Fourier Series

Definition

Exponential Form

$x(t)$ $T > 0$ $\omega_0 = \frac{2\pi}{T}$ . Then,

\tilde{x} (t) = \sum_{n = - \infty}^{\infty} X_{n} e^{j n ω_{0} t}

where,

X_{n} = \frac{1}{T} \int_{- \frac{T}{2}}^{\frac{T}{2}} x (t) e^{- j n ω_{0} t} d t

$\widetilde{x}(t)$ Fourier Series $x(t)$ .

Special Case for Real Periodic Signals

$x(t)$ $T > 0$ $\omega_0 = \frac{2\pi}{T}$ . Then,

\tilde{x} (t) = X_{0} + 2 \sum_{n = 1}^{\infty} R e (X_{n} e^{j n ω_{0} t})

where,

X_{n} = \frac{1}{T} \int_{- \frac{T}{2}}^{\frac{T}{2}} x (t) e^{- j n ω_{0} t} d t

$\widetilde{x}(t)$ Fourier Series $x(t)$ .

Trigonometric Form

$x(t)$ $T > 0$ $\omega_0 = \frac{2\pi}{T}$ . Then,

\begin{matrix} \tilde{x} (t) = \frac{1}{2} a_{0} + \sum_{n = 1}^{\infty} (a_{n} c o s (n ω_{0} t) + b_{n} s i n (n ω_{0} t)) \\ \tilde{x} (t) = \frac{1}{2} a_{0} + \sum_{n = 1}^{\infty} (a_{n} c o s (2 π n f_{0} t) + b_{n} s i n (2 π n f_{0} t)) \end{matrix}

where,

\begin{aligned} a_{0} & = \frac{1}{T} \int_{T} x (t) d t \\ a_{n} & = \frac{2}{T} \int_{T} x (t) c o s (n ω_{0} t) d t = \frac{2}{T} \int_{T} x (t) c o s (2 π n f_{0} t) d t \\ b_{n} & = \frac{2}{T} \int_{T} x (t) s i n (n ω_{0} t) d t = \frac{2}{T} \int_{T} x (t) s i n (2 π n f_{0} t) d t \end{aligned}

$\widetilde{x}(t)$ Fourier Series $x(t)$ .

Fourier Transform

Definition

$x(t)$ $\omega = 2\pi f$ . Then,

\begin{matrix} F {x (t)} = X (ω) = \int_{- \infty}^{\infty} x (t) e^{- j ω t} d t, f o r a l l - \infty < ω < \infty \\ F {x (t)} = X (f) = \int_{- \infty}^{\infty} x (t) e^{- j 2 π f t} d t, f o r a l l - \infty < f < \infty \end{matrix}

and,

\begin{matrix} x (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} X (ω) e^{j ω t} d ω, f o r a l l - \infty < t < \infty \\ x (t) = \int_{- \infty}^{\infty} X (f) e^{j 2 π f t} d f, f o r a l l - \infty < t < \infty \end{matrix}

$\mathcal{F}\{x(t)\} = X(\omega)$ $\mathcal{F}\{x(t)\} = X(f)$ Fourier Transforms $x(t)$ .

Properties

Linearity

$\Large \mathcal{F}\{c_1x_1(t)+c_2x_2(t)\} = c_1\mathcal{F}\{x_1(t)\}+c_2\mathcal{F}\{x_2(t)\}$

Duality

$\Large \mathcal{F}\{X(t)\} = 2\pi x(-\omega)$
$\Large \mathcal{F}\{X(t)\} = x(-f)$

Complex Conjugate

$\Large \mathcal{F}\{x^*(t)\} = X^*(-\omega)$
$\Large \mathcal{F}\{x^*(t)\} = X^*(-f)$

Symmetry

$\Large X(-\omega) = X^*(\omega)$
$\Large X(-f) = X^*(f)$

Time Scaling

$\Large \mathcal{F}\{x(\alpha t)\} = \frac{1}{|a|}X(\frac{\omega}{\alpha})$
$\Large \mathcal{F}\{x(\alpha t)\} = \frac{1}{|a|}X(\frac{f}{\alpha})$

Time Shift

$\Large \mathcal{F}\{x(t-t_0)\} = e^{-j\omega t_0}X(\omega)$
$\Large \mathcal{F}\{x(t-t_0)\} = e^{-j2\pi ft_0}X(f)$

Frequency Shift

$\Large \mathcal{F}\{e^{j\omega_0 t}x(t)\} = X(\omega - \omega_0)$
$\Large \mathcal{F}\{e^{j2\pi f_0t}x(t)\} = X(f - f_0)$
$\Large \mathcal{F}\{cos(\omega_0t)x(t)\} = \frac{X(\omega + \omega_0) + X(\omega - \omega_0)}{2}$
$\Large\mathcal{F}\{cos(2\pi f_0t)x(t)\} = \frac{X(f + f_0) + X(f - f_0)}{2}$
$\Large \mathcal{F}\{sin(\omega_0t)x(t)\} = j\frac{X(\omega + \omega_0) - X(\omega - \omega_0)}{2}$
$\Large \mathcal{F}\{sin(2\pi f_0t)x(t)\} = j\frac{X(f + f_0) - X(f - f_0)}{2}$

Derivatives

$\Large \mathcal{F}\{x^{(n)}(t)\} =(j\omega)^nX(\omega)$
$\Large \mathcal{F}\{x^{(n)}(t)\} =(j2\pi ft)^nX(f)$

Integrals

$\Large \mathcal{F}\{\int_{-\infty}^{t}x(\tau)d\tau\} = \frac{X(\omega)}{j\omega}+\pi X(0)\delta(\omega)$
$\Large \mathcal{F}\{\int_{-\infty}^{t}x(\tau)d\tau\} = \frac{X(f)}{j2\pi ft}+\frac{1}{2}X(0)\delta(f)$

Convolution

$\Large \mathcal{F}\{x(t)*y(t)\} = X(\omega)Y(\omega)$
$\Large \mathcal{F}\{x(t)*y(t)\} = X(f)Y(f)$

Multiplication

$\Large \mathcal{F}\{x(t)y(t)\} = \frac{1}{2\pi}(X(\omega)*Y(\omega)) = \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\lambda)Y(\omega-\lambda)d\lambda = \frac{1}{2\pi}\int_{-\infty}^{\infty}Y(\lambda)X(\omega-\lambda)d\lambda$
$\Large \mathcal{F}\{x(t)y(t)\} = X(f)*Y(f) = \int_{-\infty}^{\infty}X(\lambda)Y(f-\lambda)d\lambda = \int_{-\infty}^{\infty}Y(\lambda)X(f-\lambda)d\lambda$

$t$

$\Large \mathcal{F}\{tx(t)\} = jX'(\omega)$
$\Large \mathcal{F}\{t^nx(t)\} = j^nX^{(n)}(\omega)$
$\Large \mathcal{F}\{tx(t)\} = \frac{j}{2\pi}X'(f)$
$\Large \mathcal{F}\{t^nx(t)\} = (\frac{j}{2\pi})^nX^{(n)}(f)$

Parseval's Theorem

$\Large \int_{-\infty}^{\infty}x(t)y^*(t)dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)Y^*(\omega)d\omega$
$\Large \int_{-\infty}^{\infty}x(t)y^*(t)dt = \int_{-\infty}^{\infty}X(f)Y^*(f)df$

Fourier Transform Table

$\Large x(t)$ $\Large t\geq 0$	$\Large X(\omega)$	$\Large X(f)$
$\Large \delta(t)$	$\Large 1$	$\Large 1$
$\Large 1$	$\Large 2\pi\delta(\omega)$	$\Large \delta(f)$
$\Large \delta(t-t_0)$	$\Large e^{-j\omega t_0}$	$\Large e^{-j2\pi ft_0}$
$\Large cos(\omega_0t),\:cos(2\pi f_0t)$	$\Large \pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]$	$\Large \frac{1}{2}[\delta(f-f_0)+\delta(f+f_0)]$
$\Large sin(\omega_0t),\:sin(2\pi f_0t)$	$\Large -j\pi[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)]$	$\Large \frac{1}{2j}[\delta(f-f_0)-\delta(f+f_0)]$
$\Large u(t)$	$\Large \pi\delta(\omega)+\frac{1}{j\omega}$	$\Large \frac{1}{2}\delta(f)+\frac{1}{j2\pi f}$
$\Large \prod(t)$	$\Large \pi sinc(\frac{\omega}{2\pi})$ *	$\Large sinc(f)$ *
$\Large \Lambda(t)$	$\Large \pi^2sinc^2(\frac{\omega}{2\pi})$ *	$\Large sinc^2(f)$ *
$\Large sgn(t)$	$\Large \frac{2}{j\omega}$	$\Large \frac{1}{j\pi f}$
$\Large e^{-\alpha t}u(t),\:[\alpha>0]$	$\Large \frac{1}{\alpha+j\omega}$	$\Large \frac{1}{\alpha+j2\pi f}$
$\Large e^{-\alpha t}u(-t),\:[\alpha>0]$	$\Large \frac{1}{\alpha-j\omega}$	$\Large \frac{1}{\alpha-j2\pi f}$
$\Large e^{-\alpha \|t\|},\:[\alpha>0]$	$\Large \frac{2\alpha}{\alpha^2+\omega^2}$	$\Large \frac{2\alpha}{\alpha^2+(2\pi f)^2}$
$\Large e^{-\alpha t^2}$	$\Large \sqrt{\frac{\pi}{\alpha}}e^{\frac{-\omega^2}{4\alpha}}$	$\Large \sqrt{\frac{\pi}{\alpha}}e^{\frac{-(2\pi f)^2}{4\alpha}}$
$\Large te^{-\alpha t}u(t),\:[\alpha>0]$	$\Large \frac{1}{(\alpha+j\omega)^2}$	$\Large \frac{1}{(\alpha+j2\pi f)^2}$
$\Large t^ne^{-\alpha t}u(t),\:[\alpha>0]$	$\Large \frac{n!}{(\alpha+j\omega)^{n+1}}$	$\Large \frac{n!}{(\alpha+j2\pi ft)^{n+1}}$
$\Large \sum\limits_{n=-\infty}^{\infty}\delta(t-nT_0)$	$\Large \omega_0\sum\limits_{n=-\infty}^{\infty}\delta(\omega-n\omega_0)$	$\Large f_0\sum\limits_{n=-\infty}^{\infty}\delta(f-nf_0)$

$\Large sinc(t) = \frac{sin(\pi t)}{\pi t}$

Fourier Series

Definition

Exponential Form

Special Case for Real Periodic Signals

Trigonometric Form

Fourier Transform

Definition

Properties

Linearity

Duality

Complex Conjugate

Symmetry

Time Scaling

Time Shift

Frequency Shift

Derivatives

Integrals

Convolution

Multiplication

Multiplication by tt

Parseval's Theorem

Fourier Transform Table

$t$