Identity #6 (Noble identity) An L-fold up-sampler preceded by a linear time invariant filter H(z) is equivalent to a linear time invariant filter H(z^L) preceded by an L-fold up-sampler. Note that the fifth identity is a special case of this identity with H(z) = z^-1 and H(z^L) = z^-L.

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

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

Since equations (A) and (B) are identical the two systems are equivalent.

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