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Shifting a sequence (delaying or advancing) multiplies the z-transform by a power of z. That is to say, if x(n) has a z-transform X(z),
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Because shifting a sequence does not affect its absolute summability, shifting does not change the region of convergence. Therefore, the z-transforms 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.
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If x(n) has a z-transform X(z) with a region of convergence R, that is the annulus a < lzl < #I, the z-transform of the time-reversed sequence x(-n) is x(-n) and has a region of convergence 1/#I
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CHAP. 41
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THE z-TRANSFORM
Multiplication by an Exponential
If a sequence x(n) is multiplied by a complex exponential a n ,
This corresponds to a scaling of the z-plane. 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 z-plane.
Convolution Theorem
Perhaps the most important z-transform 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 pole-zero cancellation in the product X(z)H(z).
EXAMPLE 4.3.1
Consider the two sequences
The z-transform of x(n) is
1 X(z) = 1 -azrl
and the z-transform of h(n) is
H(z) = 1 - az-'
However. the z-transform of the convolution of x(n) with h(n) is
0 < lzl
which, due to a pole-zrro cancellation, has a region of convergence that is the entire z-plane.
Conjugation
If X(z) is the z-transform of x(n), the z-transform of the complex con.jugate of x(n) is x*(n) .Z. x*(z*) As a corollary, note that if x(n) is real-valued, x(n) = x*(n), then X(z) = X*(Z*)
Derivative
THE Z-TRANSFORM
[CHAP. 4
If X(z) is the z-transform of x(n), the z-transform of nx(n) is
Repeated application of this property allows for the evaluation of the z-transform of nkx(n) for any integer k. These properties are summarized in Table 4-2. As illustrated in the following example, these properties are useful in simplifying the evaluation of z-transforms.
Table 4-2 Properties of the z-Transform
z-Transform Linearity Shift Time reversal Exponentiation Convolution Conjugation Derivative aX(z) hY(z) z-""x(z) X(z-I) X(a-lz) x(z)y(z)
Region of Convergence Contains R, n R, Rx 1/Rx law, Contains R, n R,
Nore: Given the z-transforms X(z) and Y ( z )of x ( n ) and y ( n ) . with regions of convergence R, and R y , respectively, this table lists the z-transforms of sequences that are formed from x ( n ) and y(n).
EXAMPLE 4.3.2 Let us find the z-transform of x(n) = nal'u(-n). To find X(z), we will use the time-reversal and derivative properties. First, as we saw in Example 4.2.1,
Therefore. and, using the time-reversal property, anu(-n)
1 A - I4 < a I - a-'z
Finally, using the derivative property, it follows that the z-transform of nanu(-n) is
A property that may be used to find the initial value of a causal sequence from its z-transform 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 z-TRANSFORM
4.4 THE INVERSE Z-TRANSFORM The z-transform is a useful tool in linear systems analysis. However, just as important as techniques for finding the z-transform of a sequence are methods that may be used to invert the z-transform and recover the sequence x ( n ) from X(z). Three possible approaches are described below.
4.4.1 Partial Fraction Expansion
For z-transforms thar are rational functions of z,
a simple and straightforward approach to find the inverse z-transform 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 z-I 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 multiple-order poles, the expansion must be modified. For example, if X(z) has a second-order pole at z = ak, the expansion will include two terms,
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