ST AN3984 APPLICATION NOTE

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	Low-pass and high-pass filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	8

4.1.1 Low-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.1.2 High-pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2 Peak filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

		4.2.1	Peak filter - negative gain (cut) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	10
		4.2.2	Peak filter - positive gain (boost) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	11
	4.3	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
5	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

2/46	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

Doc ID 022240 Rev 1

3/46

List of figures	AN3984

List of figures

Figure 1. 1st-order low-pass filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Figure 2. 1st-order high-pass filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Figure 3. 2nd-order low-pass filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Figure 4. 2nd-order high-pass filter - magnitude response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Figure 5. Low-shelf filter - magnitude response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Figure 6. High-shelf filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Figure 7. Notch filter - magnitude response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Figure 8. All-pass filter - phase response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Figure 9. Band-pass filter - magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Figure 10. Code structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4/46	Doc ID 022240 Rev 1

AN3984	Overview

1 Overview

The transfer function for a first-order filter in the digital z-domain is:

Equation 1

H(z) = b0 + b1 z−1 a0 + a1 z−1

For a second-order filter (a biquad) the transfer function is:

Equation 2

H (z) = b0 + b1 z −1 + b2 z−2 a0 + a1 z −1 + a2 z− 2

This equation can be modified normalizing the a0 coefficient; the new equation is:

Equation 3

H (z) = (b0 a0 )+ (b1 a0 ) z −1 + (b2 a0 ) z −2 1 + (a1 a0 ) z−1 + (a2 a0 ) z −2

The most straightforward implementation form using Equation 3 is:

Equation 4

y[n] = (b0 a0 ) x[n] + (b1a0 ) x[n −1] + (b2 a0 ) x[n − 2] − (a1a0 ) y[n −1] − (a2 a0 ) y[n− 2]

Doc ID 022240 Rev 1

5/46

Filter stability	AN3984

2 Filter stability

2.1Definition

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.

2.2First-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

2.3Second-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

a 2 < 1

a1 < (1+ a 2 )

6/46	Doc ID 022240 Rev 1

AN3984	First-order filter design (LPF and HPF)

3 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 fc fs

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	(1 − K )
	a1	= −
			α

The numerator for an LPF can be calculated as follows:

Equation 9

K b0 = α

b1 = Kα

The numerator for an HPF can be calculated as follows:

Equation 10

1 b0 = α

b1 = − α1

The coefficient used in APWorkbench can be calculated by applying these formulas:

Equation 11

Coefficient b1 2 = (b1 2)

a0

Coefficient b2 = (b2 )

a0

Coefficient a1 2 = (− a1 2)

a0

Coefficient a2 = −aa2

0

Coefficient b0 2 = (b0 2)

a0

Doc ID 022240 Rev 1

7/46

Second-order filter design	AN3984

4 Second-order filter design

4.1Low-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

	ϑ = 2 π		fc
	c		fs
	K = tan(ωc
				2)
	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		(W − 1)
	a =		2
	1					DE
				1 −		K + W
	a2	=				Q
						DE

4.1.1Low-pass filter

The numerator coefficient for a second-order LPF can be calculated as follows:

Equation 14

W b0 = DE

W b1 = 2 DE

W b2 = DE

8/46	Doc ID 022240 Rev 1

AN3984	Second-order filter design

For a second-order LPF, the coefficients given in APWorkbench can be calculated as follows:

Equation 15

Coefficient	b1	2	=	W
				DE
Coefficient b = W
	2		DE
Coefficient	a1	2	= −1 W − 1
					DE
				1 −	K	+ W
Coefficient a2 = −					Q
					DE
Coefficient	b0	2	=	1	W
				2	DE

4.1.2High-pass filter

The numerator coefficient for a second-order HPF can be calculated as follows:

Equation 16

b0 = DE1

b1 = −2 DEW

1 b2 = DE

For a second-order HPF, the coefficients given in APWorkbench can be calculated as follows:

Equation 17

Coefficient	b1	2	= −		1
			1	DE
Coefficient b =
	2		DE
Coefficient	a1	2	= −1 W − 1
					DE
			1 −		K	+ W
Coefficient a2 = −					Q
					DE
Coefficient	b0	2	= 1		1
			2		DE

Doc ID 022240 Rev 1

9/46

Second-order filter design	AN3984

4.2Peak 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.

4.2.1Peak 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

(W − 1)

a

= 4

1

CutValue

Q

1

−

+ W

Gain

a

2

=

CutValue

+

K

1

+ W

Q

b0

=

CutValue

b

= 2

(W − 1)

1

CutValue

(1

−

K

+ W )

b2

=

Q

CutValue

10/46

Doc ID 022240 Rev 1

AN3984	Second-order filter design

The coefficients in the APWorkbench are consequently calculated as follows:

Equation 21

Coefficient

b1

2

=

W −1

CutValue

(1−

K

+W )

Coefficient b2 =

Q

CutValue

Coefficient

a1

2

=

1− W

CutValue

1 −

Q

K + W

Gain

Coefficient a2 = −

CutValue

1 +

K

+ W

Coefficient

b0

2

=

1

Q

2

CutValue

4.2.2Peak filter - positive gain (boost)

The boost value is calculated with the following equation:

Equation 22

BoostValue = 1+ KQ + W

The filter coefficient can be calculated as follows:

Equation 23

a0

= 1

(1+ K

Gain + W )

a1

=

Q

BoostValue

K

1 −

+ W

Q

a2

=

BoostValue

(1+ K Gain

+ W )

b0

= 2

Q

BoostValue

b1

= 2

W − 1

BoostValue

(1 − K

Gain + W )

b2

=

Q

BoostValue

Doc ID 022240 Rev 1

11/46

Second-order filter design	AN3984

The coefficients in the APWorkbench are consequently calculated as follows:

Equation 24

Coefficient	b1	2	=		W − 1
				BoostValue
			(1 −		Gain					K + W )
Coefficient b2 =							Q
				BoostValue
Coefficient	a1	2	=		1 − W
				BoostValue
				1 −			K	+ W
Coefficient a2 = −							Q
				BoostValue
					1 +				Gain		K + W
	b0			1						Q
Coefficient		2	=
				2				BoostValue

4.3Shelf filters

The coefficient gain is defined in Equation 25.

Equation 25

Gain = 10(GaindB 40)

The coefficients α and β are calculated as follows:

Equation 26

	sin(ϑ )				1				1
α =	c
			Gain +
	2				Gain			S
β = 2 α		Gain

− 1	+ 2

4.3.1Low-shelf filter

The coefficients for an LSF can be calculated as follows:

Equation 27

a0	= (Gain + 1)+ (Gain − 1) cos ϑc + β
a1 = −2 (Gain − 1)+ (Gain + 1) cos ϑc
a2	= (Gain + 1)+ (Gain − 1) cos ϑc − β
b0	= Gain ((Gain + 1)− (Gain − 1) cos ϑc + β )
b1 = 2 Gain ((Gain − 1)− (Gain + 1) cos ϑc )
b2	= Gain ((Gain + 1)− (Gain − 1) cos ϑc − β )

The coefficient to load in APWorkbench can be calculated by applying the calculation already shown in Equation 11.

12/46

Doc ID 022240 Rev 1

AN3984	Second-order filter design

4.3.2High-shelf filter

The coefficients for an HSF can be calculated as follows:

Equation 28

a0	= (Gain + 1)− (Gain − 1) cos ϑc + β
a1 = 2 (Gain − 1)− (Gain + 1) cos ϑc
a2	= (Gain + 1)− (Gain − 1) cos ϑc − β
b0	= Gain ((Gain + 1)+ (Gain − 1) cos ϑc + β )
b1 = −2 Gain ((Gain − 1)− (Gain + 1) cos ϑc )
b2	= Gain ((Gain + 1)+ (Gain − 1) cos ϑc − β )

The coefficient to load in APWorkbench to program a HSF can be computed by applying the formulas shown in Equation 11.

4.4Notch 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 ϑc 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

Doc ID 022240 Rev 1

13/46

Second-order filter design	AN3984

4.5All-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.

4.6Band-pass filter

Equation 29 allows calculating the constant α while Equation 32 is used to calculate the normalized gain.

Equation 32

GaindB

NormGain =10 20

The coefficients for a BPF can be calculated as follows(b):

Equation 33

a0 = 1 + α

a1 = −2 cos ϑc a2 = 1 − α

b0 = α NormGain b1 = 0

b2 = −b0 = −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.

14/46

Doc ID 022240 Rev 1