Post by Yogi B on Mar 8, 2023 14:08:37 GMT -5
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But this is pretty much only academic, as such a thing would only be possible via a pretty complex active circuit.
Mixing an unaffected signal with an all-passed version of itself results in either an LPF or HPF response (depending on the flavour of all-pass, respectively: whether the low frequencies are in phase & high frequencies out of phase, or vice-versa). Hopefully, it should be fairly obvious why this is the case (hint: think about the op-amp version of an all-pass filter as a differential amplifier). These are somewhat similar to what are achieved by the more familiar (capacitive) partial coil-split and SHOoP (series-half-out-of-phase) wirings, albeit providing 6dB/octave roll off in the stopband. (As opposed to the passive counterparts that are limited to a −6dB shelving filter, since the signal of the bypassed coil is only reduced to zero rather than inverted.) While it is true the exact equivalence to simple filters would only be true if the output from both pickups were identical, I still think this comparison is appropriate.
The variability of the in-phase/out-of-phase crossover frequency may be an interesting feature to have, but that is all the IBP offers: a (pair of) fixed phase gradient(s) swept to cover differing frequency ranges (despite Little Labs' labelling of the main control knob in degrees). This makes sense for the IBP as it is trying to correlate two signals that have a time delay (i.e. phase difference directly proportional to frequency), and an approximation of this phase difference over a limited frequency range can be had via an all-pass filter. And — although summing with a time delayed signal produces a comb filter with infinitely many 'teeth' (notches) for an ever-increasing frequency range, whereas summing with a 2nd order all-passed signal produces only a single notch — it seems that, such a region of linear-ish phase response is still enough to adequately cover the main sounding range of a particular instrument.
Rather, what we want, or at least what I think we want, is a frequency-independent phase difference that can be adjusted in magnitude. Thinking of phase versus frequency graphically: a vertically shifting flat horizontal line, as opposed to the horizontally shifting arctangent curve of the IBP. That's not to say that all-pass filters won't be useful in achieving this, we just need more of them!
A comparison between of the phase response of a variable first-order all-pass filter and the sought phase shift.
Thanks to how the addition of phasors works, a weighted sum of two signals in quadrature is all that is needed to produce any intermediary phase shift (which can be further extended to encompass a full 360° range by inverting one or both input signals). This reduces the problem from one of finding a method to produce an arbitrary phase shift to one of producing only a 90° shift. So, given that between the input and output of a first-order all-pass filter there's exactly one frequency at which a phase difference of precisely 90° occurs, one might reasonably intuit that it is possible with an n-order all-pass to achieve a 90° shift at a total of n frequencies. This is true but comes with a few caveats:
The simplest topology (and that used by phasers) of simply stacking all n filters, results in the 90° shifts of the input being of odd multiples (90°, 270°, 450°...), i.e. the absolute phase difference is 90°, yet the sign alternates (e.g. 270 = -90 (mod 360)). Additionally, for any other frequency besides those, the phase difference will be some intermediate amount, i.e., about as far away as possible from the objective of a constant 90° shift.
One way it would be theoretically possible to get close to a constant phase shift over a specific bandwidth would be to offset the pole frequencies of each filter and invert the phase response of every other filter — producing a phase gradient alternating in sign and thus produces a sequence of maxima & minima that can be positioned so as to be close to the sought phase shift. The problem here is that the second part (an inverted all-pass filter, one with positive phase gradient) is, as far as I'm aware, impossible to achieve — at least in real-time.
A workaround for this issue is to realize that we only actually care about a relative phase shift. Thus, rather than a single modified signal requiring negative & positive phase gradients paired with an unmodified signal, we can instead utilize two modified signals: one containing just the negative gradients; the other containing the would-be positive gradients, but as a standard non-inverted negative gradient. This achieves the same phase difference as the prior scenario, but possesses the definite advantage of being realizable as real world a circuit. The possible downside is that all output would be subject to an additional constant phase shift.
Besides offending phase absolutists, this presents a further hurdle in the way of the end objective: the ability to mix the phase shifted output with the output from a different pickup. For this two potential solutions are: an additional duplicated half of the phase shift circuit, purely to match the constant phase offset; alternatively (and perhaps more elegantly, assuming I have the logic correct) pre-mix the two pickup signals as both a sum and a difference, feeding each to a separate half of the phase shift circuit. This latter option is what I have presented below as a block diagram and, for the time being, is primarily what I am seeking feedback on.
