If a software-equalizer possesses GUI controls that correspond to approximate octaves, or repeated 1-2-5 sequences, it is entirely likely to be implemented as a set of bandpass filters acting in parallel. However, the simplistic bandpass filters I was contemplating, would also have required that the signal be multiplied by a factor of 4, to achieve unit gain where their low-pass and high-pass cutoff frequencies join, as I described in this posting.
Actually, the parameters which define each digital filter, are non-trivial to compute, but nevertheless computable when the translation into the digital domain has been carried out correctly. And so a type of equalizer can be achieved, based on derived bandpass-filters, on the basis that each bandpass-filter has been tuned correctly.
If the filters cross over at their -6db point, then one octave lower or higher, one filter will reach its -3db point, while the other will reach its -12db point. So instead of -12db, this combination would yield -15db.
The fact that the signal which has wandered into one adjacent band is at -3db with respect to the center of that band, does not lead to a simple summation, because there is also a phase-shift between the frequency-components that wander across.
I suppose that the user should be aware, that in such a case, the gain of the adjacent bands has not dropped to zero, at the peak of the current band, so that perhaps the signal will simplify, if the corner-frequencies have been corrected. This way, a continuous curve will result from discrete settings.
Now, if the intention is to design a digital bandpass filter with greater than 6 db /Octave falloff curves, the simplistic approach would be just to put two of the previous stages in series – into a pipeline resulting in second-order filters.
Also, the only way then to preserve the accuracy of the input samples, is to convert them into floating-point format first, for use in processing, after which they can be exported to a practical audio-format again. )
(Edit 03/25/2017 :
The way simplistic high-pass filters work, they phase-shift the signal close to +90⁰ far down along the part of the frequency-response-curve, which represents their roll-off. And simplistic low-pass filters will phase-shift the signal close to -90⁰ under corresponding conditions.
OTOH, Either type of filter is supposed to phase-shift their signal ±45⁰, at their -3db point.
What this means is that if the output from several band-pass filters is taken in parallel – i.e. summed – then the center-frequency of one band will be along the roll-off part of the curve of each adjacent band, which combined with the -3db point from either its high-pass or its low-pass component. But then if the output of this one central band is set to zero, the output from the adjacent bands will be 90⁰ apart from each other. )
(Edit 03/29/2017 :
A further conclusion of this analysis would seem to be, that even to implement an equalizer with 1 slider /Octave properly, requires that each bandpass-filter be a second-order filter instead. That way, when the signals wander across to the center-frequency of the slider for the next octave, they will be at -6db relative to the output of that slider, and 180⁰ phase-shifted with respect to each other. Then, setting the center slider to its minimum position will cause the adjacent ones to form a working Notch Filter, and will thus allow any one band to be adjusted arbitrarily low.
And, halfway between the slider-center-frequencies, the gain of each will again be -3db, resulting in a phase-shift of ±90 with respect to the other one, and achieving flat frequency-response, when all sliders are in the same position.
The problem becomes, that if a 20-band equalizer is attempted, because the 1 /Octave example already required second-order bandpass-filters, the higher one will require 4th-order filters by same token, which would be a headache to program… )
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