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3 5 . Using the Laplace transform, show that .1
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Use Eq. (3.21) and Table 3-1. ( b ) Use Eqs. (3.18) and (3.21) and Table 3-1.
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3 5 . Using the Laplace transform, redo Prob. 2.54. .2
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Hint: ( a ) Find the system function H ( s ) by Eq. (3.32) and take the inverse Laplace transform of H(s). ( 6 ) Find the ROC of H ( s ) and show that it does not contain the jo-axis.
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3.53. Find the output y ( t ) of the continuous-time LTI system with
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for the each of the following inputs: (a) x ( t ) = e-'u(t) ( b ) x ( t ) = e-'u(-t)
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Ans.
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( a ) y ( r ) = (e-'-e-")u(t) ( b ) y ( t ) = e-'u(-f)+e- 2 ' ~ ( t )
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3.54. The step response of an continuous-time LTI system is given by (1 - e-')u(t). For a certain unknown input x ( t ) , the output y ( t ) is observed to be (2 - 3e-' + e-3')u(r). Find the input x(t).
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LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
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[CHAP. 3
Fig. 3-18
Determine the overall system function H ( s ) for the system shown in Fig. 3-18.
Hint:
Use the result from Prob. 3.31 to simplify the block diagram.
Am. H ( s ) =
s3+3s2+s-2
If x ( t ) is a periodic function with fundamental period T, find the unilateral Laplace transform of x ( t ) .
Find the unilateral Laplace transforms of the periodic signals shown in Fig. 3-19.
Using the unilateral Laplace transform, find the solution of
y"(t) -y l ( t ) - 6 y ( t ) =et
with the initial conditions y ( 0 ) = 1 and y ' ( 0 )
for t 2 0.
Am. y ( t ) = - ; e l + f e - 2 ' + ;e3', t z O
Using the unilateral Laplace transform, solve the following simultaneous differential equations:
y l ( t ) +y ( t ) + x f ( r ) + x ( t ) = 1 yl(t) -y(t) -2x(t) =O
with x ( 0 ) = 0 and y ( 0 ) = 1 for t 1 0 .
Ans. x ( t ) = e-' - 1 , y ( t ) = 2 - e - ' , t
Using the unilateral Laplace transform, solve the following integral equations:
( a ) y ( t ) =eat, t 2 0 ; ( 6 ) y ( t ) =e2', t 2 0
CHAP. 31
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
F g 3-19 i.
3 6 . Consider the RC circuit in Fig. 3-20. The switch is closed at t .1 the switch closing is u,. Find the capacitor voltage for t 2 0.
Ans.
= 0 . The
capacitor voltage before
u,(t)=~,e-'/~~, 120
3 6 . Consider the RC circuit in Fig. 3-21. The switch is closed at t = 0. Before the switch closing, .2 the capacitor C , is charged to u, V and the capacitor C , is not charged.
( a ) Assuming c , = c , = c , find the current i ( t ) for t 2 0 .
( b ) Find the total energy E dissipated by the resistor R and show that E is independent of R and is equal to half of the initial energy stored in C , .
Fig. 3-20 RC circuit.
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
[CHAP. 3
Fig. 3-21
RC circuit.
Assume that R = 0 and C , = C , = C . Find the current i ( r ) for I 2 0 and voltages u,1(0+) and uC2(0+).
4
The z-Transform and Discrete-Time LTI Systems
4.1 INTRODUCTION
In Chap. 3 we introduced the Laplace transform. In this chapter we present the z-transform, which is the discrete-time counterpart of the Laplace transform. The z-transform is introduced to represent discrete-time signals (or sequences) in the z-domain ( z is a complex variable), and the concept of the system function for a discrete-time LTI system will be described. The Laplace transform converts integrodifferential equations into algebraic equations. In a similar manner, the z-transform converts difference equations into algebraic equations, thereby simplifying the analysis of discrete-time systems. The properties of the z-transform closely parallel those of the Laplace transform. However, we will see some important distinctions between the z-transform and the Laplace transform.
4.2 THE Z-TRANSFORM
In Sec. 2.8 we saw that for a discrete-time LTI system with impulse response h[n], the output y[n] of the system to the complex exponential input of the form z" is
where
Definition:
T h e function H ( z ) in Eq. (4.2) is referred to as the z-transform of h[n]. For a general discrete-time signal x[n], the z-transform X ( z ) is defined as
X(Z) =
x[n]z-"
(4.3)
T h e variable z is generally complex-valued and is expressed in polar form as
where r is the magnitude of z and R is the angle of z . The z-transform defined in Eq. ( 4 . 3 ) is often called the bilateral (or two-sided) z-transform in contrast to the unilateral (or
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