Pulse Trains

The questions below are due on Thursday April 10, 2025; 02:00:00 PM.
 
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Part 1.

Let X_1[k] represent the Discrete Fourier Transform (DFT) of x_1[n]=\delta[n] when the analysis window N_1 is 4.

Determine X_1[k].

Enter your result in the box below as a Python list of numbers of the form [X_1[0], X_1[1], X_1[2], X_1[3]].
Your list can include numbers and numeric expressions (such as fractions) but should not include variables or summations.

Part 2.

Let X_2[k] represent the DFT of x_2[n]=x_1[n\mod4] when the analysis window N_2 is 8.

Determine X_2[k].

Enter your result in the box below as a Python list of numbers of the form [X_2[0], X_2[1], X_2[2], ..., X_2[N_2{-}1].
Your list can include numbers and numeric expressions (such as fractions) but should not include variables or summations.

Part 3.

Let X_3[k] represent the DFT of x_3[n]=x_2[n\mod8] when the analysis window N_3 is 16.

Determine X_3[k].

Enter your result in the box below as a Python list of numbers of the form [X_3[0], X_3[1], X_3[2], ..., X_3[N_3{-}1].
Your list can include numbers and numeric expressions (such as fractions) but should not include variables or summations.

Part 4.

Let y_1[n], y_2[n], and y_3[n] represent periodically extended versions of x_1[n], x_2[n], and x_3[n], respectively:

  • y_1[n]=x_1[n\mod4]

  • y_2[n]=x_2[n\mod8]

  • y_3[n]=x_3[n\mod16]

Notice that y_1[n]=y_2[n]=y_3[n] for all n!

Use your result from part 1 to write an expression for y_1[n] as a sum of complex exponentials.

Enter your result as a Python expression in the box below. Your expression may contain numbers, constants (such as e or pi), and the variable n, but should not contain other variables or summations.

Use your result from part 2 to write an expression for y_2[n] as a sum of complex exponentials.

Enter your result as a Python expression in the box below. Your expression may contain numbers, constants (such as e or pi), and the variable n, but should not contain other variables or summations.

Use your result from part 3 to write an expression for y_3[n] as a sum of complex exponentials.

Enter your result as a Python expression in the box below. Your expression may contain numbers, constants (such as e or pi), and the variable n, but should not contain other variables or summations.

Check Yourself: What are the relationships between your expressions for y_1[n], y_2[n], and y_3[n]?