9.2 Common discrete time fourier transforms

Signals and systems Page 1 / 1

This module includes a listing of commonly encountered discrete time fourier transforms.

Common dtfts

Time Domain x[n]	Frequency Domain X(w)	Notes
$δ [n]$	1
$δ [n - M]$	$e^{- j w M}$	integer M
$\sum_{m = - \infty}^{\infty} δ [n - M m]$	$\sum_{m = - \infty}^{\infty} e^{- j w M m} = \frac{1}{M} \sum_{k = - \infty}^{\infty} δ (\frac{w}{2 π} - \frac{k}{M})$	integer M
$e^{- j a n}$	$2 π δ (w + a)$	real number a
$u [n]$	$\frac{1}{1 - e^{- j w}} + \sum_{k = - \infty}^{\infty} π δ (w + 2 π k)$
$a^{n} u (n)$	$\frac{1}{1 - a e^{- j w}}$	if $\| a \| < 1$
$c o s (a n)$	$π [δ (w - a) + δ (w + a)]$	real number a
$W \cdot s i n c^{2} (W n)$	$t r i (\frac{w}{2 π W})$	real number W, $0 < W \leq 0.5$
$W \cdot s i n c [W (n + a)]$	$r e c t (\frac{w}{2 π W}) \cdot e^{j a w}$	real numbers W,a $0 < W \leq 1$
$r e c t [\frac{(n - M / 2)}{M}]$	$\frac{s i n [w (M + 1) / 2]}{s i n (w / 2)} e^{- j w M / 2}$	integer M
$\frac{W}{(n + a)} {c o s [π W (n + a)] - s i n c [W (n + a)]}$	$j w \cdot r e c t (\frac{w}{π W}) e^{j} a w$	real numbers W,a $0 < W \leq 1$
$\frac{1}{π n^{2}} [{(- 1)}^{n} - 1]$	$\| w \|$
$\begin{matrix} \{\begin{matrix} 0 & n = 0 \\ \frac{{(- 1)}^{n}}{n} & elsewhere \end{matrix}) \end{matrix}$	$j w$	differentiator filter
$\begin{matrix} \{\begin{matrix} 0 & n odd \\ \frac{2}{π n} & n even \end{matrix}) \end{matrix}$	$\begin{matrix} \{\begin{matrix} j & w < 0 \\ 0 & w = 0 \\ - j & w > 0 \end{matrix}) \end{matrix}$	Hilbert Transform

rect(t) is the rectangle function for arbitrary real-valued

t

rect(t) = \begin{matrix} \{\begin{matrix} 0 & if | t | > 1 / 2 \\ 1 / 2 & if | t | = 1 / 2 \\ 1 & if | t | < 1 / 2 \end{matrix}) \end{matrix}

tri(t) is the triangle function for arbitrary real-valued $t$ .

$tri(t) = \begin{matrix} \{\begin{matrix} 1 + t & if - 1 \leq t \leq 0 \\ 1 - t & if 0 < t \leq 1 \\ 0 & otherwise \end{matrix}) \end{matrix}$

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Source: OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15

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