Discrete-Time Fourier Transforms

The questions below are due on Thursday March 13, 2025; 02:00:00 PM.
 
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Part a. Let F(\Omega) represent the Fourier transform of a discrete-time signal f[n].

F(\Omega)=\cases{1&if $|\Omega|\le\pi/2$\cr0&if $\pi/2\lt|\Omega|\lt\pi$\cr F(\Omega\mod2\pi)&if $|\Omega|\gt\pi$\cr}

Determine an expression for f[n].
Sketch f[n] versus n for -7\le n\le7 and label all important values.

Part b.

Let g[n] represent a discrete-time signal whose Fourier transform G(\Omega) is shown below.

Define a new signal h[n] as follows:

h[n] = \cases{g[n/2]&if $n$ is even\cr0&otherwise\cr}
Determine an expression for H(\Omega) in terms of G(\Omega). Sketch the magnitude and phase of H(\Omega) and label all important values.

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