(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.
(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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please assist with practice question P-3.20 letter A with details on how to do it.
Transcribed Image Text:**Problem P-3.20: Frequency Analysis of a Real Signal**
A real signal \( x(t) \) has the two-sided spectrum shown in Fig. P-3.20. The frequency axis has units of rad/s.
**Figure P-3.20: Two-sided Spectrum**
The graph demonstrates the two-sided spectrum of the real signal \( x(t) \). The horizontal axis represents the angular frequency (\( \omega \)) in rad/s, and the vertical axis denotes the amplitude of the signal components.
- At \(\omega = -90 \, \text{rad/s}\), the amplitude is \( 0.4e^{-j2} \).
- At \(\omega = -40 \, \text{rad/s}\), the amplitude is \( 0.6e^{j1.4} \).
- At \(\omega = 0 \, \text{rad/s}\), the amplitude is \( 0.5 \).
- At \(\omega = 40 \, \text{rad/s}\), the amplitude is \( 0.6e^{-j1.4} \).
- At \(\omega = 90 \, \text{rad/s}\), the amplitude is \( 0.4e^{j2} \).
### 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.**
![**Chapter 3: Spectrum Analysis**
### Periodic Signal Representation using Fourier Series
A periodic signal of this type can be represented as a Fourier series of the form:
\[ x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j\omega_0kt} \]
The \( k^{th} \) term in the series is called the \( k^{th} \) harmonic. Determine which harmonics (positive and negative) are present. In other words, for the Fourier series coefficients, \( a_k \), determine which coefficients are nonzero. List the indices of the nonzero Fourier series coefficients and their values in a table.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1296c196-158c-4a2c-9587-82b5996e9fed%2Fb006fad7-5e40-43bb-aa6e-ac40b51dc4b2%2F8hzubp4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Chapter 3: Spectrum Analysis**
### Periodic Signal Representation using Fourier Series
A periodic signal of this type can be represented as a Fourier series of the form:
\[ x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j\omega_0kt} \]
The \( k^{th} \) term in the series is called the \( k^{th} \) harmonic. Determine which harmonics (positive and negative) are present. In other words, for the Fourier series coefficients, \( a_k \), determine which coefficients are nonzero. List the indices of the nonzero Fourier series coefficients and their values in a table.
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