Digital Signal Processing Reference
In-Depth Information
The
z
-transform is
0
.
5
z
−
4
Y (z)/X(z)
=
1
+
The zeros of this transfer function do not lie on the unit circle. They can be found as
√
0
.
5
4
z
=
((π
+
2
πm)/
4
)
where
m
= 0,1,2, and 3, by making this m-code call
m = 0:1:3; angs = exp(j*(pi*(2*m +1)/4)); theZeros = 0.5 ˆ(0.25)*angs
or by making the call
theZeros = roots([1,0,0,0,0.5])
Figure 2.4, plot (b), shows the zeros returned by the above call, plotted in the complex plane, while
plot (a) shows the magnitude of the frequency response of the corresponding comb filter.
2.5
2
1.5
1
0.5
0
−1
−0.5
0
0.5
1
(a) Frequency, Units of
π
1
0.5
4
0
−0.5
−1
−4
−3
−2
−1
0
1
2
3
4
(b) Real
Figure 2.4:
(a) Magnitude of the
z
-transform
H(z)
=1 + 0.5z
−
4
over the frequency range -
π
to
π
radians;
(b) Plot of the zeros of the
z
-transform of the comb filter of (a).
Example 2.18.
Determine the roots (zeros) of the transfer function corresponding to the impulse
response
[
ones(
1
,
6
)
]
.
