P-3.20 A real signal x(1) has the two-sided spectrum shown in Fig. P-3.20. The frequency axis has units of rad/s. 0.4e-2 <-90 0.6e/1.4 -40 0 0.5 0.6e 1.4 40 0.4e/2 90 w Figure P-3.20 (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.
P-3.20 A real signal x(1) has the two-sided spectrum shown in Fig. P-3.20. The frequency axis has units of rad/s. 0.4e-2 <-90 0.6e/1.4 -40 0 0.5 0.6e 1.4 40 0.4e/2 90 w Figure P-3.20 (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
Related questions
Question
Please assist with practice question P-3.20 letter c with details on how to do it.
![### Periodic Signals and Fourier Series Representation
**Periodic Signal Representation using Fourier Series**
A periodic signal of this type can be represented as a Fourier series of the form:
\[ x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j\omega_0 k t} \]
The \( k \)th term in the series is called the \( k \)th harmonic.
**Task:**
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.
### Explanation:
This form of representation is a key concept in signal processing, allowing a complex periodic signal to be expressed in terms of its frequency components, each represented by the harmonic \( k \). By identifying and tabulating the nonzero Fourier coefficients \( a_k \), you can effectively understand the signal's frequency spectrum.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1296c196-158c-4a2c-9587-82b5996e9fed%2F0f4308bc-13de-491b-9347-28df843b245e%2Fyskal29_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Periodic Signals and Fourier Series Representation
**Periodic Signal Representation using Fourier Series**
A periodic signal of this type can be represented as a Fourier series of the form:
\[ x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j\omega_0 k t} \]
The \( k \)th term in the series is called the \( k \)th harmonic.
**Task:**
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.
### Explanation:
This form of representation is a key concept in signal processing, allowing a complex periodic signal to be expressed in terms of its frequency components, each represented by the harmonic \( k \). By identifying and tabulating the nonzero Fourier coefficients \( a_k \), you can effectively understand the signal's frequency spectrum.
Transcribed Image Text:### Analysis of Two-Sided Spectrum of a Real Signal
**P-3.20**
A real signal \( x(t) \) has the two-sided spectrum shown in Fig. P-3.20. The frequency axis has units of rad/s.
#### Spectrum Diagram (Fig. P-3.20)
- The horizontal axis represents the frequency (\( \omega \)) in radians per second (rad/s).
- Symmetrical components are plotted around the origin (0 rad/s) at frequencies \( \pm 40 \) rad/s and \( \pm 90 \) rad/s.
- The magnitudes and phases of the components at these frequencies are:
- \( \omega = -90 \) rad/s: Magnitude \( 0.4 \), Phase \( -\frac{\pi}{2} \)
- \( \omega = -40 \) rad/s: Magnitude \( 0.6 \), Phase \( 1.4 \)
- \( \omega = 0 \) rad/s: Magnitude \( 0.5 \)
- \( \omega = 40 \) rad/s: Magnitude \( 0.6 \), Phase \( -1.4 \)
- \( \omega = 90 \) rad/s: Magnitude \( 0.4 \), Phase \( \frac{\pi}{2} \)
#### Problems:
(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.**
Please refer to the diagram for the detailed frequencies and magnitudes. Evaluate the fundamental components and periods according to the illustrated spectral data.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
