Generalization As mentioned earlier, in the general case for an arbitrary M (≤ N) we have

This overall operation is known as polyphase filtering.

Type 2 polyphase decomposition Given

set m = M-1-k: as m goes from 0 to M-1, k goes from M-1 to 0. Thus

This gives us the type 2 polyphase decomposition

Type 3 polyphase decomposition Given

set m = -k: as m goes from 0 to M-1, k goes from 0 to -(M-1) to 0. Thus

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