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Important Properties of the z-Transform 109
Solution
Note that
Using the MATLAB script,
x 1(n) ={1, 2, 3} and x 2(n) ={2, 4, 3, 5}
↑
↑
>> x1 = [1,2,3]; n1 = [-1:1]; x2 = [2,4,3,5]; n2 = [-2:1];
>> [x3,n3] = conv_m(x1,n1,x2,n2)
x3 =
2 8 17 23 19 15
n3 =
-3 -2 -1 0 1 2
we have
X 3(z) =2z 3 +8z 2 +17z +23+19z −1 +15z −2
□
In passing we note that to divide one polynomial by another one, we
would require an inverse operation called deconvolution [23, Chapter 6].
In MATLAB [p,r] = deconv(b,a) computes the result of dividing b by
a in a polynomial part p and a remainder r. For example, if we divide the
polynomial X 3 (z) inExample 4.4 by X 1 (z), as follows,
>> x3 = [6,17,34,43,38,24]; x1 = [2,3,4]; [x2,r] = deconv(x3,x1)
x2 =
3 4 5 6
r =
0 0 0 0 0 0
then we obtain the coefficients of the polynomial X 2 (z) asexpected. To
obtain the sample index, we will have to modify the deconv function as
we did in the conv m function. This is explored in Problem P4.10. This
operation is useful in obtaining a proper rational part from an improper
rational function.
The second important use of the convolution property is in system
output computations as we shall see in a later section. This interpretation
is particularly useful for verifying the z-transform expression X(z) ofa
casual sequence using MATLAB. Note that since MATLAB is a numerical
processor (unless the Symbolic toolbox is used), it cannot be used for
symbolic z-transform calculations. We will now elaborate on this. Let
x(n) beasequence with a rational transform
X(z) = B(z)
A(z)
where B(z) and A(z) are polynomials in z −1 .Ifweuse the coefficients of
B(z) and A(z) asthe b and a arrays in the filter routine and excite this
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