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The questions below are due on Thursday April 10, 2025; 02:00:00 PM.

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Part 1

Consider a Linear Time Invariant (LTI) system (system 1), charactertized by a unit sample response h_1[\cdot]:

We know that when the input to this system is x[n]=\delta[n]+\delta[n+1], the output is given by:

y[n]=2\delta[n]+2\delta[n-1]+\delta[n-2]+\delta[n+1]

For each of the input signals x_i[\cdot] below, find the corresponding output, specified by y_i[\cdot] or Y_i(\cdot):

Enter your answer as a Python expression, which can depend on pi, e, j, OMEGA, and/or delta(...)(to represent \delta[...] or \delta(...), depending on the context).

x_1 [n]=2\delta[n-2]
y_1[n]:

x_2 [n]=\sin(2\pi n/3)
Y_2(\Omega):

x_3 [n]=\cos(\pi n/4)
Hint: The answer to this part can be written as a weighted sum of two delta functions,where the weights are simple numerical constants (not functions of \Omega).
Y_3(\Omega):

Part 2

Let H_2(\Omega) be the frequency response of another system (system 2), whose unit sample response is known to be:

h_2[n]=\delta[n]+\delta[n-2]+\delta[n+2]

And let H_3[k] be a sampled version of H_2(\cdot):

H_3[k]=H_2(2\pi k/4)

Compute the first four samples of h_3[\cdot], and enter your answer as a Python list (containing only single numbers) below:

[h_3[0],h_3[1],h_3[2],h_3[3]]:

Part 3

Consider a signal h_4[\cdot], whose DFT is related to H_4[\cdot]:

H_4[k]=H_3[k]\times H_3[k]

Compute the first four samples of h_4[\cdot], and enter your answer as a Python list (containing only single numbers) below:

[h_4[0],h_4[1],h_4[2],h_4[3]]:

Part 4

Finally, consider a new system h_5[\cdot], created by cascading two of system 2 as shown in the following diagram:

Find the new system's unit sample response h_5[n], Enter your answer as a Python expression, which can depend on pi, e, OMEGA, and/or delta(...)(to represent \delta[...]). :

h_5[n]: