Señales Muestreo de una señal Sea una señal continua en el tiempo x ( t ) x(t) x ( t )
x [ n ] = x ( n T ) , donde T = 1 f s x[n] = x(nT), \quad \text{donde} \quad T = \frac{1}{f_s} x [ n ] = x ( n T ) , donde T = f s 1 Equivalentemente, podemos ver el muestreo como el producto de la señal continua y un tren de impulsos unitarios p T ( t ) p_T(t) p T ( t )
x s ( t ) = x ( t ) p T ( t ) , donde p T ( t ) = ∑ k = − ∞ ∞ δ ( t − k T ) x_s(t) = x(t)\,p_T(t), \quad \text{donde} \quad p_T(t) = \sum_{k=-\infty}^{\infty} \delta(t - kT) x s ( t ) = x ( t ) p T ( t ) , donde p T ( t ) = k = − ∞ ∑ ∞ δ ( t − k T ) Representación espectral Dado que la señal muestreada x s ( t ) = x ( t ) ⋅ p T ( t ) \large x_s(t) = x(t) \cdot p_T(t) x s ( t ) = x ( t ) ⋅ p T ( t ) X s ( f ) = X ( f ) ∗ P T ( f ) X_s(f)=X(f)\;\ast\;P_T(f) X s ( f ) = X ( f ) ∗ P T ( f )
P T ( f ) = F { p T ( t ) } = 1 T ∑ m = − ∞ ∞ δ ( f − m f s ) , f s = 1 T P_T(f)=\mathcal{F}\{p_T(t)\} =\frac{1}{T}\sum_{m=-\infty}^{\infty}\delta\bigl(f - m\,f_s\bigr),\quad f_s=\tfrac{1}{T} P T ( f ) = F { p T ( t )} = T 1 m = − ∞ ∑ ∞ δ ( f − m f s ) , f s = T 1 p T ( t ) p_T(t) p T ( t )
Finalmente, al convolucionar X ( f ) X(f) X ( f )
X s ( f ) = X ( f ) ∗ ( 1 T ∑ m δ ( f − m f s ) ) = 1 T ∑ m = − ∞ ∞ X ( f − m f s ) . X_s(f)=X(f)*\Bigl(\tfrac{1}{T}\sum_{m}\delta(f-mf_s)\Bigr)=\frac{1}{T}\sum_{m=-\infty}^{\infty}X(f - m\,f_s). X s ( f ) = X ( f ) ∗ ( T 1 m ∑ δ ( f − m f s ) ) = T 1 m = − ∞ ∑ ∞ X ( f − m f s ) . Es decir, el espectro original X ( f ) X(f) X ( f ) f s f_s f s
Para evitar aliasing, se aplica un filtro antialiasing pasabajo con frecuencia de corte f c ≤ f s / 2 f_c \le f_s/2 f c ≤ f s /2
x fil ( t ) = x ( t ) ∗ h LP ( t ) , H LP ( f ) = { 1 , ∣ f ∣ ≤ f c , 0 , ∣ f ∣ > f c .
x_{\text{fil}}(t) = x(t) * h_{\text{LP}}(t),
\quad H_{\text{LP}}(f) =
\begin{cases}
1, & |f| \le f_c,\\
0, & |f| > f_c.
\end{cases}
x fil ( t ) = x ( t ) ∗ h LP ( t ) , H LP ( f ) = { 1 , 0 , ∣ f ∣ ≤ f c , ∣ f ∣ > f c . Tras el muestreo, la reconstrucción ideal se realiza con un filtro pasabajo ideal de corte f s / 2 f_s/2 f s /2
x ^ ( t ) = ∑ n = − ∞ ∞ x [ n ] s i n c ( t − n T T ) , s i n c ( u ) = sin ( π u ) π u .
\hat{x}(t) = \sum_{n=-\infty}^{\infty}x[n]\,\mathrm{sinc}\!\bigl(\tfrac{t-nT}{T}\bigr),
\quad \mathrm{sinc}(u)=\frac{\sin(\pi u)}{\pi u}.
x ^ ( t ) = n = − ∞ ∑ ∞ x [ n ] sinc ( T t − n T ) , sinc ( u ) = π u sin ( π u ) . Muestreo y espectro, simulación x ( t ) = x 1 ( t ) + x 2 ( t ) + 0.2 x(t)=x_1(t)+x_2(t) +0.2 x ( t ) = x 1 ( t ) + x 2 ( t ) + 0.2 x s ( t ) = x ( t ) ⋅ p T ( t ) \large x_s(t)=x(t) \cdot p_T(t) x s ( t ) = x ( t ) ⋅ p T ( t )
x ( t ) = cos ( 2 ⋅ π ⋅ 3 ) + cos ( 2 ⋅ π ⋅ 2 ) + 0.2 \small x(t)= \cos\left(2\cdot\pi\cdot3\right)+\cos\left(2\cdot\pi\cdot2\right)+0.2 x ( t ) = cos ( 2 ⋅ π ⋅ 3 ) + cos ( 2 ⋅ π ⋅ 2 ) + 0.2 X ( f ) = X 1 ( f ) + X 2 ( f ) + 0.2 δ ( f ) \small X(f)=X_1(f)+X_2(f) +0.2\ \delta (f) X ( f ) = X 1 ( f ) + X 2 ( f ) + 0.2 δ ( f ) X s ( f ) = X ( f ) ∗ P T ( f ) X_s(f)=X(f) \ \ast \ P_T(f) X s ( f ) = X ( f ) ∗ P T ( f )
X s ( 5 ) = ∑ n = 0 N − 1 x [ n ] e − i 2 π 5 n / f s = 2.32 + − 0.04 i \textcolor{#00cc99}{X_s}({\textcolor{#f14e0e}{5}}) = \sum_{n=0}^{N-1} x[n] e^{- i 2 \pi \textcolor{#f14e0e}{5} n/f_s} = 2.32 + -0.04i X s ( 5 ) = n = 0 ∑ N − 1 x [ n ] e − i 2 π 5 n / f s = 2.32 + − 0.04 i
X ( f ) = 1 2 δ ( ∣ f ∣ − 2 π 3 2 π ) + 1 2 δ ( ∣ f ∣ − 2 π 2 2 π ) + 0.2 δ ( f ) \small X(f)= \frac{1}{2} \delta \left( |f| - \frac{2 \pi 3}{2 \pi } \right) + \frac{1}{2} \delta \left( |f| - \frac{2 \pi 2}{2 \pi } \right) + 0.2 \delta(f) X ( f ) = 2 1 δ ( ∣ f ∣ − 2 π 2 π 3 ) + 2 1 δ ( ∣ f ∣ − 2 π 2 π 2 ) + 0.2 δ ( f ) N = f s × T = 9 × 1 f r e q = 9 × 0.33 = 3.00 = 3 N = fs \times T = 9 \times \frac{1}{freq} = 9\times 0.33 = 3.00 = 3 N = f s × T = 9 × f re q 1 = 9 × 0.33 = 3.00 = 3 Teorema de escalamiento:x ( a t ) → F 1 ∣ a ∣ X ( f a ) \quad x(a\,t) \ \xrightarrow{\mathcal{F}} \ \frac{1}{\lvert a\rvert} X\!\left( \frac{f}{a}\right) x ( a t ) F ∣ a ∣ 1 X ( a f )
Espectro de: cos ( t ) → F 1 2 δ ( ∣ f ∣ − 1 2 π ) \quad \cos(t) \ \xrightarrow{\mathcal{F}} \ \frac{1}{2} \ \delta \!\left( |f| - \frac{1}{2\pi}\right) cos ( t ) F 2 1 δ ( ∣ f ∣ − 2 π 1 )
Escalamiento del impulso: δ ( a t ) = 1 ∣ a ∣ δ ( t ) \quad \delta(at) = \frac{1}{|a|} \delta(t) δ ( a t ) = ∣ a ∣ 1 δ ( t )
cos ( t ) → F 1 2 δ ( ∣ f ∣ − 1 2 π ) \quad \cos(t) \ \xrightarrow{\mathcal{F}} \ \frac{1}{2} \ \delta \!\left( |f| - \frac{1}{2\pi}\right) cos ( t ) F 2 1 δ ( ∣ f ∣ − 2 π 1 ) cos ( a t ) → F 1 2 ∣ a ∣ δ ( ∣ f a ∣ − 1 2 π ) \quad \cos(at) \ \xrightarrow{\mathcal{F}} \ \frac{1}{2|a|} \ \delta \!\left( \left| \frac{f}{a} \right| - \frac{1}{2\pi}\right) cos ( a t ) F 2∣ a ∣ 1 δ ( a f − 2 π 1 ) cos ( a t ) → F 1 2 ∣ a ∣ δ ( 1 a ( ∣ f ∣ − a 2 π ) ) \quad \cos(at) \ \xrightarrow{\mathcal{F}} \ \frac{1}{2|a|} \ \delta \! \left( \frac{1}{a} \left( |f| - \frac{a}{2\pi} \right) \right) cos ( a t ) F 2∣ a ∣ 1 δ ( a 1 ( ∣ f ∣ − 2 π a ) ) cos ( a t ) → F a 2 ∣ a ∣ δ ( ∣ f ∣ − a 2 π ) \quad \cos(at) \ \xrightarrow{\mathcal{F}} \ \frac{a}{2|a|} \ \delta \! \left( |f| - \frac{a}{2\pi} \right) cos ( a t ) F 2∣ a ∣ a δ ( ∣ f ∣ − 2 π a ) Convolución P T ( f ) = F { p T ( t ) } = 1 T ∑ m = − ∞ ∞ δ ( f − m f s ) , f s = 1 T = 9 \Large \textcolor{#ae58ff}{P_T(f)} \normalsize =\mathcal{F}\{p_T(t)\} =\frac{1}{T}\sum_{m=-\infty}^{\infty}\delta\left( f - m\,f_s\right),\quad f_s=\tfrac{1}{T} = 9 P T ( f ) = F { p T ( t )} = T 1 m = − ∞ ∑ ∞ δ ( f − m f s ) , f s = T 1 = 9 X ~ ( f ) = F ~ { x ( t ) } \Large \textcolor{#ee00ab}{\widetilde{X}(f)} \normalsize =\widetilde{\mathcal{F}}\{x(t)\} X ( f ) = F { x ( t )} X s ( f ) = X ~ ( f ) ∗ P T ( f ) = 1 T ∑ m = − ∞ ∞ X ~ ( f ) δ ( f − m f s ) \textcolor{#00cc99}{X_s(f)} = \textcolor{#ee00ab}{\widetilde{X}(f)} \ast\textcolor{#ae58ff}{P_T(f)} =\frac{1}{T}\sum_{m=-\infty}^{\infty}\widetilde{X}(f)\delta (f - mf_s) X s ( f ) = X ( f ) ∗ P T ( f ) = T 1 m = − ∞ ∑ ∞ X ( f ) δ ( f − m f s ) Reiniciar
Recuperación de la señal Al muestrear x(t), su espectro X ( f ) X(f) X ( f ) f s f_s f s − f s 2 ≤ f ≤ f s 2 -\tfrac{f_s}{2} \le f \le \tfrac{f_s}{2} − 2 f s ≤ f ≤ 2 f s f s f_s f s 1 / T 1/T 1/ T H ( f ) = 1 T r e c t ( f f s ) \displaystyle H(f)=\frac{1}{T}\,\mathrm{rect}\left( \tfrac{f}{f_s}\right) H ( f ) = T 1 rect ( f s f ) h ( t ) = s i n c ( t / T ) = sin ( π t / T ) π t / T \displaystyle h(t) = \mathrm{sinc}\left( t/T\right) = \frac{\sin(\pi t/T)}{\pi t/T} h ( t ) = sinc ( t / T ) = π t / T sin ( π t / T )
Las funciones notadas con h ( t ) h(t) h ( t ) H ( f ) H(f) H ( f ) sistemas lineales e invariantes en el tiempo (LTI) y su análisis en el dominio de la frecuencia.
Funciones Especiales Rect rect ( t ) = { 1 , ∣ t ∣ < 1 2 , 1 2 , ∣ t ∣ = 1 2 , 0 , ∣ t ∣ > 1 2 . \textcolor{#80F700}{\text{rect}}(t) =
\begin{cases}
1, & |t| < \tfrac{1}{2},\\
\tfrac12, & |t| = \tfrac{1}{2},\\
0, & |t| > \tfrac{1}{2}.
\end{cases} rect ( t ) = ⎩ ⎨ ⎧ 1 , 2 1 , 0 , ∣ t ∣ < 2 1 , ∣ t ∣ = 2 1 , ∣ t ∣ > 2 1 . F { rect ( t ) } = sinc ( π f ) \mathcal{F}\{\textcolor{#80F700}{\text{rect}}(t)\} = \text{ \textcolor{#f14e0e}{\text{sinc}}}(\pi f) F { rect ( t )} = sinc ( π f )
Sinc sinc ( t ) = sin ( t ) t , sinc ( 0 ) = 1. \textcolor{#f14e0e}{\text{sinc}} (t) = \frac{\sin(t)}{t}, \qquad \textcolor{#f14e0e}{\text{sinc}}(0) = 1. sinc ( t ) = t sin ( t ) , sinc ( 0 ) = 1. F { sinc ( π t ) } = rect ( f ) \mathcal{F}\{ \textcolor{#f14e0e}{\text{sinc}} (\pi t)\} = \textcolor{#80F700}{\text{rect}}(f) F { sinc ( π t )} = rect ( f )
Recordar el Teorema de escalamiento:x ( a t ) → F 1 ∣ a ∣ X ( f a ) \quad \large x(a\,t)\;\xrightarrow{\mathcal{F}}\;\frac{1}{\lvert a\rvert}\;X\!\Bigl(\frac{f}{a}\Bigr) x ( a t ) F ∣ a ∣ 1 X ( a f )
r e c t ( t τ ) → F τ s i n c ( τ f ) rect\!\Bigl(\frac{t}{\tau}\Bigr) \;\xrightarrow{\mathcal{F}}\; \tau\; sinc(\tau\,f) rec t ( τ t ) F τ s in c ( τ f ) s i n c ( t τ ) → F τ r e c t ( τ f ) sinc\!\Bigl(\frac{t}{\tau}\Bigr) \;\xrightarrow{\mathcal{F}}\; \tau\; rect(\tau\,f) s in c ( τ t ) F τ rec t ( τ f ) Sistemas LTI Un sistema LTI (Lineal e Invariante en el Tiempo) es un modelo matemático que permite caracterizar completamente el comportamiento de cualquier entrada mediante convolución con su respuesta al impulso h ( t ) h(t) h ( t ) H ( f ) H(f) H ( f )
