Upsampling and Downsampling

For the DFT, we have the Stretch Theorem (Repeat Theorem) which relates upsampling (``stretch'') to spectral copies (``images'') in the DFT context (length $N$ signals and spectra).

We also have the Downsampling Theorem (Aliasing Theorem) for DFTs which relates downsampling to aliasing for finite length signals and spectra.

We now look at these relationships in the DTFT case. Thus, the signal length $N$ is extended to infinity, and the spectrum becomes defined continuously over the unit circle in the $z$ plane.