A concept that exists in radio-communications, which is derived from amplitude-modulation, and which is further derived from balanced modulation, is single-sideband modulation. And even back in the 1970s, this concept existed. Its earliest implementations required that a low-frequency signal be passed to a balanced modulator, which in turn would have the effect of producing an upper sideband (the USB) as well as an inverted lower sideband (the LSB), but zero carrier-energy. Next, the brute-force approach to achieving SSB entailed, using a radio-frequency filter to separate either the USB or the LSB.
The mere encumbrance of such high-frequency filters, especially if this method is to be used at RF frequencies higher than the frequencies, of the old ‘CB Radio’ sets, sent Engineers looking for a better approach to obtaining SSB modulation and demodulation.
And one approach that existed since the onset of SSB, was actually to operate two balanced modulators, in a scheme where one balanced modulator would modulate the original LF signal. The second balanced modulator would be fed an LF signal which had been phase-delayed 90⁰, as well as a carrier, which had either been given a +90⁰ or a -90⁰ phase-shift, with respect to whatever the first balanced modulator was being fed.
The concept that was being exploited here, is that in the USB, where the frequencies add, the phase-shifts also add, while in the LSB, where the frequencies subtract, the phase-shifts also subtract. Thus, when the outputs of the two modulators were mixed, one side-band would be in-phase, while the other would be 180⁰ out-of-phase. If the carrier had been given a +90⁰ phase-shift, then the LSB would end up 180⁰ out-of-phase – and cancel, while if the carrier had been given a -90⁰ phase-shift, the USB would end up 180⁰ out-of-phase – and cancel.
This idea hinges on one ability: To phase-shift an audio-frequency signal, spanning several octaves, so that a uniform phase-shift results, but also so that the amplitude of the derived signal be consistent over the required frequency-band. The audio signal could be filtered to reduce the number of octaves that need to be phase-shifted, but then it would need to be filtered to achieve a constrained frequency-range, before being used twice.
And so a question can arise, as to how this was achieved historically, given analog filters.
My best guess would be, that a stage which was used, involved a high-pass and a low-pass filter that acted in parallel, and which would have the same corner-frequency, the outputs of which were subtracted – with the high-pass filter negative, for -90⁰ . At the corner-frequency, the phase-shifts would have been +/- 45⁰. This stage would achieve approximately uniform amplitude-response, as well as achieving its ideal phase-shift of -90⁰ at the one center-frequency. However, this would also imply that the stage reaches -180⁰ (full inversion) at higher frequencies, because there, the high-pass component that takes over, is still being subtracted !
( … ? … )
What can in fact be done, is that a multi-band signal can be fed to a bank of 2nd-order band-pass filters, spaced 1 octave apart. The fact that the original signal can be reconstructed from their output, derives partially from the fact that at one center-frequency, an attenuated version is also passed through one-filter-up, with a phase-shift of +90⁰ , and a matching attenuated version of that signal also passed through one-filter-down, with a phase-shift of -90⁰. This means that the two vestigial signals that pass through the adjacent filters are at +/- 180⁰ with respect to each other, and cancel out, at the present center-frequency.
If the output from each band-pass filter was phase-shifted, this would need to take place in a way not frequency-dependent. And so it might seem to make sense to put an integrator at the output of each bp-filter, the time-constant of which is to achieve unit gain, that the center-frequency of that band. But what I also know, is that doing so will deform the actual frequency-response of the amplitudes, coming from the one band. What I do not know, is whether this blends well with the other bands.
If this was even to produce a semi-uniform -45⁰ shift, then the next thing to do, would be to subtract the original input-signal from the combined output.
(Edit 11/30/2017 :
It’s important to note, that the type of filter I’m contemplating does not fully achieve a phase-shift of +/- 90⁰ , at +/- 1 octave. This is just a simplification which I use to help me understand filters. According to my most recent calculation, this type only achieves a phase-shift of +/- 74⁰ , when the signal is +/- 1 octave from its center-frequency. )
Now, my main thought recently has been, if and how this problem could be solved digitally. The application could still exist, that many SSB signals are to be packed into some very high, microwave frequency-band, and that the type of filter which will not work, would be a filter that separates one audible-frequency sideband, out of the range of such high frequencies.
And as my earlier posting might suggest, the main problem I’d see, is that the discretized versions of the low-pass and high-pass filters that are available to digital technology in real-time, become unpredictable both in their frequency-response, and in their phase-shifts, close to the Nyquist Frequency. And hypothetically, the only solution that I could see to that problem would be, that the audio-frequency band would need to be oversampled first, at least 2x, so that the discretized filters become well-behaved enough, to be used in such a context. Then, the corner-frequencies of each, will actually be at 1/2 Nyquist Frequency and lower, where their behavior will start to become acceptable.
The reality of modern technology could well be such, that the need for this technique no longer exists. For example, a Quadrature Mirror Filter could be used instead, to achieve a number of side-bands that is a power of two, the sense with which each side-band would either be inverted or not inverted could be made arbitrary, and instead of achieving 2^n sub-bands at once, the QMF could just as easily be optimized, to target one specific sub-band at a time.
Again, the fact needs to be pointed out, that phase-shifting for the purposes of musical authoring, differs from phase-shifting for the purposes of signal processing, in that for musical effects, the phase-shifter does not need to have Mathematically pure properties. In fact, if it did, it might become useless for musical authoring. Yet, I’d look at the concept of how the “Phaser” works, which belongs to the DAW software named “Audacity”:
An interesting observation I can make, is that according to their documentation, the “Depth” parameter defines what the highest frequency can become, at which the phase-delay will be at its maximum. Presumably, it defines what the corner-frequency is, of each filter-pair. But note how their authors did not assign any units in Hertz to this parameter! I think it goes from (0) to (255), just like the Dry / Wet parameter does.
I take this to mean, that if we compute (ω) as (2πF/h) , and (k) as (1/(ω+1)) , we will obtain more-or-less accurate results, until (ω) reaches (1.0) . After that, to increase (ω) any further achieves no useful results. And so the Audacity programmers may have decided, not even to work with (F), but only to work with exact values of (ω) , in the range where those values work, and to let the corner-frequencies fall where they will, given the additional reality that an unknown sampling rate (h) is set.
It may be true that a digital signal can represent a 90⁰ phase-shift at 1/2 the Nyquist Frequency. But there is little interest on the part of Audacity programmers, to implement those parameters. Theirs only reach until 1/3 Nyquist Frequency.