-pass filtering, there are lots of zeros, thus these multiplications could be omitted.

For decimation, the first part is anti-aliasing low-pass filtering, and the 2nd is the down-sampling. Since only part of the low-pass filtering output is needed, it is apparent that the filtering for other points can be omitted.