Identity #3 (Noble identity) An M-fold down-sampler followed by a linear time invariant filter H(z^M) is equivalent to a linear time invariant filter H(z ) followed by an M-fold down-sampler.

Note that the second identity is a special case of this identity with H(z) = z^-1 and H(z^M ) = z^-M .

For the system consisting of the filter followed by the down-sampler we have

For the system consisting of the down sampler followed by the filter we have

Equations (A) and (B) are identical. Thus the two systems are equivalent.

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