Alternative Representations

The questions below are due on Thursday February 27, 2025; 02:00:00 PM.

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Part a. Let c_k and d_k represent the trigonometric Fourier series coefficients for a periodic function f_1(t) of continuous time t with period T:

f_1(t)=f_1(t+T)=c_0+\sum_{k=1}^\infty c_k\cos(2\pi kt/T)+\sum_{k=1}^\infty d_k\sin(2\pi kt/T)

Determine expressions for the trigonometric Fourier series coefficients c_k' and d_k' for f_2(t)={d\over dt}f_1(t):

f_2(t)={d\over dt}f_1(t)=c_0'+\sum_{k=1}^\infty c_k'\cos(2\pi kt/T)+\sum_{k=1}^\infty d_k'\sin(2\pi kt/T)

as functions of c_0, c_1, c_2,... and/or d_0, d_1, d_2, ... .

Enter expressions for c_0', c_1', d_1', c_2', d_2', c_3', and d_3' in a tabular form, as illustrated below. Your expressions may contain any combination of the original (unprimed) coefficients (c_0, c_1, c_2, ... and d_0, d_1, d_2, ...) but must not include f_1(t) or f_2(t) or any integrals or infinite sums.

Notice that d_0' is not defined, and it's box is x'd out of the table.

Briefly explain.

Part b. Let a_k represent the complex exponential Fourier series coefficients for a periodic function f_3(t) of continuous time t with period T:

f_3(t)=f_3(t+T)=\sum_{k=-\infty}^\infty a_k e^{j2\pi kt/T}

Determine expressions for the complex exponential Fourier series coefficients a_k' for f_4(t)=f_3(t)\cos(2\pi t/T)

f_4(t)=f_3(t)\cos(2\pi t/T)=\sum_{k=-\infty}^\infty a_k' e^{j2\pi kt/T}

as functions of the original (unprimed) coefficients (a_k).

Enter expressions for a_{-2}', a_{-1}', a_0', a_1', and a_2' in the table below. Your expressions may contain any combination of the original (unprimed) coefficients a_k but must not include f_3(t) or f_4(t) or any integrals or infinite sums.

Briefly explain.

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