Properties of Fourier Transform/Scaling
< Properties of Fourier Transform(Redirected from Scaling Property of Fourier Transform)
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Theorem
Let $\map f x$ be a Lebesgue integrable function.
Let $a$ be a non-zero real number.
Let $\map h x$ be a Lebesgue integrable function such that:
- $\map h x = \map f {a x}$
Then:
- $\map {\hat h} s = \dfrac 1 {\size a} \map {\hat f} {\dfrac s a}$
where $\map {\hat h} s$ and $\map {\hat f} s$ are the Fourier transforms of $\map h x$ and $\map f x$ respectively.
Proof
\(\ds \map {\hat h} s\) | \(=\) | \(\ds \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_{-\infty}^\infty e^{-2 \pi i x s} \map f {a x} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {\size a} \int_{-\infty}^\infty e^{-2 \pi i \paren {a x} \paren {s / a} } \map f {a x} \map \rd {a x}\) | Power of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {\size a} \map {\hat f} {\dfrac s a}\) |
$\blacksquare$
Sources
- Weisstein, Eric W. "Fourier Transform." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FourierTransform.html