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100% for correct answers. A CT-impulse training sampling and reconstruction system is shown in the figure. The forms / parameters of individual signals and systems are specified as follows. p(t) Ideal Low Pass Filter ☑ x(t) xp(t) H(jw) xr(t) The impulse train is represented by p(t) = sampling period. = Σn (t-nT), where n is an integer, and T is • • The Fourier transforms of the input signal x(t) and the reconstructed signal xr(t) are respectively: X(jw) and X, (jw) The sampled function is x(t). The Fourier transform of the sampled function is represented by Xp(jw). The frequency response of the ideal low-pass filter is as follows: H(jw) = G, if -wc

100% for correct answers. A CT-impulse training sampling and reconstruction system is shown in the figure. The forms / parameters of individual signals and systems are specified as follows. p(t) Ideal Low Pass Filter ☑ x(t) xp(t) H(jw) xr(t) The impulse train is represented by p(t) = sampling period. = Σn (t-nT), where n is an integer, and T is • • The Fourier transforms of the input signal x(t) and the reconstructed signal xr(t) are respectively: X(jw) and X, (jw) The sampled function is x(t). The Fourier transform of the sampled function is represented by Xp(jw). The frequency response of the ideal low-pass filter is as follows: H(jw) = G, if -wc

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
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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100% for correct answers. A CT-impulse training sampling and reconstruction system is shown in the figure. The forms / parameters of individual signals and systems are specified as follows. p(t) Ideal Low Pass Filter ☑ x(t) xp(t) H(jw) xr(t) The impulse train is represented by p(t) = sampling period. = Σn (t-nT), where n is an integer, and T is • • The Fourier transforms of the input signal x(t) and the reconstructed signal xr(t) are respectively: X(jw) and X, (jw) The sampled function is x(t). The Fourier transform of the sampled function is represented by Xp(jw). The frequency response of the ideal low-pass filter is as follows: H(jw) = G, if -wc<w< wc; H(jw) = 0, if | w | ≥ wc. Here G is the DC gain of the low-pass filter, and w, is the cutoff frequency of the low-pass filter. If exact reconstruction without aliasing is impossible, assume the low pass filter possess the following cutoff frequency and gain: w₁ = π/T, G = T. All unit for time is second, and for w is rad/s.
Transcribed Image Text:100% for correct answers. A CT-impulse training sampling and reconstruction system is shown in the figure. The forms / parameters of individual signals and systems are specified as follows. p(t) Ideal Low Pass Filter ☑ x(t) xp(t) H(jw) xr(t) The impulse train is represented by p(t) = sampling period. = Σn (t-nT), where n is an integer, and T is • • The Fourier transforms of the input signal x(t) and the reconstructed signal xr(t) are respectively: X(jw) and X, (jw) The sampled function is x(t). The Fourier transform of the sampled function is represented by Xp(jw). The frequency response of the ideal low-pass filter is as follows: H(jw) = G, if -wc<w< wc; H(jw) = 0, if | w | ≥ wc. Here G is the DC gain of the low-pass filter, and w, is the cutoff frequency of the low-pass filter. If exact reconstruction without aliasing is impossible, assume the low pass filter possess the following cutoff frequency and gain: w₁ = π/T, G = T. All unit for time is second, and for w is rad/s.
Problem 2. (40 pts) Consider another input signal. The Fourier transform of the input signal is specified as follows: X(jw) = 2, if -10 < w< 10; X(jw) = 0, if | w❘ ≥ 10. (a) (10 pts) Determine the condition on the sampling period T under which the input signal x(t) can be exactly reconstructed, i.e., x(t) = x(t). Specific the parameter values (or a range of values) of the ideal low-pass filter in order to achieve exact signal recovery, i.e., x(t) = x(t). The expression for G and we should be a function of the sampling period T. (b) (10 pts) For the sampling period of T=0.25 second, determine the form of Xp (jw). Sketch the absolute value of X, (jw) as a function of w. (c) (10 pts) For the sampling period of T=0.5 second, determine the form of Xp (jw). Sketch the absolute value of X, (jw) as a function of w. (d) (5 pts) Under the scenario in (b), can exact reconstruction be achieved using a low pass filter? Explain your reason. (e) (5 pts) Under the scenario in (c), can exact reconstruction be achieved using a low pass filter?
Transcribed Image Text:Problem 2. (40 pts) Consider another input signal. The Fourier transform of the input signal is specified as follows: X(jw) = 2, if -10 < w< 10; X(jw) = 0, if | w❘ ≥ 10. (a) (10 pts) Determine the condition on the sampling period T under which the input signal x(t) can be exactly reconstructed, i.e., x(t) = x(t). Specific the parameter values (or a range of values) of the ideal low-pass filter in order to achieve exact signal recovery, i.e., x(t) = x(t). The expression for G and we should be a function of the sampling period T. (b) (10 pts) For the sampling period of T=0.25 second, determine the form of Xp (jw). Sketch the absolute value of X, (jw) as a function of w. (c) (10 pts) For the sampling period of T=0.5 second, determine the form of Xp (jw). Sketch the absolute value of X, (jw) as a function of w. (d) (5 pts) Under the scenario in (b), can exact reconstruction be achieved using a low pass filter? Explain your reason. (e) (5 pts) Under the scenario in (c), can exact reconstruction be achieved using a low pass filter?
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