Question 4: A spectrum plot for the periodic signal x(t) is shown. The frequency axis has units of rad/s. 3e-jx14 I -8.4 4ejr13 -3.6m 7ejx 0 4e-jad3 00 x(t) = [ k=00 3.бл 3ejx14 a) Determine the fundamental frequency wo of this signal. b) Determine the fundamental period To of x(t), which is the shortest possible period. c) Determine the DC value of this signal. d) A periodic signal of this type can be represented as a Fourier series of the form akejwokt 8.4 @ , determine which If the Fourier series coefficient of x (t) are denoted by a, k = 0, ±1, ±2, ±3,.. coefficients are nonzero. List these nonzero Fourier series coefficients and their values in a table, i.e., (k, ak).
Question 4: A spectrum plot for the periodic signal x(t) is shown. The frequency axis has units of rad/s. 3e-jx14 I -8.4 4ejr13 -3.6m 7ejx 0 4e-jad3 00 x(t) = [ k=00 3.бл 3ejx14 a) Determine the fundamental frequency wo of this signal. b) Determine the fundamental period To of x(t), which is the shortest possible period. c) Determine the DC value of this signal. d) A periodic signal of this type can be represented as a Fourier series of the form akejwokt 8.4 @ , determine which If the Fourier series coefficient of x (t) are denoted by a, k = 0, ±1, ±2, ±3,.. coefficients are nonzero. List these nonzero Fourier series coefficients and their values in a table, i.e., (k, ak).
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Please assist with this practice problem 4 b,c,d with details on how to do it. Thank you.
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**Question 4:** A spectrum plot for the periodic signal \( x(t) \) is shown. The frequency axis has units of rad/s.
![Spectrum Plot]
The spectrum plot shown features several spikes at different frequencies, specifically at \(-8.4\pi\), \(-3.6\pi\), \(0\), \(3.6\pi\), and \(8.4\pi\). The heights of these spikes are indicated as follows:
- At \(-8.4\pi\): \(3e^{-j\pi/4}\)
- At \(-3.6\pi\): \(4e^{j\pi/3}\)
- At \(0\): \(7e^{j\pi}\)
- At \(3.6\pi\): \(4e^{-j\pi/3}\)
- At \(8.4\pi\): \(3e^{j\pi/4}\)
**Questions:**
a) **Determine the fundamental frequency \( \omega_0 \) of this signal.**
b) **Determine the fundamental period \( T_0 \) of \( x(t) \), which is the shortest possible period.**
c) **Determine the DC value of this signal.**
d) **A periodic signal of this type can be represented as a Fourier series of the form**
\[
x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j\omega_0 k t}
\]
**If the Fourier series coefficients of \( x(t) \) are denoted by \( a_k \), \( k = 0, \pm1, \pm2, \pm3, \ldots \), determine which coefficients are nonzero. List these nonzero Fourier series coefficients and their values in a table, i.e., \((k, a_k)\).**
---
**Detailed Diagram Explanation:**
The diagram represents a spectrum plot of the periodic signal \( x(t) \) in the frequency domain. The horizontal axis is labeled with \(\omega\) in units of rad/s. The vertical spikes indicate the magnitudes and phases of the signal's Fourier series coefficients at specific frequencies. These values represent the amplitude and phase shift corresponding to each harmonic component of the signal.
- The spectrum contains non-zero components at specific points indicated by their amplitudes and exponential terms, which include both](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1296c196-158c-4a2c-9587-82b5996e9fed%2Fa7ffeaf2-c213-43f8-8730-940246dfbbdc%2Fmuij3c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Question 4:** A spectrum plot for the periodic signal \( x(t) \) is shown. The frequency axis has units of rad/s.
![Spectrum Plot]
The spectrum plot shown features several spikes at different frequencies, specifically at \(-8.4\pi\), \(-3.6\pi\), \(0\), \(3.6\pi\), and \(8.4\pi\). The heights of these spikes are indicated as follows:
- At \(-8.4\pi\): \(3e^{-j\pi/4}\)
- At \(-3.6\pi\): \(4e^{j\pi/3}\)
- At \(0\): \(7e^{j\pi}\)
- At \(3.6\pi\): \(4e^{-j\pi/3}\)
- At \(8.4\pi\): \(3e^{j\pi/4}\)
**Questions:**
a) **Determine the fundamental frequency \( \omega_0 \) of this signal.**
b) **Determine the fundamental period \( T_0 \) of \( x(t) \), which is the shortest possible period.**
c) **Determine the DC value of this signal.**
d) **A periodic signal of this type can be represented as a Fourier series of the form**
\[
x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j\omega_0 k t}
\]
**If the Fourier series coefficients of \( x(t) \) are denoted by \( a_k \), \( k = 0, \pm1, \pm2, \pm3, \ldots \), determine which coefficients are nonzero. List these nonzero Fourier series coefficients and their values in a table, i.e., \((k, a_k)\).**
---
**Detailed Diagram Explanation:**
The diagram represents a spectrum plot of the periodic signal \( x(t) \) in the frequency domain. The horizontal axis is labeled with \(\omega\) in units of rad/s. The vertical spikes indicate the magnitudes and phases of the signal's Fourier series coefficients at specific frequencies. These values represent the amplitude and phase shift corresponding to each harmonic component of the signal.
- The spectrum contains non-zero components at specific points indicated by their amplitudes and exponential terms, which include both
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