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9.3: Transformadas de Fourier de Tiempo Discreto Común

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    86454
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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..