225 Fourier Transform
Used to transform a signal from its time domain signal to its frequency domain signal.
FFT Time-Frequency View
\[F(x) = \dfrac{1}{\sqrt{2 \pi}} \displaystyle \int_{-\infty}^{\infty} f(u) e^{ixu} \: du\]
where
*\(e^{ix} = \cos(x) + i\sin(x)\)
Derivation [3, pp.32]…
225.1 Elementary Example
225.1.1 Simple
The Fourier transform turns a time-domain sample into frequency content. On three sample values, list the discrete inputs.
\[ A = \{ 0,\ 1,\ 2 \} \]
\[ f(0) = 1,\quad f(1) = 0,\quad f(2) = -1 \]
where
- \(f\) is the time-domain function sampled on \(A\).
- the transform maps \(f\) to a frequency-domain function \(F\).
225.1.2 General
The continuous transform integrates \(f\) against a complex exponential.
\[ F(x) = \dfrac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(u)\, e^{i x u}\, du \]
\[ e^{i x} = \cos x + i \sin x \]
where
- \(F\) is the Fourier transform of \(f\).
- \(u\) is the integration variable in the time domain.
- \(x\) is the frequency variable.