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If f(t) (shown below) has Fourier transform F(w), then what is the Fourier transform of F(t) when w = 1 ? -3 -2 −1 1 f(t) 0 1 2 t

If f(t) (shown below) has Fourier transform F(w), then what is the Fourier transform of F(t) when w = 1 ? -3 -2 −1 1 f(t) 0 1 2 t

Introductory Circuit Analysis (13th Edition)
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...
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### Fourier Transform Problem on Educational Website **Problem Statement:** If \( f(t) \) (shown below) has Fourier transform \( F(\omega) \), then what is the Fourier transform of \( F(t) \) when \( \omega = 1 \)? **Graph/Diagram Description:** The graph represents the function \( f(t) \) as a piecewise linear function. The function is detailed as follows: - \( f(t) \) is 0 for \( t < -2 \). - \( f(t) \) increases linearly from 0 to 2 in the interval \( -2 \leq t \leq 0 \). - \( f(t) \) is constant at 2 for \( 0 \leq t \leq 2 \). - \( f(t) \) drops back to 0 after \( t > 2 \). The coordinates of the key points on the graph of \( f(t) \) are: - \((-2, 0)\) to \((0, 2)\) (linearly increasing segment) - \((0, 2)\) to \((2, 2)\) (horizontal line segment) - \((2, 2)\) to \((3, 0)\) (vertical drop to 0) **Multiple Choice Answers:** What is the value of the Fourier transform \( F(1) \)? - (a) \(3\pi\) - (b) \(4\pi\) - (c) \(2\) - (d) \(1\) **Explanation:** To find the Fourier transform \( F(\omega) \) at \( \omega = 1 \), one needs to apply the definition and properties of the Fourier transform on the given piecewise function \( f(t) \). After performing the integration and calculations, compare the result with the given options.
Transcribed Image Text:### Fourier Transform Problem on Educational Website **Problem Statement:** If \( f(t) \) (shown below) has Fourier transform \( F(\omega) \), then what is the Fourier transform of \( F(t) \) when \( \omega = 1 \)? **Graph/Diagram Description:** The graph represents the function \( f(t) \) as a piecewise linear function. The function is detailed as follows: - \( f(t) \) is 0 for \( t < -2 \). - \( f(t) \) increases linearly from 0 to 2 in the interval \( -2 \leq t \leq 0 \). - \( f(t) \) is constant at 2 for \( 0 \leq t \leq 2 \). - \( f(t) \) drops back to 0 after \( t > 2 \). The coordinates of the key points on the graph of \( f(t) \) are: - \((-2, 0)\) to \((0, 2)\) (linearly increasing segment) - \((0, 2)\) to \((2, 2)\) (horizontal line segment) - \((2, 2)\) to \((3, 0)\) (vertical drop to 0) **Multiple Choice Answers:** What is the value of the Fourier transform \( F(1) \)? - (a) \(3\pi\) - (b) \(4\pi\) - (c) \(2\) - (d) \(1\) **Explanation:** To find the Fourier transform \( F(\omega) \) at \( \omega = 1 \), one needs to apply the definition and properties of the Fourier transform on the given piecewise function \( f(t) \). After performing the integration and calculations, compare the result with the given options.
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