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Select all the conditions that are true for the Fourier Series of x[n] = (1/2) nl. %3D Real {X(ew)}=0 O There exist a real p such that elpwX(elw) is real Imaginary (X(elw)}=0 Periodic X(e") O S. X (cj") dw is periodic X(e) = 0

Select all the conditions that are true for the Fourier Series of x[n] = (1/2) nl. %3D Real {X(ew)}=0 O There exist a real p such that elpwX(elw) is real Imaginary (X(elw)}=0 Periodic X(e") O S. X (cj") dw is periodic X(e) = 0

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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**Problem Statement: Fourier Series Conditions** Consider the sequence \( x[n] = \left( \frac{1}{2} \right)^{|n|} \). For the corresponding Fourier Series \( X(e^{j\omega}) \), determine which of the following conditions are true. 1. \[ \text{Real} \{X(e^{j\omega}) \} = 0 \] 2. There exists a real \( p \) such that \( e^{jp\omega} X(e^{j\omega}) \) is real. 3. \[ \text{Imaginary} \{X(e^{j\omega}) \} = 0 \] 4. \[ X(e^{j\omega}) \] is periodic. 5. \[ \int_{-\pi}^{\pi} X(e^{j\omega}) \, d\omega \] is periodic. 6. \[ X(e^{0}) = 0 \] **Explanation of Problem** We are given the discrete-time signal \( x[n] = \left( \frac{1}{2} \right)^{|n|} \) and need to identify which of the listed conditions are valid for its Fourier Series representation. - The first condition checks if the real part of \( X(e^{j\omega}) \) is zero. - The second condition examines if there exists a phase shift \( p \) such that the modified Fourier transform is real. - The third condition checks if the imaginary part of \( X(e^{j\omega}) \) is zero. - The fourth condition asserts whether \( X(e^{j\omega}) \) is periodic. - The fifth condition involves the periodicity of the integral of the Fourier transform over a specific range. - The final condition evaluates if the Fourier transform at \( \omega = 0 \) equals zero. Each condition requires a thorough analysis of the properties of the Fourier Series given the form of \( x[n] \).
Transcribed Image Text:**Problem Statement: Fourier Series Conditions** Consider the sequence \( x[n] = \left( \frac{1}{2} \right)^{|n|} \). For the corresponding Fourier Series \( X(e^{j\omega}) \), determine which of the following conditions are true. 1. \[ \text{Real} \{X(e^{j\omega}) \} = 0 \] 2. There exists a real \( p \) such that \( e^{jp\omega} X(e^{j\omega}) \) is real. 3. \[ \text{Imaginary} \{X(e^{j\omega}) \} = 0 \] 4. \[ X(e^{j\omega}) \] is periodic. 5. \[ \int_{-\pi}^{\pi} X(e^{j\omega}) \, d\omega \] is periodic. 6. \[ X(e^{0}) = 0 \] **Explanation of Problem** We are given the discrete-time signal \( x[n] = \left( \frac{1}{2} \right)^{|n|} \) and need to identify which of the listed conditions are valid for its Fourier Series representation. - The first condition checks if the real part of \( X(e^{j\omega}) \) is zero. - The second condition examines if there exists a phase shift \( p \) such that the modified Fourier transform is real. - The third condition checks if the imaginary part of \( X(e^{j\omega}) \) is zero. - The fourth condition asserts whether \( X(e^{j\omega}) \) is periodic. - The fifth condition involves the periodicity of the integral of the Fourier transform over a specific range. - The final condition evaluates if the Fourier transform at \( \omega = 0 \) equals zero. Each condition requires a thorough analysis of the properties of the Fourier Series given the form of \( x[n] \).
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