Question

Use the definitions sinc(x) = \frac{\sin(\pi x)}{\pi x}, p_T(t) = \begin{cases} 1, & |t| \le T \\ 0, & |t| > T \end{cases} and the Fourier transform pairs \\ $p_T(t) \stackrel{F}{\longrightarrow} 2T\text{sinc}(\frac{\omega T}{\pi}), \frac{B}{\pi}\text{sinc}(\frac{tB}{\pi}) \stackrel{F}{\longrightarrow} P_B(\omega)$, \\ $\frac{1}{\sigma\sqrt{2\pi}}e^{-(\frac{t^2}{2\sigma^2})} \stackrel{F}{\longrightarrow} e^{-\frac{1}{2}(\sigma\omega)^2}$, \\ and other transform pairs, as needed, in all of the problems below. 6.24. The signal $f(t) = \cos(150\pi t)$ is sampled with an impulse train with period $T = 8\text{ms}$. (a) Find and sketch the sampled spectrum. (b) Can the signal be reconstructed by using a low-pass filter? Explain your answer. . Also, find a bandpass filter that can be used to reconstruct the signal f(t). Find the impulse response for this bandpass filter.

          Use the definitions sinc(x) = \frac{\sin(\pi x)}{\pi x}, p_T(t) = \begin{cases} 1, & |t| \le T \\ 0, & |t| > T \end{cases} and the Fourier transform pairs \\
$p_T(t) \stackrel{F}{\longrightarrow} 2T\text{sinc}(\frac{\omega T}{\pi}), \frac{B}{\pi}\text{sinc}(\frac{tB}{\pi}) \stackrel{F}{\longrightarrow} P_B(\omega)$, \\
$\frac{1}{\sigma\sqrt{2\pi}}e^{-(\frac{t^2}{2\sigma^2})} \stackrel{F}{\longrightarrow} e^{-\frac{1}{2}(\sigma\omega)^2}$, \\
and other transform pairs, as needed, in all of the problems below.
6.24. The signal $f(t) = \cos(150\pi t)$ is sampled with an impulse train with period $T = 8\text{ms}$.
(a) Find and sketch the sampled spectrum.
(b) Can the signal be reconstructed by using a low-pass filter? Explain your answer.
. Also, find a bandpass filter that can be used to reconstruct the signal
f(t). Find the impulse response for this bandpass filter.

Use the definitions sinc(x) = (sin(πx))/(πx), pT(t) = 1, |t| ≤T
0, |t| > T and the Fourier transform pairs

pT(t) F⟶ 2Tsinc((ω T)/(π)), (B)/(π)sinc((tB)/(π)) F⟶ PB(ω),

(1)/(σ√(2π))e^-((t^2)/(2σ^2))F⟶ e^-(1)/(2)(σω)^2,

and other transform pairs, as needed, in all of the problems below.
6.24. The signal f(t) = cos(150π t) is sampled with an impulse train with period T = 8ms.
(a) Find and sketch the sampled spectrum.
(b) Can the signal be reconstructed by using a low-pass filter? Explain your answer.
. Also, find a bandpass filter that can be used to reconstruct the signal
f(t). Find the impulse response for this bandpass filter.

Added by Audrey M.

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