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Ques. 3: Find the exponential Fourier series for the given waveform x(t) A -3T -π0 π ot

Ques. 3: Find the exponential Fourier series for the given waveform x(t) A -3T -π0 π ot

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
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**Question 3:** **Objective:** Find the exponential Fourier series for the given waveform. **Provided Waveform:** The given waveform is a triangular wave that is periodic with a period of \( 4\pi \). Below is a detailed description and explanation: **Graph Details:** - **Horizontal Axis (Time Axis, \( \omega t \)):** The x-axis ranges from \( -3\pi \) to \( 3\pi \). - At \( \omega t = -3\pi \): The waveform crosses the x-axis. - At \( \omega t = -\pi \): The waveform hits a peak at magnitude \( A \) and then decreases linearly. - At \( \omega t = 0 \): The waveform reaches its maximum value \( A \). - At \( \omega t = \pi \): The waveform reaches a minimum value back at the x-axis. - At \( \omega t = 3\pi \): The pattern between \( \omega t = -3\pi \) and \( 0 \) is mirrored. - **Vertical Axis (\( x(t) \)):** The y-axis represents the amplitude of the waveform, peaking at \( A \). **Waveform Characteristics:** - The waveform is continuous and piecewise linear, meaning it consists of straight-line segments. - The waveform ascends linearly from \( -3\pi \) to \( 0 \) and descends linearly from \( 0 \) to \( 3\pi \). - It is symmetric and has peaks at \( \pm \pi \). Given the symmetry and periodicity, you will need to derive the Fourier series using complex exponentials. This triangular wave can be broken down into its constituent sine and cosine (or exponential) components, which is essential for understanding and analyzing signals in the frequency domain. **Procedure:** 1. **Determine the period \( T \)** of the waveform: The period \( T = 4\pi \). 2. **Formulate the Fourier series**: An exponential Fourier series represent \( x(t) \) as a summation of exponentials of the form: \[ x(t) = \sum_{n=-\infty}^{\infty} C_n e^{j n \omega_0 t} \] where \( C_n \) are the Fourier coefficients, and \( \omega
Transcribed Image Text:**Question 3:** **Objective:** Find the exponential Fourier series for the given waveform. **Provided Waveform:** The given waveform is a triangular wave that is periodic with a period of \( 4\pi \). Below is a detailed description and explanation: **Graph Details:** - **Horizontal Axis (Time Axis, \( \omega t \)):** The x-axis ranges from \( -3\pi \) to \( 3\pi \). - At \( \omega t = -3\pi \): The waveform crosses the x-axis. - At \( \omega t = -\pi \): The waveform hits a peak at magnitude \( A \) and then decreases linearly. - At \( \omega t = 0 \): The waveform reaches its maximum value \( A \). - At \( \omega t = \pi \): The waveform reaches a minimum value back at the x-axis. - At \( \omega t = 3\pi \): The pattern between \( \omega t = -3\pi \) and \( 0 \) is mirrored. - **Vertical Axis (\( x(t) \)):** The y-axis represents the amplitude of the waveform, peaking at \( A \). **Waveform Characteristics:** - The waveform is continuous and piecewise linear, meaning it consists of straight-line segments. - The waveform ascends linearly from \( -3\pi \) to \( 0 \) and descends linearly from \( 0 \) to \( 3\pi \). - It is symmetric and has peaks at \( \pm \pi \). Given the symmetry and periodicity, you will need to derive the Fourier series using complex exponentials. This triangular wave can be broken down into its constituent sine and cosine (or exponential) components, which is essential for understanding and analyzing signals in the frequency domain. **Procedure:** 1. **Determine the period \( T \)** of the waveform: The period \( T = 4\pi \). 2. **Formulate the Fourier series**: An exponential Fourier series represent \( x(t) \) as a summation of exponentials of the form: \[ x(t) = \sum_{n=-\infty}^{\infty} C_n e^{j n \omega_0 t} \] where \( C_n \) are the Fourier coefficients, and \( \omega
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