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120 Chapter 4 THE z-TRANSFORM
4.4.2 TRANSFER FUNCTION REPRESENTATION
If the ROC of H(z) includes a unit circle (z = e jω ), then we can evaluate
H(z) onthe unit circle, resulting in a frequency response function or
transfer function H(e jω ). Then from (4.21)
H(e jω )=b 0 e j(N−M)ω ∏ M
1 (ejω − z l )
∏ N
1 (ejω − p k )
(4.22)
The factor (e jω −z l ) can be interpreted as a vector in the complex z-plane
from a zero z l to the unit circle at z = e jω , while the factor (e jω − p k )
can be interpreted as a vector from a pole p k to the unit circle at z = e jω .
This is shown in Figure 4.6. Hence the magnitude response function
|H(e jω )| = |b 0 | |ejω − z 1 |···|e jω − z M |
|e jω − p 1 |···|e jω − p N |
(4.23)
can be interpreted as a product of the lengths of vectors from zeros to the
unit circle divided by the lengths of vectors from poles to the unit circle
and scaled by |b 0 |. Similarly, the phase response function
̸ H(e jω ) =[0 or π]
} {{ }
Constant
+[(N − M) ω] +
} {{ }
Linear
M∑
1
(e jω − z k ) −
N∑
̸ (e jω − p k )
} {{ }
Nonlinear
(4.24)
can be interpreted as a sum of a constant factor, a linear-phase factor,
and a nonlinear-phase factor (angles from the “zero vectors” minus the
sum of angles from the “pole vectors”).
4.4.3 MATLAB IMPLEMENTATION
In Chapter 3, we plotted magnitude and phase responses in MATLAB
by directly implementing their functional forms. MATLAB also provides
1
Im{z}
p k
0
ω
Re{z}
Unit
circle
z l
FIGURE 4.6
Pole and zero vectors
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