DT Fourier Series

The questions below are due on Thursday September 25, 2025; 02:00:00 PM.

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Determine the Fourier series coefficients for each of the following DT signals, which are periodic in N=8.

Part 1

What are the Fourier series coefficients of x_1[n]?

X_1[{-8}] = X_1[0] = X_1[8] = \ldots =~

X_1[{-7}] = X_1[1] = X_1[9] = \ldots =~

X_1[{-6}] = X_1[2] = X_1[10] = \ldots =~

X_1[{-5}] = X_1[3] = X_1[{11}] = \ldots =~

X_1[{-4}] = X_1[4] = X_1[{12}] = \ldots =~

X_1[{-3}] = X_1[5] = X_1[{13}] = \ldots =~

X_1[{-2}] = X_1[6] = X_1[{14}] = \ldots =~

X_1[{-1}] = X_1[7] = X_1[{15}] = \ldots =~

Part 2

What are the Fourier series coefficients of x_2[n]? Enter a closed-form expression for X_2[k] (that may depend on k):

X_2[k] = X_2[k+8] =~

Use Python notation for multiplication, exponentiation, &c. Some mathematical constants and functions are available (pi, j, e, sin, cos, &c).

Part 3

What are the Fourier series coefficients of x_3[n]? Enter a closed-form expression for X_3[k]:

X_3[k] = X_3[k+8] =~

Part 4

What are the Fourier series coefficients of x_4[n]?

X_4[{-8}] = X_4[0] = X_4[8] = \ldots =~

X_4[{-7}] = X_4[1] = X_4[9] = \ldots =~

X_4[{-6}] = X_4[2] = X_4[10] = \ldots =~

X_4[{-5}] = X_4[3] = X_4[11] = \ldots =~

X_4[{-4}] = X_4[4] = X_4[12] = \ldots =~

X_4[{-3}] = X_4[5] = X_4[13] = \ldots =~

X_4[{-2}] = X_4[6] = X_4[14] = \ldots =~

X_4[{-1}] = X_4[7] = X_4[15] = \ldots =~