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filterfunctions.md

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Building higher order filters with Biquads

The standard filters in CamillaDSP are not of a specific type, like Butterworth. Instead they are generic, with an adjustable q-value. To make the normal filters, one or several generic filters have to be used together.

Bessel

Making a Bessel filter with a set of Biquads requires creating several Biquads, each with a unique Q and cut-off frequency.

Multiplication factor for frequency:

Order Biquad 1 Biquad 2 Biquad 3 Biquad 4
1 1.0*
2 1.27201964951
3 1.32267579991* 1.44761713315
4 1.60335751622 1.43017155999
5 1.50231627145* 1.75537777664 1.5563471223
6 1.9047076123 1.68916826762 1.60391912877
7 1.68436817927* 2.04949090027 1.82241747886 1.71635604487
8 2.18872623053 1.95319575902 1.8320926012 1.77846591177

The asterisk (*) indicates that this is a 1st order filter.

Q values:

Order Biquad 1 Biquad 2 Biquad 3 Biquad 4
1 (1st order)
2 0.57735026919
3 (1st order) 0.691046625825
4 0.805538281842 0.521934581669
5 (1st order) 0.916477373948 0.563535620851
6 1.02331395383 0.611194546878 0.510317824749
7 (1st order) 1.12625754198 0.660821389297 0.5323556979
8 1.22566942541 0.710852074442 0.559609164796 0.505991069397

Example Bessel filter

Let's make a 5th order Lowpass at 1 kHz. Loking at the tables we see that we need three filters. The first should be a 1st order while the second and third are 2nd order.

  • First filter, type LowpassFO:
    • freq = 1kHz * 1.50231627145 = 1502Hz
    • (no q-value)
  • Second filter, type Lowpass:
    • freq = 1kHz * 1.75537777664 = 1755Hz
    • q = 0.916477373948
  • Third filter, type Lowpass:
    • freq = 1kHz * 1.5563471223 = 1556Hz
    • q = 0.563535620851

Butterworth and Linkwitz-Riley

For an Nth order Butterworth you will have N/2 biquad sections if N is even, and ((N+1)/2 if N is odd. For odd filters one of the Biquads will be a first order filter. Each filter will have the same resonant frequency f0 and the second order filters will have Q according to this formula:

Q = 1/( 2*sin((pi/N)*(n + 1/2)) )

where 0 <= n < (N-1)/2

Table for q-values

Butterworth and Linkwitz-Riley filtes can easily be built with Biquads. The following table lists the most common ones. High- and lowpass use the same parameters.

Type Order Biquad 1 Biquad 2 Biquad 3 Biquad 4
Butterworth 2 0.71
4 0.54 1.31
8 0.51 0.6 0.9 2.56
Linkwitz-Riley 2 0.5
4 0.71 0.71
8 0.54 1.31 0.54 1.31

Note that a 4th order LR iconsists of two 2nd order Butterworth filters, and that an 8th order LR consists of two 4:th order Butterworth filters.