"Chapter 1 - The Op Amp's Place in the World" - HTL Wien 10

All-Pass Filter Design
16-42
2 i
1 bi 2 2 ai 2 ·e
A(s)
ja
2 i
1 bi 2 2 ai 2 ·eja This gives a constant gain of 1, and a phase shift,φ, of:
2 2 i
ai arctan
1 bi2 (16–24)
(16–25)
To transmit a signal with minimum phase distortion, the all-pass filter must have a constant
group delay across the specified frequency band. The group delay is the time by which
the all-pass filter delays each frequency within that band.
The frequency at which the group delay drops to 1 2 –times its initial value is the corner
frequency, fC. The group delay is defined through:
tgr d
d
(16–26)
To present the group delay in normalized form, refer t gr to the period of the corner frequency,
T C, of the all-pass circuit:
Tgr tgr
tgr·fc tgr·
Tc
c
2
Substituting t gr through Equation 16–26 gives:
Tgr 1 d
·
2 d
(16–27)
(16–28)