barcode generator for ssrs THE Z-TRANSFORM in Software

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THE Z-TRANSFORM
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[CHAP. 4
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A finite-length sequence has a z-transform with a region of convergence that includes the entire z-plane 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 right-sided sequence has a z-transform with a region of convergence that is the exterior of a circle:
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3. A left-sided sequence has a z-transform with a region of convergence that is the interior of a circle:
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EXAMPLE 4.2.1 Let us find the z-transform of the sequence x(n) = anu(n). Using the definition of the z-transform and the geometric series given in Table 1-I, we have
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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,
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we see that X ( z ) has a zero at z = 0 and a pole at z = a. A pole-zero diagram with the region of convergence is shown in the figure below.
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a Note that if l 1 < 1, the unit circle is included within the region of convergence, and the DTFT of x(n) exists.
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Example 4.2.1 considered the z-transform of a right-sided sequence, which led to a region of convergence that is the exterior of a circle. The following example considers the z-transform of a left-sided sequence.
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EXAMPLE 4.2.2 example. we have
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Let us find the z-transform of the sequence x(n) = -anu(-n
- I).
Proceeding as in the previous
CHAP. 41
THE z-TRANSFORM
with the sum converging if lor-lzl < 1 or lzl < lal. A pole-zero 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 z-transforms 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 z-transform of a sequence is not uniquely defined until its region of convergence has been specified.
EXAMPLE 4.2.3 Find the z-transform of x(n) = (f)"u(n) - 2"u(-n - l), and find another signal that has the same z-transform but a different region of convergence. Here we have a sum of two sequences. Therefore, we may find the z-transform of each sequence separately and add them together. From Example 4.2.1, we know that the z-transform of xl(n) = (i)"u(n) is
and from Example 4.2.2 that the z-transform of x2(n) = -2"u(-n
- 1) is
Therefore, the z-transform 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 z-transform, note that because X(z) is a sum of two z-transforms,
each term corresponds to the z-transform of either a right-sided or a left-sided sequence, depending upon the region of convergence. Therefore, choosing the right-sided sequences for both terms, it follows that
has the same z-transform as x(n), except that the region of convergence is lzl > 2.
Listed in Table 4-1 are a few common z-transform pairs. With these z-transform pairs and the z-transform properties described in the following section, most z-transforms of interest may be easily evaluated.
THE 2-TRANSFORM
[CHAP. 4
Table 4-1 Common z-lkansform Pairs
Sequence
Region of Convergence
4.3 PROPERTIES
Just as with the DTFT, there are a number of important and useful z-transform properties. A few of these properties are described below.
Linearity
As with the DTFT, the z-transform is a linear operator. Therefore, if x(n) has a z-transform X(z) with a region of convergence R,, and if y(n) has a z-transform 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 z-transform of win) = x(n) - y(n) = S(n) is the entire z-plane.
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