Consider the sequence x[n] = [-3, 0, –1, {10}, 1, 0, 3], where the number in {·} corresponds to n = 0. Write down an expression for the phase spectrum ZX (w). your answer must be in cos and/or sin terms. Is the phase spectrum even or odd?
Consider the sequence x[n] = [-3, 0, –1, {10}, 1, 0, 3], where the number in {·} corresponds to n = 0. Write down an expression for the phase spectrum ZX (w). your answer must be in cos and/or sin terms. Is the phase spectrum even or odd?
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
Related questions
Question
![### Understanding Phase Spectrum of a Sequence
Consider the sequence \( x[n] = [-3, 0, -1, \{10\}, 1, 0, 3] \), where the number in \( \{ \cdot \} \) corresponds to \( n = 0 \).
**Task:**
1. **Expression for the Phase Spectrum:**
Write down an expression for the phase spectrum \( \angle X(\omega) \). Your answer must be in cos and/or sin terms.
2. **Properties of Phase Spectrum:**
Is the phase spectrum even or odd?
### Step-by-Step Solution
#### 1. Expression for the Phase Spectrum \( \angle X(\omega) \)
The phase spectrum of a sequence \( x[n] \) is given by the argument (angle) of the Discrete-Time Fourier Transform (DTFT) of \( x[n] \). The DTFT of \( x[n] \) is defined as:
\[
X(\omega) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n}
\]
In this case, since \( x[n] \) is a finite sequence, the sum becomes finite. Substituting \( x[n] \):
\[
x[n] =
\begin{cases}
-3 & \text{for } n = -3 \\
0 & \text{for } n = -2 \\
-1 & \text{for } n = -1 \\
10 & \text{for } n = 0 \\
1 & \text{for } n = 1 \\
0 & \text{for } n = 2 \\
3 & \text{for } n = 3
\end{cases}
\]
So,
\[
X(\omega) = -3e^{j3\omega} + 0 \cdot e^{j2\omega} - e^{j\omega} + 10 + e^{-j\omega} + 0 \cdot e^{-j2\omega} + 3e^{-j3\omega}
\]
Combining terms:
\[
X(\omega) = 10 + 3e^{-j3\omega} - e^{j\omega} + e^{-j\omega} - 3e^{j3\omega}
\]
To express \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5362cbea-9427-4b2b-99cb-57a208edd853%2F71e35175-9d51-4358-9637-50722d0f4a21%2Ful6kis_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Phase Spectrum of a Sequence
Consider the sequence \( x[n] = [-3, 0, -1, \{10\}, 1, 0, 3] \), where the number in \( \{ \cdot \} \) corresponds to \( n = 0 \).
**Task:**
1. **Expression for the Phase Spectrum:**
Write down an expression for the phase spectrum \( \angle X(\omega) \). Your answer must be in cos and/or sin terms.
2. **Properties of Phase Spectrum:**
Is the phase spectrum even or odd?
### Step-by-Step Solution
#### 1. Expression for the Phase Spectrum \( \angle X(\omega) \)
The phase spectrum of a sequence \( x[n] \) is given by the argument (angle) of the Discrete-Time Fourier Transform (DTFT) of \( x[n] \). The DTFT of \( x[n] \) is defined as:
\[
X(\omega) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n}
\]
In this case, since \( x[n] \) is a finite sequence, the sum becomes finite. Substituting \( x[n] \):
\[
x[n] =
\begin{cases}
-3 & \text{for } n = -3 \\
0 & \text{for } n = -2 \\
-1 & \text{for } n = -1 \\
10 & \text{for } n = 0 \\
1 & \text{for } n = 1 \\
0 & \text{for } n = 2 \\
3 & \text{for } n = 3
\end{cases}
\]
So,
\[
X(\omega) = -3e^{j3\omega} + 0 \cdot e^{j2\omega} - e^{j\omega} + 10 + e^{-j\omega} + 0 \cdot e^{-j2\omega} + 3e^{-j3\omega}
\]
Combining terms:
\[
X(\omega) = 10 + 3e^{-j3\omega} - e^{j\omega} + e^{-j\omega} - 3e^{j3\omega}
\]
To express \(
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 4 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, electrical-engineering and related others by exploring similar questions and additional content below.Recommended textbooks for you
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:
9780133923605
Author:
Robert L. Boylestad
Publisher:
PEARSON
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:
9781337900348
Author:
Stephen L. Herman
Publisher:
Cengage Learning
Programmable Logic Controllers
Electrical Engineering
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:
9780133923605
Author:
Robert L. Boylestad
Publisher:
PEARSON
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:
9781337900348
Author:
Stephen L. Herman
Publisher:
Cengage Learning
Programmable Logic Controllers
Electrical Engineering
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
Fundamentals of Electric Circuits
Electrical Engineering
ISBN:
9780078028229
Author:
Charles K Alexander, Matthew Sadiku
Publisher:
McGraw-Hill Education
Electric Circuits. (11th Edition)
Electrical Engineering
ISBN:
9780134746968
Author:
James W. Nilsson, Susan Riedel
Publisher:
PEARSON
Engineering Electromagnetics
Electrical Engineering
ISBN:
9780078028151
Author:
Hayt, William H. (william Hart), Jr, BUCK, John A.
Publisher:
Mcgraw-hill Education,