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Ques2: For the periodic signals, find the coefficients of trigonometric Fourier series. -T/4 7/4 27

Ques2: For the periodic signals, find the coefficients of trigonometric Fourier series. -T/4 7/4 27

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 2: Fourier Series Coefficients of Periodic Signals **Problem Statement:** For the periodic signals shown in the graph, find the coefficients of the trigonometric Fourier series. **Graph Description:** The graph depicts a periodic signal that repeats every \(2\pi\) units along the horizontal time axis (t). - The signal exhibits a linear increase and decrease within a single period. - The waveform starts at \((-\pi, -1)\). - It increases linearly, crossing the time axis at \( t = -\pi/4 \), reaching a maximum value of 1 at \( t = \pi/4 \). - Then, it decreases linearly, becoming zero again at \( t = \pi \). - This pattern then repeats, creating a symmetric waveform around the time axis. The periodic waveform is characterized by alternating positive and negative slopes. The essential points along the x-axis (t) include \(-\pi\), \(-\pi/4\), \(\pi/4\), \(\pi\), and \(2\pi\). **Task:** Calculate the trigonometric Fourier series coefficients of the given periodic signal. The Fourier series representation can be expressed as: \[ x(t) = A_0 + \sum_{n=1}^{\infty} \left( A_n \cos(n\omega_0 t) + B_n \sin(n\omega_0 t) \right) \] Where: - \( A_0 \) is the average (DC) component. - \( A_n \) and \( B_n \) are the Fourier coefficients. - \( \omega_0 \) is the fundamental angular frequency. The calculation involves integrating the signal over one period to find the coefficients \( A_0 \), \( A_n \), and \( B_n \), which describe the amplitude and phase of the sinusoidal components of the waveform.
Transcribed Image Text:### Question 2: Fourier Series Coefficients of Periodic Signals **Problem Statement:** For the periodic signals shown in the graph, find the coefficients of the trigonometric Fourier series. **Graph Description:** The graph depicts a periodic signal that repeats every \(2\pi\) units along the horizontal time axis (t). - The signal exhibits a linear increase and decrease within a single period. - The waveform starts at \((-\pi, -1)\). - It increases linearly, crossing the time axis at \( t = -\pi/4 \), reaching a maximum value of 1 at \( t = \pi/4 \). - Then, it decreases linearly, becoming zero again at \( t = \pi \). - This pattern then repeats, creating a symmetric waveform around the time axis. The periodic waveform is characterized by alternating positive and negative slopes. The essential points along the x-axis (t) include \(-\pi\), \(-\pi/4\), \(\pi/4\), \(\pi\), and \(2\pi\). **Task:** Calculate the trigonometric Fourier series coefficients of the given periodic signal. The Fourier series representation can be expressed as: \[ x(t) = A_0 + \sum_{n=1}^{\infty} \left( A_n \cos(n\omega_0 t) + B_n \sin(n\omega_0 t) \right) \] Where: - \( A_0 \) is the average (DC) component. - \( A_n \) and \( B_n \) are the Fourier coefficients. - \( \omega_0 \) is the fundamental angular frequency. The calculation involves integrating the signal over one period to find the coefficients \( A_0 \), \( A_n \), and \( B_n \), which describe the amplitude and phase of the sinusoidal components of the waveform.
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