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LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
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Hence, H(s) is always rational. Note that the ROC of H(s) is not specified by Eq. (3.40) but must be inferred with additional requirements on the system such as the causality or the stability.
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D. Systems Interconnection:
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For two LTI systems [with hl(t) and h2(t), respectively] in cascade [Fig. 3-Nu)], the overall impulse response h(t) is given by [Eq. (2.811, Prob. 2.141
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Thus, the corresponding system functions are related by the product
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This relationship is illustrated in Fig. 3-8(b). Similarly, the impulse response of a parallel combination of two LTI systems [Fig. 3-9(a)] is given by (Prob. 2.53)
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This relationship is illustrated in Fig. 3-9(b).
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Fig. 3-8 Two systems in cascade. ( a ) Time-domain representation; ( b )s-domain representation.
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
[CHAP. 3
Fig. 3-9 Two systems in parallel. ( a ) Time-domain representation; Ib) s-domain representation.
3.7 THE UNILATERAL LAPLACE TRANSFORM
Definitions: The unilateral (or one-sided) Laplace transform X,(s) of a signal x ( t ) is defined as [Eq. (3.5)l
The lower limit of integration is chosen to be 0- (rather than 0 or O+) to permit x(t) to include S(t) or its derivatives. Thus, we note immediately that the integration from 0- to O + is zero except when there is an impulse function or its derivative at the origin. The unilateral Laplace transform ignores x ( t ) for t < 0. Since x ( t ) in Eq. (3.43) is a right-sided ,, signal, the ROC of X,(s) is always of the form Re(s) > u, that is, a right half-plane in the s-plane.
B. Basic Properties:
Most of the properties of the unilateral Laplace transform are the same as for the bilateral transform. The unilateral Laplace transform is useful for calculating the response of a causal system to a causal input when the system is described by a linear constantcoefficient differential equation with nonzero initial conditions. The basic properties of the unilateral Laplace transform that are useful in this application are the time-differentiation and time-integration properties which are different from those of the bilateral transform. They are presented in the following.
CHAP. 31
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
I . Differentiation in the Time Domain:
provided that lim ,, x(t )e-"' ,
= 0.
Repeated application of this property yields
where
2. Integration in the Time Domain:
C. System Function: Note that with the unilateral Laplace transform, the system function H ( s ) = Y ( s ) / X ( s ) is defined under the condition that the LTI system is relaxed, that is, all initial conditions are zero. D. Transform Circuits: The solution for signals in an electric circuit can be found without writing integrodifferential equations if the circuit operations and signals are represented with their Laplace transform equivalents. [In this subsection the Laplace transform means the unilateral Laplace transform and we drop the subscript I in X,(s).] We refer to a circuit produced from these equivalents as a transform circuit. In order to use this technique, we require the Laplace transform models for individual circuit elements. These models are developed in the following discussion and are shown in Fig. 3-10. Applications of this transform model technique to electric circuits problems are illustrated in Probs. 3.40 to 3.42. I . Signal Sources:
where u ( t ) and i ( t ) are the voltage and current source signals, respectively.
2. Resistance R:
LAPLACE TRANSFORM AND CONTINUOUS-TIMELTI SYSTEMS
[CHAP. 3
Circuit element
Representation
Voltage source
Current source
Resistance
Inductance
Capacitance
V(s)
Fig. 3-10 Representation of Laplace transform circuit-element models.
CHAP. 31
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
3. Inductance L:
di(t ) t V ( s )= sLI(s) - L i ( 0 - ) , (3.50) dt The second model of the inductance L in Fig. 3-10 is obtained by rewriting Eq. (3.50) as
~ ( t=)L -
1 i ( t )t I ( s ) = - V ( s ) , sL
4. Capacitance C:
+ -i(O-) S
(3.51)
d m i ( t ) = Ct* I ( s ) = sCV(s) - C u ( 0 - ) (3.52) dt The second model of the capacitance C in Fig. 3-10 is obtained by rewriting Eq. (3.52) as 1 u ( t )t* V ( s )= - I ( s ) sc
+ -u(O-) S
(3.53)
Solved Problems
LAPLACE TRANSFORM
Find the Laplace transform of
( a ) x ( t ) = -e-atu( - t ) (b) x(t)=ea'u(-t)
( a ) From Eq. (3.3)
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