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1) In this problem, we consider the construction of various types of block diagram representations for a causal LTI system S with input x(t), output y(t), and system function 2s2 + 4s – 6 H(s) s2 + 3s + 2 To derive the direct-form block diagram representation of S we first consider a causal LTI system S1 that has the same input x(t) as S, but whose system function is: 1 H, (s) %3D s2 + 3s + 2 With the output of S1 denoted by y1(t), the direct-form block diagram representation of S1 is shown in Figure-1. The signals e(t) and f(t) indicated in the figure represent respective inputs into the two integrators. d²y1(t) a) Express y(t) as a linear combination of y,(t), dy (t) and dt dt2 dy ( b) How is related to f (t)? dt d²y1(t) - related to e(t)? dt2 с) How is d) Express y(t) as a linear combination of e(t), f(t), and y,(t). e) Use the result from the previous part to extend the direct-form block diagram representation of S, and create a block diagram representation of S. f) Observing that 2(s – 1)\ (S + 3 H(s) s + 2 draw a block diagram representation for S as a cascade combination of two subsystems. g) Observing that 6. = 2 + 8 H(s) s + 2) draw a block diagram representation for S as a parallel combination of three subsystems. e(t) f(t) x(t) y1(t) -3 -2

1) In this problem, we consider the construction of various types of block diagram representations for a causal LTI system S with input x(t), output y(t), and system function 2s2 + 4s – 6 H(s) s2 + 3s + 2 To derive the direct-form block diagram representation of S we first consider a causal LTI system S1 that has the same input x(t) as S, but whose system function is: 1 H, (s) %3D s2 + 3s + 2 With the output of S1 denoted by y1(t), the direct-form block diagram representation of S1 is shown in Figure-1. The signals e(t) and f(t) indicated in the figure represent respective inputs into the two integrators. d²y1(t) a) Express y(t) as a linear combination of y,(t), dy (t) and dt dt2 dy ( b) How is related to f (t)? dt d²y1(t) - related to e(t)? dt2 с) How is d) Express y(t) as a linear combination of e(t), f(t), and y,(t). e) Use the result from the previous part to extend the direct-form block diagram representation of S, and create a block diagram representation of S. f) Observing that 2(s – 1)\ (S + 3 H(s) s + 2 draw a block diagram representation for S as a cascade combination of two subsystems. g) Observing that 6. = 2 + 8 H(s) s + 2) draw a block diagram representation for S as a parallel combination of three subsystems. e(t) f(t) x(t) y1(t) -3 -2

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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Can you solve options d, e and f please

1) In this problem, we consider the construction of various types of block diagram representations for a causal LTI system S with input x(t), output y(t), and system function 2s2 + 4s – 6 H(s) %3D s2 + 3s + 2 To derive the direct-form block diagram representation of S we first consider a causal LTI system S, that has the same input x(t) as S, but whose system function is: 1 H; (s) s2 + 3s + 2 With the output of S1 denoted by y1(t), the direct-form block diagram representation of S, is shown in Figure-1. The signals e(t) and f(t) indicated in the figure represent respective inputs into the two integrators. d²y1(t) a) Express y(t) as a linear combination of y,(t), dy,(t) and dt dt2 dy (t) b) How is related to f(t)? dt c) How is d?y1(t) related to e(t)? dt2 d) Express y(t) as a linear combination of e(t), f(t), and y,(t). e) Use the result from the previous part to extend the direct-form block diagram representation of S, and create a block diagram representation of S. f) Observing that (s – 1) (s +3 H(s) = s+ 2 draw a block diagram representation for S as a cascade combination of two subsystems. g) Observing that 8. H(s) = 2 + s + 2, draw a block diagram representation for S as a parallel combination of three subsystems. e(t) f(t) S x(t) y1(t) -3 -2
Transcribed Image Text:1) In this problem, we consider the construction of various types of block diagram representations for a causal LTI system S with input x(t), output y(t), and system function 2s2 + 4s – 6 H(s) %3D s2 + 3s + 2 To derive the direct-form block diagram representation of S we first consider a causal LTI system S, that has the same input x(t) as S, but whose system function is: 1 H; (s) s2 + 3s + 2 With the output of S1 denoted by y1(t), the direct-form block diagram representation of S, is shown in Figure-1. The signals e(t) and f(t) indicated in the figure represent respective inputs into the two integrators. d²y1(t) a) Express y(t) as a linear combination of y,(t), dy,(t) and dt dt2 dy (t) b) How is related to f(t)? dt c) How is d?y1(t) related to e(t)? dt2 d) Express y(t) as a linear combination of e(t), f(t), and y,(t). e) Use the result from the previous part to extend the direct-form block diagram representation of S, and create a block diagram representation of S. f) Observing that (s – 1) (s +3 H(s) = s+ 2 draw a block diagram representation for S as a cascade combination of two subsystems. g) Observing that 8. H(s) = 2 + s + 2, draw a block diagram representation for S as a parallel combination of three subsystems. e(t) f(t) S x(t) y1(t) -3 -2
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