Digital Signal Processing Reference
In-Depth Information
must be changed to 71.92 Hz to provide symmetry. The transfer function is shown
with quadratics in factored form as shown by the (2) superscript.
Ω
r
= 3.3
n
= 2.75 (3rd order)
ω
o
= 376.78 rad/sec
BW
= 137.73 rad/sec
0
73880
⋅
(
S
±
j
6
31445
)
H
E
(
S
)
=
,
3
(
S
+
0
73880
)
⋅
(
S
+
0
35753
±
j
0
93390
)
After making the substitution of (3.39) and factoring again, the following
equation emerges:
2
(
2
)
(
2
)
(
s
+
141
,
960
)
⋅
(
s
±
j
387
.
9
)
⋅
(
s
±
j
366
.
)
H
I
(
s
)
=
,
6
2
(
2
)
(
2
)
(
s
+
186
.
4
⋅
s
+
141
,
960
)
⋅
(
s
+
16
.
30
±
j
323
.
0
)
⋅
(
s
+
22
.
13
±
j
438
.
4
)
After simplification, the following transfer function results:
2
2
2
(
s
+
141
,
964
)
⋅
(
s
+
150
,
424
)
⋅
(
s
+
133
,
981
)
H
E
(
s
)
=
,
6
2
2
2
(
s
+
186
.
4
⋅
s
+
141
,
964
)
⋅
(
s
+
32
.
60
⋅
s
+
104
,
585
)
⋅
(
s
+
44
.
25
⋅
s
+
192
,
704
)
We can also use WFilter to design the bandstop elliptic filter of Example 3.8.
The results of this filter design are shown in Figures 3.8 and 3.9 that show the
coefficients and magnitude response, respectively. Again, as in the bandpass case,
a couple of the numerator coefficients are very small and can be considered zero.
3.5 ANALOG FREQUENCY RESPONSE
Up to this point we have developed the necessary foundation to design a variety of
analog filters. We have calculated the coefficients and are ready to implement the
filter in hardware. But before we address the implementation issues in the next
chapter, we need to check our design by determining the frequency response of
the filter and comparing it to our design specifications. We will discuss the
calculation of the frequency response of our filters and also view the C code for
the frequency response calculation.
3.5.1 Mathematics for Frequency Response Calculation
The filter approximation function, which we have just determined by the
calculation of the unnormalized coefficients, represents a transfer function of a
linear system in the
s-
domain. In order to determine the frequency response of the
transfer function, we must substitute
j
ω for each of the
s
-variables in that transfer
