Determine the CFT of p(t)=∑n=-∞^∞ δ(t-n Ts). Given, xs(t)=x(t) p(t), derive the following,

Step 1: Understand the problem statement. We are given a periodic impulse train ( p(t) = sum_{

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Determine the CFT of $p(t)=\sum_{n=-\infty}^{\infty} \delta\left(t-n T_s\right)$. Given, $x_s(t)=x(t) p(t)$, derive the following, $$ \begin{aligned} & X_s(\omega)=\frac{1}{T_s} \sum_{k=-\infty}^{\infty} X\left(\omega-k \omega_s\right) \\ & x(t)=\sum_{n=-\infty}^{\infty} x\left(n T_s\right) \operatorname{sinc}\left(B\left(t-n T_s\right)\right), \end{aligned} $$ where $X(\omega)$ and $X_s(\omega)$ are the spectra of ideally bandlimited and uniformly sampled signals, respectively, and $\omega_s=2 \pi / T_s$. (Refer to Figure 2.8 for variable definitions.)

   Determine the CFT of $p(t)=\sum_{n=-\infty}^{\infty} \delta\left(t-n T_s\right)$. Given, $x_s(t)=x(t) p(t)$, derive the following,
$$
\begin{aligned}
& X_s(\omega)=\frac{1}{T_s} \sum_{k=-\infty}^{\infty} X\left(\omega-k \omega_s\right) \\
& x(t)=\sum_{n=-\infty}^{\infty} x\left(n T_s\right) \operatorname{sinc}\left(B\left(t-n T_s\right)\right),
\end{aligned}
$$
where $X(\omega)$ and $X_s(\omega)$ are the spectra of ideally bandlimited and uniformly sampled signals, respectively, and $\omega_s=2 \pi / T_s$. (Refer to Figure 2.8 for variable definitions.)

Audio Signal Processing and Coding

Andreas Spanias, Ted… 1st Edition

Chapter 2, Problem 4

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Step 1

We are given a periodic impulse train $ p(t) = \sum_{n=-\infty}^{\infty} \delta(t - nT_s) $ and a sampled signal $ x_s(t) = x(t) p(t) $. We need to find the continuous-time Fourier transform (CFT) of $ x_s(t) $ and derive the given expressions for \( Show more…

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Determine the CFT of $p(t)=\sum_{n=-\infty}^{\infty} \delta\left(t-n T_s\right)$. Given, $x_s(t)=x(t) p(t)$, derive the following, $$ \begin{aligned} & X_s(\omega)=\frac{1}{T_s} \sum_{k=-\infty}^{\infty} X\left(\omega-k \omega_s\right) \\ & x(t)=\sum_{n=-\infty}^{\infty} x\left(n T_s\right) \operatorname{sinc}\left(B\left(t-n T_s\right)\right), \end{aligned} $$ where $X(\omega)$ and $X_s(\omega)$ are the spectra of ideally bandlimited and uniformly sampled signals, respectively, and $\omega_s=2 \pi / T_s$. (Refer to Figure 2.8 for variable definitions.)

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