Where in "Discrete Time Signal Processing" (Oppenheim et al.) can I find the scale-change theorem?

Question

I'm sorry to bother you with this question, but somehow I fail to find the corresponding passage?

Answer 1

The scale-change theorem could be better known as the scaling property or scaling formula, a property of the continuous Fourier transform. It is one avatar of the "invariance" properties ("scale invariance") of some kernel transformations.

It tells you about some proto-version of the Gabor-Weyl-Heisenberg limit: if you compress a signal along the time axis, it will expand or dilate in frequency, and vice-versa. This is the main duality aspect of the time and frequency variables.

More precisely, if $s(t)\mapsto S(f)$, then $\forall \alpha \neq 0$, and $ S(f)$ is the Fourier transform, then for the compressed signal:

$$ s(\alpha t) \mapsto \frac{1}{|\alpha|} S \left( \frac{f}{|\alpha|}\right) $$

For discrete signals $s[n]$, not all real $\alpha$s dilations make sense in general, because $\alpha n$ may not be a valid integer index.

Discrete equivalents exist for the time reversal ($\alpha = -1$) and integers or inverses of integers ($\alpha = k$ or $\alpha = 1/k$, $k\in \mathbb{Z}$), and fractions in general, somehow corresponding to downsampling and upsampling, addressed in Chapter 4.6, Changing the sampling rate using discrete-time processing, page 167 sq., with a related lecture at Rice.

A recent reference: Generalized Rational Sampling Rate Conversion Polyphase FIR Filter, IEEE Signal Processing Letters, Nov. 2017. For a more historical one, On upsampling, downsampling, and rational sampling rate filter banks, R. Gopinath, 1994.