Slow Down

The questions below are due on Thursday March 20, 2025; 02:00:00 PM.
 
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Let x[n] represent a discrete time signal whose DTFT is given by

X(\Omega) = \cases{1&if $|\Omega|\lt{\pi\over5}$\cr0&if ${\pi\over5}\lt|\Omega|\lt\pi$\cr}

and is periodic in \Omega with period 2\pi as shown below.

Part a. Determine an expression for x[n]. Plot x[n] and label the important features of your plot.

Part b. A new signal y_0[n] is derived by stretching x[n] as follows:

y_0[n] = \cases{x\left[{n\over2}\right]&if $n$ is even\cr0&otherwise\cr}

Make a plot of y_0[n] and label its key features.

Part c. Determine an expression for Y_0(\Omega) in terms of X(\Omega). Sketch the magnitude and angle of Y_0(\Omega) on the axes below. Label all important parameters of your plots.

Briefly describe the important differences between X(\Omega) and Y_0(\Omega).

Part d. The y_0[n] signal alternates between non-zero and zero values. To reduce the effect of the zero values, we define

y_1[n]={1\over2}y_0[n{-}1]+y_0[n]+{1\over2}y_0[n{+}1]

Plot y_1[n] and label the important features of your plot. Briefly describe the relation between y_0[n] and y_1[n].

Part e. Determine an expression for Y_1(\Omega) (the Fourier transform of y_1[n]) in terms of Y_0(\Omega). Briefly describe the relation between Y_0(\Omega) and Y_1(\Omega).

Please upload a single pdf file that contains your answers to all parts of this problem:

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