We decided to write a script that will check online filters for digital noise. One method can be implemented using the following algorithm:

• calculate filter output using single precision
• calculate filter output using double precision and assume that it is precise
• find digital noise at the output of the filter when single precision is used
• extrapolate the result to the double precision filter dividing by 2^D-S ~ 10⁷, D - number of bits used in double precision mantissa, S - in single precision

Restriction: Single precision filter internal variables must be checked for overflows.

I applied this method to filtering a 1 Hz sine wave with a notch filter. Precise output should also be a 1 Hz wave => at other frequencies we see noise => digital noise spectrum should coincide with filter output. The plot shows the method worked out for this example.

Using this method I estimated digital noise of butter("LowPass", 2, 0.001) applied to white noise. Sampling frequency was 16 kHz.