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All-Pass Filter Design
Inserting the ϕ term in Equation 16–25 into Equation 16–28 and completing the derivation,
results in:
T gr 1 i
a i
1 bi 2
1 a1 2 2b 1
· 2 b 1
2
4
(16–29)
Setting Ω = 0 in Equation 16–29 gives the group delay for the low frequencies, 0 < Ω < 1,
which is:
T gr0
1 i
a i
(16–30)
The values for T gr0 are listed in Table 16–10, Section 16.9, from the first to the tenth order.
In addition, Figure 16–42 shows the group delay response versus the frequency for the
first ten orders of all-pass filters.
T gr — Normalized Group Delay — s/s
3.5
3
2.5
2
1.5
1
0.5
10 th Order
9 th Order
8 th Order
7 th Order
6 th Order
5 th Order
4 th Order
3 rd Order
2 nd Order
1 st Order
0
0.01 0.1 1 10
Frequency — Ω
100
Figure 16–42.
Frequency Response of the Group Delay for the First 10 Filter Orders
Active Filter Design Techniques
16-43