Properties of Fourier Transform/Modulation

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Theorem

Let $\map f x$ be a Lebesgue integrable function.


Let $s_0$ be a real number.

Let $\map h x$ be a Lebesgue integrable function such that:

$\map h x = e^{2 \pi i x s_0} \map f x$


Then:

$\map {\hat h} s = \map {\hat f} {s - s_0}$

where $\map {\hat h} s$ and $\map {\hat f} s$ are the Fourier transforms of $\map h x$ and $\map f x$ respectively.


Proof

\(\ds \map {\hat h} s\) \(=\) \(\ds \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x\)
\(\ds \) \(=\) \(\ds \int_{-\infty}^\infty e^{-2 \pi i x s} e^{2 \pi i x s_0} \map f x \rd x\)
\(\ds \) \(=\) \(\ds \int_{-\infty}^\infty e^{-2 \pi i x \paren {s - s_0} } \map f x \rd x\) Product of Powers
\(\ds \) \(=\) \(\ds \map {\hat f} {s - s_0}\)

$\blacksquare$


Sources