 # 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:

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:

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.