8.3: Transformadas comunes de Fourier
- Page ID
- 86328
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Propiedades comunes de CTFT
| Señal de dominio de tiempo | Señal de dominio de frecuencia | Condición |
|---|---|---|
| \(e^{-(a t)} u(t)\) | \(\frac{1}{a+j \omega}\) | \(a>0\) |
| \(e^{at}u(−t)\) | \(\frac{1}{a-j \omega}\) | \(a>0\) |
| \(e^{−(a|t|)}\) | \(\frac{2a}{a^2+\omega^2}\) | \(a>0\) |
| \(te^{−(at)}u(t)\) | \(\frac{1}{(a+j \omega)^2}\) | \(a>0\) |
| \(t^ne^{−(at)}u(t)\) | \(\frac{n !}{(a+j \omega)^{n+1}}\) | \(a>0\) |
| \(\delta(t)\) | \(1\) | |
| \(1\) | \(2 \pi \delta(\omega)\) | |
| \(e^{j \omega_0 t}\) | \(2 \pi \delta\left(\omega-\omega_{0}\right)\) | |
| \( \cos (\omega_0 t) \) | \(\pi\left(\delta\left(\omega-\omega_{0}\right)+\delta\left(\omega+\omega_{0}\right)\right)\) | |
| \(\sin (\omega_0 t)\) | \(j \pi\left(\delta\left(\omega+\omega_{0}\right)-\delta\left(\omega-\omega_{0}\right)\right)\) | |
| \(u(t)\) | \(\pi \delta(\omega)+\frac{1}{j \omega}\) | |
| sgn (\(t)\) | \(\frac{2}{j \omega}\) | |
| \(\cos \left(\omega_{0} t\right) u(t)\) | \(\frac{\pi}{2}\left(\delta\left(\omega-\omega_{0}\right)+\delta\left(\omega+\omega_{0}\right)\right)+\frac{j \omega}{\omega_{0}^{2}-\omega^{2}}\) | |
| \(\sin \left(\omega_{0} t\right) u(t)\) | \(\frac{\pi}{2 j}\left(\delta\left(\omega-\omega_{0}\right)-\delta\left(\omega+\omega_{0}\right)\right)+\frac{\omega_{0}}{\omega_{0}^{2}-\omega^{2}}\) | |
| \(e^{-(a t)} \sin \left(\omega_{0} t\right) u(t)\) | \(\frac{\omega_{0}}{(a+j \omega)^{2}+\omega_{0}^{2}}\) | \(a>0\) |
| \(e^{-(a t)} \cos \left(\omega_{0} t\right) u(t)\) | \(\frac{a+j \omega}{(a+j \omega)^{2}+\omega_{0}^{2}}\) | \(a>0\) |
| \(u(t+\tau)-u(t-\tau)\) | \(2 \tau \frac{\sin (\omega \tau)}{\omega \tau}=2 \tau \operatorname{sinc}(\omega t)\) | |
| \(\frac{\omega_{0}}{\pi} \frac{\sin \left(\omega_{0} t\right)}{\omega_{0} t}=\frac{\omega_{0}}{\pi} \operatorname{sinc}\left(\omega_{0}\right)\) | \(u\left(\omega+\omega_{0}\right)-u\left(\omega-\omega_{0}\right)\) | |
| \ (\ begin {array} {l} \ izquierda (\ frac {t} {\ tau} +1\ derecha)\ izquierda (u\ izquierda (\ frac {t} {\ tau} +1\ derecha) -u\ izquierda (\ frac {t} {\ tau}\ derecha)\ derecha)\ \ izquierda (-\ frac {t} {\ tau}\ +1 derecha)\ izquierda (u\ izquierda (\ frac {t} {\ tau}\ derecha) -u\ izquierda (\ frac {t} {\ tau} -1\ derecha)\ derecha) =\\ \ operatorname {triag}\ left (\ frac {t} {2\ tau}\ derecha) \ end {array}\) |
\(\tau \operatorname{sinc}^{2}\left(\frac{\omega \tau}{2}\right)\) | |
| \(\frac{\omega_{0}}{2 \pi} \operatorname{sinc}^{2}\left(\frac{\omega_{0} t}{2}\right)\) | \ (\ begin {array} {l} \ izquierda (\ frac {\ omega} {\ omega_ {0}} +1\ derecha)\ izquierda (u\ izquierda (\ frac {\ omega} {\ omega_ {0}} +1\ derecha) -u\ izquierda (\ frac {\ omega} {\ omega_ {0}}\ derecha)\ derecha) +\\ izquierda (- \ frac {\ omega} {\ omega_ {0}} +1\ derecha)\ izquierda (u\ izquierda (\ frac {\ omega} {\ omega_ {0}}\ derecha) -u\ izquierda (\ frac {\ omega} {\ omega} {\ omega_ {0}} -1\ derecha)\ derecha) =\\ \ nombreoperador {triag}\ izquierda (\ frac {\ omega} {2\ omega_ {0}}\ derecha) \ end {array}\) |
|
| \(\sum_{n=-\infty}^{\infty} \delta(t-n T)\) | \(\omega_{0} \sum_{n=-\infty}^{\infty} \delta\left(\omega-n \omega_{0}\right)\) | \(\omega_0 = \frac{2 \pi}{T}\) |
| \(e^{-\frac{t^{2}}{2 \sigma^{2}}}\) | \(\sigma \sqrt{2 \pi} e^{-\frac{\sigma^{2} \omega^{2}}{2}}\) |
triag [n] es la función triangular para valores reales arbitrarios\(n\).
\ [\ nombreoperador {triag} [\ mathrm {n}] =\ left\ {\ begin {array} {ll}
1+n &\ text {if} -1\ leq n\ leq 0\\
1-n &\ text {if} 0<n\ leq 1\\
0 &\ text {de lo contrario}
\ end {array}\ derecho. \ nonumber\]