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barcode print in asp net THE ZTRANSFORM AND DISCRETETIME LTI SYSTEMS in .NET
THE ZTRANSFORM AND DISCRETETIME LTI SYSTEMS Recognize Denso QR Bar Code In .NET Framework Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Painting QR In .NET Framework Using Barcode creation for VS .NET Control to generate, create QR Code ISO/IEC18004 image in .NET framework applications. [CHAP. 4
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Hence, we obtain
Inferring the ROC for each term in Eq. (4.35) from the overall ROC of X(z) and using Table 41, we can then invert each term, producing thereby the overall inverse ztransform (see Probs. 4.19 to 4.23). If rn > n in Eq. (4.321, then a polynomial of z must be added to the righthand side of Eq. (4.351, the order of which is (m  n). Thus for rn > n, the complete partialfraction CHAP. 41
THE zTRANSFORM AND DISCRETETIME LTI SYSTEMS
expansion would have the form
If X(Z) has multipleorder poles, say pi is the multiple pole with multiplicity r, then the expansion of X(z)/z will consist of terms of the form where
THE SYSTEM FUNCTION OF DISCRETETIME LTI SYSTEMS
A. The System Function: In Sec. 2.6 we showed that the output y[n] of a discretetime LTI system equals the convolution of the input x[n] with the impulse response h[n]; that is [Eq. (2.3511, Applying the convolution property (4.26) of the ztransform, we obtain
where Y(z), X(z), and H(z) are the ztransforms of y[n], x[n], and h[n], respectively. Equation (4.40) can be expressed as The ztransform H(z) of h[n] is referred to as the system function (or the transfer function) of the system. By Eq. (4.41) the system function H(z) can also be defined as the ratio of the ztransforms of the output y[n] and the input x[n.l. The system function H ( z ) completely characterizes the system. Figure 43 illustrates the relationship of Eqs. (4.39) and (4.40). X(Z) H(z) Y(z)=X(z)H(z) Fig. 43 Impulse response and system function.
THE ZTRANSFORMAND DISCRETETIME LTI SYSTEMS
[CHAP. 4
B. Characterization of DiscreteTime LTI Systems: Many properties of discretetime LTI systems can be closely associated with the characteristics of H(z) in the zplane and in particular with the pole locations and the ROC. 1. Causality: For a causal discretetime LTI system, we have [Eq. (2.4411 since h[n] is a rightsided signal, the corresponding requirement on H(z) is that the ROC of H ( z ) must be of the form That is, the ROC is the exterior of a circle containing all of the poles of H ( z ) in the zplane. Similarly, if the system is anticausal, that is, then h[n] is leftsided and the ROC of H ( z ) must be of the form
That is, the ROC is the interior of a circle containing no poles of H ( z ) in the zplane. 2. Stability: In Sec. 2.7 we stated that a discretetime LTI system is BIB0 stable if and only if [Eq. (2.4911 The corresponding requirement on H(z) is that the ROC of H(z1 contains the unit circle (that is, lzl= 1). (See Prob. 4.30.) 3. Ctzusal and Stable Systems: If the system is both causal and stable, then all of the poles of H ( z ) must lie inside the unit circle of the zplane because the ROC is of the form lzl> r,,, and since the unit circle is included in the ROC, we must have r,, < 1. C. System Function for LTI Systems Described by Linear ConstantCoefficient Difference Equations: ! Sec. 2.9 we considered a discretetime LTI system for which input x[n] and output n y[n] satisfy the general linear constantcoefficient difference equation of the form

