4.7 Noble identities

The Noble identities (illustrated in and ) describe when it is possible to reverse the order of upsampling/downsampling and filtering.We prove the Noble identities showing the equivalence of each pair of block diagrams.

The Noble identity for interpolation can be depicted as in :

For the left side of the diagram, we have $$Y(z)=H(z^L)V_1(z)$$ where $V_1(z)=X(z^L)$ $$Y(z)=H(z^L)X(z^L)$$ while for the right side, $$Y(z)=V_2(z^L)$$ where $V_2(z)=H(z)X(z)$ $$Y(z)=H(z^L)X(z^L)$$ Thus we have established the Noble identity for interpolation.

The Noble identity for decimation can be depicted as in :

For the left side of the preceding diagram, we have $$V_1(z)=\frac{1}{M}\sum_{k=0}^{M-1} X(e^{-i\frac{2\pi }{M}k}z^{\left(\frac{1}{M}\right)})$$