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Fourier Transform

$F (ν) = \int_{- \infty}^{\infty} f (t) \exp (- j 2 π ν t) d t$	Fourier Transform
$f (t) = \int_{- \infty}^{\infty} F (ν) \exp (j 2 π ν t) d ν$	Inverse Fourier Transform
$f (t) = \int_{- \infty}^{\infty} f_{1} (τ) f_{2} (t - τ) d τ$	Convolution
$f (t) = \int_{- \infty}^{\infty} f_{1}^{*} (τ) f_{2} (t + τ) d τ$	Correlation

Properties:

Original Function	Fourier Transform
$a f (t) + b g (t)$	$a F (ν) + b G (ν)$	Linearity
$f (t / τ)$	$\| τ \| F (τ ν)$	Scaling
$f (- t)$	$F (- ν)$
$f (t - τ)$	$e^{- j 2 π ν τ} F (ν)$	Time Translation
$e^{j 2 π ν_{0} τ} f (t)$	$F (ν - ν_{0})$	Frequency Translation
$f (t) = \int_{- \infty}^{\infty} f_{1} (τ) f_{2} (t - τ) d τ$	$F (ν) = F_{1} (ν) F_{2} (ν)$	Convolution
$f (t) = \int_{- \infty}^{\infty} f_{1}^{*} (τ) f_{2} (t + τ) d τ$	$F (ν) = F_{1}^{*} (ν) F_{2} (ν)$	Correlation

$f^{} (t) = f (t) \Rightarrow F (- ν) = F^{} (ν)$	Symmetry
$\int_{- \infty}^{\infty} {\| f (t) \|}^{2} d t = \int_{- \infty}^{\infty} {\| F (ν) \|}^{2} d ν$	Parseval's Theorem

Transforms:

Function

f(t)

F(ν)

Uniform

1

δ(ν)

fourier 1

Impulse

function 2

δ(t)

1

Rectangular

function 3

rect(t)

sinc(ν)

Exponential

exp(-|t|)

2

1+(2πν)²

fourier 4

Gaussian

exp(-πt²)

exp(-πt^ν)

Chirp

exp(jπt²)

e^jπ/4exp(-jπν²)

Sum of M=2S+1 impulses

S
Σ	δ(t-n)
n=-S

sin(Mπν)

sin(πν)

Infinite sum of impulses

∞
Σ	δ(t-n)
n=-∞

∞
Σ	δ(ν-n)
n=-∞