9.36. 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 H₁(s) = 1 s² + 3s + 2* With the output of S₁ denoted by y₁(t), the direct-form block diagram representation of S₁ is shown in Figure P9.36. The signals e(t) and f(t) indicated in the figure represent respective inputs into the two integrators. (a) Express y(t) (the output of S) as a linear combination of y₁(t), dy₁(t)/dt, and d² y₁(t)/dt². (b) How is dy₁(t)/dt related to f(t)? (c) How is d²y₁(t)/dt² related to e(t)? (d) Express y(t) as a linear combination of e(t), ƒ(t), and y₁(t). e(t) + -S 1 x(t) f(t) - 3 + 1 -S -2 y₁(t) Figure P9.36 (e) Use the result from the previous part to extend the direct-form block diagram representation of S₁ and create a block diagram representation of S. (f) Observing that 2(s – 1) - H(s) ==== S+3 5+2 s + 1 draw a block diagram representation for S as a cascade combination of two subsystems. (g) Observing that 6 8 H(s) = 2+ S+2 s + 1' draw a block-diagram representation for S as a parallel combination of three subsystems.
9.36. 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 H₁(s) = 1 s² + 3s + 2* With the output of S₁ denoted by y₁(t), the direct-form block diagram representation of S₁ is shown in Figure P9.36. The signals e(t) and f(t) indicated in the figure represent respective inputs into the two integrators. (a) Express y(t) (the output of S) as a linear combination of y₁(t), dy₁(t)/dt, and d² y₁(t)/dt². (b) How is dy₁(t)/dt related to f(t)? (c) How is d²y₁(t)/dt² related to e(t)? (d) Express y(t) as a linear combination of e(t), ƒ(t), and y₁(t). e(t) + -S 1 x(t) f(t) - 3 + 1 -S -2 y₁(t) Figure P9.36 (e) Use the result from the previous part to extend the direct-form block diagram representation of S₁ and create a block diagram representation of S. (f) Observing that 2(s – 1) - H(s) ==== S+3 5+2 s + 1 draw a block diagram representation for S as a cascade combination of two subsystems. (g) Observing that 6 8 H(s) = 2+ S+2 s + 1' draw a block-diagram representation for S as a parallel combination of three subsystems.
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This is a practice problem from the Signals and Systems course in my Electrical Engineering program.
Could you please walk me through the steps of solving it so that I can solve more complicated ones in the future?
Thank you for your assistance.
Transcribed Image Text:9.36. 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
H₁(s)
=
1
s² + 3s + 2*
With the output of S₁ denoted by y₁(t), the direct-form block diagram representation
of S₁ is shown in Figure P9.36. The signals e(t) and f(t) indicated in the figure
represent respective inputs into the two integrators.
(a) Express y(t) (the output of S) as a linear combination of y₁(t), dy₁(t)/dt, and
d² y₁(t)/dt².
(b) How is dy₁(t)/dt related to f(t)?
(c) How is d²y₁(t)/dt² related to e(t)?
(d) Express y(t) as a linear combination of e(t), ƒ(t), and y₁(t).
e(t)
+
-S
1
x(t)
f(t)
- 3
+
1
-S
-2
y₁(t)
Figure P9.36
(e) Use the result from the previous part to extend the direct-form block diagram
representation of S₁ and create a block diagram representation of S.
(f) Observing that
2(s – 1)
-
H(s)
====
S+3
5+2 s + 1
draw a block diagram representation for S as a cascade combination of two
subsystems.
(g) Observing that
6
8
H(s) = 2+
S+2
s + 1'
draw a block-diagram representation for S as a parallel combination of three
subsystems.
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