A Word Of Compliment To Audacity

One of the open-source applications which can be used as a Sound-Editor, is named ‘Audacity’. And in an earlier posting, I had written that this application may apply certain effects, which first involve performing a Fourier Transform of some sort on sampling-windows, which then manipulate the frequency-coefficients, and which then invert the Fourier Transform, to result in time-domain sound samples again.

On closer inspection of Audacity, I’ve recently come to realize that its programmers have avoided going that route, as often as possible. They may have designed effects which sound more natural as a result, but which follow how traditional analog methods used to process sound.

In some places, this has actually led to criticism of Audacity, let’s say because the users have discovered, that a low-pass or a high-pass filter would not maintain phase-constancy. But in traditional audio work, low-pass or high-pass filters always used to introduce phase-shifts. Audacity simply brings this into the digital realm.

I just seem to be remembering certain other sound editors, that used the Fourier Transforms extensively.

Dirk

 

A Practical Application, that calls for A Uniform Phase-Shift: SSB Modulation

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.

Continue reading A Practical Application, that calls for A Uniform Phase-Shift: SSB Modulation

About +90⁰ Phase-Shifting

I have run into people, who believe that a signal cannot be phase-advanced in real-time, only phase-delayed. And as far as I can tell, this idea stems from the misconception, that in order for a signal to be given a phase-advance, some form of prediction would be needed. The fact that this is not true can best be visualized, when we take an analog signal, and derive another signal from it, which would be the short-term derivative of the first signal. ( :1 ) Because the derivative would be most-positive at points in its waveform where the input had the most-positive slope, and zero where the input was at its peak, we would already have derived a sine-wave for example, that will be phase-advanced 90⁰ with respect to an input sine-wave.

90-deg-phase-y

But the main reason this is not done, is the fact that a short-term derivative also acts as a high-pass filter, which progressively doubles in output amplitude, for every octave of frequencies.

What can be done in the analog domain however, is that a signal can be phase-delayed 90⁰, and the frequency-response kept uniform, and then simply inverted. The phase-diagram of each of the signal’s frequency-components will then show, the entire signal has been phase-advanced 90⁰.

90-deg-phase

(Updated 11/29/2017 : )

Continue reading About +90⁰ Phase-Shifting

Quantum Mechanics is Falsifiable.

One concept which exists in Science, is that certain theories are Falsifiable. This means that a given hypothesis will predict some sort of experimental outcome, which other theories would not predict, and then an experiment can be performed to test whether this outcome is according to the theory. If it is not, then this test will break the theory, and will thus falsify it.

Quantum Mechanics is often Falsifiable. If the reader thinks it is not, then maybe the reader is confusing Quantum Mechanics with String Theory, which is supposedly not falsifiable? And thinking that String Theory is just the same thing as Quantum-Mechanics, is a bit like thinking that Cosmology is just the same thing as Astronomy.

(Edit 02/03/2018 :

There is an aspect to a theory being Falsifiable, which I did not spell out above, assuming that the reader could infer it. But certain conversations I’ve had with people I personally know, suggest that those people do not understand this concept.

The result of a physical experiment can easily be, that the outcome is according to the theory. Just as much as the inverse situation would falsify the theory, such an outcome can eventually confirm the theory, and without confirming the theory, there is no real way in which Scientists can know, whether a new theory is in fact valid.

There is no specific imperative to prove a theory wrong, in the theory being Falsifiable. )

(Edit 02/15/2018 :

One aspect to how this posting should be read, which some readers might infer, but which other readers might not infer, is that it begins by stating a hypothesis. At first, I declared this hypothesis as distinct from several other theoretical explanations of light.

But it would break the flow of a blog-posting, if every paragraph which I wrote after that, began with a redeclaration, stating that the truth of the paragraph depends on the initial hypothesis.

This dependency should be assumed, and belongs to my intended meaning. )

According to Quantum Mechanics, light can be polarized, just as it can according to the classical, wave-based theory of light. Only, because according to Quantum-Mechanics light is driven by particles – by photons – its explanation of polarization is quite different from polarized light, according to the classical, electrodynamic explanation.

According to wave-based light, plane-polarized light is the primary phenomenon, and circular-polarized light is secondary. Circular-polarized light would follow, when waves of light are polarized in two planes at right-angles to each other, but when these waves also have a 90⁰ phase-shift.

(Edit 02/20/2018: A Hypothesis which I’ve just disproved, but which this whole posting’s validity depends on.) According to Quantum-Mechanics, the photon is in itself a circular-polarized quantum of light, of which there can trivially be left- and right-handed examples. According to Quantum-Mechanics, plane-polarized light forms, when left- and right-handed photons pair up, so that their electrostatic components form constructive interference in one plane, while canceling at right-angles to that plane.

From a thermodynamic point of view, there is little reason to doubt that photons could do this, since the particles which make up matter are always agitated, and since the photons in an original light-source also have some random basis. So a conventional plane-polarizing filter, of the kind that we used to attach to our film-cameras, would not be so hard to explain. It would just need to phase-shift the present left-handed photons in one way, while phase-shifting the present right-handed ones oppositely, until they line up.

But there exists one area in which the predictions of Quantum-Mechanics do not match those of classical wave-mechanics. If we are given a digital camera that accepts lens-attachments, we will want to attach circular polarizing filters, instead of plane-polarizing filters. And the classical explanation of what a circular polarizer does, is first to act as a plane-polarizer, which thereby selects a plane of polarization which we want our camera to be sensitive to, but the output of which is next circularly-polarized, so that light reaches the autofocus mechanism of the camera, which is still not plane-polarized. Apparently, fully plane-polarized light will cause the autofocus to fail.

This behavior of a polarizer is easily explained according to Quantum-Mechanics. The plane-polarized light which is at first admitted by our filter, already possesses left- and right-handed photons. After that, we could visualize sorting out the photons that are circular-polarized in the wrong direction.

But the opposite behavior of a filter would not be predicted by Quantum-Mechanics. According to that, if we first pass randomly-polarized light through a circular polarizer, and if we then pass the resulting beam into a plane-polarizer, we should not be able to obtain plane-polarized output from the last polarizer.

According to the classical explanation of light, this should still be an easy thing to do. Our circularly-polarized light is supposed to have two components at right-angles, and our plane-polarizer should only allow vibration in one plane. But according to Quantum-Mechanics, if the incident beam is already circularly-polarized, it should only consist of either left-handed or right-handed photons, and then a simple filter should not be able to conjure photons that are not present in the original beam. And so our circularly-polarized light should not be convertible into plane-polarized light.

Continue reading Quantum Mechanics is Falsifiable.