Compute the convolution f(t) * g(t) for f(t) = e-3, g(t) = 2t.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Problem 2.** Compute the convolution \( f(t) * g(t) \) for \( f(t) = e^{-3t}, g(t) = 2t \).
In this problem, you are asked to compute the convolution of two functions. The function \( f(t) \) is given as an exponential decay function \( e^{-3t} \), and the function \( g(t) \) is a linear function \( 2t \).
### Detailed Explanation:
Convolution is an integral that expresses the amount of overlap of one function as it is shifted over another function. In mathematical terms, the convolution of two functions \( f(t) \) and \( g(t) \) is given by:
\[
(f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) \, d\tau
\]
For the given \( f(t) \) and \( g(t) \):
1. Substitute the expressions for \( f(t) \) and \( g(t) \) into the convolution integral:
\[
(f * g)(t) = \int_{-\infty}^{\infty} e^{-3\tau} \cdot 2(t - \tau) \, d\tau
\]
2. The limits of integration are generally from \(-\infty\) to \(\infty\), but for practical computations, these may be adjusted based on the behavior of the functions involved.
3. Solve the integral to find the convolution result.
The problem provides a foundation for understanding how convolution is used in various applications such as signal processing, where one waveform is modified by another.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac565010-669d-49c5-8c58-19dca7d8b9f0%2Fa2846746-2781-4e7a-aa38-e001f9700a6a%2F978x35e_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 2.** Compute the convolution \( f(t) * g(t) \) for \( f(t) = e^{-3t}, g(t) = 2t \).
In this problem, you are asked to compute the convolution of two functions. The function \( f(t) \) is given as an exponential decay function \( e^{-3t} \), and the function \( g(t) \) is a linear function \( 2t \).
### Detailed Explanation:
Convolution is an integral that expresses the amount of overlap of one function as it is shifted over another function. In mathematical terms, the convolution of two functions \( f(t) \) and \( g(t) \) is given by:
\[
(f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) \, d\tau
\]
For the given \( f(t) \) and \( g(t) \):
1. Substitute the expressions for \( f(t) \) and \( g(t) \) into the convolution integral:
\[
(f * g)(t) = \int_{-\infty}^{\infty} e^{-3\tau} \cdot 2(t - \tau) \, d\tau
\]
2. The limits of integration are generally from \(-\infty\) to \(\infty\), but for practical computations, these may be adjusted based on the behavior of the functions involved.
3. Solve the integral to find the convolution result.
The problem provides a foundation for understanding how convolution is used in various applications such as signal processing, where one waveform is modified by another.
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