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I am rather new to the polyphase structure and thought I had understood, but apparently I havent.

My question is I guess very simple: what is the commutator at the input of a polyphase decimator? Is it a memory-less switch or doe it have memory at each phase output of the commutator.

More explicitly: assuming I have a polyphase decomposition of 2 phases. So that I have the P0(z) and P1(z). Assume we are at time t=0 so that the commutator is at position 0 and at this time the input of P0(z) is some value x[0]. Between t=0 and t=1, P0(z) should be computed. at time t=1 the commutator switched and P1(z) (x[1] as input) should compute between t=1 and t=2. So here is where I am not understanding. Between t=1 and t=2 what happens to the INPUT to P0(z)? Is it 0 (as in the commutator is a simple switch without memory) or is the the previous value seen at t=0 (i.e. x[0]).

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I think you are referring to the symbol shown at the left of the structure: the arm that moves between different phases? See slide 4 of this PDF.

The point the Polyphase representation is trying to get across is that we give different samples of the input to different Phases of the Polyphase filter. Polyphase Filtering is a Digital Signal Processing approach, so your input would only be defined at discrete locations. Between the samples, we do not know where the signal was (unless it is bandlimited...).

So you can think of it as a memoryless switch.

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