# Coherent averaging of polyphase components of a signal

I want to subsample a multi-band signal to introduce controlled aliasing. My frequency components are uniformly distributed across the spectrum (comb structure) and occupy a relatively narrow bandwidth each. The spectrum can be said to be sparse.

I achieve decimation through picking every $$n$$-th sample from the original signal. This of course does not take advantage of the remaining samples, which are simply rejected. By adding a phase (sample) shift in my decimation routine, I can get $$n$$ under-sampled signals in total, which are known as polyphase components. I can also incoherently average them (periodogram averaging) in the spectral domain, since the spectrum is of my interest rather then the time-domain representation.

Is it possible to coherently (in phase) average such undersampled polyphase components of my signal? What phase shift should be applied after down-sampling with a factor $$n$$? In other words, can we simultaneously undersample (decimate) and oversample (average) to take advantage of the remaining samples for quantization noise suppression?

A typical scenario would be sampling a 1 GHz wide comb signal with a spacing of 10 MHz between the teeth, which we can decimate 10 times to have a spacing of 1 MHz and a bandwidth of 100 MHz. If we sample the signal with an 8-bit analog-to-digital converter, can we use the remaining 9 polyphase components to enhance the vertical resolution of the signal after subsampling?

Thank you, Lukasz S.