Compute the convolution L ★ II of the functions L and II, which are defined below: 1 if |æ|
Compute the convolution L ★ II of the functions L and II, which are defined below: 1 if |æ|
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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Question
![### Problem Statement
**Objective:** Compute the convolution \( L \ast \Pi \) of the functions \( L \) and \( \Pi \), which are defined below:
### Definitions
#### Function \(\Pi(x)\):
\[
\Pi(x) =
\begin{cases}
1 & \text{if } |x| \leq a/2 \\
0 & \text{otherwise}
\end{cases}
\]
#### Function \(L(x)\):
\[
L(x) = \frac{1}{1 + x^2/w^2}
\]
### Assumptions
- \( a \) and \( w \) are positive real numbers.
### Explanation
The problem requires the computation of the convolution of two functions: a rectangular function \(\Pi(x)\) and a Lorentzian function \(L(x)\). The function \(\Pi(x)\) behaves like a rectangular window with a width \(a\), while \(L(x)\) is a Lorentzian distribution with a width dependent on the parameter \(w\). The convolution of these functions will combine their effects in the spatial domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F65ba51bd-93fe-4637-8f7c-e52d3fdaa23e%2F0506c608-ed76-42c4-84f1-335ef926b71c%2Ffcnze9_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Objective:** Compute the convolution \( L \ast \Pi \) of the functions \( L \) and \( \Pi \), which are defined below:
### Definitions
#### Function \(\Pi(x)\):
\[
\Pi(x) =
\begin{cases}
1 & \text{if } |x| \leq a/2 \\
0 & \text{otherwise}
\end{cases}
\]
#### Function \(L(x)\):
\[
L(x) = \frac{1}{1 + x^2/w^2}
\]
### Assumptions
- \( a \) and \( w \) are positive real numbers.
### Explanation
The problem requires the computation of the convolution of two functions: a rectangular function \(\Pi(x)\) and a Lorentzian function \(L(x)\). The function \(\Pi(x)\) behaves like a rectangular window with a width \(a\), while \(L(x)\) is a Lorentzian distribution with a width dependent on the parameter \(w\). The convolution of these functions will combine their effects in the spatial domain.
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