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barcode print in asp net THE ZTRANSFORM AND DISCRETETIME LTI SYSTEMS in VS .NET
THE ZTRANSFORM AND DISCRETETIME LTI SYSTEMS Recognize QR Code In Visual Studio .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Encode Denso QR Bar Code In .NET Framework Using Barcode printer for VS .NET Control to generate, create QR Code 2d barcode image in Visual Studio .NET applications. [CHAP. 4
Decoding QR Code ISO/IEC18004 In .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in VS .NET applications. Drawing Barcode In VS .NET Using Barcode creator for Visual Studio .NET Control to generate, create bar code image in .NET applications. onesided) ztransform, which is defined as
Barcode Reader In Visual Studio .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Print QR Code 2d Barcode In C# Using Barcode creator for Visual Studio .NET Control to generate, create QR Code ISO/IEC18004 image in VS .NET applications. Clearly the bilateral and unilateral ztransforms are equivalent only if x[n] = 0 for n < 0. The unilateral ztransform is discussed in Sec. 4.8. We will omit the word "bilateral" except where it is needed to avoid ambiguity. As in the case of the Laplace transform, Eq. (4.3) is sometimes considered an operator that transforms a sequence x[n] into a function X ( z ) , symbolically represented by Painting Quick Response Code In Visual Studio .NET Using Barcode drawer for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. Drawing QR Code In VB.NET Using Barcode generator for .NET framework Control to generate, create QR Code image in VS .NET applications. The x[n] and X ( z ) are said to form a ztransform pair denoted as
Matrix Barcode Creator In .NET Framework Using Barcode printer for .NET framework Control to generate, create 2D Barcode image in Visual Studio .NET applications. Draw UCC128 In Visual Studio .NET Using Barcode generator for VS .NET Control to generate, create UCC  12 image in .NET applications. B. The Region of Convergence: As in the case of the Laplace transform, the range of values of the complex variable z for which the ztransform converges is called the region of convergence. T o illustrate the ztransform and the associated R O C let us consider some examples. Barcode Maker In VS .NET Using Barcode maker for Visual Studio .NET Control to generate, create bar code image in Visual Studio .NET applications. Print International Standard Book Number In Visual Studio .NET Using Barcode generator for .NET framework Control to generate, create ISBN image in Visual Studio .NET applications. EXAMPLE 4.1. Consider the sequence
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Bar Code Generator In ObjectiveC Using Barcode encoder for iPhone Control to generate, create bar code image in iPhone applications. Create Bar Code In ObjectiveC Using Barcode drawer for iPhone Control to generate, create bar code image in iPhone applications. Then by Eq. (4.3) the ztransform of x [ n ] is
Data Matrix Creation In None Using Barcode generation for Font Control to generate, create DataMatrix image in Font applications. Bar Code Scanner In .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET applications. For the convergence of X(z) we require that
Thus, the ROC is the range of values of z for which laz 'I < 1 or, equivalently, lzl > lal. Then
Alternatively, by multiplying the numerator and denominator of Eq. (4.9) by z, we may write X(z) as X(z) = za Izl > la1
Both forms of X ( z ) in Eqs. ( 4 . 9 ) and (4.10) are useful depending upon the application. From Eq. (4.10) we see that X ( z ) is a rational function of z. Consequently, just as with rational Laplace transforms, it can be characterized by its zeros (the roots of the numerator polynomial) and its poles (the roots of the denominator polynomial). From Eq. (4.10) we see that there is one zero at z = 0 and one pole at z = a . The ROC and the polezero plot for this example are shown in Fig. 41. In ztransform applications, the complex plane is commonly referred to as the zplane. CHAP. 4 1
THE zTRANSFORM AND DISCRETETIME LTI SYSTEMS
Unit circle
I ca<O
a<I
Fig.41 ROCofthe form lzl>lal.
EXAMPLE 4.2. Consider the sequence
x [ n ] =  a n u [  n  11 Its ztransform X(z) is given by (Prob. 4.1) THE ZTRANSFORMAND DISCRETETIME LTI SYSTEMS
[CHAP. 4
Again, as before, X ( z ) may be written as X(z) Izl < la1
Thus, the ROC and the polezero plot for this example are shown in Fig. 42. Comparing Eqs. (4.9) and (4.12) [or Eqs. (4.10) and (4.13)], we see that the algebraic expressions of X ( z ) for two different sequences are identical except for the ROCs. Thus, as in the Laplace Fig. 42 ROC of the form I z I < lal.
CHAP. 41
THE ZTRANSFORM AND DISCRETETIME LTI SYSTEMS
transform, specification of the ztransform requires both the algebraic expression and the ROC.
C. Properties of the ROC: As we saw in Examples 4.1 and 4.2, the ROC of X ( z ) depends on the nature of x [ n ] . The properties of the ROC are summarized below. We assume that X ( Z ) is a rational function of z. Property 1 : Property 2: The ROC does not contain any poles. If x [ n ] is a finite sequence (that is, x [ n ] = 0 except in a finite interval N l ~ n s N,, where N , and N , are finite) and X(z) converges for some value of z, then the ROC is the entire zplane except possibly z = 0 or z = co. If x [ n ] is a rightsided sequence (that is, x [ n ] = 0 for n < N, < 03) and X(z) converges for some value of z, then the ROC is of the form Property 3: where r,, equals the largest magnitude of any of the poles of X(z). Thus, the ROC is the exterior of the circle lzl= r,, in the zplane with the possible exception of z = m. Property 4: If x [ n ] is a leftsided sequence (that is, x [ n l = 0 for n > N , > for some value of z, then the ROC is of the form  03) and
X(z) converges
where r,, is the smallest magnitude of any of the poles of X(z). Thus, the ROC is the interior of the circle lzlE rminin the zplane with the possible exception of z = 0. Property 5: If x [ n ] is a twosided sequence (that is, x [ n ] is an infiniteduration sequence that is neither rightsided nor leftsided) and X ( z ) converges for some value of z, then the ROC is of the form where r , and r, are the magnitudes of the two poles of X(z). Thus, the ROC is an annular ring in the zplane between the circles lzl= r , and lzl = r2 not containing any poles. Note that Property 1 follows immediately from the definition of poles; that is, X(z) is infinite at a pole. For verification of the other properties, see Probs. 4.2 and 4.5. zTRANSFORMS OF SOME COMMON SEQUENCES From definition (1.45) and ( 4 . 3 ) A. Unit Impulse Sequence 61 nl: X ( z )=

