High-Pass Filter Design

10

A0 A∞

0
Lowpass Highpass

|A| — Gain — dB
–10

–20

–30
0.1 1 10
Frequency — Ω

Figure 16–24. Developing The Gain Response of a High-Pass Filter

The general transfer function of a high-pass filter is then:

AR
A(s) + (16–4)
P ǒ
ai bi
i 1 ) s ) s2 Ǔ
with A∞ being the passband gain.

Since Equation 16–4 represents a cascade of second-order high-pass filters, the transfer
function of a single stage is:

AR
A i(s) + (16–5)
ǒ1 ) ai
s
b
) s 2i Ǔ
With b=0 for all first-order filters, the transfer function of a first-order filter simplifies to:

A0
A(s) + ai
(16–6)
1) s

16-22
 High-Pass Filter Design

16.4.1 First-Order High-Pass Filter

Figure 16–25 and 16–26 show a first-order high-pass filter in the noninverting and the in-
verting configuration.
C1
VIN
VOUT
R1

R2
R3

Figure 16–25. First-Order Noninverting High-Pass Filter

R2
C1 R1
VIN
VOUT

Figure 16–26. First-Order Inverting High-Pass Filter

The transfer functions of the circuits are:

R R2
1 ) R2 R1
A(s) + 3
and A(s) + *
1 ) w R1 C · 1s 1)w 1
· 1s
c 1 1 cR 1 C 1

The negative sign indicates that the inverting amplifier generates a 180° phase shift from
the filter input to the output.

The coefficient comparison between the two transfer functions and Equation 16–6 pro-
vides two different passband gain factors:

R2 R2
AR + 1 ) and AR + *
R3 R1

while the term for the coefficient a1 is the same for both circuits:

a1 + 1
w cR 1C 1

Active Filter Design Techniques 16-23

High-Pass Filter Design

To dimension the circuit, specify the corner frequency (fC), the dc gain (A∞), and capacitor
(C1), and then solve for R1 and R2:

R1 + 1
2pf ca 1C 1

R 2 + R 3(A R * 1) and R2 + * R1 AR

16.4.2 Second-Order High-Pass Filter

High-pass filters use the same two topologies as the low-pass filters: Sallen-Key and Mul-
tiple Feedback. The only difference is that the positions of the resistors and the capacitors
have changed.

16.4.2.1 Sallen-Key Topology

The general Sallen-Key topology in Figure 16–27 allows for separate gain setting via
A0 = 1+R4/R3.
R2
C1 C2
VIN
VOUT
R1

R4
R3

Figure 16–27. General Sallen-Key High-Pass Filter

The transfer function of the circuit in Figure 16–27 is:

A(s) + a R4
R 2ǒC 1)C 2Ǔ)R 1C 2(1*a) 1 with a+1)
1) w cR 1 R 2 C 1 C 2
·s ) 1
·1
w c 2 R 1R 2C 1C 2 s 2 R3

The unity-gain topology in Figure 16–28 is usually applied in low-Q filters with high gain
accuracy.
R2
C C
VIN
VOUT
R1

Figure 16–28. Unity-Gain Sallen-Key High-Pass Filter

16-24
 High-Pass Filter Design

To simplify the circuit design, it is common to choose unity-gain (α = 1), and C1 = C2 = C.

The transfer function of the circuit in Figure 16–28 then simplifies to:

A(s) + 1
1) 2
·1
w cR 1 C s
)w 1
2R
1
2 · s2
c 1R 2C

The coefficient comparison between this transfer function and Equation 16–5 yields:

AR + 1

a1 + 2
w cR 1C
b1 + 2 1
w c R 1R 2C 2

Given C, the resistor values for R1 and R2 are calculated through:

R1 + 1
pf cCa 1
a1
R2 +
4pf cCb 1

16.4.2.2 Multiple Feedback Topology

The MFB topology is commonly used in filters that have high Qs and require a high gain.

To simplify the computation of the circuit, capacitors C1 and C3 assume the same value
(C1 = C3 = C) as shown in Figure 16–29.
C2

C1=C C3=C
R1
VIN
VOUT
R2

Figure 16–29. Second-Order MFB High-Pass Filter

The transfer function of the circuit in Figure 16–29 is:

* CC
A(s) + 2C 2)C 1
2

1) ·
w cR 1 C 2 C s
) w 2R 1
· s12
c 2R 1C 2C

Active Filter Design Techniques 16-25

High-Pass Filter Design

Through coefficient comparison with Equation 16–5, obtain the following relations:

AR + C
C2
2C ) C 2
a1 +
w cR 1CC 2
2C ) C 2
b1 +
w cR 1CC 2

Given capacitors C and C2, and solving for resistors R1 and R2:
1 * 2A R
R1 +
2pf c·C·a 1
a1
R2 +
2pf c·b 1C 2(1 * 2A R)

The passband gain (A∞) of a MFB high-pass filter can vary significantly due to the wide
tolerances of the two capacitors C and C2. To keep the gain variation at a minimum, it is
necessary to use capacitors with tight tolerance values.

16.4.3 Higher-Order High-Pass Filter

Likewise, as with the low-pass filters, higher-order high-pass filters are designed by cas-
cading first-order and second-order filter stages. The filter coefficients are the same ones
used for the low-pass filter design, and are listed in the coefficient tables (Tables 16–4
through 16–10 in Section 16.9).

Example 16–4. Third-Order High-Pass Filter with fC = 1 kHz

The task is to design a third-order unity-gain Bessel high-pass filter with the corner fre-
quency fC = 1 kHz. Obtain the coefficients for a third-order Bessel filter from Table 16–4,
Section 16.9:

ai bi
Filter 1 a1 = 0.756 b1 = 0

Filter 2 a2 = 0.9996 b2 = 0.4772

and compute each partial filter by specifying the capacitor values and calculating the re-
quired resistor values.

First Filter

With C1 = 100 nF,

16-26
 Band-Pass Filter Design

R1 + 1 + 1 + 2.105 kW
2pf ca 1C 1 2p·10 3Hz·0.756·100·10 *9F

Closest 1% value is 2.1 kΩ.

Second Filter

With C = 100nF,

R1 + 1 + 1 + 3.18 kW
pf cCa 1 p·10 3·100·10 *9·0.756

Closest 1% value is 3.16 kΩ.

a1 0.9996
R2 + + + 1.67 kW
4pf cCb 1 4p·10 3·100·10 *9·0.4772

Closest 1% value is 1.65 kΩ.

Figure 16–30 shows the final filter circuit.

1.65k
100n
VIN 100n 100n

2.10k VOUT
3.16k

Figure 16–30. Third-Order Unity-Gain Bessel High-Pass

16.5 Band-Pass Filter Design

In Section 16.4, a high-pass response was generated by replacing the term S in the low-
pass transfer function with the transformation 1/S. Likewise, a band-pass characteristic
is generated by replacing the S term with the transformation:

DW
ǒ
1 s)1
s
Ǔ (16–7)

In this case, the passband characteristic of a low-pass filter is transformed into the upper
passband half of a band-pass filter. The upper passband is then mirrored at the mid fre-
quency, fm (Ω=1), into the lower passband half.

Active Filter Design Techniques 16-27

Band-Pass Filter Design

|A| [dB] |A| [dB]

0 0
–3 –3

∆Ω

0 1 Ω 0 Ω1 1 Ω2 Ω

Figure 16–31. Low-Pass to Band-Pass Transition

The corner frequency of the low-pass filter transforms to the lower and upper –3 dB fre-
quencies of the band-pass, Ω1 and Ω2. The difference between both frequencies is de-
fined as the normalized bandwidth ∆Ω:

DW + W 2 * W 1

The normalized mid frequency, where Q = 1, is:

W m + 1 + W 2·W 1

In analogy to the resonant circuits, the quality factor Q is defined as the ratio of the mid
frequency (fm) to the bandwidth (B):

fm fm 1
Q+ + + + 1 (16–8)
B f2 * f1 W2 * W1 DW

The simplest design of a band-pass filter is the connection of a high-pass filter and a low-
pass filter in series, which is commonly done in wide-band filter applications. Thus, a first-
order high-pass and a first-order low-pass provide a second-order band-pass, while a
second-order high-pass and a second-order low-pass result in a fourth-order band-pass
response.

In comparison to wide-band filters, narrow-band filters of higher order consist of cascaded

second-order band-pass filters that use the Sallen-Key or the Multiple Feedback (MFB)
topology.

16-28
 Band-Pass Filter Design

16.5.1 Second-Order Band-Pass Filter

To develop the frequency response of a second-order band-pass filter, apply the trans-
formation in Equation 16–7 to a first-order low-pass transfer function:
A0
A(s) +
1)s

Replacing s with
1 s)1
DW s
ǒ Ǔ
yields the general transfer function for a second-order band-pass filter:
A 0·DW·s
A(s) + (16–9)
1 ) DW·s ) s 2
When designing band-pass filters, the parameters of interest are the gain at the mid fre-
quency (Am) and the quality factor (Q), which represents the selectivity of a band-pass
filter.
Therefore, replace A0 with Am and ∆Ω with 1/Q (Equation 16–7) and obtain:
Am
Q
·s
A(s) + (16–10)
1 ) Q1 ·s ) s2

Figure 16–32 shows the normalized gain response of a second-order band-pass filter for
different Qs.
0

–5
Q=1
–10
|A| — Gain — dB

–15

–20
Q = 10
–25

–30

–35

–45
0.1 1 10
Frequency — Ω

Figure 16–32. Gain Response of a Second-Order Band-Pass Filter

Active Filter Design Techniques 16-29

Band-Pass Filter Design

The graph shows that the frequency response of second-order band-pass filters gets
steeper with rising Q, thus making the filter more selective.

16.5.1.1 Sallen-Key Topology

R

R C
VIN
VOUT
C 2R

R2
R1

Figure 16–33. Sallen-Key Band-Pass

The Sallen-Key band-pass circuit in Figure 16–33 has the following transfer function:

G·RCw m·s
A(s) +
1 ) RCw m(3 * G)·s ) R 2C 2w m 2·s 2

Through coefficient comparison with Equation 16–10, obtain the following equations:
1
mid-frequency: f m +
2pRC

R2
inner gain: G+1)
R1

Am + G
gain at fm:
3*G

Q+ 1
filter quality:
3*G

The Sallen-Key circuit has the advantage that the quality factor (Q) can be varied via the
inner gain (G) without modifying the mid frequency (fm). A drawback is, however, that Q
and Am cannot be adjusted independently.

Care must be taken when G approaches the value of 3, because then Am becomes infinite
and causes the circuit to oscillate.

To set the mid frequency of the band-pass, specify fm and C and then solve for R:

R+ 1
2pf mC

16-30
 Band-Pass Filter Design

Because of the dependency between Q and Am, there are two options to solve for R2: ei-
ther to set the gain at mid frequency:
2A m * 1
R2 +
1 ) Am

or to design for a specified Q:

R 2 + 2Q * 1
Q

16.5.1.2 Multiple Feedback Topology

C

R1 C
R2
VIN
VOUT

R3

Figure 16–34. MFB Band-Pass

The MFB band-pass circuit in Figure 16–34 has the following transfer function:
R 2R 3
*R Cw m·s
1)R 3
A(s) + 2R 1R 3 R R R
1)R Cw m·s ) R1 )R
2 3
C 2·w m 2·s 2
1)R 3 1 3

The coefficient comparison with Equation 16–9, yields the following equations:

mid-frequency: f m +
1
2pC
Ǹ R1 ) R3
R 1R 2R 3

R2
gain at fm: * Am +
2R 1

filter quality: Q + pf mR 2C

B+ 1
bandwidth:
pR 2C

The MFB band-pass allows to adjust Q, Am, and fm independently. Bandwidth and gain
factor do not depend on R3. Therefore, R3 can be used to modify the mid frequency with-

Active Filter Design Techniques 16-31

Band-Pass Filter Design

out affecting bandwidth, B, or gain, Am. For low values of Q, the filter can work without R3,
however, Q then depends on Am via:

* A m + 2Q 2

Example 16–5. Second-Order MFB Band-Pass Filter with fm = 1 kHz

To design a second-order MFB band-pass filter with a mid frequency of fm = 1 kHz, a quali-
ty factor of Q = 10, and a gain of Am = –2, assume a capacitor value of C = 100 nF, and
solve the previous equations for R1 through R3 in the following sequence:

R2 + Q + 10 + 31.8 kW
pf mC p·1 kHz·100 nF
R2
R1 + + 31.8 kW + 7.96 kW
* 2A m 4
* A mR 1
R3 + + 2·7.96 kW + 80.4 W
2Q 2 ) A m 200 * 2

16.5.2 Fourth-Order Band-Pass Filter (Staggered Tuning)

Figure 16–32 shows that the frequency response of second-order band-pass filters gets
steeper with rising Q. However, there are band-pass applications that require a flat gain
response close to the mid frequency as well as a sharp passband-to-stopband transition.
These tasks can be accomplished by higher-order band-pass filters.
Of particular interest is the application of the low-pass to band-pass transformation onto
a second-order low-pass filter, since it leads to a fourth-order band-pass filter.
Replacing the S term in Equation 16–2 with Equation 16–7 gives the general transfer func-
tion of a fourth-order band-pass:
2
s 2·A 0(DW)
b1
A(s) + (16–11)
1)
a1
b1
DW·s ƪ
) 2)
(DW)
b1
2
ƫ ·s 2 )
a1
b1
DW·s 3 ) s4

Similar to the low-pass filters, the fourth-order transfer function is split into two second-or-
der band-pass terms. Further mathematical modifications yield:
A mi A mi s
Qi
·as Qi a
·
A(s) + · (16–12)
ƪ1 ) as
Q1
) (as) ƫ ƪ1 )
2 1 ǒsǓ
Qi a
) ƫ
ǒas Ǔ 2

Equation 16–12 represents the connection of two second-order band-pass filters in se-
ries, where

16-32
 Band-Pass Filter Design

D Ami is the gain at the mid frequency, fmi, of each partial filter
D Qi is the pole quality of each filter
D α and 1/α are the factors by which the mid frequencies of the individual filters, fm1
and fm2, derive from the mid frequency, fm, of the overall bandpass.

In a fourth-order band-pass filter with high Q, the mid frequencies of the two partial filters
differ only slightly from the overall mid frequency. This method is called staggered tuning.

Factor α needs to be determined through successive approximation, using equation

16–13:

ƪ ƫ
2
a·DW·a 1 (DW) 2
a2 ) ) 12 * 2 * +0 (16–13)
b 1ǒ1 ) a 2Ǔ a b1

with a1 and b1 being the second-order low-pass coefficients of the desired filter type.

To simplify the filter design, Table 16–2 lists those coefficients, and provides the α values
for three different quality factors, Q = 1, Q = 10, and Q = 100.

Table 16–2. Values of α For Different Filter Types and Different Qs

Bessel Butterworth Tschebyscheff

a1 1.3617 a1 1.4142 a1 1.0650
b1 0.6180 b1 1.0000 b1 1.9305
Q 100 10 1 Q 100 10 1 Q 100 10 1
∆Ω 0.01 0.1 1 ∆Ω 0.01 0.1 1 ∆Ω 0.01 0.1 1
α 1.0032 1.0324 1.438 α 1.0035 1.036 1.4426 α 1.0033 1.0338 1.39

Active Filter Design Techniques 16-33

Band-Pass Filter Design

After α has been determined, all quantities of the partial filters can be calculated using the
following equations:

The mid frequency of filter 1 is:

f
f m1 + am (16–14)

the mid frequency of filter 2 is:

f m2 + f m·a (16–15)

with fm being the mid frequency of the overall forth-order band-pass filter.

The individual pole quality, Qi, is the same for both filters:

ǒ1 ) a 2Ǔb 1
Q i + Q· a·a 1
(16–16)

with Q being the quality factor of the overall filter.

The individual gain (Ami) at the partial mid frequencies, fm1 and fm2, is the same for both
filters:

A mi +
Qi
Q
· Ǹ
Am
B1
(16–17)

with Am being the gain at mid frequency, fm, of the overall filter.

Example 16–6. Fourth-Order Butterworth Band-Pass Filter

The task is to design a fourth-order Butterworth band-pass with the following parameters:
D mid frequency, fm = 10 kHz
D bandwidth, B = 1000 Hz
D and gain, Am = 1

From Table 16–2 the following values are obtained:

D a1 = 1.4142
D b1 = 1
D α = 1.036

16-34
 Band-Pass Filter Design

In accordance with Equations 16–14 and 16–15, the mid frequencies for the partial filters
are:

f mi + 10 kHz + 9.653 kHz and f m2 + 10 kHz·1.036 + 10.36 kHz

1.036

The overall Q is defined as Q + f mńB , and for this example results in Q = 10.

Using Equation 16–16, the Qi of both filters is:

ǒ1 ) 1.036 2Ǔ·1
Q i + 10· + 14.15
1.036·1.4142

With Equation 16–17, the passband gain of the partial filters at fm1 and fm2 calculates to:

A mi + 14.15 ·
10
Ǹ11 + 1.415
The Equations 16–16 and 16–17 show that Qi and Ami of the partial filters need to be inde-
pendently adjusted. The only circuit that accomplishes this task is the MFB band-pass fil-
ter in Paragraph 16.5.1.2.

To design the individual second-order band-pass filters, specify C = 10 nF, and insert the
previously determined quantities for the partial filters into the resistor equations of the
MFB band-pass filter. The resistor values for both partial filters are calculated below.

Filter 1: Filter 2:
Qi 14.15 Qi 14.15
R 21 + + + 46.7 kW R 22 + + + 43.5 kW
pf m1C p·9.653 kHz·10 nF pf m2C p·10.36 kHz·10 nF

R 21 46.7 kW R 22 43.5 kW
R 11 + + + 16.5 kW R 12 + + + 15.4 kW
* 2A mi * 2· * 1.415 * 2A mi * 2· * 1.415

* A miR 11 kW + 58.1 W R + * A miR 12 + 1.415·15.4 kW + 54.2 W

R 31 + + 1.415·16.5
2 ) 1.415
2
2Q i ) A mi 2·14.15 32 2
2Q i ) A mi 2·14.15 2 ) 1.415

Figure 16–35 compares the gain response of a fourth-order Butterworth band-pass filter
with Q = 1 and its partial filters to the fourth-order gain of Example 16–4 with Q = 10.

Active Filter Design Techniques 16-35

Band-Rejection Filter Design

5
A2
A1
0
Q=1

–5
Q = 10

|A| — Gain — dB
–10

–15

–20

–25

–30

–35
100 1k 10 k 100 k 1M
f — Frequency — Hz

Figure 16–35. Gain Responses of a Fourth-Order Butterworth Band-Pass and its Partial Filters

16.6 Band-Rejection Filter Design

A band-rejection filter is used to suppress a certain frequency rather than a range of fre-
quencies.

Two of the most popular band-rejection filters are the active twin-T and the active Wien-
Robinson circuit, both of which are second-order filters.

To generate the transfer function of a second-order band-rejection filter, replace the S

term of a first-order low-pass response with the transformation in 16–18:

DW (16–18)
s ) 1s

which gives:

A 0ǒ1 ) s 2Ǔ
A(s) + (16–19)
1 ) DW·s ) s 2

Thus the passband characteristic of the low-pass filter is transformed into the lower pass-
band of the band-rejection filter. The lower passband is then mirrored at the mid frequen-
cy, fm (Ω=1), into the upper passband half (Figure 16–36).

16-36
 Band-Rejection Filter Design

|A| [dB] |A| [dB]

0 0 ∆Ω
–3 –3

0 1 Ω 0 Ω1 1 Ω2 Ω

Figure 16–36. Low-Pass to Band-Rejection Transition

The corner frequency of the low-pass transforms to the lower and upper –3-dB frequen-
cies of the band-rejection filter Ω1 and Ω2. The difference between both frequencies is the
normalized bandwidth ∆Ω:
DW + W max * W min

Identical to the selectivity of a band-pass filter, the quality of the filter rejection is defined
as:
fm
Q+ + 1
B DW
Therefore, replacing ∆Ω in Equation 16–19 with 1/Q yields:
A 0ǒ1 ) s 2Ǔ
A(s) + (16–20)
1 ) Q1 ·s ) s 2

16.6.1 Active Twin-T Filter

The original twin-T filter, shown in Figure 16–37, is a passive RC-network with a quality
factor of Q = 0.25. To increase Q, the passive filter is implemented into the feedback loop
of an amplifier, thus turning into an active band-rejection filter, shown in Figure 16–38.
C C

R/2
VIN VOUT

R R

2C

Figure 16–37. Passive Twin-T Filter

Active Filter Design Techniques 16-37

Band-Rejection Filter Design

C C

R/2
VIN

R R
VOUT
2C

R2
R1

Figure 16–38. Active Twin-T Filter

The transfer function of the active twin-T filter is:

kǒ1 ) s 2Ǔ
A(s) + (16–21)
1 ) 2(2 * k)·s ) s 2

Comparing the variables of Equation 16–21 with Equation 16–20 provides the equations
that determine the filter parameters:
1
mid-frequency: f m +
2pRC
R2
inner gain: G+1)
R1

passband gain: A 0 + G

1
rejection quality: Q +
2 ( 2 * G)

The twin-T circuit has the advantage that the quality factor (Q) can be varied via the inner
gain (G) without modifying the mid frequency (fm). However, Q and Am cannot be adjusted
independently.

To set the mid frequency of the band-pass, specify fm and C, and then solve for R:

R+ 1
2pf mC

Because of the dependency between Q and Am, there are two options to solve for R2: ei-
ther to set the gain at mid frequency:

R 2 + ǒA 0 * 1 ǓR 1

16-38
 Band-Rejection Filter Design

or to design for a specific Q:

ǒ
R2 + R1 1 * 1
2Q
Ǔ
16.6.2 Active Wien-Robinson Filter
The Wien-Robinson bridge in Figure 16–39 is a passive band-rejection filter with differen-
tial output. The output voltage is the difference between the potential of a constant voltage
divider and the output of a band-pass filter. Its Q-factor is close to that of the twin-T circuit.
To achieve higher values of Q, the filter is connected into the feedback loop of an amplifier.
VIN

R 2R1

C VOUT

R C R1

Figure 16–39. Passive Wien-Robinson Bridge

R3

R2 R1 2R1
R4
VIN C R
VOUT

Figure 16–40. Active Wien-Robinson Filter

The active Wien-Robinson filter in Figure 16–40 has the transfer function:
b
ǒ1 ) s 2Ǔ
1)a
A(s) + * (16–22)
1 ) 1)a
3
·s ) s 2

R2 R2
with a + and b+
R3 R4

Comparing the variables of Equation 16–22 with Equation 16–20 provides the equations
that determine the filter parameters:

Active Filter Design Techniques 16-39

Band-Rejection Filter Design

1
mid-frequency: f m +
2pRC

b
passband gain: A 0 + *
1)a

1)a
rejection quality: Q +
3

To calculate the individual component values, establish the following design procedure:

1) Define fm and C and calculate R with:

R+ 1
2pf mC

2) Specify Q and determine α via:

a + 3Q * 1

3) Specify A0 and determine β via:

b + * A 0·3Q

4) Define R2 and calculate R3 and R4 with:

R
R 3 + a2

and

R2
R4 +
b

In comparison to the twin-T circuit, the Wien-Robinson filter allows modification of the
passband gain, A0, without affecting the quality factor, Q.

If fm is not completely suppressed due to component tolerances of R and C, a fine-tuning

of the resistor 2R2 is required.

Figure 16–41 shows a comparison between the filter response of a passive band-rejec-
tion filter with Q = 0.25, and an active second-order filter with Q = 1, and Q = 10.

16-40
 All-Pass Filter Design

–5
Q = 10

|A| — Gain — dB
Q=1

–10 Q = 0.25

–15

–20
1 10 100 1k 10 k
Frequency — Ω

Figure 16–41. Comparison of Q Between Passive and Active Band-Rejection Filters

16.7 All-Pass Filter Design

In comparison to the previously discussed filters, an all-pass filter has a constant gain
across the entire frequency range, and a phase response that changes linearly with fre-
quency.

Because of these properties, all-pass filters are used in phase compensation and signal
delay circuits.

Similar to the low-pass filters, all-pass circuits of higher order consist of cascaded first-or-
der and second-order all-pass stages. To develop the all-pass transfer function from a
low-pass response, replace A0 with the conjugate complex denominator.

The general transfer function of an allpass is then:

Pǒ
i 1 * a is ) b is Ǔ
2
A(s) + (16–23)
Pǒ
i 1 ) a is ) b is Ǔ
2

with ai and bi being the coefficients of a partial filter. The all-pass coefficients are listed in
Table 16–10 of Section 16.9.

Expressing Equation 16–23 in magnitude and phase yields:

Active Filter Design Techniques 16-41

 Ǹǒ1 * b W Ǔ
P
i i
2
2
) a i 2W 2 ·e *ja
A(s) + (16–24)
P Ǹǒ 2
i 1*bW Ǔ i
2 ) a i W 2 ·e )ja
2

This gives a constant gain of 1, and a phase shift,φ, of:

f + * 2a + * 2 ȍ arctan 1 *aiWb W2 (16–25)

i i

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 1ń Ǹ2 –times its initial value is the corner
frequency, fC.

The group delay is defined through:

df
t gr + * (16–26)
dw

To present the group delay in normalized form, refer tgr to the period of the corner frequen-
cy, TC, of the all-pass circuit:
t gr w
T gr + + t gr·f c + t gr· c (16–27)
Tc 2p

Substituting tgr through Equation 16–26 gives:

df
T gr + * 1 · (16–28)
2p dW