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All-Pass Filter Design
16.7.1 First-Order All-Pass Filter
Figure 16–43 shows a first-order all-pass filter with a gain of +1 at low frequencies and
a gain of –1 at high frequencies. Therefore, the magnitude of the gain is 1, while the phase
changes from 0° to –180°.
R 1
R 1
V IN
V OUT
R
C
Figure 16–43.
First-Order All-Pass
The transfer function of the circuit above is:
A(s) 1 RC c·s
1 RC c·s
The coefficient comparison with Equation 16–23 (b 1 =1), results in:
a i
RC·2f c
(16–31)
To design a first-order all-pass, specify f C and C and then solve for R:
R
a i
2f c·C
(16–32)
Inserting Equation 16–31 into 16–30 and substituting ω C with Equation 16–27 provides
the maximum group delay of a first-order all-pass filter:
t gr0
2RC
(16–33)
16.7.2 Second-Order All-Pass Filter
Figure 16–44 shows that one possible design for a second-order all-pass filter is to subtract
the output voltage of a second-order band-pass filter from its input voltage.
C
V IN
R 1
C
R 2
R 3
R
V OUT
R
Figure 16–44.
Second-Order All-Pass Filter
16-44