ST AN2874 APPLICATION NOTE

AN2874

Applications note

BQD filter design equations

1 Introduction

A generalized set of equations is formulated for the design of first-order and second-order low-pass and high-pass filters. A specialized set of equations is then given for the design of parametric biquad EQ filters.

The parameters governing the characteristics of the filter are:

!filter cut-off frequency, fC (-3 dB corner frequency)

!sampling rate (fS).

!Q factor

!boost/cut gain value at fC.

These parameters can be used to determine the coefficients of the filter transfer function.

The transfer function for a first-order filter in the digital z-domain can be written as: H(z) = (b0 + b1.z-1) / (a0 + a1.z-1).

And for a second-order filter as:

H(z) = (b0 + b1.z-1 + b2.z-2) / (a0 + a1.z-1+ a2.z-2).

This document can be used in conjunction with all the ST digital audio devices having EAQ filters onboard. These include STA309A, STA320, STA321, STA33xBW and several others.

February 2009

Rev 1

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Low-pass and high-pass filter design	AN2874

2 Low-pass and high-pass filter design

First-order filter design

As a first step to obtain the coefficients for the 1st-order low-pass or high-pass filter equation on page 1, the following equations can be used:

θC = 2 π fC / fS (= the normalized cut-off frequency) K = tan(θC / 2)

α = 1 + K.

The denominator coefficients are identical for both low-pass and high-pass filters designed for the same cut-off frequency and are computed as follows:

a0 = 1

a1 = -(1 - K) / α.

The numerator coefficients for a low-pass filter can be calculated as follows:

b0 = K / α b1 = K / α.

The numerator coefficients for a high-pass filter can be calculated as follows:

b0 = 1 / α b1 = -1 / α.

Second-order filter design

As a first step to obtain the coefficients for the 2nd-order low-pass or high-pass filter equation on page 1, the following equations can be used:

θC = 2 π fC / fS K = tan(θC / 2)

W = K2

α = 1 + K / Q + W.

The denominator coefficients are the same for both low-pass and high-pass filters designed for the same cut-off frequency and are computed as follows:

a0 = 1

a1 = 2 * (W - 1) / α

a2 = (1 - K / Q + W) / α.

The numerator coefficients for a low-pass filter can be calculated as follows: b0 = W / α

b1 = 2 * W / α b2 = b0.

The numerator coefficients for a high-pass filter can be calculated as follow: b0 = 1 / α

b1 = -2 / α b2 = b0.

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