9.2 Common discrete time fourier transforms

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

Common dtfts

 Time Domain x[n] Frequency Domain X(w) Notes $\delta \left[n\right]$ 1 $\delta \left[n-M\right]$ ${e}^{-jwM}$ integer M ${\sum }_{m=-\infty }^{\infty }\delta \left[n-Mm\right]$ ${\sum }_{m=-\infty }^{\infty }{e}^{-jwMm}=\frac{1}{M}{\sum }_{k=-\infty }^{\infty }\delta \left(\frac{w}{2\pi }-\frac{k}{M}\right)$ integer M ${e}^{-jan}$ $2\pi \delta \left(w+a\right)$ real number a $u\left[n\right]$ $\frac{1}{1-{e}^{-jw}}+{\sum }_{k=-\infty }^{\infty }\pi \delta \left(w+2\pi k\right)$ ${a}^{n}u\left(n\right)$ $\frac{1}{1-a{e}^{-jw}}$ if $|a|<1$ $cos\left(an\right)$ $\pi \left[\delta \left(w-a\right)+\delta \left(w+a\right)\right]$ real number a $W·sin{c}^{2}\left(Wn\right)$ $tri\left(\frac{w}{2\pi W}\right)$ real number W, $0 $W·sinc\left[W\left(n+a\right)\right]$ $rect\left(\frac{w}{2\pi W}\right)·{e}^{jaw}$ real numbers W,a $0 $rect\left[\frac{\left(n-M/2\right)}{M}\right]$ $\frac{sin\left[w\left(M+1\right)/2\right]}{sin\left(w/2\right)}{e}^{-jwM/2}$ integer M $\frac{W}{\left(n+a\right)}\left\{cos\left[\pi W\left(n+a\right)\right]-sinc\left[W\left(n+a\right)\right]\right\}$ $jw·rect\left(\frac{w}{\pi W}\right){e}^{j}aw$ real numbers W,a $0 $\frac{1}{\pi {n}^{2}}\left[{\left(-1\right)}^{n}-1\right]$ $|w|$ $\begin{array}{c}\left\{\begin{array}{cc}0\hfill & n=0\hfill \\ \frac{{\left(-1\right)}^{n}}{n}\hfill & \text{elsewhere}\hfill \end{array}\right)\hfill \end{array}$ $jw$ differentiator filter $\begin{array}{c}\left\{\begin{array}{cc}0\hfill & n\text{odd}\hfill \\ \frac{2}{\pi n}\hfill & n\text{even}\hfill \end{array}\right)\hfill \end{array}$ $\begin{array}{c}\left\{\begin{array}{cc}j\hfill & w<0\hfill \\ 0\hfill & w=0\hfill \\ -j\hfill & w>0\hfill \end{array}\right)\hfill \end{array}$ Hilbert Transform

Notes

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

$\text{rect(t)}=\begin{array}{c}\left\{\begin{array}{cc}0\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}|t|>1/2\hfill \\ 1/2\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}|t|=1/2\hfill \\ 1\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}|t|<1/2\hfill \end{array}\right)\hfill \end{array}$

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

$\text{tri(t)}=\begin{array}{c}\left\{\begin{array}{cc}1+t\hfill & \text{if}-1\le t\le 0\hfill \\ 1-t\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}0

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