9.3: Transformadas de Fourier de Tiempo Discreto Común
- Page ID
- 86454
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DTFT comunes
| Dominio del Tiempo\(x[n]\) | Dominio de frecuencia\(X(w)\) | Notas |
|---|---|---|
| \ (x [n]\)” class="lt-eng-22895">\(\delta[n]\) | \ (X (w)\)” class="lt-eng-22895">\(1\) | |
| \ (x [n]\)” class="lt-eng-22895">\(\delta[n-M]\) | \ (X (w)\)” class="lt-eng-22895">\(e^{−j w M}\) | entero\(M\) |
| \ (x [n]\)” class="lt-eng-22895">\(\sum_{m=-\infty}^{\infty} \delta[n-M m]\) | \ (X (w)\)” class="lt-eng-22895">\(\sum_{m=-\infty}^{\infty} e^{-j w M m}=\frac{1}{M} \sum_{k=-\infty}^{\infty} \delta\left(\frac{w}{2 \pi}-\frac{k}{M}\right)\) | entero\(M\) |
| \ (x [n]\)” class="lt-eng-22895">\(e^{−jan}\) | \ (X (w)\)” class="lt-eng-22895">\(2 \pi \delta (w+a)\) | número real\(a\) |
| \ (x [n]\)” class="lt-eng-22895">\(u[n]\) | \ (X (w)\)” class="lt-eng-22895">\(\frac{1}{1-e^{-j w}}+\sum_{k=-\infty}^{\infty} \pi \delta(w+2 \pi k)\) | |
| \ (x [n]\)” class="lt-eng-22895">\(a^n u(n)\) | \ (X (w)\)” class="lt-eng-22895">\(\frac{1}{1-a e^{-j w}}\) | si\(|a|<1\) |
| \ (x [n]\)” class="lt-eng-22895">\(\cos(an)\) | \ (X (w)\)” class="lt-eng-22895">\(\pi[\delta(w-a)+\delta(w+a)]\) | número real\(a\) |
| \ (x [n]\)” class="lt-eng-22895">\(W \cdot \operatorname{sinc}^{2}(W n)\) | \ (X (w)\)” class="lt-eng-22895">\(\operatorname{tri}\left(\frac{w}{2 \pi W}\right)\) | número real\(W\),\(0<W≤0.50\) |
| \ (x [n]\)” class="lt-eng-22895">\(W \cdot \operatorname{sinc}[W(n+a)]\) | \ (X (w)\)” class="lt-eng-22895">\(\operatorname{rect}\left(\frac{w}{2 \pi W}\right) \cdot e^{j a w}\) | números reales\(W\),\(a\)\(0<W≤1\) |
| \ (x [n]\)” class="lt-eng-22895">\(\operatorname{rect}\left[\frac{(n-M / 2)}{M}\right]\) | \ (X (w)\)” class="lt-eng-22895">\(\frac{\sin [w(M+1) / 2]}{\sin (w / 2)} e^{-j w M / 2}\) | entero\(M\) |
| \ (x [n]\)” class="lt-eng-22895">\(\frac{W}{(n+a)}\{\cos [\pi W(n+a)]-\operatorname{sinc}[W(n+a)]\}\) | \ (X (w)\)” class="lt-eng-22895">\(j w \cdot \operatorname{rect}\left(\frac{w}{\pi W}\right) e^{j} a w\) | números reales\(W\),\(a\)\(0<W≤1\) |
| \ (x [n]\)” class="lt-eng-22895">\(\frac{1}{\pi n^{2}}\left[(-1)^{n}-1\right]\) | \ (X (w)\)” class="lt-eng-22895">\(|w|\) | |
| \ (x [n]\)” class="lt-eng-22895">\ (\ left\ {\ begin {array} {ll} 0 & n=0\\ \ frac {(-1) ^ {n}} {n} &\ text {en otra parte} \ end {array}\ right.\) |
\ (X (w)\)” class="lt-eng-22895">\(jw\) | filtro diferenciador |
| \ (x [n]\)” class="lt-eng-22895">\ (\ left\ {\ begin {array} {ll} 0 &\ quad n\ text {impar}\\ \ frac {2} {\ pi n} &\ quad n\ text {par} \ end {array}\ right.\) |
\ (X (w)\)” class="lt-eng-22895">\ (\ left\ {\ begin {array} {cc} j & w<0\\ 0 & w=0\\ -j & w>0 \ end {array}\ right.\) |
Transformación de Hilbert |
Notas
rect (\(t\)) es la función rectangular para valores reales arbitrarios\(t\).
\ [\ nombreoperador {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}\ derecho. \ nonumber\]
tri (\(t\)) es la función triangular para valores reales arbitrarios\(t\).
\ [\ 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 {de lo contrario}
\ end {array}\ derecho. \ nonumber\]