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Properties of the Fourier Series

February 2, 2019 by 3200 Creative

Properties of the Fourier Series

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Question 1 of 4

1. Question
A periodic signal $x(t)$ has Fourier series coefficients $X[k]$ . If $y(t)=x(t+5)$ , what are $Y[k]$ ? The period of both signals is $T=20$ sec.
- $Y[k]=X[k+5]$
- $Y[k]=e^{jk5}X[k]$
- $Y[k]=e^{jk5\pi}X[k]$
- $Y[k]=e^{jk\pi/2}X[k]$
- $Y[k]=e^{jk\pi/4}X[k]$
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Question 2 of 4

2. Question
You form the Fourier series coefficients $Z[k]$ for a signal $z(t)$ as $Z[k]=0.5X[k+1]+0.5X[k-1]$ . How is $z(t)$ related to $x(t)$ ? Assume both $z(t)$ and $x(t)$ have the same period of $T=1$ sec.
- $z(t) = 0.5x(t+1) + 0.5x(t-1)$
- $z(t) = \sin(2\pi t)x(t)$
- $z(t) = 0.5\sin(2\pi t)x(t)$
- $z(t) = \cos(2\pi t)x(t)$
- $z(t) = 0.5\cos(2\pi t)x(t)$
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Question 3 of 4

3. Question
You form a signal $z(t)=x^2(t)$ . How are the Fourier series coefficients $Z[k]$ related to $X[k]$ ?
- $Z[k] = X^{1/2}[k]$
- $Z[k] = X[k]X^*[k]$
- $Z[k] = 2X[k]$
- $Z[k] = \sum_{p=-\infty}^\infty X[p]X[k-p]$
- $Z[k] = \sum_{p=-\infty}^\infty X[p]X[k+p]$
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Question 4 of 4

4. Question
A signal with period $T=2$ has Fourier series coefficients $X[k] = \left\{\begin{array}{cl} 1,&k=\pm 2, \pm 4\\0& k\ne\pm 2, \pm 4 \end{array}\right.$ . What is the value of $c=\int_0^2|x(t)|^2 dt$ ?
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Course Lessons

The Fourier Series: Continuous-Time Periodic Signals
Square Wave Fourier Series and the Sinc Function
Fourier Series Properties
The Fourier Transform: Linking Time and Frequency Domains
Properties of the Fourier Transform
The Discrete-Time Fourier Transform
Discrete-Time Fourier Transform Properties
Fourier Transforms and Discrete-Time Fourier Transforms for Periodic Signals
Frequency-Domain Descriptions for Continuous-Time Linear Time-Invariant Systems
Frequency-Domain Descriptions for Discrete-Time Linear Time-Invariant Systems
Two-Dimensional Signal Processing: Continuous Space

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Time Domain LTI Systems
Fourier Series and Transforms
Sampling and Reconstruction
The DFT and Applications
The Z-Transform
Intro to Filter Design
IIR Filter Design
FIR Filter Design
Random Signal Characterization
Basis Representations of Signals
Estimation of Power Spectra and Coherence
Introduction to Signal Estimation and Detection Theory
MMSE Filtering and Least-Squares Problems

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