Butterworth Filters

There exists a basic type of low-pass filter, called a Butterworth Filter, which is a 2nd-order filter, which therefore has a falloff-rate of -12db /Octave, far above the corner frequency, and this is its general diagram:

basic_1

Even though it is clear from this diagram that the two capacitors, or the two resistors, are allowed to have different values, the way the design of this filter is mainly taught today, both resistors are made equal, as are both capacitors, thus simplifying the computation of each, once the other has been determined according to what seems practical, applying the same principle as what would be applied for a 1st-order filter.

One basic weakness of this filter, especially in modern applications, is the fact that it will attenuate frequency-components considerably, which are below its corner-frequency. There have historically been two approaches taken to reduce this effect, if any attempt has been made to do so at all:

  1. C1 can be given twice the value of C2, but R1 and R2 kept equal. This poses the question of whether the corner-frequency will still be correct. And my estimation is that because of the way Electrical Engineers have defined the corner-frequency, the specific frequency-response at that frequency should remain the square root of 1/2 (or, -3db). But, if C1 is larger than C2, then the frequency-response will not be the same at any other point in the curve. I.e., the curve could be flatter, with response-values closer to unity, at frequencies considerably lower than the corner-frequency.
  2. The operational amplifier stage, which in the basic design is just a voltage-follower, can be transformed into a gain-stage, with a gain slightly higher than one. This is done by placing a voltage-divider from the output of an operational amplifier, to yield the feedback voltage, fed to its inverting input. What needs to be stressed here, is that significantly high gain leads to an unstable circuit.

While either approach can be taken, it is important not to apply both at the same time, as the amount of feedback given by C1 would be exaggerated, and would lead to a hot-spot somewhere in the pass-band of this filter. In general, the trend today would be to use approach (2).

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About The Applicability of Over-Sampling Theory

One fact which I have described in my blog, is that when Audio Engineers set the sampling rate at 44.1kHz, they were taking into account a maximum perceptible frequency of 20kHz, but that if the signal was converted from analog to digital format, or the other way around, directly at that sampling rate, they would obtain strong aliasing as their main feature. And so a concept which once existed was called ‘over-sampling’, in which then, the sample-rate was quadrupled, and by now, could simply be doubled, so that all the analog filters still have to be able to do, is suppress a frequency which is twice as high, as the frequencies which they need to pass.

The interpolation of the added samples, exists digitally as a low-pass filter, the highest-quality variety of which would be a sinc-filter.

All of this fun and wonderful technology has a main weakness. It actually needs to be incorporated into the devices, in order to have any bearing on them. That MP3-player, which you just bought at the dollar-store? It has no sinc-filter. And therefore, whatever a sinc-filter would have done, gets lost on the consumer.

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Modern Consumer Sound Appreciation

Over recent months, I have been racking my brain, trying to answer questions I have, about how sound that was compressed in the frequency-domain, may or may not be able to preserve phase-information. This does not mean that I, personally, can hear phase-information, nor that specific MP3 Files I have been listening to, would even be good examples of how well modern MP3s compress sound. I suspect that in order to stay in business, the developers of MP3 have in fact been improving their codec, so that when played back correctly, the quality of MP3s will stay in line with more-recent formats that exist, such as OGG Vorbis…

But I think that people under-appreciate my intellectual point of view.

For many months and years, I had my doubts, that MP3 Files can in fact encode ± 180⁰ phase-shifts, i.e. a stereo-difference channel that has the correct polarity with respect to the stereo-sum channel, over a range of frequencies. What my own musings have only taught me in recent days, is that in fact, MP3 is capable of ± 180⁰ phase-separation.

Further, similar types of compression should be capable of better phase-separation than that, If their bit-rates are set high enough, that not too many of their frequency-coefficients get chopped down – according to what I have reasoned out today.

What I also know, is that the sound-formats AC3 and AAC have as an explicit feature, to store surround-sound. MPEG-2 Video Files more-or-less require the use of the AC3 codec for sound, and MP4 Files absolutely require the use of the AAC codec. And, stored in its compressed format, the surround-effect only requires ± 180⁰ phase-accuracy.

This subject is orthogonal to debate which exists, about whether it is of benefit to human listeners, to have sound reproduced at very high sample-rates, or at great bit-depths. Furthermore, I do not fully know what good a very high sample-rate – such as “192kHz” – is supposed to do any listener, if his sound has been MP3-compressed. As far as I am concerned, ultra-high sample-rates have to do with lossless compression, or no compression, which also happen to produce the same file-sizes at that signal-format.

What I did was just check, in what format iTunes downloads music by default. And it downloads its music in AAC Format. All this does for me, is corroborate a claim a friend of mine made, that he can hear his music with full positioning, since that is also the main feature of AAC, and not of MP3.

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A Note on Sample-Rate Conversion Filters

One type of (low-pass) filter which I had learned about some time ago, is a Sinc Filter. And by now, I have forgiven the audio industry, for placing the cutoff frequencies of various sinc filters, directly equal to a relevant Nyquist Frequency. Apparently, it does not bother them that a sinc filter will pass the cutoff frequency itself, at an amplitude of 1/2, and that therefore a sampled audio stream can result, with signal energy directly at its Nyquist Frequency.

There are more details about sinc filters to know, that are relevant to the Digital Audio Workstation named ‘QTractor‘, as well as to other DAWs. Apparently, if we want to resample an audio stream from 44.1 kHz to 48 kHz, in theory this corresponds to a “Rational” filter of 147:160, which means that if our Low-Pass Filter is supposed to be a sinc filter, it would need to have 160 * (n) coefficients in order to work ideally.

But, since no audio experts are usually serious about devising such a filter, what they will try next in such a case, is just to oversample the original stream by some reasonable factor, such as by a factor of 4 or 8, then to apply the sinc filter to this sample-rate, and after that to achieve a down-sampling, by just picking samples out, the sample-numbers of which have been rounded down. This is also referred to as an “Arbitrary Sample-Rate Conversion”.

Because 1 oversampled interval then corresponds to only 1/4 or 1/8 the real sampling interval of the source, the artifacts can be reduced in this way. Yet, this use of a sinc filter is known to produce some loss of accuracy, due to the oversampling, which sets a limit in quality.

Now, I have read that a type of filter also exists, which is called a “Farrow Filter”. But personally, I know nothing about Farrow Filters.

As an alternative to cherry-picking samples in rounded-down positions, it is possible to perform a polynomial smoothing of the oversampled stream (after applying a sinc filter if set to the highest quality), and then to ‘pick’ points along the (now continuous) polynomial that correspond to the output sampling rate. This can be simplified into a system of linear equations, where the exponents of the input-stream positions conversely become the constants, multipliers of which reflect the input stream. At some computational penalty, it should be possible to reduce output artifacts greatly.

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