Homework Help is Here – Start Your Trial Now!
Engineering
Electrical Engineering
(c) (10 pts) Repeat part (b), but the sampling period is now T=0.5 second. 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) ☑ x(t) xp(t) Ideal Low Pass Filter H(jw) xr(t) • • • • The impulse train is represented by p(t) = Σn 8(t - nT), where n is an integer, and T is sampling period. 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 xp (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 1. (25 pts) Suppose the input signal is given by x(t) = 1 + 4cos(5t) sin(3t). (a) (5 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). (b) (10 pts) If the sampling period is T=0.25 second, determine whether the input signal can be exactly reconstructed. If exact reconstruction is possible, determine the constraints on the parameters of the low pass filter under which one can achieve exact reconstruction. If exact reconstruction is impossible, derive the form of x(t), assuming the cutoff frequency to be wc =π/T and the gain to be G = T.

(c) (10 pts) Repeat part (b), but the sampling period is now T=0.5 second. 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) ☑ x(t) xp(t) Ideal Low Pass Filter H(jw) xr(t) • • • • The impulse train is represented by p(t) = Σn 8(t - nT), where n is an integer, and T is sampling period. 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 xp (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 1. (25 pts) Suppose the input signal is given by x(t) = 1 + 4cos(5t) sin(3t). (a) (5 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). (b) (10 pts) If the sampling period is T=0.25 second, determine whether the input signal can be exactly reconstructed. If exact reconstruction is possible, determine the constraints on the parameters of the low pass filter under which one can achieve exact reconstruction. If exact reconstruction is impossible, derive the form of x(t), assuming the cutoff frequency to be wc =π/T and the gain to be G = T.

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...
See similar textbooks
icon
Related questions
Question
(c) (10 pts) Repeat part (b), but the sampling period is now T=0.5 second.
Transcribed Image Text:(c) (10 pts) Repeat part (b), but the sampling period is now T=0.5 second.
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) ☑ x(t) xp(t) Ideal Low Pass Filter H(jw) xr(t) • • • • The impulse train is represented by p(t) = Σn 8(t - nT), where n is an integer, and T is sampling period. 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 xp (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 1. (25 pts) Suppose the input signal is given by x(t) = 1 + 4cos(5t) sin(3t). (a) (5 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). (b) (10 pts) If the sampling period is T=0.25 second, determine whether the input signal can be exactly reconstructed. If exact reconstruction is possible, determine the constraints on the parameters of the low pass filter under which one can achieve exact reconstruction. If exact reconstruction is impossible, derive the form of x(t), assuming the cutoff frequency to be wc =π/T and the gain to be G = T.
Transcribed Image Text: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) ☑ x(t) xp(t) Ideal Low Pass Filter H(jw) xr(t) • • • • The impulse train is represented by p(t) = Σn 8(t - nT), where n is an integer, and T is sampling period. 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 xp (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 1. (25 pts) Suppose the input signal is given by x(t) = 1 + 4cos(5t) sin(3t). (a) (5 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). (b) (10 pts) If the sampling period is T=0.25 second, determine whether the input signal can be exactly reconstructed. If exact reconstruction is possible, determine the constraints on the parameters of the low pass filter under which one can achieve exact reconstruction. If exact reconstruction is impossible, derive the form of x(t), assuming the cutoff frequency to be wc =π/T and the gain to be G = T.
Expert Solution
steps

Step by step

Solved in 2 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:
9780133923605
Author:
Robert L. Boylestad
Publisher:
PEARSON
Delmar's Standard Textbook Of Electricity
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:
9781337900348
Author:
Stephen L. Herman
Publisher:
Cengage Learning
Programmable Logic Controllers
Programmable Logic Controllers
Electrical Engineering
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
Fundamentals of Electric Circuits
Fundamentals of Electric Circuits
Electrical Engineering
ISBN:
9780078028229
Author:
Charles K Alexander, Matthew Sadiku
Publisher:
McGraw-Hill Education
Electric Circuits. (11th Edition)
Electric Circuits. (11th Edition)
Electrical Engineering
ISBN:
9780134746968
Author:
James W. Nilsson, Susan Riedel
Publisher:
PEARSON
Engineering Electromagnetics
Engineering Electromagnetics
Electrical Engineering
ISBN:
9780078028151
Author:
Hayt, William H. (william Hart), Jr, BUCK, John A.
Publisher:
Mcgraw-hill Education,
SEE MORE TEXTBOOKS