**Fourier Transform and Signal Processing**### Problem StatementDetermine the Fourier transform for the signal \( x(t) \) given in **Figure Q1(a)**.An arbitrary system has an impulse response given by \( h(t) = e^{-2t}u(t) \). If the signal \( x(t) \) is passed through the system, determine the output signal \( y(t) \) using Fourier analysis.**Hint:**\[ \text{sinc}\left(\frac{\omega}{2}\right) = \frac{e^{ \frac{j\omega}{2}} - e^{ -\frac{j\omega}{2}}}{j \omega} \]### Figure Explanation**Figure Q1(a)** presents a graphical representation of the signal \( x(t) \) as a function of time \( t \).- The signal \( x(t) \) is defined as follows: - For \( t < 0 \), \( x(t) = 0 \). - At \( t = 0 \), \( x(t) = 1 \). - For \( 0 < t \leq 1 \), \( x(t) \) maintains a constant value of 1. - For \( 1 < t \leq 2 \), \( x(t) \) linearly decreases from 1 to 0. - For \( t > 2 \), \( x(t) = 0 \).The \( x \)-axis represents time ( \( t \) ), and the \( y \)-axis represents the signal amplitude ( \( x(t) \) ).**Figure Q1(a):**Graph of the signal \( x(t) \)This problem requires finding the Fourier transform of \( x(t) \), determining how an arbitrary system with impulse response \( h(t) = e^{-2t}u(t) \) affects the signal, and then computing the output signal \( y(t) \) via Fourier analysis. The hint provides a useful mathematical identity involving the sinc function to facilitate the Fourier transform calculation.

Determine the Fourier transform for the signal x(t) given in Figure Q1(a). An arbitrary system has impulse response given h(t) = e¯²tu(t). If the signal x(t) is passed through the system, determine the output signal y(t) using Fourier analysis.

Determine the Fourier transform for the signal x(t) given in Figure Q1(a). An arbitrary system has impulse response given h(t) = e¯²tu(t). If the signal x(t) is passed through the system, determine the output signal y(t) using Fourier analysis.

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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Concept explainers

Fourier Transform

In mathematics, the Fourier transformation is a mathematical transformation that rotates responsibilities by using region or time into tasks depending on the local or temporal frequency, such as the rendering of a musical track in step with existing volumes and frequencies. The term Fourier amendment refers to each illustration of the frequency range and the mathematical functionality related to the representation of the frequency domain and the function of space or time.

Question

**Fourier Transform and Signal Processing**

### Problem Statement

Determine the Fourier transform for the signal \( x(t) \) given in **Figure Q1(a)**.

An arbitrary system has an impulse response given by \( h(t) = e^{-2t}u(t) \). If the signal \( x(t) \) is passed through the system, determine the output signal \( y(t) \) using Fourier analysis.

**Hint:**
\[ \text{sinc}\left(\frac{\omega}{2}\right) = \frac{e^{ \frac{j\omega}{2}} - e^{ -\frac{j\omega}{2}}}{j \omega} \]

### Figure Explanation

**Figure Q1(a)** presents a graphical representation of the signal \( x(t) \) as a function of time \( t \).

- The signal \( x(t) \) is defined as follows:
- For \( t < 0 \), \( x(t) = 0 \).
- At \( t = 0 \), \( x(t) = 1 \).
- For \( 0 < t \leq 1 \), \( x(t) \) maintains a constant value of 1.
- For \( 1 < t \leq 2 \), \( x(t) \) linearly decreases from 1 to 0.
- For \( t > 2 \), \( x(t) = 0 \).

The \( x \)-axis represents time ( \( t \) ), and the \( y \)-axis represents the signal amplitude ( \( x(t) \) ).

**Figure Q1(a):**
Graph of the signal \( x(t) \)

This problem requires finding the Fourier transform of \( x(t) \), determining how an arbitrary system with impulse response \( h(t) = e^{-2t}u(t) \) affects the signal, and then computing the output signal \( y(t) \) via Fourier analysis. The hint provides a useful mathematical identity involving the sinc function to facilitate the Fourier transform calculation.

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