9.3: Common Discrete Time Fourier Transforms
- Page ID
- 22895
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| Time Domain \(x[n]\) | Frequency Domain \(X(w)\) | Notes |
|---|---|---|
| \(\delta[n]\) | \(1\) | |
| \(\delta[n-M]\) | \(e^{−j w M}\) | integer \(M\) |
| \(\sum_{m=-\infty}^{\infty} \delta[n-M m]\) | \(\sum_{m=-\infty}^{\infty} e^{-j w M m}=\frac{1}{M} \sum_{k=-\infty}^{\infty} \delta\left(\frac{w}{2 \pi}-\frac{k}{M}\right)\) | integer \(M\) |
| \(e^{−jan}\) | \(2 \pi \delta (w+a)\) | real number \(a\) |
| \(u[n]\) | \(\frac{1}{1-e^{-j w}}+\sum_{k=-\infty}^{\infty} \pi \delta(w+2 \pi k)\) | |
| \(a^n u(n)\) | \(\frac{1}{1-a e^{-j w}}\) | if \(|a|<1\) |
| \(\cos(an)\) | \(\pi[\delta(w-a)+\delta(w+a)]\) | real number \(a\) |
| \(W \cdot \operatorname{sinc}^{2}(W n)\) | \(\operatorname{tri}\left(\frac{w}{2 \pi W}\right)\) | real number \(W\), \(0<W≤0.50\) |
| \(W \cdot \operatorname{sinc}[W(n+a)]\) | \(\operatorname{rect}\left(\frac{w}{2 \pi W}\right) \cdot e^{j a w}\) | real numbers \(W\), \(a\) \(0<W≤1\) |
| \(\operatorname{rect}\left[\frac{(n-M / 2)}{M}\right]\) | \(\frac{\sin [w(M+1) / 2]}{\sin (w / 2)} e^{-j w M / 2}\) | integer \(M\) |
| \(\frac{W}{(n+a)}\{\cos [\pi W(n+a)]-\operatorname{sinc}[W(n+a)]\}\) | \(j w \cdot \operatorname{rect}\left(\frac{w}{\pi W}\right) e^{j} a w\) | real numbers \(W\), \(a\) \(0<W≤1\) |
| \(\frac{1}{\pi n^{2}}\left[(-1)^{n}-1\right]\) | \(|w|\) | |
| \(\left\{\begin{array}{ll} 0 & n=0 \\ \frac{(-1)^{n}}{n} & \text { elsewhere } \end{array}\right.\) |
\(jw\) | differentiator filter |
| \(\left\{\begin{array}{ll} 0 & \quad n \text { odd } \\ \frac{2}{\pi n} & \quad n \text { even } \end{array}\right.\) |
\(\left\{\begin{array}{cc} j & w<0 \\ 0 & w=0 \\ -j & w>0 \end{array}\right.\) |
Hilbert Transform |
Notes
rect(\(t\)) is the rectangle function for arbitrary real-valued \(t\).
\[\operatorname{rect}(\mathrm{t})=\left\{\begin{array}{ll}
0 & \text { if }|t|>1 / 2 \\
1 / 2 & \text { if }|t|=1 / 2 \\
1 & \text { if }|t|<1 / 2
\end{array}\right. \nonumber \]
tri(\(t\)) is the triangle function for arbitrary real-valued \(t\).
\[\operatorname{tri}(\mathrm{t})=\left\{\begin{array}{ll}
1+t & \text { if }-1 \leq t \leq 0 \\
1-t & \text { if } 0<t \leq 1 \\
0 & \text { otherwise }
\end{array}\right. \nonumber \]