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108 Chapter 4 THE z-TRANSFORM
6. Differentiation in the z-domain:
Z [nx(n)] = −z dX(z) ; ROC: ROC x (4.9)
dz
This property is also called the multiplication-by-a-ramp property.
7. Multiplication:
Z [x 1 (n)x 2 (n)] = 1 ∮
X 1 (ν)X 2 (z/ν) ν −1 dν;
2πj C
(4.10)
ROC: ROC x1 ∩ Inverted ROC x2
where C is a closed contour that encloses the origin and lies in the
common ROC.
8. Convolution:
Z [x 1 (n) ∗ x 2 (n)] = X 1 (z)X 2 (z); ROC: ROC x1 ∩ ROC x2 (4.11)
This last property transforms the time-domain convolution operation
into a multiplication between two functions. It is a significant property
in many ways. First, if X 1 (z) and X 2 (z) are two polynomials, then their
product can be implemented using the conv function in MATLAB.
□ EXAMPLE 4.4 Let X 1(z) =2+3z −1 +4z −2 and X 2(z) =3+4z −1 +5z −2 +6z −3 . Determine
X 3(z) =X 1(z)X 2(z).
Solution
From the definition of the z-transform, we observe that
x 1(n) ={2, 3, 4} and x 2(n) ={3, 4, 5, 6}
↑
↑
Then the convolution of these two sequences will give the coefficients of the
required polynomial product.
MATLAB script:
>> x1 = [2,3,4]; x2 = [3,4,5,6]; x3 = conv(x1,x2)
x3 = 6 17 34 43 38 24
Hence
X 3(z) =6+17z −1 +34z −2 +43z −3 +38z −4 +24z −5
Using the conv m function developed in Chapter 2, we can also multiply
two z-domain polynomials corresponding to noncausal sequences. □
□ EXAMPLE 4.5 Let X 1(z) =z +2+3z −1 and X 2(z) =2z 2 +4z +3+5z −1 . Determine X 3(z) =
X 1(z)X 2(z).
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