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signal, x(t), is sampled (multiplied) by a sampling function, p(t), to yield an output nction, y(t) [i.e., x(t)p(t) = y(t)]. e Fourier transform of x(t), X(jw) is as shown below and is = 0 for |w| > Wm= 2π×5000 d/sec. -WM X(jw) WM 3

signal, x(t), is sampled (multiplied) by a sampling function, p(t), to yield an output nction, y(t) [i.e., x(t)p(t) = y(t)]. e Fourier transform of x(t), X(jw) is as shown below and is = 0 for |w| > Wm= 2π×5000 d/sec. -WM X(jw) WM 3

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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A signal, x(t), is sampled (multiplied) by a sampling function, p(t), to yield an output function, y(t) [i.e., x(t)p(t) = y(t)]. The Fourier transform of x(t), X(jw) is as shown below and is = 0 for |w| > wm = 2 x 5000 rad/sec. -WM X(jw) ∞ WM The function p(t) is a train of continuous impulse functions given by the formula: p(t) = S(t - KT) k=-∞ (a) What is the minimum sampling frequency, ws (in rad/sec), and maximum period, T (in seconds), for p(t) such that the output function, y(t), (a sampled version of x(t)) will contain no aliasing? (b) Sketch P(jw) and Y(jw), the FT of p(t) and y(t) respectively, if the sampling period T is 1.5 times the value you found in part a). (c) Sketch P(jw) and Y(jw), the FT of p(t) and y(t) respectively, if the sampling period T is 0.75 times the value you found in part a). (d) Can x(t) be recovered from y(t) if the sampling period is as stated in part b)? If so, sketch a block diagram of the recovery system and label any parameters for the parts of your system (modulation frequencies, modulator gains, filter cutoff frequencies, filter gains, etc.). If it cannot, state why not. (e) Can x(t) be recovered from y(t) if the sampling period is as stated in part c)? If so, sketch a block diagram of the recovery system and label any parameters for the parts of your system (modulation frequencies, modulator gains, filter cutoff frequencies, filter gains, etc.). If it cannot, state why not.
Transcribed Image Text:A signal, x(t), is sampled (multiplied) by a sampling function, p(t), to yield an output function, y(t) [i.e., x(t)p(t) = y(t)]. The Fourier transform of x(t), X(jw) is as shown below and is = 0 for |w| > wm = 2 x 5000 rad/sec. -WM X(jw) ∞ WM The function p(t) is a train of continuous impulse functions given by the formula: p(t) = S(t - KT) k=-∞ (a) What is the minimum sampling frequency, ws (in rad/sec), and maximum period, T (in seconds), for p(t) such that the output function, y(t), (a sampled version of x(t)) will contain no aliasing? (b) Sketch P(jw) and Y(jw), the FT of p(t) and y(t) respectively, if the sampling period T is 1.5 times the value you found in part a). (c) Sketch P(jw) and Y(jw), the FT of p(t) and y(t) respectively, if the sampling period T is 0.75 times the value you found in part a). (d) Can x(t) be recovered from y(t) if the sampling period is as stated in part b)? If so, sketch a block diagram of the recovery system and label any parameters for the parts of your system (modulation frequencies, modulator gains, filter cutoff frequencies, filter gains, etc.). If it cannot, state why not. (e) Can x(t) be recovered from y(t) if the sampling period is as stated in part c)? If so, sketch a block diagram of the recovery system and label any parameters for the parts of your system (modulation frequencies, modulator gains, filter cutoff frequencies, filter gains, etc.). If it cannot, state why not.
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