Fourier Series Representation of Signals

What is the fundamental period of the signal x(t) = 1+cosπt+sin2πt?

What is the Fourier series coefficient sequence ck for the signal x(t) = [cos4t][sint]?

What is the condition for a complex periodic signal x(t) to be even?

What is the derivative of the signal x(t) with Fourier series coefficient sequence ck given by ck = 1 for k = 0, ck = 1/j for k =±1, ck = 0 otherwise?

What is the period of the signal x(t) in Figure (a)?

What is the property of the Fourier series coefficient sequence ck of a real signal?

What is the condition for a periodic signal x(t) to be an odd harmonic signal?

If x(t) = -x(t - T/2) for all t, what can be concluded about the signal x(t)?

What is the Fourier series coefficient sequence of the signal Even{x(t)} in terms of ck?

What is the frequency response of the LTI system in the given problem?

What is the output y(t) of the LTI system if the input x(t) is given by x(t) = 1 + 2cos(2t) + 2cos(4t) + 1/2 cos(6t)?

What is the period of the periodic signal x(t) shown in the figure?

Fourier Series Representation

The Fourier series representation can be written in complex exponential form for each signal
To find the Fourier series representation, identify the fundamental period and Fourier series coefficient sequence ck

Signals and Their Representations

Signal x(t) = 1 + cos(πt) + sin(2πt)
- Fundamental period: T = 2
- Fourier series coefficient sequence ck: ck = 1 for k = 0, ck = 1/2 for k = ±1, ck = -i/2 for k = ±2
Signal x(t) = [cos(4t)][sin(t)]
- Fundamental period: T = π
- Fourier series coefficient sequence ck: ck = 1/4 for k = ±1, ck = -i/4 for k = ±3
Signal x(t) = |sin(2πt)|
- Fundamental period: T = 1
- Fourier series coefficient sequence ck: ck = 2/π for k = ±1, ck = 4/(3π) for k = ±3

Properties of Fourier Series

If x(t) is a periodic signal with Fourier series coefficient sequence ck, then:
- x(t) is real if ck = ck* (complex conjugate)
- x(t) is even if ck = c-k
- d/dt x(t) is even if ck = -ck
If x(t) is odd harmonic, then ck = 0 for all even k
If x(t) is odd harmonic, then x(t) = -x(t - T/2) for all t

Fourier Series Coefficients

The Fourier series coefficient sequence dk of the signal d/dt x(t) can be determined from the Fourier series coefficient sequence ck of x(t)
The Fourier series coefficient sequence of Even{x(t)} and Re{x(t)} can be determined from the Fourier series coefficient sequence ck of x(t)

Periodic Signals and Their Spectra

A periodic signal x(t) with fundamental period T and Fourier series coefficient sequence ck has a frequency spectrum
The frequency spectrum of a signal can be plotted, including the first five harmonics

Find the Fourier series representation of various signals, including trigonometric functions and absolute value functions, and identify the fundamental period and Fourier series coefficients.