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LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
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Use of Tables of Laplace Transform Pairs: In the second method for the inversion of X(s), we attempt to express X(s) as a sum X(s) = X,(s) + . . . +Xn(s) (3.26) where X,(s), . . ., Xn(s) are functions with known inverse transforms xl(t),. . . ,xn(t). From the linearity property (3.15) it follows that x(t) =xl(t)+
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C. Partial-Fraction Expansion: If X(s) is a rational function, that is, of the form
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a simple technique based on partial-fraction expansion can be used for the inversion of
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When X(s) is a proper rational function, that is, when m < n:
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1. Simple Pole Case:
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If all poles of X(s), that is, all zeros of D(s), are simple (or distinct), then X(s) can be written as
where coefficients ck are given by
If D(s) has multiple roots, that is, if it contains factors of the form ( s -pi)', we say that pi is the multiple pole of X(s) with multiplicity r . Then the expansion of X(s) will consist of terms of the form
where
( b ) When X(s) is an improper rational function, that is, when m 2 n: If m 2 n, by long division we can write X(s) in the form
where N(s) and D(s) are the numerator and denominator polynomials in s, respectively, of X(s), the quotient Q(s) is a polynomial in s with degree rn - n, and the remainder R(s)
CHAP. 31
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
is a polynomial in s with degree strictly less than n. The inverse Laplace transform of X ( s ) can then be computed by determining the inverse Laplace transform of Q ( s ) and the inverse Laplace transform of R ( s ) / D ( s ) .Since R ( s ) / D ( s ) is proper, the inverse Laplace transform of R ( s ) / D ( s ) can be computed by first expanding into partial fractions as given above. The inverse Laplace transform of Q ( s ) can be computed by using the transform pair
3.6 THE SYSTEM FUNCTION
A. The System Function:
In Sec. 2.2 we showed that the output y ( t ) of a continuous-time LTI system equals the convolution of the input x ( t ) with the impulse response h ( t ) ; that is, Applying the convolution property (3.23), we obtain where Y ( s ) , X ( s ) , and H ( s ) are the Laplace transforms of y ( t ) , x ( t ) , and h ( t ) , respectively. Equation (3.36) can be expressed as
The Laplace transform H ( s ) of h ( t ) is referred to as the system function (or the transfer function) of the system. By Eq. (3.37), the system function H ( s ) can also be defined as the ratio of the Laplace transforms of the output y ( t ) and the input x ( t ) . The system function H ( s ) completely characterizes the system because the impulse response h ( t ) completely characterizes the system. Figure 3-7 illustrates the relationship of Eqs. (3.35) and (3.36). B. Characterization of LTI Systems: Many properties of continuous-time LTI systems can be closely associated with the characteristics of H ( s ) in the s-plane and in particular with the pole locations and the ROC.
Fig. 3-7 Impulse response and system function.
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
[CHAP. 3
Causality:
For a causal continuous-time LTI system, we have h(t)
Since h(t) is a right-sided signal, the corresponding requirement on H(s) is that the ROC of H ( s ) must be of the form R e W > amax That is, the R O C is the region in the s-plane to the right of all of the system poles. Similarly, if the system is anticausal, then h(t)
and h ( t ) is left-sided. Thus, the R O C of H(s) must be of the form Re( s ) < %in That is, the ROC is the region in the s-plane to the left of all of the system poles.
2. Stabilio:
In Sec. 2.3 we stated that a continuous-time LTI system is B I B 0 stable if and only if [Eq. (2-2111
The corresponding requirement on H(s) is that the R O C of H ( s ) contains the jw-axis (that is, s = j w ) (Prob. 3.26).
3. Causal and Stable Systems:
If the system is both causal and stable, then all the poles of H(s) must lie in the left half of the s-plane; that is, they all have negative real parts because the ROC is of the , form Re(s) >amax,and since the jo axis is included in the ROC, we must have a, < 0.
C. System Function for LTI Systems Described by Linear Constant-Coefficient Differential Equations:
In Sec. 2.5 we considered a continuous-time LTI system for which input x ( t ) and output y(t) satisfy the general linear constant-coefficient differential equation of the form
Applying the Laplace transform and using the differentiation property (3.20) of the Laplace transform, we obtain
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