Where in “Discrete Time Signal Processing” (Oppenheim et al.) can I find the scale-change theorem? [closed]

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

closed as unclear what you're asking by lennon310, Peter K.♦Nov 1 '17 at 1:56

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• Sorry but what's scale-change theorem? Such a name does not appear (that I remember so) in that book. – Fat32 Oct 31 '17 at 11:19

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.