Fourier Scaling Property
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The scaling property (also called the similarity theorem) describes how stretching or compressing a function in the time/space domain affects its Fourier transform in the frequency domain.
Statement
If $\mathcal{F}\{g(x)\} = G(f)$, then:
$$ \mathcal{F}\{g(ax)\} = \frac{1}{|a|} G\left(\frac{f}{a}\right) $$where $a \neq 0$ is a real constant.
Interpretation
| Spatial Domain | Frequency Domain |
|---|---|
| Compress by factor $a > 1$ | Expand by factor $a$, reduce amplitude by $1/a$ |
| Expand by factor $a < 1$ | Compress by factor $1/a$, increase amplitude by $1/ |
Key insight: Narrow functions have wide spectra, and wide functions have narrow spectra. You cannot be simultaneously compact in both domains.
Proof
Starting from the definition:
$$ \mathcal{F}\{g(ax)\} = \int_{-\infty}^{\infty} g(ax) e^{-j2\pi f x} dx $$Substitute $u = ax$, so $du = a \, dx$:
$$ = \int_{-\infty}^{\infty} g(u) e^{-j2\pi f (u/a)} \frac{du}{|a|} $$$$ = \frac{1}{|a|} \int_{-\infty}^{\infty} g(u) e^{-j2\pi (f/a) u} du = \frac{1}{|a|} G\left(\frac{f}{a}\right) $$The absolute value handles negative $a$ (which reverses the integration limits).
2D Version
For 2D functions with independent scaling:
$$ \mathcal{F}\{g(ax, by)\} = \frac{1}{|ab|} G\left(\frac{f_x}{a}, \frac{f_y}{b}\right) $$For uniform scaling in 2D:
$$ \mathcal{F}\{g(ar)\} = \frac{1}{a^2} G\left(\frac{\rho}{a}\right) $$where $r = \sqrt{x^2 + y^2}$ and $\rho = \sqrt{f_x^2 + f_y^2}$.
Examples
Rectangle Function
$$ \text{rect}(x) \leftrightarrow \text{sinc}(f) $$$$ \text{rect}(x/W) \leftrightarrow W \, \text{sinc}(Wf) $$Wider rectangle → narrower sinc (but taller central lobe).
Gaussian
$$ e^{-\pi x^2} \leftrightarrow e^{-\pi f^2} $$$$ e^{-\pi (x/\sigma)^2} \leftrightarrow \sigma \, e^{-\pi (\sigma f)^2} $$Wider Gaussian in space → narrower Gaussian in frequency.
Physical Applications
| Application | Implication |
|---|---|
| Diffraction | Smaller aperture → wider diffraction pattern |
| Pulses | Shorter pulse → broader spectrum |
| Sampling | Finer sampling → wider representable bandwidth |
| Uncertainty | Product of widths is constant |
Connection to Uncertainty
The scaling property underlies the uncertainty principle: if you make a function more localized (smaller effective width), its spectrum must become less localized (larger effective bandwidth), and vice versa.