Digital Signal Processing Reference
In-Depth Information
The discrete-time transfer function
H
(
z
) can now be determined by removing
the (1 −
z
−1
) factor relating to the step input, and we can make the substitution of
the different sampling periods in the general form to find the two distinct transfer
functions:
−
1
−
0
707
⋅
T
−
1
z
+
e
[(sin(
0
707
⋅
T
)
−
cos(
0
707
⋅
T
)]
⋅
z
H
(
z
)
=
−
0
707
⋅
T
−
1
−
1
414
⋅
T
−
2
[
−
2
⋅
e
cos(
0
707
⋅
T
)
⋅
z
+
e
z
]
−
1
414
⋅
T
−
2
−
0
707
⋅
T
−
2
e
z
−
e
[sin(
0
707
⋅
T
)
+
cos(
0
707
⋅
T
)]
⋅
z
+
−
0
707
⋅
T
−
1
−
1
414
⋅
T
−
2
[
−
2
⋅
e
cos(
0
707
⋅
T
)
⋅
z
+
e
z
]
−
1
−
2
0
.
94546
⋅
z
−
0
.
45205
⋅
z
H
(
z
)
=
T
=
1
.
0
−
1
−
2
1
−
0
.
74971
⋅
z
+
0
.
24312
⋅
z
−
1
−
2
0
.
13643
⋅
z
−
0
.
12711
⋅
z
H
(
z
)
=
T
=
0
.
1
−
1
−
2
1
−
1
.
8588
⋅
z
+
0
.
86812
⋅
z
Note that the pole locations are the same as for the impulse invariant design
but that the zero locations have changed. We can compare the frequency responses
of these two discrete-time filters as shown in Figure 6.2.
1.4
T
= 1.0 (frequency range 0 - 0.5 Hz)
Mag.
T
= 0.1 (frequency range 0 - 5.0 Hz)
0.0
π
Frequency
0.0
, 0.5, 5.0
Figure 6.2
Frequency responses for Example 6.4.
As we can see, there is significant difference between the two
implementations, but the differences are again a result of the frequency axis
having two different scales. In the step invariant design method, there is no need
