E. Find the Fourier Transform in both f format and omega format. 1, 0₂ = -1≤t≤1 otherwise
E. Find the Fourier Transform in both f format and omega format. 1, 0₂ = -1≤t≤1 otherwise
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
![**Question 2: Fourier Transform**
**Problem Statement:**
Find the Fourier Transform of the function \( f(t) \) in both frequency \( f \) format and angular frequency \( \omega \) format.
**Function Definition:**
\[
f(t) =
\begin{cases}
1, & -1 \leq t \leq 1 \\
0, & \text{otherwise}
\end{cases}
\]
**Explanation:**
This piecewise function describes a rectangular pulse, where the function \( f(t) \) is equal to 1 for the time interval \([-1, 1]\) and is 0 outside this interval. The task is to compute the Fourier Transform of this signal, which involves converting it from the time domain to the frequency domain using two formats:
1. **Frequency format (f):** In terms of Hertz.
2. **Angular frequency format (\(\omega\)):** In terms of radians per second.
To solve this, apply the Fourier Transform formulas for each format to the defined function \( f(t) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2828890-e758-4d3f-889f-4bdd3ee1fb27%2F08dd505d-0d80-4b3b-955c-461b2804bb67%2Fwggwbfm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 2: Fourier Transform**
**Problem Statement:**
Find the Fourier Transform of the function \( f(t) \) in both frequency \( f \) format and angular frequency \( \omega \) format.
**Function Definition:**
\[
f(t) =
\begin{cases}
1, & -1 \leq t \leq 1 \\
0, & \text{otherwise}
\end{cases}
\]
**Explanation:**
This piecewise function describes a rectangular pulse, where the function \( f(t) \) is equal to 1 for the time interval \([-1, 1]\) and is 0 outside this interval. The task is to compute the Fourier Transform of this signal, which involves converting it from the time domain to the frequency domain using two formats:
1. **Frequency format (f):** In terms of Hertz.
2. **Angular frequency format (\(\omega\)):** In terms of radians per second.
To solve this, apply the Fourier Transform formulas for each format to the defined function \( f(t) \).
Expert Solution
Introduction:
Fourier Transform:
A signal and its frequency domain, Fourier transform is
defined as:
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
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,
