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2d barcode generator vb.net LAPLACE TRANSFORM AND CONTINUOUSTIME LTI SYSTEMS in Visual Studio .NET
LAPLACE TRANSFORM AND CONTINUOUSTIME LTI SYSTEMS QRCode Scanner In Visual Studio .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET applications. QRCode Generator In Visual Studio .NET Using Barcode generation for Visual Studio .NET Control to generate, create QR Code image in .NET applications. [CHAP. 3
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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 CONTINUOUSTIME 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 continuoustime 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 37 illustrates the relationship of Eqs. (3.35) and (3.36). B. Characterization of LTI Systems: Many properties of continuoustime LTI systems can be closely associated with the characteristics of H ( s ) in the splane and in particular with the pole locations and the ROC. Fig. 37 Impulse response and system function.
LAPLACE TRANSFORM AND CONTINUOUSTIME LTI SYSTEMS
[CHAP. 3
Causality: For a causal continuoustime LTI system, we have h(t) Since h(t) is a rightsided 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 splane to the right of all of the system poles. Similarly, if the system is anticausal, then h(t) and h ( t ) is leftsided. 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 splane to the left of all of the system poles. 2. Stabilio: In Sec. 2.3 we stated that a continuoustime LTI system is B I B 0 stable if and only if [Eq. (22111 The corresponding requirement on H(s) is that the R O C of H ( s ) contains the jwaxis (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 splane; 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 ConstantCoefficient Differential Equations: In Sec. 2.5 we considered a continuoustime LTI system for which input x ( t ) and output y(t) satisfy the general linear constantcoefficient differential equation of the form Applying the Laplace transform and using the differentiation property (3.20) of the Laplace transform, we obtain

