Scaling Property

Fourier Analysis · Axiom Academy

LESSON Scaling Property and Time-Frequency Tradeoff Understanding how compression in time leads to expansion in frequency The scaling property describes what happens to the Fourier transform when we compress or stretch a function in time. Where a is the scaling factor. If |a| > 1, the signal is compressed in time. If |a| < 1, the signal is stretched in time. 2. Intuition: The Time-Frequency Duality Think of this as a fundamental law of nature: you cannot compress a signal in time without expanding its frequency content, and vice versa. 3. Proof of the Scaling Property Let's derive why this property holds using the definition of the Fourier transform and a substitution of variables. Starting with the Fourier transform of f(at): Substitute u = at, so t = u/a and dt = du/a: Factor out the constant (1/|a|): 4. The Uncertainty Principle Connection The scaling property is the foundation of the Heisenberg-Gabor uncertainty principle : a signal cannot be arbitrarily localized in both time and frequency simultaneously. Where T is the time duration and is the frequency bandwidth. This product has a lower bound - you can never make both arbitrarily small.

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