The z-Transform - Digital Signal Processing: Fundamentals and Applications - page 156

Notice that

Y ðzÞþ0:1Y ðzÞz 1 0:2Y ðzÞz 2 ¼ X ðzÞþX ðzÞz 1

Then the z-transform of the output sequence yðnÞ can be obtained as

z
z 1

1 þ z 1
1 þ 0:1z 1 0:2z 2

z 2 ðz þ 1Þ
ðz 1Þðz 0:4Þðz þ 0:5Þ

Y ðzÞ¼ ¼

Using the partial fraction expansion method as before gives

Y ðzÞ¼ 2:2222 z
z 1

þ 1:0370 z
z 0:4

þ 0:1852 z
z þ 0:5

and the system response is found by using Table 5.1 :
y n ¼ 2:2222u n 1:0370ð0:4Þ

n u n 0:1852ð0:5Þ n u n

5.5 SUMMARY
1. The one-sided (unilateral) z-transform, which can be used to transform the causal sequence to the
z-transform domain, was defined.
2. The lookup table of the z-transform determines the z-transform for a simple causal sequence, or the
causal sequence from a simple z-transform function.
3. The important properties of the z-transform, such as linearity, the shift theorem, and
convolution were introduced. The shift theorem can be used to solve a difference equation.
The z-transform of a digital convolution of two digital sequences is equal to the product of
their z-transforms.
4. Methods to determine the inverse of the z-transform, such as partial fraction expansion, invert the
complicated z-transform function, which can have first-order real poles, multiple-order real poles,
and first-order complex poles assuming that the z-transform function is proper. The MATLAB
techniques to determine the inverse were introduced.
5. The z-transform can be applied to solve linear difference equations with nonzero initial conditions
and zero initial conditions.

5.6 PROBLEMS
5.1. Find the z-transform for each of the following sequences:

a. xðnÞ¼ 4 uðnÞ
b. xðnÞ¼ð 0 : 7 Þ n uðnÞ
c. xðnÞ¼ 4 e 2 n uðnÞ
d. xðnÞ¼ 4 ð 0 : 8 Þ n cos ð 0 : 1 pnÞuðnÞ
e. xðnÞ¼ 4 e 3 n sin ð 0 : 1 pnÞuðnÞ