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Frequency-domain sampling using Fourier series

Frequency-domain sampling. Consider the following discrete-time signal

$$ x(n)= \begin{cases}a^{|n|}, & |n| \leq L \\ 0, & |n|>L\end{cases} $$

where \(a=0.95\) and \(L=10\).

(a) Compute and plot the signal \(x(n)\).

(b) Show that

$$ X(\omega)=\sum_{n=-\infty}^{\infty} x(n) e^{-j \omega n}=x(0)+2 \sum_{n-1}^{L} x(n) \cos \omega n $$

Plot \(X(\omega)\) by computing it at \(\omega=\pi k / 100, k=0,1, \ldots, 100\).

(c) Compute

$$ c_{k}=\frac{1}{N} X\left(\frac{2 \pi}{N} K\right), \quad k=0,1, \ldots, N-1 $$

for \(N=30\).

(d) Determine and plot the signal

$$ \tilde{x}(n)=\sum_{k=0}^{N-1} c k e^{j(2 \pi / N) k n} $$

What is the relation between the signals \(x(n)\) and \(\tilde{x}(n)\) ? Explain.

(e) Compute and plot the signal \(\tilde{x}_{1}(n)=\sum_{l=-\infty}^{\infty} x(n-l N),-L \leq n \leq L\) for \(N=30\).

Compare the signals \(\tilde{x}(n)\) and \(\tilde{x}_{1}(n)\).

(f) Repeat parts (c) to (e) for \(N=15\).

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