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barcode generator for ssrs THE ZTRANSFORM in Software
THE ZTRANSFORM Code 128A Scanner In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code128 Encoder In None Using Barcode creation for Software Control to generate, create USS Code 128 image in Software applications. [CHAP. 4
USS Code 128 Reader In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. Make Code128 In C# Using Barcode drawer for Visual Studio .NET Control to generate, create Code128 image in .NET framework applications. A finitelength sequence has a ztransform with a region of convergence that includes the entire zplane except, possibly, z = 0 and z = ca.The point z = ca will be included if x(n) = 0 for n < 0, and the point z = 0 will be included if x(n) = 0 for n > 0. 2. A rightsided sequence has a ztransform with a region of convergence that is the exterior of a circle: Making Code 128B In VS .NET Using Barcode creation for ASP.NET Control to generate, create Code 128 image in ASP.NET applications. Encode Code 128A In Visual Studio .NET Using Barcode drawer for .NET framework Control to generate, create Code 128 Code Set B image in Visual Studio .NET applications. 3. A leftsided sequence has a ztransform with a region of convergence that is the interior of a circle: Code 128 Code Set B Encoder In Visual Basic .NET Using Barcode generator for Visual Studio .NET Control to generate, create Code128 image in Visual Studio .NET applications. Painting EAN13 In None Using Barcode creator for Software Control to generate, create European Article Number 13 image in Software applications. EXAMPLE 4.2.1 Let us find the ztransform of the sequence x(n) = anu(n). Using the definition of the ztransform and the geometric series given in Table 1I, we have UPCA Creator In None Using Barcode generator for Software Control to generate, create UCC  12 image in Software applications. Code 3 Of 9 Generator In None Using Barcode printer for Software Control to generate, create USS Code 39 image in Software applications. with the sum converging if laz'\ < 1. Therefore the region of convergence is the exterior of a circle defined by the set of points Izl z la\.Expressing X ( z ) in terms of positive powers of z, Painting Data Matrix 2d Barcode In None Using Barcode maker for Software Control to generate, create DataMatrix image in Software applications. Bar Code Creator In None Using Barcode maker for Software Control to generate, create barcode image in Software applications. we see that X ( z ) has a zero at z = 0 and a pole at z = a. A polezero diagram with the region of convergence is shown in the figure below. Generate USD3 In None Using Barcode generator for Software Control to generate, create Code 93 Extended image in Software applications. Painting Bar Code In None Using Barcode printer for Word Control to generate, create barcode image in Office Word applications. a Note that if l 1 < 1, the unit circle is included within the region of convergence, and the DTFT of x(n) exists. Barcode Creation In .NET Using Barcode creator for ASP.NET Control to generate, create barcode image in ASP.NET applications. UPC Code Reader In VS .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Example 4.2.1 considered the ztransform of a rightsided sequence, which led to a region of convergence that is the exterior of a circle. The following example considers the ztransform of a leftsided sequence. Bar Code Encoder In Visual C#.NET Using Barcode creator for VS .NET Control to generate, create bar code image in VS .NET applications. Create Code 39 In VS .NET Using Barcode creator for .NET Control to generate, create Code 39 image in .NET framework applications. EXAMPLE 4.2.2 example. we have
UCC  12 Decoder In VB.NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in VS .NET applications. Generating UCC  12 In ObjectiveC Using Barcode printer for iPad Control to generate, create UPCA image in iPad applications. Let us find the ztransform of the sequence x(n) = anu(n
 I). Proceeding as in the previous
CHAP. 41
THE zTRANSFORM
with the sum converging if lorlzl < 1 or lzl < lal. A polezero diagram with the region of convergence indicated is given in the figure below. Note that if lor1 5 1, the unit circle is not included within the region of convergence, and the DTFT of x(n) does not exist. Comparing the ztransforms of the signals in Examples 4.2.1 and 4.2.2, we see that they are the same, differing only in their regions of convergence. Thus, the ztransform of a sequence is not uniquely defined until its region of convergence has been specified. EXAMPLE 4.2.3 Find the ztransform of x(n) = (f)"u(n)  2"u(n  l), and find another signal that has the same ztransform but a different region of convergence. Here we have a sum of two sequences. Therefore, we may find the ztransform of each sequence separately and add them together. From Example 4.2.1, we know that the ztransform of xl(n) = (i)"u(n) is and from Example 4.2.2 that the ztransform of x2(n) = 2"u(n
 1) is
Therefore, the ztransform of x(n) = xl(n) +x2(n) is
with a region of convergence < lzl < 2, which is the set of all points that are in the ROC of both Xl(z) and X2(z). To find another sequence that has the same ztransform, note that because X(z) is a sum of two ztransforms, each term corresponds to the ztransform of either a rightsided or a leftsided sequence, depending upon the region of convergence. Therefore, choosing the rightsided sequences for both terms, it follows that has the same ztransform as x(n), except that the region of convergence is lzl > 2.
Listed in Table 41 are a few common ztransform pairs. With these ztransform pairs and the ztransform properties described in the following section, most ztransforms of interest may be easily evaluated. THE 2TRANSFORM
[CHAP. 4
Table 41 Common zlkansform Pairs
Sequence
Region of Convergence
4.3 PROPERTIES
Just as with the DTFT, there are a number of important and useful ztransform properties. A few of these properties are described below. Linearity
As with the DTFT, the ztransform is a linear operator. Therefore, if x(n) has a ztransform X(z) with a region of convergence R,, and if y(n) has a ztransform Y (z) with a region of convergence R,,, and the ROC of w(n) will include the intersection of R, and R,, that is, R , contains R , n R, However, the region of convergence of W(z) may be larger. For example, if x(n) = u(n) and yin) = u(n  I ) , the ROC of X(z) and Y(z) is Izl > 1. However, the ztransform of win) = x(n)  y(n) = S(n) is the entire zplane.

