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Time Domain x[n] | Frequency Domain X(w) | Notes |
$\delta \left[n\right]$ | 1 | |
$\delta [n-M]$ | ${e}^{-jwM}$ | integer M |
${\sum}_{m=-\infty}^{\infty}\delta [n-Mm]$ | ${\sum}_{m=-\infty}^{\infty}{e}^{-jwMm}=\frac{1}{M}{\sum}_{k=-\infty}^{\infty}\delta (\frac{w}{2\pi}-\frac{k}{M})$ | integer M |
${e}^{-jan}$ | $2\pi \delta (w+a)$ | real number a |
$u\left[n\right]$ | $\frac{1}{1-{e}^{-jw}}+{\sum}_{k=-\infty}^{\infty}\pi \delta (w+2\pi k)$ | |
${a}^{n}u\left(n\right)$ | $\frac{1}{1-a{e}^{-jw}}$ | if $\left|a\right|<1$ |
$cos\left(an\right)$ | $\pi \left[\delta \right(w-a)+\delta (w+a\left)\right]$ | real number a |
$W\xb7sin{c}^{2}\left(Wn\right)$ | $tri\left(\frac{w}{2\pi W}\right)$ | real number W, $0<W\le 0.5$ |
$W\xb7sinc\left[W\right(n+a\left)\right]$ | $rect\left(\frac{w}{2\pi W}\right)\xb7{e}^{jaw}$ | real numbers W,a $0<W\le 1$ |
$rect\left[\frac{(n-M/2)}{M}\right]$ | $\frac{sin\left[w\right(M+1)/2]}{sin(w/2)}{e}^{-jwM/2}$ | integer M |
$\frac{W}{(n+a)}\{cos\left[\pi W(n+a)\right]-sinc\left[W(n+a)\right]\}$ | $jw\xb7rect\left(\frac{w}{\pi W}\right){e}^{j}aw$ | real numbers W,a $0<W\le 1$ |
$\frac{1}{\pi {n}^{2}}[{(-1)}^{n}-1]$ | $\left|w\right|$ | |
$\begin{array}{c}\left\{\begin{array}{cc}0\hfill & n=0\hfill \\ \frac{{(-1)}^{n}}{n}\hfill & \text{elsewhere}\hfill \end{array}\right)\hfill \end{array}$ | $jw$ | 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 |
tri(t) is the triangle function for arbitrary real-valued $t$ .
$$\text{tri(t)}=\begin{array}{c}\left\{\begin{array}{cc}1+t\hfill & \text{if}-1\le t\le 0\hfill \\ 1-t\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}0<t\le 1\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}\right)\hfill \end{array}$$
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