Fourier Transform

Definition

An integral transform that converts a function into form that describes the frequencies present in the original function

Continuous Time Fourier Transform

$F(u)={\int }_{-\infty }^{\infty }f(t)e^{-i2\pi ut}dt={\int }_{-\infty }^{\infty }f(t)\cos (2\pi ut)+if(x)\sin (2\pi ut)dt$

Magnitude

$|F(u)|=\sqrt{|F_R(u){|}^2+|F_I(u){|}^2}$

Phase

$\measuredangle F(u)=atan2(F_I(u),F_R(u))$

Continuous Time Inverse Fourier Transform

$f(t)={\int }_{-\infty }^{\infty }F(u)e^{i2\pi ut}du={\int }_{-\infty }^{\infty }F(u)\cos (2\pi ut)+iF(u)\sin (2\pi ut)du$

Discrete Time Fourier Transform

$X(\omega )=\sum _{-\infty }^{\infty }x[n]e^{-i\omega n}$ where $\omega$ is a frequency variable

Discrete Time Inverse Fourier Transform

$x[n]=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }X(\omega )e^{i\omega n}d\omega$ where $\omega$ is a frequency variable

Discrete Fourier Transform

$X[k]=\sum _{n=0}^{N-1}x[n]e^{-i\frac{2\pi }{N}kn}$ where $k$ is the frequency index, $N$ is the length of the sequence.

DFT can be seen as a sampled version of DTFT $X[k]=X(k)$

Discrete Inverse Fourier Transform

$x[n]=\frac{1}{N}\sum _{n=0}^{N-1}X[k]e^{i\frac{2\pi }{N}kn}$ where $k$ is the frequency index, $N$ is the length of the sequence.

Facts

With a magnitude and phase, the original time domain function can be reconstructed

$x(t)\ast h(t)\iff X(\omega )H(\omega )$

When using CTFT, convolution in the time domain becomes multiplication in the frequency domain

$x[n]⊛h[n]\iff X[k]H[k]$

When using DFT, cyclic convolution in the time domain becomes multiplication in the frequency domain

The frequency axis of the DFT result is $k\frac{f_s}{N}$ where $k=0,1,\dots ,N-1$, $f_s$ is sampling frequency, and $N$ is the number of samples