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Properties of the Fourier Transform

February 2, 2019 by 3200 Creative

Properties of the Fourier Transform

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Question 1 of 5

1. Question
If signals $x(t)$ and $y(t)$ have Fourier transforms $X(\Omega) = \frac{1}{j\Omega + 1}$ and $Y(\Omega) = \frac{1}{j\Omega +3}$ , what is the Fourier transform of $z(t) = x(t)-2y(t)$ ?
- $Z(\Omega) = \frac{-2}{(j\Omega+3)( j\Omega + 1)}$
- $Z(\Omega) \frac{-2( j\Omega + 1)}{j\Omega+3}$
- $Z(\Omega) = \frac{-2}{j\Omega+3}$
- $Z(\Omega) = \frac{-2}{j\Omega+3} + \frac{1}{j\Omega + 1}$
- $Z(\Omega) = \frac{1}{j\Omega + 1}$
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Question 2 of 5

2. Question
A signal $x(t) = e^{-t}u(t)$ is applied to a linear time-invariant system with impulse response $h(t) = e^{-3t}u(t)$ . What is the Fourier transform of the output $y(t) = x(t)*h(t)$ ? Hint: recall that $e^{-at}u(t)$ has Fourier transform $\frac{1}{j\Omega + a}$ .
- $Z(\Omega) = \frac{1}{(j\Omega+3)( j\Omega + 1)}$
- $Z(\Omega) = \frac{ j\Omega + 1}{j\Omega+3}$
- $Z(\Omega) = \frac{1}{j\Omega+3}$
- $Z(\Omega) = \frac{1}{j\Omega+3} + \frac{1}{j\Omega + 1}$
- $Z(\Omega) = \frac{1}{j\Omega + 1}$
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Question 3 of 5

3. Question
A signal $x(t) = e^{-t}u(t)$ has Fourier transform $X(\Omega) = \frac{1}{j\Omega + 1}$ . Use the properties of the Fourier transform to find the signal $z(t)$ with Fourier transform $Z(\Omega) = \frac{e^{-j2\Omega}}{j\Omega + 1}$ .
- $z(t) = e^{-j2t} e^{-t}u(t)$
- $z(t) = \frac{d}{dt} e^{-t}u(t)$
- $z(t) = e^{-(t-2)}u(t-2)$
- $z(t) = e^{-(t+2)}u(t+2)$
- $x(t) = e^{-2t}u(t)$
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Question 4 of 5

4. Question
A signal $x(t) = e^{-t}u(t)$ has Fourier transform $X(\Omega) = \frac{1}{j\Omega + 1}$ . Use the properties of the Fourier transform to find the signal $z(t)$ with Fourier transform $Z(\Omega) = \frac{-j\Omega}{j\Omega + 1}$ .
- $z(t) = e^{-jt} e^{-t}u(t)$
- $z(t) = -\frac{d}{dt} e^{-t}u(t)$
- $z(t) = e^{-(t-1)}u(t-1)$
- $z(t) = \frac{d}{dt} e^{-t}u(t)$
- $x(t) = e^{-2t}u(t)$
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Question 5 of 5

5. Question
A signal $x(t) = e^{-t}u(t)$ has Fourier transform $X(\Omega) = \frac{1}{j\Omega + 1}$ . Use the properties of the Fourier transform to find the signal $z(t)$ with Fourier transform $Z(\Omega) = \frac{1}{j(\Omega-2) + 1}$ .
- $z(t) = e^{-j2t} e^{-t}u(t)$
- $z(t) = -\frac{d}{dt} e^{-2t}u(t)$
- $z(t) = e^{-(t-2)}u(t-2)$
- $z(t) = e^{j2t} e^{-t}u(t)$
- $x(t) = e^{-2t}u(t)$
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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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Basis Representations of Signals
Estimation of Power Spectra and Coherence
Introduction to Signal Estimation and Detection Theory
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