CT Fourier Series

The questions below are due on Thursday February 20, 2025; 02:00:00 PM.
 
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Part 1

A continuous-time periodic signal x(t) is real-valued and has a fundamental period of T = 8. The nonzero Fourier series coefficients for x(t) are

X[1] = X[-1] = 2
X[3] = X^*[-3] = 4j

Express x(t) in the form:

x(t) = \sum_{k=0}^\infty A_k\cos(\omega_kt + \phi_k)

You do not need to find a closed form for this sum. For this part, you should enter your answers in terms of cos, pi, and any other constants necessary (but not sin, e, or j).

x(t) =~

Part 2

Use the Fourier series analysis equation to calculate the coefficients X[k] for the continuous-time periodic signal:

x(t) = x(t+2) = \begin{cases} 1.5, & \text{if}~0\leq t\lt 1\\ -1.5 & \text{if}~1\leq t\lt 2\\ \end{cases}

with a fundamental period T=2.

X[0] =~

For k\neq 0, X[k] =~

Part 3

Consider the following continuous-time periodic signal:

x(t) = 2 + \cos\left(\frac{2\pi}{3}t\right) + 4\sin\left(\frac{5\pi}{3}t\right)

What is the fundamental period T of this signal?

Using the fundamental period from above, what are the Fourier series coefficients X[k] of this signal? Enter your answer as a Python dictionary mapping values of k to the associated X[k] values. You may include as many entries as you want, but you only need to include entries where X[k] is nonzero.

Dictionary mapping k\to X[k]:

Part 4

Suppose the periodic signal x(t) has fundamental period T and Fourier coefficients X[k]. In a variety of situations, it is easier to calculate the Fourier series coefficients G[k] for g(t) = \frac{dx(t)}{dt} as opposed to calculating X[k] directly. Given that:

\int_T^{2T} x(t)dt = 2

find an expression for X[k] in terms of G[k] and T.

Enter your answers in terms of g_k (representing G[k]), k and/or T (and/or other common mathematical constants/operations such as e, j, and pi).

X[0] =~

For k\neq 0, X[k] =~