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)
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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 \( x[n] \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9de2b0ed-01a2-4a8a-9b30-8cd619de91d2%2F39e13919-42c7-478c-860c-962d6a71f3bc%2Fdxbpwog8_processed.png&w=3840&q=75)
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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