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z(t) = 3. dx(t) dt +x(t) (3) where x(t) is input, and z(t) is output. (a) (10 pts) Based on the differential equation itself, prove that system 1 is equivalent to the series cascade configuration shown in Fig. (c). (b) (10 pts) Calculate the frequency response function Ha(jw) of system 1, i.e., Fig. (a). (c) (10 pts) Calculate the frequency response Hb (jw) of the system configuration in Fig. (b), as well as the frequency response function Hc(jw) of the system configuration in Fig. (c). Demonstrate Ha(jw) = H₁(jw) = Hc(jw). (d) (10 pts) If the input signal for the system in Fig. (a) is x(t) = 8(t), calculate its Fourier transform X(jw) and the Fourier transform of the output Y(jw). (e) (10 pts) Based on the inverse Fourier transform of Y(jw) in (d), determine the impulse response function of the CT-LTI system described by Eq. (1). (f) (10 pts) The input signal for the system in Fig. (b) is x(t) = 8(t). Calculate the Fourier transform of w₁(t) and y(t). (g) (10 pts) The input signal for the system in Fig. (c) is x(t) = 8(t).Calculate the Fourier transform of w2(t) and y(t). Another way of looking at this problem can be found in textbook Problem P2.55 and P2.56, page 157 to page 160. (a) x(t). Eq (1) y(t) W1(t) (b) x(t) Eq (2) Eq (3) y(t) W2(t) (c) x(t) Eq (3) Eq (2) y(t) Problem 2: Consider a CT-LTI system described by: dy(t) dt +2y(t) = 3 dx(t) dt +x(t) (1) where the system input is x(t) and the output is y(t). We also consider two related sub-systems: system 2 and 3. In particular, system 2 is described by dy(t) dt +2y(t) = z(t) (2) where z(t) is system input and y(t) is its output. System 3 is described by:

z(t) = 3. dx(t) dt +x(t) (3) where x(t) is input, and z(t) is output. (a) (10 pts) Based on the differential equation itself, prove that system 1 is equivalent to the series cascade configuration shown in Fig. (c). (b) (10 pts) Calculate the frequency response function Ha(jw) of system 1, i.e., Fig. (a). (c) (10 pts) Calculate the frequency response Hb (jw) of the system configuration in Fig. (b), as well as the frequency response function Hc(jw) of the system configuration in Fig. (c). Demonstrate Ha(jw) = H₁(jw) = Hc(jw). (d) (10 pts) If the input signal for the system in Fig. (a) is x(t) = 8(t), calculate its Fourier transform X(jw) and the Fourier transform of the output Y(jw). (e) (10 pts) Based on the inverse Fourier transform of Y(jw) in (d), determine the impulse response function of the CT-LTI system described by Eq. (1). (f) (10 pts) The input signal for the system in Fig. (b) is x(t) = 8(t). Calculate the Fourier transform of w₁(t) and y(t). (g) (10 pts) The input signal for the system in Fig. (c) is x(t) = 8(t).Calculate the Fourier transform of w2(t) and y(t). Another way of looking at this problem can be found in textbook Problem P2.55 and P2.56, page 157 to page 160. (a) x(t). Eq (1) y(t) W1(t) (b) x(t) Eq (2) Eq (3) y(t) W2(t) (c) x(t) Eq (3) Eq (2) y(t) Problem 2: Consider a CT-LTI system described by: dy(t) dt +2y(t) = 3 dx(t) dt +x(t) (1) where the system input is x(t) and the output is y(t). We also consider two related sub-systems: system 2 and 3. In particular, system 2 is described by dy(t) dt +2y(t) = z(t) (2) where z(t) is system input and y(t) is its output. System 3 is described by:

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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z(t) = 3. dx(t) dt +x(t) (3) where x(t) is input, and z(t) is output. (a) (10 pts) Based on the differential equation itself, prove that system 1 is equivalent to the series cascade configuration shown in Fig. (c). (b) (10 pts) Calculate the frequency response function Ha(jw) of system 1, i.e., Fig. (a). (c) (10 pts) Calculate the frequency response Hb (jw) of the system configuration in Fig. (b), as well as the frequency response function Hc(jw) of the system configuration in Fig. (c). Demonstrate Ha(jw) = H₁(jw) = Hc(jw). (d) (10 pts) If the input signal for the system in Fig. (a) is x(t) = 8(t), calculate its Fourier transform X(jw) and the Fourier transform of the output Y(jw). (e) (10 pts) Based on the inverse Fourier transform of Y(jw) in (d), determine the impulse response function of the CT-LTI system described by Eq. (1). (f) (10 pts) The input signal for the system in Fig. (b) is x(t) = 8(t). Calculate the Fourier transform of w₁(t) and y(t). (g) (10 pts) The input signal for the system in Fig. (c) is x(t) = 8(t).Calculate the Fourier transform of w2(t) and y(t). Another way of looking at this problem can be found in textbook Problem P2.55 and P2.56, page 157 to page 160. (a) x(t). Eq (1) y(t) W1(t) (b) x(t) Eq (2) Eq (3) y(t) W2(t) (c) x(t) Eq (3) Eq (2) y(t)
Transcribed Image Text:z(t) = 3. dx(t) dt +x(t) (3) where x(t) is input, and z(t) is output. (a) (10 pts) Based on the differential equation itself, prove that system 1 is equivalent to the series cascade configuration shown in Fig. (c). (b) (10 pts) Calculate the frequency response function Ha(jw) of system 1, i.e., Fig. (a). (c) (10 pts) Calculate the frequency response Hb (jw) of the system configuration in Fig. (b), as well as the frequency response function Hc(jw) of the system configuration in Fig. (c). Demonstrate Ha(jw) = H₁(jw) = Hc(jw). (d) (10 pts) If the input signal for the system in Fig. (a) is x(t) = 8(t), calculate its Fourier transform X(jw) and the Fourier transform of the output Y(jw). (e) (10 pts) Based on the inverse Fourier transform of Y(jw) in (d), determine the impulse response function of the CT-LTI system described by Eq. (1). (f) (10 pts) The input signal for the system in Fig. (b) is x(t) = 8(t). Calculate the Fourier transform of w₁(t) and y(t). (g) (10 pts) The input signal for the system in Fig. (c) is x(t) = 8(t).Calculate the Fourier transform of w2(t) and y(t). Another way of looking at this problem can be found in textbook Problem P2.55 and P2.56, page 157 to page 160. (a) x(t). Eq (1) y(t) W1(t) (b) x(t) Eq (2) Eq (3) y(t) W2(t) (c) x(t) Eq (3) Eq (2) y(t)
Problem 2: Consider a CT-LTI system described by: dy(t) dt +2y(t) = 3 dx(t) dt +x(t) (1) where the system input is x(t) and the output is y(t). We also consider two related sub-systems: system 2 and 3. In particular, system 2 is described by dy(t) dt +2y(t) = z(t) (2) where z(t) is system input and y(t) is its output. System 3 is described by:
Transcribed Image Text:Problem 2: Consider a CT-LTI system described by: dy(t) dt +2y(t) = 3 dx(t) dt +x(t) (1) where the system input is x(t) and the output is y(t). We also consider two related sub-systems: system 2 and 3. In particular, system 2 is described by dy(t) dt +2y(t) = z(t) (2) where z(t) is system input and y(t) is its output. System 3 is described by:
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