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# 9.3: Transformadas de Fourier de Tiempo Discreto Común

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## DTFT comunes

Mesa$$\PageIndex{1}$$
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\]

This page titled 9.3: Transformadas de Fourier de Tiempo Discreto Común is shared under a CC BY license and was authored, remixed, and/or curated by Richard Baraniuk et al..