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Discrete Time Fourier Transform Properties

February 2, 2019 by 3200 Creative

Discrete Time Fourier Transform Properties

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Question 1 of 4

1. Question
If signals $x[n]$ and $y[n]$ have discrete-time Fourier transforms (DTFTs) $X(e^{j\omega}) = \frac{1}{1- 0.5e^{j\omega}}$ and $Y(e^{j\omega}) = \frac{1}{1+ 0.25e^{j\omega}}$ , what is the DTFT of $z[n] = -x[n] + 2y[n]$ ??
- $Z(e^{j\omega}) = \frac{2}{(1+ 0.25e^{j\omega})( 1- 0.5e^{j\omega})}$
- $Z(e^{j\omega}) \frac{-2(1- 0.5e^{j\omega})}{1+ 0.25e^{j\omega}}$
- $Z(e^{j\omega}) = \frac{2}{1+ 0.25e^{j\omega}}-\frac{1}{1- 0.5e^{j\omega}}$
- $Z(e^{j\omega}) = \frac{-2}{1- 0.5e^{j\omega}}$
- $Z(e^{j\omega}) = \frac{1}{1+ 0.25e^{j\omega}}$
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Question 2 of 4

2. Question
A signal $x[n] = 0.5^n u[n]$ is applied to a linear time-invariant system with impulse response $h[n] = 0.25^n u[n]$ . What is the DTFT of the output $y[n] = x[n]*h[n]$ ? Hint: recall that $a^n u[n]$ has DTFT $\frac{1}{1- ae^{-j\omega}}$ .
- $Z(e^{j\omega}) = \frac{1- 0.5e^{-j\omega}}{1- 0.25e^{-j\omega}}$
- $Z(e^{j\omega})= \frac{1}{1-0.5e^{-j\omega}}$
- $Z(e^{j\omega})= \frac{1}{1- 0.5e^{-j\omega}} + \frac{1}{1-0.25e^{-j\omega}}$
- $Z(e^{j\omega})= \frac{1}{(1- 0.5e^{-j\omega})( 1-0.25e^{-j\omega})}$
- $Z(e^{j\omega})= \frac{1}{1-0.25e^{-j\omega}}$
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Question 3 of 4

3. Question
A signal $x[n] = 0.8^n u[n]$ has DTFT $X(e^{j\omega}) = \frac{1}{1-0.8e^{-j\omega}}$ . Use the properties of the DTFT to find the signal $z[n]$ with DTFT $Z(e^{j\omega}) = \frac{e^{j4\omega}}{1-0.8e^{-j\omega}}$ .
- $z[n] = e^{-j4n} 0.8^n u[n]$
- $z[n] = 3.2^nu[n]$
- $z[n] = 0.8^{(n-4)} u[n-4]$
- $z[n] = 0.8^{(n+4)} u[n+4]$
- $x(t) = e^{-2t}u(t)$
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Question 4 of 4

4. Question
A signal $x[n] = 0.8^n u[n]$ has DTFT $X(e^{j\omega}) = \frac{1}{1- 0.8e^{-j\omega}}$ . Use the properties of the DTFT to find the DTFT of the signal $z[n] = e^{j\pi n/8} 0.8^n u[n]$ .
- $Z(e^{j\omega})= \frac{j\omega}{1-0.8e^{-j\omega}}$
- $Z(e^{j\omega})= \frac{d}{d\omega}\frac{1}{1-0.8e^{-j\omega}}$
- $Z(e^{j\omega})= \frac{1 }{1-0.8e^{-j(\omega+\pi/8)}}$
- $Z(e^{j\omega})= \left(\frac{1 }{1-0.8e^{-j(\omega-\pi/8)}}\right)^2$
- $Z(e^{j\omega})= \frac{1 }{1-0.8e^{-j(\omega-\pi/8)}}$
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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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