Fourier Series Representations of Modulation

The questions below are due on Thursday September 17, 2026; 02:00:00 PM.
 
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The frequency content of a signal can be (reversibly) altered by a process called modulation. Let g(t) represent a modulated version of the signal f(t), where

f(t)=\cos(2\pi10t)+2\sin(2\pi30t)
and
g(t)=f(t)\cos(2\pi1000t)\,.

As we will see later in the semester, modulation is widely used in communications systems based on electromagnetic (radio) waves, including cellular systems, wifi, bluetooth, GPS, and navigation systems.

Part a. Is the function g(t) periodic? If yes, determine the fundamental period of g(t). If not, explain how you know that it is not periodic.

Part b. Find a Fourier series representation of f(t) with the following form:

f(t)=c_0+\sum_{k=1}^\infty \Big(c_k\cos(k\omega_ot)+d_k\sin(k\omega_ot)\Big)
where \omega_o represents the fundamental frequency of the Fourier series.

Determine the value of \omega_o.

Determine the nonzero values of c_k and d_k and enter them in the table below. Use a separate line for each value of k for which either c_k or d_k or both are nonzero. Leave unused rows of the table empty.

Part c. Find a Fourier series representation of g(t) with the following form:

g(t)=c_0+\sum_{k=1}^\infty \Big(c_k\cos(k\omega_ot)+d_k\sin(k\omega_ot)\Big)
where \omega_o represents the fundamental frequency of the Fourier series.

Determine the value of \omega_o:

Determine the nonzero values of c_k and d_k and enter them in the table below. Use a separate line for each value of k for which either c_k or d_k or both are nonzero. Leave unused rows of the table empty.

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