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Deterministic Sampling and Reconstruction

x(t) bandlimited to $\pm W Hz$
Sample at at least
1/2W second intervals and x(t) can be obtained from samples.WHY?

no aliasing


\begin{displaymath}\frac{1}{T}\ge 2W\rightarrow T \le \frac{1}{2W} \mbox{\qquad NYQUIST RATE}
\end{displaymath}

aside


\begin{displaymath}x(t)s(t)\Rightarrow X(f)\ast S(f)
\end{displaymath}


\begin{displaymath}s(t)=\sum_n\delta(t-nT) \mbox{is periodic so t has FS}
\end{displaymath}

Therefore,


\begin{displaymath}S(f)=\sum_k s_k\delta(f-kf_0)
\end{displaymath}


\begin{displaymath}s_k=\frac{1}{T}\int_{-T/2}^{T/2}\delta(t)e^{-j\frac{2\pi}{T}kt}dt=\frac{1}{T}
\end{displaymath}


\begin{displaymath}S(f)=\frac{1}{T}\sum_k\delta(f-\frac{k}{T})
\end{displaymath}

Thererfore


\begin{displaymath}X(f)\ast S(f)=\frac{1}{T}\sum_kX(f-\frac{k}{T})
\end{displaymath}

Recovery $\Rightarrow$ *******FIG.7*********




1999-02-01