dm00038862

AN3984
Application note
IIR filter design equations for Sound Terminal® devices
Introduction
The purpose of this document is to provide a tool to calculate the IIR filter coefficients to
program the Sound Terminal® devices from STMicroelectronics.
For each filter the procedure and the formulas to calculate the coefficient will be described;
the Matlab code is given in Appendix A: Matlab code (functions) on page 24.
A generalized set of equations can be formulated for the design of first-order low-pass and
high-pass filters and of second-order filters.
A specialized set of equations is devised for designing parametric biquad EQ filters. As with
any other filter design procedure, the desired characteristics of the filter are to be made
available.
The parameters governing the characteristics of each filter are:
■
fc: filter cutoff frequency which is the -3dB corner frequency or the midpoint frequency in
a peak or notch filter
■
fs: sampling frequency
■
Q: quality factor (not applicable for low and high-shelf filters)
■
Slope: applicable only for low and high-shelf filters
■
Gain: the boost or the attenuation at f = fc
These parameters can be used to determine the coefficients of the digital filter transfer
function.
September 2011
Doc ID 022240 Rev 1
1/46
www.st.com
Contents
AN3984
Contents
1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2
Filter stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2
First-order filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3
Second-order filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
First-order filter design (LPF and HPF) . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4
Second-order filter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.1
4.2
4.3
5
2/46
Low-pass and high-pass filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.1.1
Low-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.1.2
High-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Peak filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2.1
Peak filter - negative gain (cut) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2.2
Peak filter - positive gain (boost) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Shelf filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.3.1
Low-shelf filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.3.2
High-shelf filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.4
Notch filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.5
All-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.6
Band-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.1
1st-order low-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.2
1st-order high-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.3
2nd-order low-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.4
2nd-order high-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.5
Low-shelf filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.6
High-shelf filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.7
Notch filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.8
All-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.9
Band-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Doc ID 022240 Rev 1
AN3984
Contents
Appendix A Matlab code (functions). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A.1
Code structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A.2
Peak filter (PeakFilterAPW.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
A.3
Low-pass and high-pass filter (LHPassFilterAPW.m) . . . . . . . . . . . . . . . . 27
A.4
Low and high-shelf filter (ShelfFilterAPW.m) . . . . . . . . . . . . . . . . . . . . . . . 30
A.5
Notch filter (NotchFilterAPW.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
A.6
All-pass filter (AllPassFilterAPW.m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
A.7
Band-pass filter (BandPassFilterAPW.m) . . . . . . . . . . . . . . . . . . . . . . . . . 37
A.8
Float to hex conversion (myFloat2Hex.m) . . . . . . . . . . . . . . . . . . . . . . . . . 39
A.9
Max coefficient limit value calculator (LimitVal.m) . . . . . . . . . . . . . . . . . . . 40
A.10
Display coefficient and error messages (Display_Coeff.m). . . . . . . . . . . . 41
Appendix B Abbreviations and acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6
Revision history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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3/46
List of figures
AN3984
List of figures
Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
Figure 7.
Figure 8.
Figure 9.
Figure 10.
4/46
1st-order low-pass filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1st-order high-pass filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2nd-order low-pass filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2nd-order high-pass filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Low-shelf filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
High-shelf filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Notch filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
All-pass filter - phase response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Band-pass filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Code structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Doc ID 022240 Rev 1
AN3984
1
Overview
Overview
The transfer function for a first-order filter in the digital z-domain is:
Equation 1
H(z) =
b 0 + b1 ⋅ z −1
a 0 + a1 ⋅ z −1
For a second-order filter (a biquad) the transfer function is:
Equation 2
H ( z) =
b 0 + b1 ⋅ z −1 + b2 ⋅ z −2
a 0 + a 1 ⋅ z −1 + a 2 ⋅ z − 2
This equation can be modified normalizing the a0 coefficient; the new equation is:
Equation 3
H ( z) =
(b0
a 0 ) + (b1 a 0 ) ⋅ z −1 + (b 2 a 0 ) ⋅ z −2
1 + (a1 a 0 )⋅ z −1 + (a 2 a 0 ) ⋅ z −2
The most straightforward implementation form using Equation 3 is:
Equation 4
y[n] = (b0 a0 )⋅ x[n] + (b1 a0 ) ⋅ x[n − 1] + (b2 a0 ) ⋅ x[n − 2] − (a1 a0 ) ⋅ y[n − 1] − (a2 a0 )⋅ y[n − 2]
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Filter stability
2
Filter stability
2.1
Definition
AN3984
A filter is said to be stable in the z-domain if the roots (or poles) of the filter lie inside the unit
circle.
This definition of stability can be translated in terms of the filter coefficients.
2.2
First-order filter
For a first-order filter, the stability condition that needs to be satisfied is that the pole of the
filter lies within the unit circle.
In terms of the coefficients, the condition can be given as:
Equation 5
a1 < 1
2.3
Second-order filter
For a 2nd-order filter, two conditions must be satisfied to ensure filter stability and translated
in terms of the filter coefficients they are:
Equation 6
a2 <1
a 1 < (1 + a 2 )
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AN3984
3
First-order filter design (LPF and HPF)
First-order filter design (LPF and HPF)
The preliminary step to obtain the coefficients for the first-order low-pass filter or high-pass
filter is to define three constants obtained from the filter parameters:
Equation 7
ωc = 2 ⋅ π ⋅ f c f s
K = tan (ω c 2 )
α = 1+ K
In a first-order filter both the coefficients a2 and b2 are null.
The denominator coefficients are identical for both an LPF and an HPF designed for the
same cutoff frequency and they are computed as follows:
Equation 8
a0 = 1
a1 = −
(1 − K )
α
The numerator for an LPF can be calculated as follows:
Equation 9
b0 =
b1 =
K
α
K
α
The numerator for an HPF can be calculated as follows:
Equation 10
b0 =
1
α
b1 = −
1
α
The coefficient used in APWorkbench can be calculated by applying these formulas:
Equation 11
(b1 2)
2 = a
0
(
b2 )
Coefficient b2 =
a0
(− a1 2 )
Coefficient a1 =
2
a0
Coefficient
b1
Coefficient a2 =
Coefficient
b0
− a2
a0
2 =
Doc ID 022240 Rev 1
(b0 2 )
a0
7/46
Second-order filter design
AN3984
4
Second-order filter design
4.1
Low-pass and high-pass filters
The preliminary step to obtain the coefficients for a second-order filter is the calculation of
these coefficients obtained from the filter parameters:
Equation 12
fc
fs
K = tan (ωc 2 )
ϑc = 2 ⋅ π ⋅
W = K2
α =1+ K
DE = 1 +
K
+W
Q
The denominator coefficients are the same for both an LPF and an HPF if designed for the
same cutoff frequency. They are computed as follows:
Equation 13
a0 = 1
− 1)
DE
K
1−
+W
Q
a2 =
DE
a1 = 2 ⋅
4.1.1
(W
Low-pass filter
The numerator coefficient for a second-order LPF can be calculated as follows:
Equation 14
W
DE
W
b1 = 2 ⋅
DE
W
b2 =
DE
b0 =
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AN3984
Second-order filter design
For a second-order LPF, the coefficients given in APWorkbench can be calculated as
follows:
Equation 15
W
2 = DE
W
Coefficient b2 =
DE
W −1
a
Coefficient 1 = −1 ⋅
2
DE
K
1− +W
Q
Coefficient a 2 = −
DE
W
1
b
Coefficient 0 = ⋅
2 2 DE
Coefficient
4.1.2
b1
High-pass filter
The numerator coefficient for a second-order HPF can be calculated as follows:
Equation 16
b0 =
1
DE
b1 = −2 ⋅
b2 =
W
DE
1
DE
For a second-order HPF, the coefficients given in APWorkbench can be calculated as
follows:
Equation 17
Coefficient b1
2
=−
Coefficient b2 =
1
DE
1
DE
W −1
Coefficient a1 2 = −1 ⋅
DE
K
+W
Q
Coefficient a 2 = −
DE
1 1
Coefficient b0 = ⋅
2 2 DE
1−
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Second-order filter design
4.2
AN3984
Peak filters
The first step is the calculation of the constant gain obtained from the gain filter parameter
(GdB is expressed in dB).
Equation 18
Gain = exp(GaindB ⋅ 0 .115129254)
The filter coefficients are different if the gain is positive or negative.
4.2.1
Peak filter - negative gain (cut)
The cut value is calculated with the following equation:
Equation 19
⎛ Q ⎞
CutValue = 1 + K ⋅ ⎜
⎟+W
⎝ Gain ⎠
The filter coefficient can be calculated as follows:
Equation 20
a0 = 1
a1 = 4 ⋅
(W − 1)
CutValue
⎛
Q
+W
⎜1 −
Gain
a2 = ⎜
⎜ CutValue
⎜
⎝
⎛
⎞
K
⎜⎜ 1 +
+ W ⎟⎟
Q
⎠
b0 = ⎝
CutValue
(W − 1)
b1 = 2 ⋅
CutValue
K
(1 − + W )
Q
b2 =
CutValue
10/46
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⎞
⎟
⎟
⎟
⎟
⎠
AN3984
Second-order filter design
The coefficients in the APWorkbench are consequently calculated as follows:
Equation 21
W −1
2 = CutValue
K
(1 − + W )
Q
Coefficient b2 =
CutValue
1− W
a
Coefficient 1 2 =
CutValue
Q
1−
⋅K +W
Coefficient a 2 = − Gain
CutValue
K
1+ +W
1
Q
b
Coefficient 0 2 = ⋅
2 CutValue
Coefficient
4.2.2
b1
Peak filter - positive gain (boost)
The boost value is calculated with the following equation:
Equation 22
BoostValue = 1 +
K
+W
Q
The filter coefficient can be calculated as follows:
Equation 23
a0 = 1
Gain
+W )
Q
a1 =
BoostValue
⎛
⎞
K
⎜⎜ 1 −
+ W ⎟⎟
Q
⎠
a2 = ⎝
BoostValue
Gain
+W )
(1 + K ⋅
Q
b0 = 2 ⋅
BoostValue
W −1
b1 = 2 ⋅
BoostValue
Gain
+W)
(1 − K ⋅
Q
b2 =
BoostValue
(1 + K ⋅
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Second-order filter design
AN3984
The coefficients in the APWorkbench are consequently calculated as follows:
Equation 24
W −1
BoostValue
Gain
(1 −
⋅ K +W )
Q
Coefficient b2 =
BoostValue
1−W
a
Coefficient 1 =
2 BoostValue
K
1− +W
Q
Coefficient a 2 = −
BoostValue
Gain
1+
⋅ K +W
1
Q
Coefficient b0 = ⋅
2 2
BoostValue
Coefficient b1
4.3
2
=
Shelf filters
The coefficient gain is defined in Equation 25.
Equation 25
Gain = 10
Gain dB
(
40 )
The coefficients α and β are calculated as follows:
Equation 26
α=
sin (ϑc )
⋅
2
⎞
⎛⎛
⎜ ⎜ Gain + ⎛⎜ 1 ⎞⎟ ⎞⎟ ⋅ ⎛⎜ 1 − 1⎞⎟ + 2 ⎟
⎟
⎟
⎜⎜
⎝ Gain ⎠ ⎠ ⎝ S ⎠
⎠
⎝⎝
β = 2 ⋅ α ⋅ Gain
4.3.1
Low-shelf filter
The coefficients for an LSF can be calculated as follows:
Equation 27
a 0 = (Gain + 1) + (Gain − 1) ⋅ cos ϑ c + β
a1 = − 2 ⋅ (Gain − 1) + (Gain + 1) ⋅ cos ϑ c
a 2 = (Gain + 1) + (Gain − 1) ⋅ cos ϑ c − β
b0 = Gain ⋅ ((Gain + 1) − (Gain − 1) ⋅ cos ϑ c + β )
b1 = 2 ⋅ Gain ⋅ ((Gain − 1) − (Gain + 1) ⋅ cos ϑ c )
b2 = Gain ⋅ ((Gain + 1) − (Gain − 1) ⋅ cos ϑ c − β )
The coefficient to load in APWorkbench can be calculated by applying the calculation
already shown in Equation 11.
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AN3984
4.3.2
Second-order filter design
High-shelf filter
The coefficients for an HSF can be calculated as follows:
Equation 28
a 0 = (Gain + 1) − (Gain − 1) ⋅ cos ϑ c + β
a1 = 2 ⋅ (Gain − 1) − (Gain + 1) ⋅ cos ϑc
a 2 = (Gain + 1) − (Gain − 1) ⋅ cos ϑ c − β
b0 = Gain ⋅ ((Gain + 1) + (Gain − 1) ⋅ cos ϑ c + β )
b1 = − 2 ⋅ Gain ⋅ ((Gain − 1) − (Gain + 1) ⋅ cos ϑ c )
b2 = Gain ⋅ ((Gain + 1) + (Gain − 1) ⋅ cos ϑ c − β )
The coefficient to load in APWorkbench to program a HSF can be computed by applying the
formulas shown in Equation 11.
4.4
Notch filter
The first step is to define the constant α(a):
Equation 29
α=
sin (ϑc )
2 ⋅Q
The coefficients for a notch filter can be calculated as follows:
Equation 30
a0 = 1 + α
a1 = − 2 ⋅ cos ϑ c
a2 = 1− α
b0 = 1
b1 = − 2 ⋅ cos ϑ
b2 = 1
The coefficients to load in APWorkbench can be calculated using Equation 11.
a. ϑc is defined in Equation 12
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Second-order filter design
4.5
AN3984
All-pass filter
Equation 29 allows calculating the constant α.
The coefficients for an APF can be calculated as follows:
Equation 31
a0 = 1 + α
a1 = − 2 ⋅ cos ϑ c
a2 = 1 − α
b0 = 1 − α
b1 = − 2 ⋅ cos ϑ c = a1
b2 = 1 + α
The coefficients to load in APWorkbench can be calculated using Equation 11.
4.6
Band-pass filter
Equation 29 allows calculating the constant α while Equation 32 is used to calculate the
normalized gain.
Equation 32
⎛ Gain dB ⎞
⎜
⎟
20 ⎠
NormGain = 10⎝
The coefficients for a BPF can be calculated as follows(b):
Equation 33
a0 = 1 + α
a1 = − 2 ⋅ cos ϑ c
a2 = 1 − α
b0 = α ⋅ NormGain
b1 = 0
b2 = −b 0 = −α ⋅ NormGain
The coefficients to load in APWorkbench can be calculated using Equation 11.
b. α is defined in Equation 29, ϑc is defined in Equation 12.
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AN3984
Examples
5
Examples
5.1
1st-order low-pass filter
Input data:
●
Cutoff freq: 1 kHz
●
Coefficient range: 4
●
Processing frequency: 96 kHz
Output data:
Filter coefficients
'Coeff 1: b1/2'
'Coeff 2: b2'
'Coeff 3: -a1/2'
'Coeff 4: -a2'
'Coeff 5: b0/2'
'0081d6'
'000000'
'0efc52'
'000000'
'0081d6'
1st-order low-pass filter - magnitude response
Figure 1.
Magnitude Response (dB)
0
Magnitude (dB)
-5
-10
-15
-20
-1
10
0
10
Frequency (kHz)
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1
10
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Examples
5.2
AN3984
1st-order high-pass filter
Input data:
●
Cutoff freq: 1 kHz
●
Coefficient range: 4
●
Processing frequency: 96 kHz
Output data:
Filter coefficients
'Coeff 1: b1/2'
'Coeff 2: b2'
'Coeff 3: -a1/2'
'Coeff 4: -a2'
'Coeff 5: b0/2'
'f081d6'
'000000'
'0efc52'
'000000'
'0f7e29'
1st-order high-pass filter - magnitude response
Figure 2.
Magnitude Response (dB)
0
Magnitude (dB)
-5
-10
-15
-20
-1
10
16/46
0
10
Frequency (kHz)
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1
10
AN3984
2nd-order low-pass filter
Input data:
●
Cutoff freq: 1 kHz
●
Coefficient range: 4
●
Quality factor (Q): 2
●
Processing frequency: 96 kHz
Output data:
Filter coefficients
'Coeff 1: b1/2'
'Coeff 2: b2'
'Coeff 3: -a1/2'
'Coeff 4: -a2'
'Coeff 5: b0/2'
'0008a0'
'0008a0'
'1f6af3'
'e10794'
'000450'
2nd-order low-pass filter - magnitude response
Figure 3.
Magnitude Response (dB)
15
10
5
0
-5
Magnitude (dB)
5.3
Examples
-10
-15
-20
-25
-30
-35
-40
-1
10
0
10
Frequency (kHz)
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1
10
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Examples
5.4
AN3984
2nd-order high-pass filter
Input data:
●
Cutoff freq: 1 kHz
●
Coefficient range: 4
●
Quality factor (Q): 2
●
Processing frequency: 96 kHz
Output data:
Filter coefficients
'Coeff 1: b1/2'
'Coeff 2: b2'
'Coeff 3: -a1/2'
'Coeff 4: -a2'
'Coeff 5: b0/2'
'e08c6b'
'1f7394'
'1f6af3'
'e10794'
'0fb9ca'
2nd-order high-pass filter - magnitude response
Figure 4.
Magnitude Response (dB)
20
10
Magnitude (dB)
0
-10
-20
-30
-40
-1
10
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0
10
Frequency (kHz)
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1
10
AN3984
Low-shelf filter
Input data:
●
Cutoff freq: 1 kHz
●
Gain: -10 dB
●
Coefficient range: 4
●
Slope: 2
●
Processing frequency: 96 kHz
Output data:
Filter coefficients
'Coeff 1: b1/2'
'Coeff 2: b2'
'Coeff 3: -a1/2'
'Coeff 4: -a2'
'Coeff 5: b0/2'
'e0f9f2'
'1e8e49'
'1efbb2'
'e1cc06'
'0fc87d'
Figure 5.
Low-shelf filter - magnitude response
Magnitude Response (dB)
5
0
-5
Magnitude (dB)
5.5
Examples
-10
-15
-20
-1
10
0
10
Frequency (kHz)
Doc ID 022240 Rev 1
1
10
19/46
Examples
5.6
AN3984
High-shelf filter
Input data:
●
Cutoff freq: 1 kHz
●
Gain: -10 dB
●
Coefficient range: 4
●
Slope: 2
●
Processing frequency: 96 kHz
Output data:
Filter coefficients
'Coeff 1: b1/2'
'Coeff 2: b2'
'Coeff 3: -a1/2'
'Coeff 4: -a2'
'Coeff 5: b0/2'
'f61151'
'09aea8'
'1f732a'
'e1063e'
'052110'
Figure 6.
High-shelf filter - magnitude response
Magnitude Response (dB)
5
Magnitude (dB)
0
-5
-10
-15
-1
10
20/46
0
10
Frequency (kHz)
Doc ID 022240 Rev 1
1
10
AN3984
Notch filter
Input data:
●
Cutoff freq: 1 kHz
●
Quality factor: 0.5
●
Coefficient range: 4
●
Processing frequency: 96 kHz
Output data:
Filter coefficients
'Coeff 1: b1/2'
'Coeff 2: b2'
'Coeff 3: -a1/2'
'Coeff 4: -a2'
'Coeff 5: b0/2'
'e2075a'
'1e091b'
'1df8a5'
'e3edc8'
'0f048d'
Figure 7.
Notch filter - magnitude response
Magnitude Response (dB)
0
-10
-20
Magnitude (dB)
5.7
Examples
-30
-40
-50
-60
-1
10
0
10
Frequency (kHz)
Doc ID 022240 Rev 1
1
10
21/46
Examples
5.8
AN3984
All-pass filter
Input data:
●
Cutoff freq: 1 kHz
●
Quality factor: 5
●
Coefficient range: 4
●
Processing frequency: 96 kHz
Output data:
Filter coefficients
'Coeff 1: b1/2'
'Coeff 2: b2'
'Coeff 3: -a1/2'
'Coeff 4: -a2'
'Coeff 5: b0/2'
'e046a7'
'200000'
'1fb958'
'e06a75'
'0fcac5'
Figure 8.
All-pass filter - phase response
Phase Response
0
-1
Phase (radians)
-2
-3
-4
-5
-6
-1
10
22/46
0
10
Frequency (kHz)
Doc ID 022240 Rev 1
1
10
AN3984
Band-pass filter
Input data:
●
Cutoff freq: 1 kHz
●
Gain: +6 dB
●
Quality factor: 3
●
Coefficient range: 4
●
Processing frequency: 96 kHz
Output data:
Filter coefficients
'Coeff 1: b1/2'
'Coeff 2: b2'
'Coeff 3: -a1/2'
'Coeff 4: -a2'
'Coeff 5: b0/2'
'000000'
'ff4fc0'
'1f9650'
'e0b0ab'
'00581f'
Figure 9.
Band-pass filter - magnitude response
Magnitude Response (dB)
10
5
0
-5
-10
Magnitude (dB)
5.9
Examples
-15
-20
-25
-30
-35
-40
-1
10
0
10
Frequency (kHz)
Doc ID 022240 Rev 1
1
10
23/46
Matlab code (functions)
Appendix A
A.1
AN3984
Matlab code (functions)
Code structure
Figure 10. Code structure
Start
Filter data
(cutoff freq, gain,…)
Coefficient
calculation
Coefficient limit
calculation
Float to Hex
Conversion
Coefficients and Filter
stability check
Display coefficient
End
24/46
Doc ID 022240 Rev 1
AN3984
A.2
Matlab code (functions)
Peak filter (PeakFilterAPW.m)
%-------------------------------------------------------------------------%
%
function [Coeff_Hex, CoeffAPW, LimitVal] = PeakFilterAPW(Fc, Gain, Q,
%
CoeffRange, Fs)
%
%
Args:Fc -> Cutoff Frequency
%
Gain -> Gain
%
Q -> Quality factor
%
CoeffRange -> Coefficient Range (1, 2 or 4)
%
Fs -> Sample frequency
%
Return: Coeff_Hex -> APW filter Coeff
%
CoeffAPW -> APW filter Coeff
%
LimitVal -> Limit coeff value
%
- Hex
- Floating Point
Description: Generates the APWorkbench coeff for a Peak Filter
%
%
STMicroelectronics - Agrate (ITALY)
%
MSH - Audio & Sound BU
%
Revision: 1.1
%
Date: 23 June 2011
%-------------------------------------------------------------------------%
%% Function code
function [Coeff_Hex, CoeffAPW, LimitValue] = PeakFilterAPW(Fc, Gain, Q, ...
CoeffRange, Fs)
format long
if (nargin <5)
Fs = 96000;
end
Teta = (2*pi*Fc)/Fs;
%Angle from frequency
K = tan(Teta/2);
W = K*K;
%% Process Gain
Gain = Gain* 0.115129254;
NormGain = exp(Gain);
Doc ID 022240 Rev 1
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Matlab code (functions)
AN3984
%% Coefficint Calculation
if NormGain<1
%
Negative NormGain - Cut
fCutValue = 1+(1/NormGain/Q)*K+W;
% Boost/NormGain
Coeff_4 = ((1+(1/Q)*K+W)/fCutValue)/2.0;
% b0/2
Coeff_0 = (W-1)/fCutValue;
% b1/2
Coeff_1 = (1-(1/Q)*K+W)/fCutValue;
% b2
Coeff_3 = ((1-(1/NormGain/Q)*K+W)/fCutValue)*-1.0;
% -a2
Coeff_2 = (Coeff_0)*-1.0;
% -a1/2
else
%
Positive NormGain - Boost
fBoostValue = 1+(1/Q)*K+W;
% Boost/NormGain
Coeff_4 = ((1+(NormGain/Q)*K+W)/fBoostValue)/2.0;
% b0/2
Coeff_0 = (W-1)/fBoostValue;
% b1/2
Coeff_1 = (1-(NormGain/Q)*K+W)/fBoostValue;
% b2
Coeff_3 = ((1-(1/Q)*K+W)/fBoostValue)*-1.0;
% -a2
Coeff_2 =(Coeff_0)*-1.0;
% -a1/2
end
%% Coefficient Matrix
CoeffAPW = [Coeff_0 Coeff_1 Coeff_2 Coeff_3 Coeff_4];
%% Coefficient Limit Value
LimitValue = LimitVal(CoeffRange);
%% Coefficient Matrix - Hex format
Coeff_Hex = myFloat2Hex(CoeffAPW, CoeffRange);
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Doc ID 022240 Rev 1
AN3984
A.3
Matlab code (functions)
Low-pass and high-pass filter (LHPassFilterAPW.m)
%-------------------------------------------------------------------------%
%
function [Coeff_Hex, CoeffAPW] = LHPassFilterAPW(CutOff_Freq, Q,
%
FType, Order, CoeffRange, Fc)
%
Args:Fc -> Cutoff Frequency
%
Q -> Quality factor
%
FType -> 0->LowPassFilter; 1->HighPass Filter
%
Order -> 1=1st order; 2=2nd order
%
CoeffRange -> Coefficient Range (1, 2 or 4)
%
Fc -> Sample frequency
%
Return: Coeff_Hex -> APW filter Coeff
%
CoeffAPW -> APW filter Coeff
%
LimitVal -> Limit coeff value
%
- Hex
- Floating point
Description: Generates the APWorkbench coeff for a LHPassFilter
%
%
STMicroelectronics - Agrate (ITALY)
%
MSH - Audio & Sound BU
%
Revision: 1.1
%
Date: 23 June 2011
%-------------------------------------------------------------------------%
%% Function code
function [Coeff_Hex, CoeffAPW, LimitValue] = LHPassFilterAPW(Fc, Q, ...
FType, Order, CoeffRange, Fs)
format long
if (nargin <6)
Fs = 96000;
end
Teta = (2*pi*Fc)/Fs;
%Angle from frequency
K = tan(Teta/2);
alpha = 1+K;
a2 = 0;
b2 = 0;
a0 = 1.0;
a1 = -(1-K)/alpha;
Doc ID 022240 Rev 1
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Matlab code (functions)
AN3984
if Order == 1 %1st Order Filter
if FType ==0 % Low Pass Filter
b0 = K/alpha;
b1 = b0;
Coeff_0 = (b1/2.0)/a0;
Coeff_1 = (b2)/a0;
% Always =0!!!
Coeff_2 = (-a1/2.0)/a0;
Coeff_3 = (-a2)/a0;
% Always =0!!!
Coeff_4 = (b0/2.0)/a0;
else
% High Pass Filter
b0 = 1/alpha;
b1 = -b0;
Coeff_0 = (b1/2.0)/a0;
Coeff_1 = (b2)/a0;
% Always =0!!!
Coeff_2 = (-a1/2.0)/a0;
Coeff_3 = (-a2)/a0;
% Always =0!!!
Coeff_4 = (b0/2.0)/a0;
end
else % 2nd Order Filter
Teta = (2*pi*Fc)/Fs;
%Angle from frequency
K = tan(Teta/2);
W = K*K;
DE = 1+(1/Q)*K+W;
Coeff_3 = ((1-(1/Q)*K+W)/DE)*-1.0; % -a2
Coeff_2 = ((W-1)/DE)*-1.0;
% -a1/2
if FType ==0 % Low Pass Filter 2nd Order
Coeff_4 = (W/DE)/2.0;
% b0/2
Coeff_0 = W/DE;
% b1/2
Coeff_1 = W/DE;
% b2
else
% High Pass Filter 2nd Order
28/46
Coeff_4 = (1/DE)/2.0;
% b0/2
Coeff_0 = -1/DE;
% b1/2
Doc ID 022240 Rev 1
AN3984
Matlab code (functions)
Coeff_1 = 1/DE;
% b2
end
end
%% Coefficient Matrix
CoeffAPW = [Coeff_0 Coeff_1 Coeff_2 Coeff_3 Coeff_4];
%% Coefficient Limit Value
LimitValue = LimitVal(CoeffRange);
%% Coefficient Matrix - Hex format
Coeff_Hex = myFloat2Hex(CoeffAPW, CoeffRange);
Doc ID 022240 Rev 1
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Matlab code (functions)
A.4
AN3984
Low and high-shelf filter (ShelfFilterAPW.m)
%-------------------------------------------------------------------------%
%
function [CoeffAPW] = ShelfFilterAPW(Fc, Gain, Slope, FType,
%
%
CoeffRange, Fs)
Args:Fc -> Cutoff Frequency
%
Gain -> Gain
%
Slope -> Slope
%
FType -> Filter type (Low or High Shelf)
%
CoeffRange -> Coefficient Range (1, 2 or 4)
%
Fs -> Sample frequency
%
Return: Coeff_Hex -> APW filter Coeff
%
CoeffAPW -> APW filter Coeff
%
LimitVal -> Limit coeff value
%
- Hex
- Floating Point
Description: Generates APWorkbench coeff for a Low or a High
%
Shelf Filter
%
%
STMicroelectronics - Agrate (ITALY)
%
MSH - Audio & Sound BU
%
Revision: 1.1
%
Date: 23 June 2011
%-------------------------------------------------------------------------%
%% Function code
function [Coeff_Hex, CoeffAPW, LimitValue] = ShelfFilterAPW(Fc, Gain, ...
Slope, FType,...
CoeffRange, Fs)
format long
%
if FType=0 =>LowShelf
%
if FType=1 =>HighShelf
if (nargin < 6)
Fs = 96000;
end
Teta = (2*pi*Fc)/Fs;
%Angle from frequency
SinTeta = sin(Teta);
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Doc ID 022240 Rev 1
AN3984
Matlab code (functions)
CosTeta = cos(Teta);
%
Normalized Gain
NormGain = 10^(Gain/40);
%
alpha and beta
alpha = (SinTeta/2)*sqrt((NormGain+(1/NormGain))*(1.0/Slope-1.0)+2.0);
beta = 2*sqrt(NormGain)*alpha;
%% Coefficient Calculation
if FType == 0
%
FType = 0 => LowShelf
b0 = NormGain*((NormGain+1)-(NormGain-1)*CosTeta + beta);
b1 = 2*NormGain*((NormGain-1)-(NormGain+1)*CosTeta);
b2 = NormGain*((NormGain+1)-(NormGain-1)*CosTeta - beta);
a0 = (NormGain+1)+(NormGain-1)*CosTeta + beta;
a1 = -2*((NormGain-1)+(NormGain+1)*CosTeta);
a2 = (NormGain+1)+(NormGain-1)*CosTeta-beta;
else
%
FType = 1 => HighShelf
b0 = NormGain*((NormGain+1)+(NormGain-1)*CosTeta + beta);
b1 = -2*NormGain*((NormGain-1)+(NormGain+1)*CosTeta);
b2 = NormGain*((NormGain+1)+(NormGain-1)*CosTeta - beta);
a0 = (NormGain+1)-(NormGain-1)*CosTeta + beta;
a1 = 2*((NormGain-1)-(NormGain+1)*CosTeta);
a2 = (NormGain+1)-(NormGain-1)*CosTeta-beta;
end
%
APW Coefficient - Reworked coefficient
Coeff_0 = (b1/2.0)/a0;
Coeff_1 = (b2)/a0;
Coeff_2 = (-a1/2.0)/a0;
Coeff_3 = (-a2)/a0;
Coeff_4 = (b0/2.0)/a0;
Doc ID 022240 Rev 1
31/46
Matlab code (functions)
AN3984
%% Coefficient Matrix
CoeffAPW = [Coeff_0 Coeff_1 Coeff_2 Coeff_3 Coeff_4];
%% Coefficient Limit Value
LimitValue = LimitVal(CoeffRange);
%% Coefficient Matrix - Hex format
Coeff_Hex = myFloat2Hex(CoeffAPW, CoeffRange);
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AN3984
A.5
Matlab code (functions)
Notch filter (NotchFilterAPW.m)
%-------------------------------------------------------------------------%
%
function [Filter_Coeff, CoeffAPW] = NotchFilterAPW(Fc, Q, CoeffRange,
%
Fs)
%
Args:Fc -> Cutoff Frequency
%
Gain -> Gain
%
Q -> Quality factor
%
CoeffRange -> Coefficient Range (1, 2 or 4)
%
Fs -> Sample frequency
%
Return: Coeff_Hex -> APW filter Coeff
%
CoeffAPW -> APW filter Coeff
%
LimitVal -> Limit coeff value
- Hex
- Floating Point
%
%
Description: Generates the APWorkbench coeff for a Notch Filter
%
%
STMicroelectronics - Agrate (ITALY)
%
MSH - Audio & Sound BU
%
Revision: 1.1
%
Date: 23 June 2011
%-------------------------------------------------------------------------%
%% Function code
function [Coeff_Hex, CoeffAPW, LimitValue] = NotchFilterAPW(Fc, Q, ...
CoeffRange, Fs)
format long
if (nargin == 3)
Fs = 96000;
end
Teta = (2*pi*Fc)/Fs;
%Angle from frequency
SinTeta = sin(Teta);
CosTeta = cos(Teta);
alpha = SinTeta/(2*Q);
%% Coefficint Calculation
b0 = 1;
b1 = -2*CosTeta;
Doc ID 022240 Rev 1
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Matlab code (functions)
AN3984
b2 = 1;
a0 = 1+alpha;
a1 = -2*CosTeta;
a2 = 1-alpha;
%
APW Coefficient - Reworked coefficient
Coeff_0 = (b1/2.0)/a0;
Coeff_1 = (b2)/a0;
Coeff_2 = (-a1/2.0)/a0;
Coeff_3 = (-a2)/a0;
Coeff_4 = (b0/2.0)/a0;
%% Coefficient Matrix
CoeffAPW = [Coeff_0 Coeff_1 Coeff_2 Coeff_3 Coeff_4];
%% Coefficient Limit Value
LimitValue = LimitVal(CoeffRange);
%% Coefficient Matrix - Hex format
Coeff_Hex = myFloat2Hex(CoeffAPW, CoeffRange);
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AN3984
A.6
Matlab code (functions)
All-pass filter (AllPassFilterAPW.m)
%-------------------------------------------------------------------------%
%
function [Coeff_Hex, CoeffAPW] = AllPassFilterAPW(Fc, Q, CoeffRange,
%
Fs)
%
Args:Fc -> Cutoff Frequency
%
Q -> Quality factor
%
CoeffRange -> Coefficient Range (1, 2 or 4)
%
Fs -> Sample frequency
%
Return: Coeff_Hex -> APW filter Coeff
%
CoeffAPW -> APW filter Coeff
%
LimitVal -> Limit coeff value
- Hex
- Floating Point
%
%
Description: Generates the APWorkbench coeff for a All Pass Filter
%
%
STMicroelectronics - Agrate (ITALY)
%
MSH - Audio & Sound BU
%
Revision: 1.1
%
Date: 23 June 2011
%-------------------------------------------------------------------------%
%% Function code
function [Coeff_Hex, CoeffAPW, LimitValue] = AllPassFilterAPW(Fc, Q, ...
CoeffRange, Fs)
format long
if (nargin < 4)
Fs = 96000;
end
Teta = (2*pi*Fc)/Fs;
%Angle from frequency
SinTeta = sin(Teta);
CosTeta = cos(Teta);
alpha = SinTeta/(2*Q);
%% Coefficint Calculation
b0 = 1-alpha;
b1 = -2*CosTeta;
b2 = 1+alpha;
Doc ID 022240 Rev 1
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Matlab code (functions)
AN3984
a0 = 1+alpha;
a1 = b1;
a2 = 1-alpha;
%
APW Coefficient - Reworked coefficient
Coeff_0 = (b1/2.0)/a0;
Coeff_1 = (b2)/a0;
Coeff_2 = (-a1/2.0)/a0;
Coeff_3 = (-a2)/a0;
Coeff_4 = (b0/2.0)/a0;
%% Coefficient Matrix
CoeffAPW = [Coeff_0 Coeff_1 Coeff_2 Coeff_3 Coeff_4];
%% Coefficient Limit Value
LimitValue = LimitVal(CoeffRange);
%% Coefficient Matrix - Hex format
Coeff_Hex = myFloat2Hex(CoeffAPW, CoeffRange);
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AN3984
A.7
Matlab code (functions)
Band-pass filter (BandPassFilterAPW.m)
%-------------------------------------------------------------------------%
%
function [Filter_Coeff, CoeffAPW] = BandPassFilterAPW(Fc, Q, CoeffRange,
%
%
Fs)
Args:Fc -> Cutoff Frequency
%
Gain -> Gain
%
Q -> Quality factor
%
CoeffRange -> Coefficient Range (1, 2 or 4)
%
Fs -> Sample frequency
%
Return: Coeff_Hex -> APW filter Coeff
%
CoeffAPW -> APW filter Coeff
%
LimitVal -> Limit coeff value
- Hex
- Floating Point
%
%
Description: Generates the APWorkbench coeff for a Band Pass Filter
%
%
STMicroelectronics - Agrate (ITALY)
%
MSH - Audio & Sound BU
%
Revision: 1.1
%
Date: 23 June 2011
%-------------------------------------------------------------------------%
%% Function code
function [Coeff_Hex, CoeffAPW, LimitValue] = BandPassFilterAPW(Fc, Gain,...
Q, CoeffRange, Fs)
format long
if (nargin == 3)
Fs = 96000;
end
Teta = (2*pi*Fc)/Fs;
%Angle from frequency
SinTeta = sin(Teta);
CosTeta = cos(Teta);
alpha = SinTeta/(2*Q);
NormGain = 10^(Gain/20);
%% Coefficint Calculation
Doc ID 022240 Rev 1
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Matlab code (functions)
AN3984
b0 = alpha*NormGain;
b1 = 0;
b2 = -b0;
a0 = 1+alpha;
a1 = -2*CosTeta;
a2 = 1-alpha;
%
APW Coefficient - Reworked coefficient
Coeff_0 = (b1/2.0)/a0;
Coeff_1 = (b2)/a0;
Coeff_2 = (-a1/2.0)/a0;
Coeff_3 = (-a2)/a0;
Coeff_4 = (b0/2.0)/a0;
%% Coefficient Matrix
CoeffAPW = [Coeff_0 Coeff_1 Coeff_2 Coeff_3 Coeff_4];
%% Coefficient Limit Value
LimitValue = LimitVal(CoeffRange);
%% Coefficient Matrix - Hex format
Coeff_Hex = myFloat2Hex(CoeffAPW, CoeffRange);
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AN3984
A.8
Matlab code (functions)
Float to hex conversion (myFloat2Hex.m)
%-------------------------------------------------------------------------%
%
function [floatN] = myFloat2Hex(hexN, range)
%
Args:hexN -> hexadecimal number to be converted in string format
%
without the 0x, i.e. 0x123456 => '123456' (24 bits
%
only)
%
%
range -> coefficients range 4, 2, 1
Return: floatN -> floating point notation number
%
%
Description: converts a fixed point hexadecimal number into a
%
floating point one
%
%
STMicroelectronics - Agrate (ITALY)
%
MSH - Audio & Sound BU
%
Revision: 1.1
%
Date: 23 June 2011
%-------------------------------------------------------------------------%
function [hexN] = myFloat2Hex(floatN, range)
format long
quantizerSetup.mode = 'fixed';
% quantizerSetup.roundmode = 'nearest';
quantizerSetup.roundmode = 'ceil';
quantizerSetup.overflowmode = 'saturate';
%Quantizer to translate from hex to num
if(range == 1)
quantizerquantizerSetup.format = ([24 23]);
elseif(range == 2)
quantizerquantizerSetup.format = ([24 22]);
elseif(range == 4);
quantizerquantizerSetup.format = ([24 21]);
end
q = quantizer(quantizerquantizerSetup);
Doc ID 022240 Rev 1
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Matlab code (functions)
AN3984
hexN = num2hex(q,floatN);
A.9
Max coefficient limit value calculator (LimitVal.m)
%-------------------------------------------------------------------------%
%
function [LimitValue] = LimitVal(CoeffRange)
%
%
Args:CoeffRange -> APW filter Coeff Range
%
%
Return: LimitValue -> APW filter limit value
%
%
Description: From the CoeffRange it calculates the LimitValue
%
%
STMicroelectronics - Agrate (ITALY)
%
MSH - Audio & Sound BU
%
Revision: 1.1
%
Date: 23 June 2011
%-------------------------------------------------------------------------%
%% Function code
function [LimitValue] = LimitVal(CoeffRange)
format long
switch CoeffRange
case 1
% Coefficient +/- 1
LimitValue = 0.99999;
case 2
% Coefficient +/- 2
LimitValue = 1.99999;
case 4
% Coefficient +/- 4
LimitValue = 3.99999;
end
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AN3984
A.10
Matlab code (functions)
Display coefficient and error messages (Display_Coeff.m)
%-------------------------------------------------------------------------%
%
function []=Display_Coeff(Filter_Coeff, CoeffAPW, LimitValue)
%
Args:Filter_Coeff -> APW filter Coeff
%
CoeffAPW -> APW filter Coeff
%
LimitVal -> Limit coeff value
%
Return: Display -> APW filter Coeff
- Hex
- Floating
- Hex
%
%
Description: Display Filter Coefficient (HEX)
%
%
STMicroelectronics - Agrate (ITALY)
%
MSH - Audio & Sound BU
%
Revision: 1.1
%
Date: 23 june 2011
%-------------------------------------------------------------------------%
function
[]=Display_Coeff(Filter_Coeff, CoeffAPW, LimitValue)
a0 = 1;
a1 =-2*CoeffAPW(3);
a2 = -CoeffAPW(4);
b0 =2*CoeffAPW(5);
b1 =2*CoeffAPW(1);
b2 = CoeffAPW(2);
Coeff = [b0 b1 b2 a0 a1 a2];
%% Check for stability and Limit
Error = 0;
if ((abs(b0)>=LimitValue) || (abs(b1)>=LimitValue)|| (abs(b2)>=LimitValue))
Error = 1;
end
if abs(a2)>1 && (abs(a1)>1+a2)
Error = 2;
end
Doc ID 022240 Rev 1
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Matlab code (functions)
AN3984
counter = 1;
while counter<=5
if imag(CoeffAPW(counter))~=0
Error = 3;
end
counter = counter+1;
end
% Filter coefficiners or Error message.
switch Error
case 0
% No error
h=fvtool(Coeff(1:3),Coeff(4:6));
disp('
');
disp('
');
disp('******************************************************************************
*****************');
disp('
Filter Coefficients');
disp('******************************************************************************
*****************');
Label = {'Coeff 1: b1/2', 'Coeff 2: b2', 'Coeff 3: -a1/2', 'Coeff 4: -a2',
'Coeff 5: b0/2'};
TABLE_data = {Filter_Coeff(1,:)
Filter_Coeff(2,:) Filter_Coeff(3,:) ...
Filter_Coeff(4,:) Filter_Coeff(5,:)};
% TABLE_data = num2cell(TABLE_data);
TABLE = [Label; TABLE_data];
disp (TABLE);
case 1
% The coefficient range must be changed
disp('
');
disp('
');
disp('******************************************************************************
*****************');
disp('
disp('
Error!!!');
The coefficient range must be increased');
disp('******************************************************************************
*****************');
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Matlab code (functions)
case 2
% The filter is not stable
disp('
');
disp('
');
disp('******************************************************************************
*****************');
disp('
Error!!! The Filter is not stable!');
disp('
Please check the filter parameters');
disp('******************************************************************************
*****************');
case 3
% A CoeffAPW coeff is not real
disp('
');
disp('
');
disp('******************************************************************************
*****************');
disp('
disp('
Error!!!');
Please check the filter parameters');
disp('******************************************************************************
*****************');
end
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Abbreviations and acronyms
Appendix B
AN3984
Abbreviations and acronyms
The abbreviations and acronyms used throughout this application note are defined as
follows:
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●
fc: cutoff frequency
●
fs: sampling frequency
●
Q: filter quality factor
●
G: gain
●
LPF: low-pass filter
●
HPF: high-pass filter
●
LSF: low-shelf filter
●
HSF: high-shelf filter
●
APF: all-pass filter
●
BPF: band-pass filter
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6
Revision history
Revision history
Table 1.
Document revision history
Date
Revision
26-Sep-2011
1
Changes
Initial release.
Doc ID 022240 Rev 1
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AN3984
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