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Questions and Answers
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?
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What is the period of the signal x(t) in Figure (a)?
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What is the property of the Fourier series coefficient sequence ck of a real signal?
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What is the condition for a periodic signal x(t) to be an odd harmonic signal?
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If x(t) = -x(t - T/2) for all t, what can be concluded about the signal x(t)?
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What is the Fourier series coefficient sequence of the signal Even{x(t)} in terms of ck?
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What is the frequency response of the LTI system in the given problem?
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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)?
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What is the period of the periodic signal x(t) shown in the figure?
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Study Notes
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(Ï€t) + sin(2Ï€t)
- 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(2Ï€t)|
- 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
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Description
Find the Fourier series representation of various signals, including trigonometric functions and absolute value functions, and identify the fundamental period and Fourier series coefficients.
