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The Fourier Series Continuous Time Periodic Signals

February 2, 2019 by 3200 Creative

The Fourier Series Continuous Time Periodic Signals

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Question 1 of 5

1. Question
The primary role of the Fourier series is analysis of periodic continuous-time signals and their interactions with systems.
- True
- False
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Question 2 of 5

2. Question
The Fourier series expresses a periodic time signal $x(t)$ as an integral of weighted complex sinusoids.
- True
- False
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Question 3 of 5

3. Question
A signal $x(t)$ has fundamental period $T=0.5$ sec. Which of the following could be the Fourier series expansion for the signal? Select all that apply.
- $x(t) = 2e^{-j4\pi t} + e^{-j2\pi t} + e^{j2\pi t} + 2e^{j4\pi t}$
- $x(t) = -2e^{-j16\pi t} + e^{-j12\pi t} + e^{j12\pi t}-2e^{j16\pi t}$
- $x(t) = 1+ e^{j4\pi t} + 2e^{j8\pi t} + e^{j16\pi t} + 2e^{j32\pi t}$
- $x(t) = 1+ e^{j4\pi t} + 2e^{j8\pi t} + e^{j10\pi t} + 2e^{j16\pi t}$
- $x(t) = \sum_{k=-\infty}^\infty 0.5^{|k|}e^{jk\pi t}$
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Question 4 of 5

4. Question
The Fourier series coefficients $X[k]$ for $x(t) = 2\cos(2\pi t) + 2\sin(2\pi t)$ are
- $X[k] = \left\{\begin{array}{cl} 1-j,&k=1\\1-j,&k=-1\\0, &k\ne\pm 1\end{array}\right.$
- $X[k] = \left\{\begin{array}{cl} 1+j,&k=1\\1-j,&k=-1\\0, &k\ne\pm 1\end{array}\right.$
- $X[k] = \left\{\begin{array}{cl} 1-j,&k=1\\1+j,&k=-1\\0, &k\ne\pm 1\end{array}\right.$
- $X[k] = \left\{\begin{array}{cl} 1-j,&k=2\\1+j,&k=-2\\0, &k\ne\pm 2\end{array}\right.$
- $X[k] = \left\{\begin{array}{cl} 1+j,&k=2\\1-j,&k=-2\\0, &k\ne\pm 2\end{array}\right.$
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Question 5 of 5

5. Question
Which of the following are true of the Fourier series coefficients $X[k]$ if $x(t)$ is real valued? Select all that apply.
- Re{ $X[k]$ } = Im{ $X[-k]$ }
- Im{ $X[k]$ } = Im{ $X[-k]$ }
- Re{ $X[k]$ } = Re{ $X[-k]$ }
- $|X[k]|$ } = - $|X[-k]|$
- arg{ $X[k]$ } = -arg{ $X[-k]$ }
- $|X[k]|$ } = $|X[-k]|$ }
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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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Foundations
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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