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barcode generator for ssrs Shifting Property in Software
Shifting Property Decode Code 128 Code Set C In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. ANSI/AIM Code 128 Maker In None Using Barcode drawer for Software Control to generate, create Code 128C image in Software applications. Shifting a sequence (delaying or advancing) multiplies the ztransform by a power of z. That is to say, if x(n) has a ztransform X(z), USS Code 128 Reader In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. Code128 Creation In Visual C#.NET Using Barcode printer for .NET Control to generate, create Code 128 image in VS .NET applications. Because shifting a sequence does not affect its absolute summability, shifting does not change the region of convergence. Therefore, the ztransforms of s ( n ) and x(n  no) have the same region of convergence, with the possible exception of adding or deleting the points z = 0 and z = oo. Code 128 Code Set A Maker In VS .NET Using Barcode creation for ASP.NET Control to generate, create Code 128 image in ASP.NET applications. Make Code 128 Code Set C In .NET Framework Using Barcode creator for Visual Studio .NET Control to generate, create USS Code 128 image in Visual Studio .NET applications. Time Reversal
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Bar Code Encoder In .NET Framework Using Barcode drawer for VS .NET Control to generate, create barcode image in .NET applications. Code128 Maker In Java Using Barcode generation for BIRT reports Control to generate, create Code128 image in BIRT applications. THE zTRANSFORM
Multiplication by an Exponential
If a sequence x(n) is multiplied by a complex exponential a n , This corresponds to a scaling of the zplane. If the region of convergence of X(z) is r < lzl < r,, which will be denoted by R,, the region of convergence of ~ ( a  ' z is lair < IzI < lair+, which is denoted by lalR,. As ) a special case, note that if x(n) is multiplied by a complex exponential. eJnwcl, which corresponds to a rotation of the zplane.
Convolution Theorem
Perhaps the most important ztransform property is the convolution theorem, which states that convolution in the time domain is mapped into multiplication in the frequency domain, that is, y(n) = x(n) * h(n) Y(z) = X(z)H(z) The region of convergence of Y(z) includes the intersection of R, and R,, R, contains R, f' R , However, the region of convergence of Y(z) may be larger, if there is a polezero cancellation in the product X(z)H(z). EXAMPLE 4.3.1
Consider the two sequences
The ztransform of x(n) is
1 X(z) = 1 azrl
and the ztransform of h(n) is
H(z) = 1  az' However. the ztransform of the convolution of x(n) with h(n) is
0 < lzl
which, due to a polezrro cancellation, has a region of convergence that is the entire zplane.
Conjugation
If X(z) is the ztransform of x(n), the ztransform of the complex con.jugate of x(n) is x*(n) .Z. x*(z*) As a corollary, note that if x(n) is realvalued, x(n) = x*(n), then X(z) = X*(Z*) Derivative
THE ZTRANSFORM
[CHAP. 4
If X(z) is the ztransform of x(n), the ztransform of nx(n) is
Repeated application of this property allows for the evaluation of the ztransform of nkx(n) for any integer k. These properties are summarized in Table 42. As illustrated in the following example, these properties are useful in simplifying the evaluation of ztransforms. Table 42 Properties of the zTransform
zTransform Linearity Shift Time reversal Exponentiation Convolution Conjugation Derivative aX(z) hY(z) z""x(z) X(zI) X(alz) x(z)y(z) Region of Convergence Contains R, n R, Rx 1/Rx law, Contains R, n R, Nore: Given the ztransforms X(z) and Y ( z )of x ( n ) and y ( n ) . with regions of convergence R, and R y , respectively, this table lists the ztransforms of sequences that are formed from x ( n ) and y(n). EXAMPLE 4.3.2 Let us find the ztransform of x(n) = nal'u(n). To find X(z), we will use the timereversal and derivative properties. First, as we saw in Example 4.2.1, Therefore. and, using the timereversal property, anu(n) 1 A  I4 < a I  a'z
Finally, using the derivative property, it follows that the ztransform of nanu(n) is
A property that may be used to find the initial value of a causal sequence from its ztransform is the initial value theorem. Initial Value Theorem
If x(n) is equal to zero for n < 0, the initial value, x(O), may be found from X(z) as follows: x(0) = lim X(z) Z'OO
This property is a consequence of the fact that if x(n) = 0 for n < 0, Therefore, if we let z + oo.each term in X ( z ) goes to zero except the first.
CHAP. 41
THE zTRANSFORM
4.4 THE INVERSE ZTRANSFORM The ztransform is a useful tool in linear systems analysis. However, just as important as techniques for finding the ztransform of a sequence are methods that may be used to invert the ztransform and recover the sequence x ( n ) from X(z). Three possible approaches are described below. 4.4.1 Partial Fraction Expansion
For ztransforms thar are rational functions of z, a simple and straightforward approach to find the inverse ztransform is to perform a partial fraction expansion of X(z). Assuming that p > q , and that all of the roots in the denominator are simple, a, # a for i # k, X(z) k may be expanded as follows: for some constants Ak for k = 1,2, . . . , p. The coefficients Ak may be found by multiplying both sides of Eq. (4.5) by (1  ak ') and setting z = a k . The result is If p ( q , the partial fraction expansion must include a polynomial in zI of order ( p q). The coefficients of this polynomial may be found by long division (i.e., by dividing the numerator polynomial by the denominator). For multipleorder poles, the expansion must be modified. For example, if X(z) has a secondorder pole at z = ak, the expansion will include two terms,

