An interesting experience. The choice of FIR or IIR is the most primary. As the filtering is modelling classic filters, the shorter coefficient varieties of IIR are the best choice for me. The fact of an infinite impulse response is not of concern with a continuous stream of data, and coefficient rounding is not really an issue when using doubles. IIR also has the advantage of an easy Sallen-Key implementation, due to the subtraction and re-adding of the feedback component, with a very simple CR processing.
The most interesting choices are to do with the anti-alias filtering, as the interpolation filter, on up-sampling is an easy choice. As the ear is not really responsive to phase, all the effort should be on the pass band response levels, and a good stop band non response. A Legendre or Butterworth are the candidates. The concept of a characteristic sound enters the design process at this point, as the cascading of SK filter sections is conceptually useful to improve the -6 dB response at cut off. This is a trade off of 20 kHz to 22.05 kHz in the alias pass band, and greater attenuation in the above 22.05 kHz infinite stop desire. The slight greater desire of alias attenuation above pass band maximal flatness (for audio harmony) implies the Legendre filter is better for the purpose than Butterworth.
In the end, the final choice is one of convenience. and a 9th order filter was decided upon, with 4 times oversampling. The use of 4 times oversampling instead of 8 times oversampling increases the alias by an octave reduction. This fact under the assumption of at least a linear reduction in the amplitude of the frequency of the generator of an alias frequency, with frequency increase, just requires a -12 dB extra gain reduction in the alias filter for an effective equivalence to 8 times oversampling (the up to and the reflection back down to 6 + 6). The amount of GHz processing also halves. These facts then become constructive in the design, with the bulk alias close to the cut off, and the minor reflected alias-alias limit, not being too relevant to overall alias inharmonic distortion.
A triple chain of 3 pole Legendre filter sections is the decided design. The approximate -9 dB at the corner, allows for slightly shifting up the cut off and still maintaining a very effective stop band. Code reuse also aids in the I-cache usage for CPU effective use. A single 3 pole Legendre is the interpolation up sample filter. The roll off for not using Butterworth does cut some high frequency content from the maximally flat, hence the concept of maximally flat, but it out performs a Bessel filter in this regard. It’s not as though a phasor or flanger needs to operate almost perfectly in the alias band.
Perhaps there is improvement to be made in the up sampling filter, by post up sample 88.2 kHz noise shaped injection to eliminate all error at 44.1 kHz. This may have a potential advantage to map the alias noise into the low frequencies, instead of encroaching from the higher frequencies to the lower, and for creating the alias as a reduction in signal to noise, instead of at certain inharmonic peaks. The main issue with this is the 44.1 kHz wave fundamental, seen as the amplitude ring modulation of the injected phase noise, by the 44.1 kHz stepped waveform between samples input. The 88.2 kHz “carrier” and the sidebands are higher in frequency, and of the same amplitude magnitude.
But as this is following for no 44.1 kHz error, the 88.2 kHz and sidebands are the induced noise, the magnitude of which is of the order of 1 octave up from the -3 dB roll at the corner, plus approximately the octave for a 3 pole filter, or about 36 dB cut of a signal 3/4 of the input amplitude. I’d estimate about -37 dB at 88.1 kHz, and -19 dB at 44.1 kHz. Post processing with a 9 pole filter, provides an extra -54 dB on down sampling, for an estimate of around -73 dB or greater on the noise. That would be about 12 bit resolution at 44.1 kHz increasing with frequency. All estimates, likely errors, but in general not a good idea from first principals. Given that the 44.1 kHz content would be very small though post the interpolation filter, -73 dB down from this would be good, although I don’t think achievable in a sensible manor.
Using the last filtered sample in as the reference for the present sample filtered in as a base line, the signal at 22.05 kHz would be smoothed. It would have a notch filter effect, by injecting quantization offset ringing noise at 88.2 kHz to cancel 22.05 kHz. The notch would likely extend down in frequency for maybe -6 dB at about 11 kHz. Perhaps in the end it is just better to subtract the multiplied difference between two up sample filters using different sinc spreading of a 1000 and a 1100 sample occupancy zero inter fill. Subtracting the alternates up conversion delta as it were.
There is potentially also an argument for having a second order section with damping factor near 0.68 and corner 22.05 kHz to achieve some normalisation from sinc up-sampling. This adds in an amount of Q such as to peak the filter cancelling the sinc droop, which would be about 3% at 4 times oversampling.
EDIT: Some of you may have noticed that the required frequencies for stable filtering are too high at 4 times oversampling. So unfortunate for the CPU load an 8 times oversample has to be used. The sinc error is less than 1% at this oversample, but still corrected in a similar way, and a benefit of 2 extra poles. Following this by a 0.1 dB 3 pole Chebyshev high pass which has been inverted, gives a reasonable 5 pole up sampling filter. The down sampling filter for code efficiency is a triple instance of the sample inverse Chebyshev, with the corner frequencies slightly offset to produce more individual zeros, and some spreading of the “ringing”. These 9 poles are enough to get the stop band ripple to be lower than a 16 bit resolution. Odd order inverse Chebyshev are essential for the reflected spectra to be continually decreasing in amplitude.
