Glossary

Last updated
Save as PDF

Page ID: 126580

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\dsum}{\displaystyle\sum\limits} $

$ \newcommand{\dint}{\displaystyle\int\limits} $

$ \newcommand{\dlim}{\displaystyle\lim\limits} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$\newcommand{\longvect}{\overrightarrow}$

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

Example and Directions
Words (or words that have the same definition)	The definition is case sensitive	(Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pages]	(Optional) Caption for Image	(Optional) External or Internal Link	(Optional) Source for Definition
(Eg. "Genetic, Hereditary, DNA ...")	(Eg. "Relating to genes or heredity")		The infamous double helix	https://bio.libretexts.org/	CC-BY-SA; Delmar Larsen

Glossary Entries
Word(s)	Definition	Image	Caption	Link	Source

Important Discrete Formulas & Tables:

Discrete-Time Convolution Equation

\[y[n] = \sum_{k=-\infty}^{\infty} x[k]\,h[n-k]\]

DTFS Synthesis (build $x[n]$ from $a_k$)

\[x[n] = \sum_{k=0}^{N-1} a_k\,e^{j\frac{2\pi}{N}kn}$$

DTFS Analysis (compute $a_k$ from $x[n]$)

\[a_k = \frac{1}{N}\sum_{n=0}^{N-1} x[n]\,e^{-j\frac{2\pi}{N}kn}$$

Table 1.7.1 Properties of the Discrete-Time Fourier Series (DTFS)

Periodic Signal	Fourier Series Coefficients
$x[n]$, $y[n]$ periodic with period $N$ and fundamental frequency $\omega_0=\frac{2\pi}{N}$	$a_k$, $b_k$ periodic with period $N$
Parseval’s Relation for Periodic Signals
$\displaystyle \frac{1}{N}\sum_{n=(N)}\|x[n]\|^2$ = $\displaystyle \sum_{k=(N)}\|a_k\|^2$
Linearity
$\displaystyle A\,x[n] + B\,y[n]$	$\displaystyle A\,a_k + B\,b_k$
Time Shifting
$\displaystyle x[n-n_0]$	$\displaystyle a_k\,e^{-j\frac{2\pi}{N}kn_0}$
Frequency Shifting
$\displaystyle e^{j\frac{2\pi}{N}mn}\,x[n]$	$\displaystyle a_{k-m}$
Conjugation
$\displaystyle x^*[n]$	$\displaystyle a_{-k}^*$
Time Reversal
$\displaystyle x[-n]$	$\displaystyle a_{-k}$
Time Scaling
$\displaystyle x_m[n]= \begin{cases} x[n/m], & \text{if } n \text{ is a multiple of } m\\[2pt] 0, & \text{if } n \text{ is not a multiple of } m \end{cases} $ $\displaystyle \text{(periodic with period } mN)$	$\displaystyle \frac{1}{m}a_k$ $\displaystyle \text{(viewed as periodic with period } mN)$
Periodic Convolution
$\displaystyle \sum_{r=(N)} x[r]\,y[n-r]$	$\displaystyle N\,a_k\,b_k$
Multiplication
$\displaystyle x[n]\,y[n]$	$\displaystyle \sum_{\ell=(N)} a_\ell\,b_{k-\ell}$
Conjugate Symmetry for Real Signals
$\displaystyle x[n]\ \text{real}$	$\displaystyle a_k=a_{-k}^*$ $\displaystyle \Re\{a_k\}=\Re\{a_{-k}\}$ $\displaystyle \Im\{a_k\}=-\Im\{a_{-k}\}$ $\displaystyle \|a_k\|=\|a_{-k}\|$ $\displaystyle \angle a_k=-\angle a_{-k}$
Real and Even Signals
$\displaystyle x[n]\ \text{real and even}$	$\displaystyle a_k\ \text{real and even}$
Real and Odd Signals
$\displaystyle x[n]\ \text{real and odd}$	$\displaystyle a_k\ \text{purely imaginary and odd}$

DTFT Analysis Equation (Definition)

DTFT pair (Analysis and Synthesis):

\[ X\!\left(e^{j\omega}\right)=\sum_{n=-\infty}^{\infty} x[n]\,e^{-j\omega n} \quad\Longleftrightarrow\quad x[n]=\frac{1}{2\pi}\int_{-\pi}^{\pi} X\!\left(e^{j\omega}\right)e^{j\omega n}\,d\omega. \]

Table 1.10.1 Properties of the Discrete-Time Fourier Transform (DTFT)

Discrete-Time Signal	Fourier Transform
$x[n]$, $y[n]$	$X(e^{j\omega})$, $Y(e^{j\omega})$ periodic with period $2\pi$
Parseval’s Relation for Aperiodic Signals
$\displaystyle \sum_{n=-\infty}^{\infty} \|x[n]\|^2$ = $\displaystyle \frac{1}{2\pi}\int_{2\pi} \|X(e^{j\omega})\|^2\,d\omega$
Linearity
$\displaystyle a\,x[n] + b\,y[n]$	$\displaystyle a\,X(e^{j\omega}) + b\,Y(e^{j\omega})$
Time Shifting
$\displaystyle x[n-n_0]$	$\displaystyle e^{-j\omega n_0}X(e^{j\omega})$
Frequency Shifting
$\displaystyle e^{j\omega_0 n}\,x[n]$	$\displaystyle X\!\left(e^{j(\omega-\omega_0)}\right)$
Conjugation
$\displaystyle x^*[n]$	$\displaystyle X^*(e^{-j\omega})$
Time Reversal
$\displaystyle x[-n]$	$\displaystyle X(e^{-j\omega})$
Time Expansion
$\displaystyle x_k[n]= \begin{cases} x[n/k], & \text{if } n \text{ is a multiple of } k\\[2pt] 0, & \text{if } n \text{ is not a multiple of } k \end{cases} $	$\displaystyle X(e^{jk\omega})$
Convolution
$\displaystyle x[n]*y[n]$	$\displaystyle X(e^{j\omega})\,Y(e^{j\omega})$
Multiplication
$\displaystyle x[n]\,y[n]$	$\displaystyle \frac{1}{2\pi}\int_{2\pi} X(e^{j\theta})\,Y(e^{j(\omega-\theta)})\,d\theta$
Differentiation in Frequency
$\displaystyle n\,x[n]$	$\displaystyle j\,\frac{dX(e^{j\omega})}{d\omega}$

Table 1.10.2 Basic Discrete-Time Fourier Transform (DTFT) Pairs

Signal	Fourier Transform	Fourier Series Coefficients (if periodic)
$\displaystyle \sum_{k=-\infty}^{\infty} a_k e^{j\frac{2\pi}{N}kn}$	$\displaystyle 2\pi \sum_{k=-\infty}^{\infty} a_k\,\delta\!\left(\omega-\frac{2\pi k}{N}\right)$	$\displaystyle a_k$
$\displaystyle e^{j\omega_0 n}$	$\displaystyle 2\pi\sum_{\ell=-\infty}^{\infty}\delta(\omega-\omega_0-2\pi\ell)$	(a) $\displaystyle \omega_0=\frac{2\pi m}{N}$ \[ a_k= \begin{cases} 1, & k=m,\, m\pm N,\, m\pm 2N,\,\ldots\\ 0, & \text{otherwise} \end{cases} \] (b) $\displaystyle \frac{\omega_0}{2\pi}$ irrational → The signal is aperiodic
$\displaystyle \cos(\omega_0 n)$	$\displaystyle \pi\sum_{\ell=-\infty}^{\infty}\Big[\delta(\omega-\omega_0-2\pi\ell)+\delta(\omega+\omega_0-2\pi\ell)\Big]$	(a) $\displaystyle \omega_0=\frac{2\pi m}{N}$ \[ a_k= \begin{cases} \frac{1}{2}, & k=m,\, m\pm N,\, m\pm 2N,\,\ldots\\ \frac{1}{2}, & k=-m,\, -m\pm N,\, -m\pm 2N,\,\ldots\\ 0, & \text{otherwise} \end{cases} \] (b) $\displaystyle \frac{\omega_0}{2\pi}$ irrational → The signal is aperiodic
$\displaystyle \sin(\omega_0 n)$	$\displaystyle \frac{\pi}{j}\sum_{\ell=-\infty}^{\infty}\Big[\delta(\omega-\omega_0-2\pi\ell)-\delta(\omega+\omega_0-2\pi\ell)\Big]$	(a) $\displaystyle \omega_0=\frac{2\pi r}{N}$ \[ a_k= \begin{cases} -\frac{j}{2}, & k=r,\, r\pm N,\, r\pm 2N,\,\ldots\\ \frac{j}{2}, & k=-r,\, -r\pm N,\, -r\pm 2N,\,\ldots\\ 0, & \text{otherwise} \end{cases} \] (b) $\displaystyle \frac{\omega_0}{2\pi}$ irrational → The signal is aperiodic
$\displaystyle x[n]=1$	$\displaystyle 2\pi\sum_{\ell=-\infty}^{\infty}\delta(\omega-2\pi\ell)$	\[ a_k= \begin{cases} 1, & k=0,\, \pm N,\, \pm 2N,\,\ldots\\ 0, & \text{otherwise} \end{cases} \]
Periodic square wave $ x[n]= \begin{cases} 1, & \|n\|\le N_1\\ 0, & N_1<\|n\|\le N/2 \end{cases} $ $x[n+N]=x[n]$	$\displaystyle 2\pi\sum_{k=-\infty}^{\infty}a_k\,\delta\!\left(\omega-\frac{2\pi k}{N}\right)$	\[ a_k= \begin{cases} \dfrac{\sin\!\left(\dfrac{2\pi k}{N}\left(N_1+\frac{1}{2}\right)\right)}{N\sin\!\left(\dfrac{\pi k}{N}\right)}, & k\ne 0,\, \pm N,\, \pm 2N,\ldots\\[10pt] \dfrac{2N_1+1}{N}, & k=0,\, \pm N,\, \pm 2N,\ldots \end{cases} \]
$\displaystyle \sum_{k=-\infty}^{\infty}\delta[n-kN]$	$\displaystyle \frac{2\pi}{N}\sum_{k=-\infty}^{\infty}\delta\!\left(\omega-\frac{2\pi k}{N}\right)$	$\displaystyle a_k=\frac{1}{N}\ \text{for all }k$
$\displaystyle a^n u[n],\ \ \|a\|<1$	$\displaystyle \frac{1}{1-ae^{-j\omega}}$	—
$ x[n]= \begin{cases} 1, & \|n\|\le N_1\\ 0, & \|n\|>N_1 \end{cases} $	$\displaystyle \frac{\sin\!\left(\omega\left(N_1+\frac{1}{2}\right)\right)}{\sin(\omega/2)}$	—
$ \frac{\sin\left(\frac{\omega_m}{\pi}n\right)}{n} \;=\; \frac{\omega_m}{\pi}\,\text{sinc}\!\left(\frac{\omega_m}{\pi}n\right) $ $0<\omega_m<\pi$	$ X(\omega)= \begin{cases} 1, & 0\le \|\omega\| \le \omega_m\\ 0, & \omega_m<\|\omega\|\le \pi \end{cases} $ $X(\omega)$ periodic with period $2\pi$	—
$\displaystyle \delta[n]$	$\displaystyle 1$	—
$\displaystyle u[n]$	$\displaystyle \frac{1}{1-e^{-j\omega}} + \pi\sum_{k=-\infty}^{\infty}\delta(\omega-2\pi k)$	—
$\displaystyle \delta[n-n_0]$	$\displaystyle e^{-j\omega n_0}$	—

Definition: z-Transform

\[ X(z)=\sum_{n=-\infty}^{\infty}x[n]\,z^{-n}. \]

Important Continuous Formulas & Tables:

Definition: Convolution Integral

\[ y(t)=\int_{-\infty}^{+\infty} x(\tau)\,h_{\tau}(t)\,d\tau \]

\[ y(t)=x(t)*h(t) \]

Figure 2.3.2: Animation showing the convolution between two rectangular functions.

Fourier Series (Synthesis and Analysis Equations)

Synthesis equation

\[ x(t)=\sum_{k=-\infty}^{\infty} a_k e^{jk\omega_0 t} \]

Analysis equation

\[ a_k=\frac{1}{T}\int_T x(t)e^{-jk\omega_0 t}\,dt =\frac{1}{T}\int_T x(t)e^{-jk\frac{2\pi}{T}t}\,dt \]

Fourier Transform (Analysis and Synthesis Equations)

$x(t)$ → Fourier Transform → $ X(j\omega)=\int_{-\infty}^{+\infty} x(t)\,e^{-j\omega t}\,dt $ → Inverse Fourier Transform → $ x(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} X(j\omega)\,e^{j\omega t}\,d\omega $

Laplace Transform \[ X(s)\triangleq \int_{-\infty}^{+\infty} x(t)e^{-st}\,dt \]

Laplace Transform Properties

\[ X(s)=\int_{-\infty}^{+\infty} x(t)\,e^{-st}\,dt \]

Time	Laplace
$y(t)=x(t-t_0)$	$Y(s)=e^{-st_0}X(s)$ same ROC
$y(t)=e^{s_0 t}\,x(t)$	$Y(s)=X(s-s_0)$ shift ROC by $s_0$
$y(t)=x(t)*h(t)$	$Y(s)=X(s)\cdot H(s)$ ROC of $Y$ includes $ROC_X\cap ROC_H$ (intersection)
$y(t)=\dfrac{dx(t)}{dt}$	$Y(s)=s\,H(s)$
$y(t)=-t\,x(t)$	$Y(s)=\dfrac{d}{ds}X(s)$
$y(t)=\displaystyle\int_{-\infty}^{t}x(z)\,dz$	$Y(s)=\dfrac{1}{s}X(s)$

If $x(t)=0$ for $t<0$, $x(0^+)=\displaystyle\lim_{s\to\infty} sX(s)$ Initial Value Theorem

If $x(t)=0$ for $t<0$, $\displaystyle\lim_{t\to\infty}x(t)=\lim_{s\to 0} sX(s)$

Laplace Transforms of Elementary Functions

Signal	Transform	ROC
$\delta(t)$	$1$	All $s$
$u(t)$	$\dfrac{1}{s}$	$\operatorname{Re}\{s\}>0$
$-u(-t)$	$\dfrac{1}{s}$	$\operatorname{Re}\{s\}<0$
$e^{-at}u(t)$	$\dfrac{1}{s+a}$	$\operatorname{Re}\{s\}>-a$
$-e^{-at}u(-t)$	$\dfrac{1}{s+a}$	$\operatorname{Re}\{s\}<-a$
$\delta(t-T)$	$e^{-sT}$	All $s$
$[\cos \omega_0 t]u(t)$	$\dfrac{s}{s^2+\omega_0^2}$	$\operatorname{Re}\{s\}>0$
$[\sin \omega_0 t]u(t)$	$\dfrac{\omega_0}{s^2+\omega_0^2}$	$\operatorname{Re}\{s\}>0$
$[e^{-at}\cos \omega_0 t]u(t)$	$\dfrac{s+a}{(s+a)^2+\omega_0^2}$	$\operatorname{Re}\{s\}>-a$
$[e^{-at}\sin \omega_0 t]u(t)$	$\dfrac{\omega_0}{(s+a)^2+\omega_0^2}$	$\operatorname{Re}\{s\}>-a$
$u_n(t)=\dfrac{d^n\delta(t)}{dt^n}$	$s^n$	All $s$
$u_{-n}(t)=u(t)\cdotsu(t)$ (n times)	$\dfrac{1}{s^n}$	$\operatorname{Re}\{s\}>0$

Periodic Signal	Fourier Series Coefficients
\(x[n]\), \(y[n]\) periodic with period \(N\) and fundamental frequency \(\omega_0=\frac{2\pi}{N}\)	\(a_k\), \(b_k\) periodic with period \(N\)
Parseval’s Relation for Periodic Signals
\(\displaystyle \frac{1}{N}\sum_{n=(N)}\|x[n]\|^2\) = \(\displaystyle \sum_{k=(N)}\|a_k\|^2\)
Linearity
\(\displaystyle A\,x[n] + B\,y[n]\)	\(\displaystyle A\,a_k + B\,b_k\)
Time Shifting
\(\displaystyle x[n-n_0]\)	\(\displaystyle a_k\,e^{-j\frac{2\pi}{N}kn_0}\)
Frequency Shifting
\(\displaystyle e^{j\frac{2\pi}{N}mn}\,x[n]\)	\(\displaystyle a_{k-m}\)
Conjugation
\(\displaystyle x^*[n]\)	\(\displaystyle a_{-k}^*\)
Time Reversal
\(\displaystyle x[-n]\)	\(\displaystyle a_{-k}\)
Time Scaling
\(\displaystyle x_m[n]= \begin{cases} x[n/m], & \text{if } n \text{ is a multiple of } m\\[2pt] 0, & \text{if } n \text{ is not a multiple of } m \end{cases} \) \(\displaystyle \text{(periodic with period } mN)\)	\(\displaystyle \frac{1}{m}a_k\) \(\displaystyle \text{(viewed as periodic with period } mN)\)
Periodic Convolution
\(\displaystyle \sum_{r=(N)} x[r]\,y[n-r]\)	\(\displaystyle N\,a_k\,b_k\)
Multiplication
\(\displaystyle x[n]\,y[n]\)	\(\displaystyle \sum_{\ell=(N)} a_\ell\,b_{k-\ell}\)
Conjugate Symmetry for Real Signals
\(\displaystyle x[n]\ \text{real}\)	\(\displaystyle a_k=a_{-k}^*\) \(\displaystyle \Re\{a_k\}=\Re\{a_{-k}\}\) \(\displaystyle \Im\{a_k\}=-\Im\{a_{-k}\}\) \(\displaystyle \|a_k\|=\|a_{-k}\|\) \(\displaystyle \angle a_k=-\angle a_{-k}\)
Real and Even Signals
\(\displaystyle x[n]\ \text{real and even}\)	\(\displaystyle a_k\ \text{real and even}\)
Real and Odd Signals
\(\displaystyle x[n]\ \text{real and odd}\)	\(\displaystyle a_k\ \text{purely imaginary and odd}\)

Discrete-Time Signal	Fourier Transform
\(x[n]\), \(y[n]\)	\(X(e^{j\omega})\), \(Y(e^{j\omega})\) periodic with period \(2\pi\)
Parseval’s Relation for Aperiodic Signals
\(\displaystyle \sum_{n=-\infty}^{\infty} \|x[n]\|^2\) = \(\displaystyle \frac{1}{2\pi}\int_{2\pi} \|X(e^{j\omega})\|^2\,d\omega\)
Linearity
\(\displaystyle a\,x[n] + b\,y[n]\)	\(\displaystyle a\,X(e^{j\omega}) + b\,Y(e^{j\omega})\)
Time Shifting
\(\displaystyle x[n-n_0]\)	\(\displaystyle e^{-j\omega n_0}X(e^{j\omega})\)
Frequency Shifting
\(\displaystyle e^{j\omega_0 n}\,x[n]\)	\(\displaystyle X\!\left(e^{j(\omega-\omega_0)}\right)\)
Conjugation
\(\displaystyle x^*[n]\)	\(\displaystyle X^*(e^{-j\omega})\)
Time Reversal
\(\displaystyle x[-n]\)	\(\displaystyle X(e^{-j\omega})\)
Time Expansion
\(\displaystyle x_k[n]= \begin{cases} x[n/k], & \text{if } n \text{ is a multiple of } k\\[2pt] 0, & \text{if } n \text{ is not a multiple of } k \end{cases} \)	\(\displaystyle X(e^{jk\omega})\)
Convolution
\(\displaystyle x[n]*y[n]\)	\(\displaystyle X(e^{j\omega})\,Y(e^{j\omega})\)
Multiplication
\(\displaystyle x[n]\,y[n]\)	\(\displaystyle \frac{1}{2\pi}\int_{2\pi} X(e^{j\theta})\,Y(e^{j(\omega-\theta)})\,d\theta\)
Differentiation in Frequency
\(\displaystyle n\,x[n]\)	\(\displaystyle j\,\frac{dX(e^{j\omega})}{d\omega}\)

Signal	Fourier Transform	Fourier Series Coefficients (if periodic)
\(\displaystyle \sum_{k=-\infty}^{\infty} a_k e^{j\frac{2\pi}{N}kn}\)	\(\displaystyle 2\pi \sum_{k=-\infty}^{\infty} a_k\,\delta\!\left(\omega-\frac{2\pi k}{N}\right)\)	\(\displaystyle a_k\)
\(\displaystyle e^{j\omega_0 n}\)	\(\displaystyle 2\pi\sum_{\ell=-\infty}^{\infty}\delta(\omega-\omega_0-2\pi\ell)\)	(a) \(\displaystyle \omega_0=\frac{2\pi m}{N}\) \[ a_k= \begin{cases} 1, & k=m,\, m\pm N,\, m\pm 2N,\,\ldots\\ 0, & \text{otherwise} \end{cases} \] (b) \(\displaystyle \frac{\omega_0}{2\pi}\) irrational → The signal is aperiodic
\(\displaystyle \cos(\omega_0 n)\)	\(\displaystyle \pi\sum_{\ell=-\infty}^{\infty}\Big[\delta(\omega-\omega_0-2\pi\ell)+\delta(\omega+\omega_0-2\pi\ell)\Big]\)	(a) \(\displaystyle \omega_0=\frac{2\pi m}{N}\) \[ a_k= \begin{cases} \frac{1}{2}, & k=m,\, m\pm N,\, m\pm 2N,\,\ldots\\ \frac{1}{2}, & k=-m,\, -m\pm N,\, -m\pm 2N,\,\ldots\\ 0, & \text{otherwise} \end{cases} \] (b) \(\displaystyle \frac{\omega_0}{2\pi}\) irrational → The signal is aperiodic
\(\displaystyle \sin(\omega_0 n)\)	\(\displaystyle \frac{\pi}{j}\sum_{\ell=-\infty}^{\infty}\Big[\delta(\omega-\omega_0-2\pi\ell)-\delta(\omega+\omega_0-2\pi\ell)\Big]\)	(a) \(\displaystyle \omega_0=\frac{2\pi r}{N}\) \[ a_k= \begin{cases} -\frac{j}{2}, & k=r,\, r\pm N,\, r\pm 2N,\,\ldots\\ \frac{j}{2}, & k=-r,\, -r\pm N,\, -r\pm 2N,\,\ldots\\ 0, & \text{otherwise} \end{cases} \] (b) \(\displaystyle \frac{\omega_0}{2\pi}\) irrational → The signal is aperiodic
\(\displaystyle x[n]=1\)	\(\displaystyle 2\pi\sum_{\ell=-\infty}^{\infty}\delta(\omega-2\pi\ell)\)	\[ a_k= \begin{cases} 1, & k=0,\, \pm N,\, \pm 2N,\,\ldots\\ 0, & \text{otherwise} \end{cases} \]
Periodic square wave \( x[n]= \begin{cases} 1, & \|n\|\le N_1\\ 0, & N_1<\|n\|\le N/2 \end{cases} \) \(x[n+N]=x[n]\)	\(\displaystyle 2\pi\sum_{k=-\infty}^{\infty}a_k\,\delta\!\left(\omega-\frac{2\pi k}{N}\right)\)	\[ a_k= \begin{cases} \dfrac{\sin\!\left(\dfrac{2\pi k}{N}\left(N_1+\frac{1}{2}\right)\right)}{N\sin\!\left(\dfrac{\pi k}{N}\right)}, & k\ne 0,\, \pm N,\, \pm 2N,\ldots\\[10pt] \dfrac{2N_1+1}{N}, & k=0,\, \pm N,\, \pm 2N,\ldots \end{cases} \]
\(\displaystyle \sum_{k=-\infty}^{\infty}\delta[n-kN]\)	\(\displaystyle \frac{2\pi}{N}\sum_{k=-\infty}^{\infty}\delta\!\left(\omega-\frac{2\pi k}{N}\right)\)	\(\displaystyle a_k=\frac{1}{N}\ \text{for all }k\)
\(\displaystyle a^n u[n],\ \ \|a\|<1\)	\(\displaystyle \frac{1}{1-ae^{-j\omega}}\)	—
\( x[n]= \begin{cases} 1, & \|n\|\le N_1\\ 0, & \|n\|>N_1 \end{cases} \)	\(\displaystyle \frac{\sin\!\left(\omega\left(N_1+\frac{1}{2}\right)\right)}{\sin(\omega/2)}\)	—
\( \frac{\sin\left(\frac{\omega_m}{\pi}n\right)}{n} \;=\; \frac{\omega_m}{\pi}\,\text{sinc}\!\left(\frac{\omega_m}{\pi}n\right) \) \(0<\omega_m<\pi\)	\( X(\omega)= \begin{cases} 1, & 0\le \|\omega\| \le \omega_m\\ 0, & \omega_m<\|\omega\|\le \pi \end{cases} \) \(X(\omega)\) periodic with period \(2\pi\)	—
\(\displaystyle \delta[n]\)	\(\displaystyle 1\)	—
\(\displaystyle u[n]\)	\(\displaystyle \frac{1}{1-e^{-j\omega}} + \pi\sum_{k=-\infty}^{\infty}\delta(\omega-2\pi k)\)	—
\(\displaystyle \delta[n-n_0]\)	\(\displaystyle e^{-j\omega n_0}\)	—

Time	Laplace
\(y(t)=x(t-t_0)\)	\(Y(s)=e^{-st_0}X(s)\) same ROC
\(y(t)=e^{s_0 t}\,x(t)\)	\(Y(s)=X(s-s_0)\) shift ROC by \(s_0\)
\(y(t)=x(t)*h(t)\)	\(Y(s)=X(s)\cdot H(s)\) ROC of \(Y\) includes \(ROC_X\cap ROC_H\) (intersection)
\(y(t)=\dfrac{dx(t)}{dt}\)	\(Y(s)=s\,H(s)\)
\(y(t)=-t\,x(t)\)	\(Y(s)=\dfrac{d}{ds}X(s)\)
\(y(t)=\displaystyle\int_{-\infty}^{t}x(z)\,dz\)	\(Y(s)=\dfrac{1}{s}X(s)\)

Signal	Transform	ROC
\(\delta(t)\)	\(1\)	All \(s\)
\(u(t)\)	\(\dfrac{1}{s}\)	\(\operatorname{Re}\{s\}>0\)
\(-u(-t)\)	\(\dfrac{1}{s}\)	\(\operatorname{Re}\{s\}<0\)
\(e^{-at}u(t)\)	\(\dfrac{1}{s+a}\)	\(\operatorname{Re}\{s\}>-a\)
\(-e^{-at}u(-t)\)	\(\dfrac{1}{s+a}\)	\(\operatorname{Re}\{s\}<-a\)
\(\delta(t-T)\)	\(e^{-sT}\)	All \(s\)
\([\cos \omega_0 t]u(t)\)	\(\dfrac{s}{s^2+\omega_0^2}\)	\(\operatorname{Re}\{s\}>0\)
\([\sin \omega_0 t]u(t)\)	\(\dfrac{\omega_0}{s^2+\omega_0^2}\)	\(\operatorname{Re}\{s\}>0\)
\([e^{-at}\cos \omega_0 t]u(t)\)	\(\dfrac{s+a}{(s+a)^2+\omega_0^2}\)	\(\operatorname{Re}\{s\}>-a\)
\([e^{-at}\sin \omega_0 t]u(t)\)	\(\dfrac{\omega_0}{(s+a)^2+\omega_0^2}\)	\(\operatorname{Re}\{s\}>-a\)
\(u_n(t)=\dfrac{d^n\delta(t)}{dt^n}\)	\(s^n\)	All \(s\)
\(u_{-n}(t)=u(t)\cdotsu(t)\) (n times)	\(\dfrac{1}{s^n}\)	\(\operatorname{Re}\{s\}>0\)

Search

Text Color

Text Size

Margin Size

Font Type

Definition: z-Transform

Definition: Convolution Integral