Question

Given that a signal x(t) has the following complex exponential Fourier Series

parameters - where Cn is the nth component of the complex exponential Fourier series
- find the x(t) in time-domain,
w0=π,C0=2,C1=1,C3=ejπ,C−3=e−jπ

a. x(t)=2+e−²jπt+cos(3πt−π)
b. x(t)=2+e²jπt+cos(3πt+π)
c. x(t)=2+ejπt+cos(3πt−π)
d. x(t)=2+ejπt+cos(3πt+π)

Answer 1

Using the given complex exponential Fourier series coefficients, the time-domain signal x(t) can be reconstructed, resulting in option d: x(t) = 2 + ejπt + cos(3πt + π).

Step-by-step explanation:

The student asks to find the time-domain representation of a signal x(t) given its complex exponential Fourier series coefficients Cn, with a specific fundamental frequency w0. Based on the provided coefficients, the time-domain signal x(t) can be reconstructed using the inverse Fourier series formula:

1. C0 is the constant term, which is simply added as it is.
2. C1 and C-1 would correspond to the first harmonic with frequency w0, but C-1 is not provided and generally C-1 = C1* for real signals.
3. C3 and C-3 correspond to the third harmonic with frequency 3w0, therefore contributing a cosine function with a phase shift, since C3 = ejπ which has a phase shift of π radians.

Combining these elements, and considering that w0 = π and C0 = 2, C1 = 1, C3 = ejπ, C-3 = e-jπ, the equation with the terms reconstructed from the Fourier series would be:

x(t) = 2 + e-jπt + ejπtcos(3πt - π)

Since ejπt + e-jπt = 2cos(πt) by Euler's formula, we can rewrite the signal as:

x(t) = 2 + 2cos(πt) + cos(3πt - π)

Therefore, the correct time-domain representation of x(t) in the given options is:

d. x(t) = 2 + ejπt + cos(3πt + π)