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1. Consider a continuous-time periodic signal x(t) = x(t +T) with Fourier series representation where (1)x 1 0.8 0.6 0.2 0 -12 XK -10 It is straightforward to show that these Fourier series coefficients correspond to a continuous- time periodic trapezoidal signal, plotted below for T=4 seconds. wwwwwww W -2 0 t = -8 8 x(t) = Σ Xhelankt/T k=-00 Xo = 0.5 0, k even 6 π²k² -6 sin sin πk k odd 2 4 6 8 10 12

1. Consider a continuous-time periodic signal x(t) = x(t +T) with Fourier series representation where (1)x 1 0.8 0.6 0.2 0 -12 XK -10 It is straightforward to show that these Fourier series coefficients correspond to a continuous- time periodic trapezoidal signal, plotted below for T=4 seconds. wwwwwww W -2 0 t = -8 8 x(t) = Σ Xhelankt/T k=-00 Xo = 0.5 0, k even 6 π²k² -6 sin sin πk k odd 2 4 6 8 10 12

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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1. Consider a continuous-time periodic signal x(t) = x(t + T) with Fourier series representation where 1 0.8 0.6 0.4 It is straightforward to show that these Fourier series coefficients correspond to a continuous- time periodic trapezoidal signal, plotted below for T = 4 seconds. 0.2 0 Xx = { XK wwww -12 -10 ∞ x(t) = Σ Xxeizmkt/T k=-∞0 -8 Xo = 0.5 0, k even 6 π²k² 17²1K² Sin (777) sin (7), -6 -4 -2 , k odd 10 t 2 4 6 8 10 12
Transcribed Image Text:1. Consider a continuous-time periodic signal x(t) = x(t + T) with Fourier series representation where 1 0.8 0.6 0.4 It is straightforward to show that these Fourier series coefficients correspond to a continuous- time periodic trapezoidal signal, plotted below for T = 4 seconds. 0.2 0 Xx = { XK wwww -12 -10 ∞ x(t) = Σ Xxeizmkt/T k=-∞0 -8 Xo = 0.5 0, k even 6 π²k² 17²1K² Sin (777) sin (7), -6 -4 -2 , k odd 10 t 2 4 6 8 10 12
(a) (b) (c) Suppose another periodic continuous-time signal z(t) is created having Fourier series coefficients given by Zk = Xkeink. Using the Time Shift property of the continuous-time Fourier Series, sketch z(t), where T= 4 seconds. Suppose the original periodic signal x(t) is input to a continuous-time LTI system with frequency response H() shown below. With the signal period still T = 4 seconds, find an expression for the LTI system output signal y(t) and sketch y(t) -3π/4 1 H(Q) 3π/4 (rad/s) [Extra Credit] Write a routine (you are encouraged to use sq_wave.m as a guide) to numerically compute and plot several periods of the partial sums of the continuous-time Fourier series. Use enough terms so that you obtain a close approximation to x(t).
Transcribed Image Text:(a) (b) (c) Suppose another periodic continuous-time signal z(t) is created having Fourier series coefficients given by Zk = Xkeink. Using the Time Shift property of the continuous-time Fourier Series, sketch z(t), where T= 4 seconds. Suppose the original periodic signal x(t) is input to a continuous-time LTI system with frequency response H() shown below. With the signal period still T = 4 seconds, find an expression for the LTI system output signal y(t) and sketch y(t) -3π/4 1 H(Q) 3π/4 (rad/s) [Extra Credit] Write a routine (you are encouraged to use sq_wave.m as a guide) to numerically compute and plot several periods of the partial sums of the continuous-time Fourier series. Use enough terms so that you obtain a close approximation to x(t).
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