Is Bessel filter linear phase?
Thread Summary
The user inquired about the method used to generate digital Bessel filters in the General 2nd Order block, noting that the bilinear transform can distort phase linearity. The final answer confirmed that Bessel, Butterworth, Chebyshev, and Linkwitz-Riley filters in SigmaStudio are based on the Bilinear Transform, which can introduce phase response errors at higher cutoff frequencies. The accompanying answer mentioned that SigmaStudio's other filter types are implemented using Robert Bristow-Johnson's Audio EQ Cookbook, which is also based on the bilinear transform.AI Generated Content
In the General 2nd Order block, you can select a Bessel LPF. Is this digital Bessel filter generated using bilinear transform or some other transform? As I understand, the bilinear transform does not produce a true linear-phase filter:
Note that the digital Bessel filter is designed using BLT. That introduces a phase response error due to the frequency warping. In the other words, the linear phase will be considerably distorted if Fc is higher then Fs/4 approximately. You should use impulse invariant approach or other phase preserving technique to design Bessel filter for high frequencies.
http://freeverb3.sourceforge.net/iir_filter.shtml
The only advantage of a Bessel design over a Butterworth or Chebyshev is that the phase response is nearly linear throughout the passband, i.e. the group delay is almost constant throughout the passband. The ``warping'' inherent in the bilinear transform method upsets this linearity.
http://www-users.cs.york.ac.uk/~fisher/mkfilter/mzt.html
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SigmaStudio's LPF, HPF, BPF, notch, APF, peaking, low shelf, and high shelf were implemented based on Robert Bristow-Johnson's famous Audio EQ Cookbook, which was based on the bilinear transform.
However, I'm not sure where we derived our Butterworth, Bessel, Chebyshev, and Linkwitz-Riley implementations. I will check with the software team and see if they have some insight.
I confirmed that the Bessel/Butterworth/Chebyshev/Linkwitz-Riley filters are also based on the Bilinear Transform, so the phase linearity of the analog prototype is not preserved after the transformation due to frequency warping.
By the way, you can easily check the magnitude and phase responses of filters in SigmaStudio using the built-in Stimulus and Probe tools in the System toolbox.
Here's a screenshot of a Bessel LPF with a 1 kHz cutoff.
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