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= Suppose f(t) has Fourier transform F(w) rect (w9) + rect(w +9). If G(w) is the Fourier ransform of g(t) = f(-t/3), then what is G(T)? (a) 3 (b) 3T (c) T

= Suppose f(t) has Fourier transform F(w) rect (w9) + rect(w +9). If G(w) is the Fourier ransform of g(t) = f(-t/3), then what is G(T)? (a) 3 (b) 3T (c) 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 **Problem Statement:** Suppose \( f(t) \) has Fourier transform \( F(\omega) = \text{rect}(\omega - 9) + \text{rect}(\omega + 9) \). If \( G(\omega) \) is the Fourier transform of \( g(t) = f(-t/3) \), then what is \( G(\pi) \)? **Choices:** (a) \( 3 \) (b) \( 3\pi \) (c) \( \pi \) **Solution Explanation:** To find \( G(\pi) \), we first need to understand the relationship between the functions \( f(t) \) and \( g(t) \) and their Fourier transforms. 1. **Fourier Transform of Time-Scaled Function:** - When \( f(t) \) is scaled in time as \( f(at) \), its Fourier transform \( F(\omega) \) scales in frequency by \( 1/|a| \) and is stretched by a factor of \( |a| \). Specifically, if \( h(t) = f(at) \), then \( H(\omega) = \frac{1}{|a|} F\left( \frac{\omega}{a} \right) \). 2. **Given Functions:** - Here, \( g(t) = f(-t/3) \). This implies \( a = -1/3 \), so \( |a| = \frac{1}{3} \). 3. **Fourier Transform of \( g(t) \):** - Using the scaling property, we have: \[ G(\omega) = \frac{1}{|a|} F\left( \frac{\omega}{a} \right) = 3 F(-3 \omega). \] 4. **Calculating Specific Value:** - We are interested in \( G(\pi) \): \[ G(\pi) = 3 F(-3\pi). \] 5. **Evaluating \( F(\omega) \):** - Given \( F(\omega) = \text{rect}(\omega - 9) + \text{rect}(\omega + 9) \), we need to evaluate \( F(-3\pi) \). **rect Function:** The rect function
Transcribed Image Text:### Fourier Transform Problem **Problem Statement:** Suppose \( f(t) \) has Fourier transform \( F(\omega) = \text{rect}(\omega - 9) + \text{rect}(\omega + 9) \). If \( G(\omega) \) is the Fourier transform of \( g(t) = f(-t/3) \), then what is \( G(\pi) \)? **Choices:** (a) \( 3 \) (b) \( 3\pi \) (c) \( \pi \) **Solution Explanation:** To find \( G(\pi) \), we first need to understand the relationship between the functions \( f(t) \) and \( g(t) \) and their Fourier transforms. 1. **Fourier Transform of Time-Scaled Function:** - When \( f(t) \) is scaled in time as \( f(at) \), its Fourier transform \( F(\omega) \) scales in frequency by \( 1/|a| \) and is stretched by a factor of \( |a| \). Specifically, if \( h(t) = f(at) \), then \( H(\omega) = \frac{1}{|a|} F\left( \frac{\omega}{a} \right) \). 2. **Given Functions:** - Here, \( g(t) = f(-t/3) \). This implies \( a = -1/3 \), so \( |a| = \frac{1}{3} \). 3. **Fourier Transform of \( g(t) \):** - Using the scaling property, we have: \[ G(\omega) = \frac{1}{|a|} F\left( \frac{\omega}{a} \right) = 3 F(-3 \omega). \] 4. **Calculating Specific Value:** - We are interested in \( G(\pi) \): \[ G(\pi) = 3 F(-3\pi). \] 5. **Evaluating \( F(\omega) \):** - Given \( F(\omega) = \text{rect}(\omega - 9) + \text{rect}(\omega + 9) \), we need to evaluate \( F(-3\pi) \). **rect Function:** The rect function
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