### 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

Answered: Image /qna-images/answer/de223f5a-5f0b-40af-a27d-071a60e29c65.jpg

Answered: = Suppose f(t) has Fourier transform…

= 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)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

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Question

### 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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