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128 Chapter 4 THE z-TRANSFORM
1.5
Magnitude Response
Magnitude
1
0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
frequency in π units
0.5
Phase Response
Phase in π units
0
−0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
frequency in π units
FIGURE 4.10 Frequency response plots for Example 4.13
4.5 SOLUTIONS OF THE DIFFERENCE EQUATIONS
In Chapter 2 we mentioned two forms for the solution of linear constant
coefficient difference equations. One form involved finding the particular
and the homogeneous solutions, while the other form involved finding
the zero-input (initial condition) and the zero-state responses. Using
z-transforms, we now provide a method for obtaining these forms. In addition,
we will also discuss the transient and the steady-state responses.
In digital signal processing, difference equations generally evolve in the
positive n direction. Therefore our time frame for these solutions will be
n ≥ 0. For this purpose we define a version of the bilateral z-transform
called the one-sided z-transform.
DEFINITION 4
The One-sided z Transform
The one-sided z-transform of a sequence x(n) is given by
∞∑
Z + [x(n)] = △ Z [x(n)u(n)] = △ X + [z] = x(n)z −n (4.25)
n=0
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