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

An Observation about the Daubechies Wavelet and PQF

In an earlier posting, I had written about what a wonderful thing Quadrature Mirror Filter was, and that it is better to apply the Daubechies Wavelet than the older Haar Wavelet. But the question remains less obvious, as to how the process can be reversed.

The concept was clear, that an input stream in the Time-Domain could first be passed through a low-pass filter, and then sub-sampled at (1/2) its original sampling rate. Simultaneously, the same stream can be passed through the corresponding band-pass filter, and then sub-sampled again, so that only frequencies above half the Nyquist Frequency are sub-sampled, thereby reversing them to below the new Nyquist Frequency.

A first approximation for how to reverse this might be, to duplicate each sample of the lower sub-band once, before super-sampling them, and to invert each sample of the upper side-band once, after expressing it positively, but we would not want playback-quality to drop to that of a Haar wavelet again ! And so we would apply the same wavelets to recombine the sub-bands. There is a detail to that which I left out.

We might want to multiply each sample of each sub-band by its entire wavelet, but only once for every second output-sample. And then one concern we might have could be, that the output-amplitude might not be constant. I suspect that one of the constraints which each of these wavelets satisfies would be, that their output-amplitude will actually be constant, if they are applied once per second output-sample.

Now, in the case of ‘Polyphase Quadrature Filter’, Engineers reduced the amount of computational effort, by not applying a band-pass filter, but only the low-pass filter. When encoding, the low sub-band is produced as before, but the high sub-band is simply produced as the difference between every second input-sample, and the result that was obtained when applying the low-pass filter. The question about this which is not obvious, is ‘How does one recombine that?’

And the best answer I can think of would be, to apply the low-pass wavelet to the low sub-band, and then to supply the sample from the high sub-band for two operations:

  1. The first sample from the output of the low-pass wavelet, plus the input sample.
  2. The second sample from the output of the low-pass wavelet, minus the same input sample, from the high sub-band.

Continue reading An Observation about the Daubechies Wavelet and PQF

An Update about MP3-Compressed Sound

In many of my earlier postings, I stated what happens in MP3-compressed sound somewhat inaccurately. One reason is the fact that an overview requires that information be combined from numerous sources. While earlier WiKiPedia articles tended to be quite incomplete on this subject, it happens that more-recent WiKi-coverage has become quite complete, yet still requires that users click deeper and deeper, into subjects such as the Type 4 Discrete Cosine Transform, the Modified Discrete Cosine Transform, and Polyphase Quadrature Filters.

What seems to happen with MP3 compression, which is also known as MPEG-2, Layer 3, is that the Discrete Cosine Transform is not applied to the audio directly, but that rather, the audio stream is divided down to 32 sub-bands in fact, and that the MDCT is applied to each sub-band individually.

Actually, after the coefficients are computed, a specific filter is applied to them, to reduce the aliasing that happened, just because of the PQF Filter-bank.

I cannot be sure that this was always how MP3 was implemented, because if we take into account the fact that with PQF, every second sub-band is frequency-inverted, we may be able to obtain equivalent results just by performing the Discrete Cosine Transform which is needed, directly on the audio. But apparently, there is some advantage in subdividing the spectrum into its 32 sub-bands first.

One advantage could be, that doing so reduces the amount of computation required. Another advantage could be the reduction of round-off errors. Computing many smaller Fourier Transforms has generally accomplished both.

Also, if the spectrum is first subdivided in this way, it becomes easier to extract the parameters from each sub-band, that will determine how best to quantize its coefficients, or to cull ones either deemed to be inaudible, or aliased artifacts.

Continue reading An Update about MP3-Compressed Sound

An Elaboration on Quadrature Mirror Filter

This was an earlier posting of mine, in which I wrote about a “Quadrature Mirror Filter”. But the above posting may not make it clear to all readers, why a QMF approach will actually result in two streams, each of which has half the sample-rate of the original stream.

A basic premise which gets used, is the Daubechies Wavelet, according to which there exists a Scaling Function that later gets named ‘H1′, and a corresponding Wavelet which gets named ‘H0′. It could also be thought that H1 is a low-pass filter with a corner frequency of 1/2 the Nyquist Frequency, while H0 is a Band-Pass Filter derived from H1. Also, because the upper cutoff frequency of H0 is the Nyquist Frequency, it is not clear to me either, why we would not just call that a High-Pass Filter. But the WiKi page calls that the Band-Pass Filter.

Alright, So we can start with a stream sampled at 44.1 kHz and derive two output streams, one which contains the lower half of frequencies, and the other of which contains the upper half. How do the sample-rates of either get halved?

The answer is that after we have filtered the original stream both ways, we pick out every second sample of each.

This is also what would get done if we were to use a (more expensive) Half-Band Filter based on ‘the Sinc Function’, to down-sample a stream. In contrast, if we are over-sampling a stream to the highest level of accuracy, we first repeat each sample once, and then apply the (better) low-pass filter.  (It should be noted however, that a 4-coefficient Daubechies Wavelet would be considered ‘deficient’. Those start to become interesting, at maybe 8 coefficients.)

But when it comes to Quadrature Mirror Filters, when we have down-sampled the stream, we have also halved its Nyquist Frequency – both times. But then in the case of ‘H0′ above, original frequency components above the Nyquist Frequency are subject to the phenomenon I mentioned in another posting, according to which they get mirrored back down, from the new, lower Nyquist Frequency, all the way to zero (DC). Hence, the output of H0 gets inverted in frequencies, when it is subsequently down-sampled.

Dirk

Continue reading An Elaboration on Quadrature Mirror Filter