User:Vanished user fijw983kjaslkekfhj45/DTFT Tables

Definition

$X(\omega )=\sum _{n=-\infty }^{\infty }x[n]\,e^{-i\omega n}.$

Inverse transform

$x[n]={\frac {1}{2\pi }}\int _{-\pi }^{\pi }X(\omega )\cdot e^{i\omega n}\,d\omega$

Properties

Property	Time domain $x[n]\!$	Frequency domain $X(\omega )\!$
Linearity	$ax[n]+by[n]\!$	$aX(e^{i\omega })+bY(e^{i\omega })\!$
Shift in time	$x[n-k]\!$	$X(e^{i\omega })e^{-i\omega k}\!$
Shift in frequency (modulation)	$x[n]e^{ian}\!$	$X(e^{i(\omega -a)})\!$
Time reversal	$x[-n]\!$	$X(e^{-i\omega })\!$
Time conjugation	$x[n]^{*}\!$	$X(e^{-i\omega })^{*}\!$
Time reversal & conjugation	$x[-n]^{*}\!$	$X(e^{i\omega })^{*}\!$
Derivative in frequency	${\frac {n}{i}}x[n]\!$	${\frac {dX(e^{i\omega })}{d\omega }}\!$
Integral in frequency	${\frac {i}{n}}x[n]\!$	$\int _{-\pi }^{\omega }X(e^{i\vartheta })d\vartheta \!$
Convolve in time	$x[n]*y[n]\!$	$X(e^{i\omega })\cdot Y(e^{i\omega })\!$
Multiply in time	$x[n]\cdot y[n]\!$	${\frac {1}{2\pi }}\int _{-\pi }^{\pi }{X(e^{i\vartheta })\cdot Y(e^{i(\omega -\vartheta )})d\vartheta }\!$
Cross correlation	$\rho _{xy}[n]=x[-n]^{*}*y[n]\!$	$R_{xy}(\omega )=X(e^{i\omega })^{*}\cdot Y(e^{i\omega })\!$
Parseval's theorem	$E=\sum _{n=-\infty }^{\infty }{x[n]\cdot y^{*}[n]}\!$	$E={\frac {1}{2\pi }}\int _{-\pi }^{\pi }{X(e^{i\omega })\cdot Y^{*}(e^{i\omega })d\omega }\!$

Table of discrete-time Fourier transforms

Time domain $x[n]\,$	Frequency domain $X(\omega )\,$	Remarks
$\delta [n]\!$	$1\!$
$\delta [n-M]\!$	$e^{-i\omega M}\!$	integer M
$\sum _{m=-\infty }^{\infty }\delta [n-Mm]\!$	$\sum _{m=-\infty }^{\infty }e^{-i\omega Mm}={\frac {1}{M}}\sum _{k=-\infty }^{\infty }\delta \left({\frac {\omega }{2\pi }}-{\frac {k}{M}}\right)\,$	integer M
$u[n]\!$	${\frac {1}{1-e^{-i\omega }}}+\sum \limits _{k=-\infty }^{\infty }{\pi \delta (\omega -2\pi k)}\!$	The $1/(1-e^{-i\omega })$ term must be interpreted as a distribution in the sense of a Cauchy principal value around its poles at $\omega =2\pi k$.
$a^{n}u[n]\!$	${\frac {1}{1-ae^{-i\omega }}}\!$	$|a|<1$
$e^{-ian}\!$	$2\pi \delta (\omega +a)\,$	real number a
$\cos(an)\!$	$\pi \left[\delta (\omega -a)+\delta (\omega +a)\right]$	real number a
$\sin(an)\!$	${\frac {\pi }{i}}\left[\delta (\omega -a)-\delta (\omega +a)\right]$	real number a
$\mathrm {rect} \left[{(n-M/2) \over M}\right]$	${\sin[\omega (M+1)/2] \over \sin(\omega /2)}\,e^{-i\omega M/2}\!$	integer M
$\operatorname {sinc} [(a+n)]$	$e^{ia\omega }\!$	real number a
$W\cdot \operatorname {sinc} ^{2}(Wn)\,$	$\operatorname {tri} \left({\omega \over 2\pi W}\right)$	real number W $0<W\leq 0.5$
$W\cdot \operatorname {sinc} (Wn)$	$\operatorname {rect} \left({\omega \over 2\pi W}\right)$	real numbers W $0<W\leq 1$
${\begin{cases}0&n=0\\{\frac {(-1)^{n}}{n}}&{\mbox{elsewhere}}\end{cases}}$	$j\omega$	it works as a differentiator filter
${\frac {W}{(n+a)}}\left\{\cos[\pi W(n+a)]-\operatorname {sinc} [W(n+a)]\right\}$	$j\omega \cdot \operatorname {rect} \left({\omega \over \pi W}\right)e^{ja\omega }$	real numbers W, a $0<W\leq 1$
${\frac {1}{\pi n^{2}}}[(-1)^{n}-1]\!$	$|\omega |\!$
${\begin{cases}0;&n{\mbox{ even}}\\{\frac {2}{\pi n}};&n{\mbox{ odd}}\end{cases}}$	${\begin{cases}j&\omega <0\\0&\omega =0\\-j&\omega >0\end{cases}}$	Hilbert transform
${\frac {C(A+B)}{2\pi }}\cdot \operatorname {sinc} \left[{\frac {A-B}{2\pi }}n\right]\cdot \operatorname {sinc} \left[{\frac {A+B}{2\pi }}n\right]$		real numbers A, B complex C

• $n\!$ is an integer representing the discrete-time domain (in samples)
• $\omega \!$ is a real number in $(-\pi ,\ \pi )$, representing continuous angular frequency (in radians per sample).
  • The remainder of the transform $(|\omega |>\pi \,)$ is defined by: $X(\omega +2\pi k)=X(\omega )\,$
• $u[n]\!$ is the discrete-time unit step function
• $\operatorname {sinc} (t)\!$ is the normalized sinc function
• $\delta (\omega )\!$ is the Dirac delta function
• $\delta [n]\!$ is the Kronecker delta $\delta _{n,0}\!$
• $\operatorname {rect} (t)$ is the rectangle function for arbitrary real-valued t:

$\mathrm {rect} (t)=\sqcap (t)={\begin{cases}0&{\mbox{if }}|t|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|t|<{\frac {1}{2}}\end{cases}}$

• $\operatorname {tri} (t)$ is the triangle function for arbitrary real-valued t:

$\operatorname {tri} (t)=\land (t)={\begin{cases}1+t;&-1\leq t\leq 0\\1-t;&0<t\leq 1\\0&{\mbox{otherwise}}\end{cases}}$