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Application Report
SLOA152–December 2010
Analog Active Audio Filters
Stephen Crump .......................................................................................................... Audio Products
ABSTRACT
Analog active audio filters can be used to compensate frequency response problems in a variety of
systems. Their responses are examined here to simplify the filter design task for design engineers.
Contents
1 Introduction .................................................................................................................. 1
2 First Order Filters ........................................................................................................... 2
3 Second Order Filters ....................................................................................................... 3
4 Biquadratic Filters ........................................................................................................... 4
5 Analog Filter Implementations: Second-Order Filters. ................................................................. 5
6 Analog Filter Implementations: Biquadratic Filters. ..................................................................... 7
1 Introduction
Active audio filters may be used to compensate problems in frequency response of audio systems and
loudspeakers. This paper deals with analog filters. These filters can produce a response that is
approximately the inverse of a system response or a loudspeaker acoustic response so that when the two
are summed the result is nearly flat. They may also be used to produce the difference of a target response
minus the system or loudspeaker response, so that when the two are summed the result is nearly the
target. In either case the final response is more accurate or more pleasing.
Cell phone and PDA speaker responses like the one shown below often have annoying peaks that reduce
intelligibility. This is compensated with the response of a biquad filter, also shown. The sum, the
compensated response, is much more pleasing and far more intelligible than the original.
1SLOA152–December 2010 Analog Active Audio Filters
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Biquad
Valley
Filter
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43kW
22nF22nF
10kW 5.6kW
1.2kW1.5kW
First Order Filters www.ti.com
SPACER
The schematic for the biquad filter is at right.
This and other filters are discussed in detail in
this paper.
This paper begins with the relationship between filter singularities and their responses. Then it considers a
number of possible filter implementations. (It’s typically easier to adjust a set of filter singularities to
achieve the best result than to adjust the numerous component values in a circuit. Once the singularities
are decided the filter usually can be implemented relatively easily.) It provides equations for responses
and the parameters in them and discusses optimizing component choices.
Audio filters may be first, second or higher order. First and second order analog filters are generally well
understood and their audio uses are somewhat limited, so they are examined briefly. The paper also
examines biquadratic filters, or biquads, in more depth because they are more powerful tools for response
compensation or EQ. Bridged-T filters will be added in a later version.
2 First Order Filters
First-order filters implement responses with single poles as their denominators. They have limited
response bands, either low-pass or high-pass, which are described below.
• Low pass: H(s) = Ho wo / ( s + wo ).
For small s, this is Ho, flat low-frequency response; for large s, it is Ho wo / s, a first-order rolloff.
• High pass: H(s) = Ho s / ( s + wo ).
For large s, this is Ho, flat high-frequency response; for small s, it is Ho s / wo, a first-order rollup.
2 Analog Active Audio Filters SLOA152–December 2010
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2nd-ORDERFILTERRESPONSES,H =1, =0.2/0.5/1/2O a
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www.ti.com Second Order Filters
3 Second Order Filters
Second-order filters implement responses with quadratic terms as their denominators. They have limited
response bands, either low-pass, band-pass or high-pass, which are described below. (1)
• Low pass: H(s) = Ho wo 2 / (s2 + a wo s + wo 2).
For small s, this is Ho, flat low-frequency response; for large s, it is Ho wo 2 / s2, a second-order rolloff.
Response magnitude is G(w) = sqrt(Ho 2 wo 4 / (w4 + w2 wo 2 (a2 – 2) + wo 4) ).
Phase is Φ(w) = p/2 – arctan( a wo w / (wo 2 – w2)).
SPACER
• Band pass: H(s) = Ho a wo s / ( s2 + a wo s + wo 2 ).
For small s, this is Ho a s / wo, an increasing first-order response or rollup; for large s, it is Ho a wo / s, a
decreasing first-order response or rolloff. At s = jwo, it is Ho, band-center response.
Response magnitude is G(w) = sqrt(Ho 2 a2 w2 wo 2 / (w4 + w2 wo 2 (a2 – 2) + wo 4) ).
Phase is Φ(w) – arctan( a wo w / (wo 2 – w2)).
SPACER
• High pass: H(s) = Ho s2 / (s2 + a wo s + wo 2).
For large s, this is Ho, flat high-frequency response; for small s, it is Ho s2 / wo 2, a second-order rollup.
Response magnitude is G(w) = sqrt(Ho 2 w4 / (w4 + w2 wo 2 (a2 – 2) + wo 4) ).
Phase is Φ(w) = p – arctan( a wow / (wo 2 – w2)).
SPACER
Ho scales response magnitude, while wo sets the characteristic frequency, the frequency at which the filter
operates. The variable a sets the sharpness of the peak the filter produces, which varies inversely with a
(sharper peak with smaller a).
(1) Reference: Operational Amplifiers, Design and Applications, Graeme, Tobey and Huelsman, Burr-Brown, McGraw-Hill Book Company,
1971, ISBN 07-064917-0, pages 284-286.
3SLOA152–December 2010 Analog Active Audio Filters
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BIQUAD RESPONSES vs. FILTER FORM
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Low Frequency Boost High Frequency Boost
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Biquadratic Filters www.ti.com
4 Biquadratic Filters
Biquadratic or biquad filters implement responses with quadratic terms for both their numerators and their
denominators. They have wide responses, with a single peak or valley or with low or high frequency boost
with or without peaking. Their responses have the following form.
H(s) = Ho ( s2 + aZ wZ s + wZ 2 ) / ( s2 + aP wP s + wP 2 ).
Filter response is influenced by relationships between wZ and wP and between aZ and aP.
• Single peak or valley. The filter produces this response when wZ and wP are equal.
If aZ is greater than aP the filter produces a peak. If aP is greater than aZ the filter produces a valley.
SPACER
• Low frequency boost. The filter produces this response when wZ is greater than wP.
If aZ is less than about 1, the response includes a valley above the boost frequency. If aP is less than
about 1, the response includes a peak below the boost frequency.
SPACER
• High frequency boost. The filter produces this response when wP is greater than wZ.
If aZ is less than about 1, the response includes a valley below the boost frequency. If aP is less than
about 1, the response includes a peak above the boost frequency.
SPACER
Response magnitude is G(w) = sqrt( (w4 + w2 wz 2 (az 2 – 2) + wz 4) / (w4 + w2 wp 2 (ap 2 – 2) + wp 4) ).
Phase is Φ(w) = arctan( aZ wZ w / (wZ 2 – w 2) ) – arctan( aP wP w / (wP 2 – w2) ).
Traces in the graph that follows illustrate something of the range of responses a biquad filter can
generate. The responses are arranged as follows for clarity.
• Single peak and valley responses are presented in the order of decreasing peaks, or decreasing aZ
with respect to aP. For all these responses Ho is 1 and wZ and wP are 1 kHz.
SPACER
• Low and high frequency boost responses are presented in the order of decreasing peaks and valleys,
or increasing aZ and aP. Also, aZ and aP are made equal, wZ and wP are placed symmetrically around
1kHz, and Ho is set to 0.5 for low frequency boost and 2 for high frequency boost, to make the
responses symmetrical around zero dB and 1kHz.
SPACER
4 Analog Active Audio Filters SLOA152–December 2010
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BIQUADRESPONSESvs.H , , , ,o z z p pa v a v
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www.ti.com Analog Filter Implementations: Second-Order Filters.
Note that aZ and aP do not have to be equal! Varying aZ and aP can create responses that range among
and beyond the extremes in the graph.
5 Analog Filter Implementations: Second-Order Filters.
Second-order filters may be non-inverting or inverting. The schematics below show single-ended forms,
both non-inverting and inverting, and an inverting, differential form, with equations for their Ho, a and wo.
Low-Pass Filters
Non-Inverting Inverting Differential Inverting
Ho
wo
a
5SLOA152–December 2010 Analog Active Audio Filters
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© 2010, Texas Instruments Incorporated
-
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VI
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R1
C1
C2
Inverting
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C3
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Analog Filter Implementations: Second-Order Filters. www.ti.com
Band-Pass Filters
Non-Inverting Inverting Differential Inverting
Ho
wo
a
High-Pass Filters
Non-Inverting Inverting Differential Inverting
Ho
wo
a
6 Analog Active Audio Filters SLOA152–December 2010
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© 2010, Texas Instruments Incorporated
-
+
R1
R2
R3
R4
C1 C2
VO
VI R5
K=-R5/R4
Inverting
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R4
C1 C2
V -
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V +
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2
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Differential
+
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R3
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(R1 C1 C2)
www.ti.com Analog Filter Implementations: Biquadratic Filters.
6 Analog Filter Implementations: Biquadratic Filters.
Biquadratic filters also may be non-inverting, inverting, or differential inverting. The schematics below
show all these forms with equations for their Ho, a and wo. The equations begin with a factor K, the gain of
the inner opamp circuit, for each of the filter forms. The remaining quantities, D, Ho, wZ, aZ, wP and aP, are
common to all the filter forms. D is a multiplier used to simplify the following equations.
Biquad Filters – High Frequency Boost
Non-Inverting Inverting Differential Inverting
K
D Multplier for R2 and K
Ho K Response at high frequency
wz
wP
az
ap
Schematics for low frequency boost filters follow. The input circuit in each filter includes a series chain of
capacitors to ground or virtual ground. This load could destabilize either the opamp in the filter or an
opamp driving the input, so a small value resistor, maybe 100Ω, is added in series with the final cap in the
chain. Also, feedback elements in the inverting and differential filters are capacitors. These provide no
path for DC bias, so a large value resistor, maybe 1 to 10 M, is added in parallel with the feedback cap. Of
course, these resistances will have a small effect on filter responses, but they should not degrade them
significantly.
7SLOA152–December 2010 Analog Active Audio Filters
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-
+
Inverting
S
R1
R2
C1 C2 C3 C4
VO
VI
L
R3
K = -C4/C3
L = Large-Value
Resistor 3 Places
Differential
-
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+
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2
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L
C4
L
S
S
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V +
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+
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Non-Inverting
R1
R2
R3
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C3
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VI
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C4
(NEGATIVE)
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f
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( )
( )
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R1 DR2 C1 C2 + C1C3+C2C3
´ ´
´
( ) ( )
C1 + C2 DR2
(R1 C1 C2)
Analog Filter Implementations: Biquadratic Filters. www.ti.com
Biquad Filters – Low Frequency Boost
Non-Inverting Inverting Differential Inverting
K
D Multplier for R2 and K
Ho Response at low frequency
wz wP
az
ap
8 Analog Active Audio Filters SLOA152–December 2010
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+
-
Non-Inverting
ThereareNOInvertingor
Differentialversions
becauseeachofthese
loadstheoutputofthe
inputnetworkand
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R2
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VO
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f
s
R
1+
R
( )
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R2 + R3
1
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1
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( )
DR2 (C1 + C2)+ R1 C2 (1-KD)
R1 DR2 C1 C2
´ ´
( )
( )
C1 + C2 DR2
R1C1C2
´
www.ti.com Analog Filter Implementations: Biquadratic Filters.
Biquad Filters – Single Peak or Valley
Non-Inverting
K
D Multiplier for R2 and K
Ho K Response at very low and high frequencies
wz wP (= wz)
az
ap
The most complicated quantities are aZ and aP, so we will look at these in some detail. As we will see, it is
more difficult to achieve low values of aZ and aP than high, so we will concentrate on reducing these
quantities. We will consider how to produce values as small as about 0.5, a value that provides significant
peaking.
aZ is the same for all 3 filter configurations. The ratio (C1+C2) / √(C1C2) in aZ ranges from about 3.5 to 2
to about 3.5 again as (C1/C2) ranges from 0.1 to 1 to 10, so aZ is reduced by making C1 and C2 different
in value. So it is typically best to make C1 and C2 similar in value. aZ can be controlled by varying the
ratio √(DR2/R1). If R1 = 20×DR2 and C1 = C2, aZ = 2 / √(20) = 0.45, probably close to the lowest value
needed.
For aP we face similar constraints with the first term or two in the numerators, but we have the advantage
of the last term, which is negative in non-inverting and differential filters for any KD product greater than 1.
So we can use this term to reduce aP if we need to do so. (Beware, however: if KD is made large enough,
the last term will cancel the rest of the numerator, aP will equal zero and the filter will oscillate at wP!)
Note that we do NOT have this advantage in inverting biquads! In those, since K is negative, the sum (1-
KD) is always greater than 1. As a result, it is likely to be very difficult to achieve low values of aP. For this
reason inverting biquads are not likely to be generally useful in EQ work.
Single peak or valley filters are a special case. In these, aP = aZ + R1C2×(1-KD) / √(R1DR2C1C2). If C1
and C2 are similar in value, aP ~= aZ + (1-KD) √(R1/DR2) ~= aZ + 2×(1-KD)/aZ. If KD is less than 1, (1-KD)
is positive, aP is greater than aZ and the filter creates a valley. If KD equals 1, aP equals aZ and there is no
peak or valley. If KD is greater than 1, the second term is negative, aP is smaller than aZ and the filter
creates a peak.
Note that, for single peak or valley filters, reducing aZ increases the magnitude of the second term in aP.
So in valley filters, with KD < 1, reducing aZ tends to increase aP. This tends to make the resulting valley
broad and deep. Make (1-KD) smaller to narrow or reduce the valley. In peak filters, with KD > 1, reducing
aZ tends to reduce aP by making the last term in aP more negative. This tends to make the resulting peak
narrow and high. Make (1-KD) less negative to broaden or reduce the peak.
9SLOA152–December 2010 Analog Active Audio Filters
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