Phase Value

We already know that a digital waveform can be transformed, using a Fourier Transform, into a different representation where each data point represents a certain particular frequency, and the magnitude of the transform at that data point represents the amount of that frequency that is present in the original signal.

This is interesting, because we humans are able to perceive both of these aspects of a sound’s frequency content.  If the frequency itself changes - increases or decreases - we perceive the pitch to go up or down.  And if the magnitude changes - increases or decreases - we perceive the volume to get louder or quieter.  Between them, these two things would appear to totally define how we perceive (or, if you prefer, “hear”) audio signals.  Interestingly enough, a physical analysis of how the human hearing system actually works suggests that it is those separate individual frequencies, rather than the waveform itself in its full complexity, that our ears respond to.

If we take all the frequencies in the Fourier Transform and create a sine wave from each one, whose magnitude is the magnitude of the Fourier Transform, and add them all together, the sum total of all these sine waves will be the exact original waveform.  But there are a couple of wrinkles to bear in mind.  The first is that this is only strictly true if the original waveform used to create the Fourier Transform was of infinite duration, producing a Fourier Transform with an infinite number of frequencies.  For the purposes of this post we can safely ignore that limitation.  The second is that we need to know the relative phase of each frequency component.

I wrote in a previous post how we can decompose a square wave into its constituent frequency components and use those to reconstruct the square wave.  However, if we change the phase of these individual frequency components - which describes how the individual sine waves “line up” against each other - then we end up changing the shape of the original square wave.  Indeed, the change can be rather dramatic.  In other words, changing the phases of a waveform’s component frequencies can significantly alter the waveform’s shape without changing any of its component frequencies or their magnitudes.  To a first approximation, changes in the phase response of an audio system are considered not to be audible.  However, at the bleeding edge where audiophiles live that is not so clear.

The Fourier Transform I mentioned in fact encodes both the magnitude and the phase information because the transformation actually produces complex numbers (numbers having two components which we term Real and Imaginary).  We can massage these two components to yield both the phase and the magnitude.  This is one example of how the phase and frequency responses of an audio system are tightly intertwined.

We are used to demanding that anything which affects an audio system has a frequency response that meets our objectives.  This applies equally in the analog domain (whether we apply it to circuits such as amplifiers or components such as transistors) as in the digital domain (where we can apply it to simple filters or elaborate MP3 encoders).  We are familiar with the common requirement for flat frequency response across the audio bandwidth because we know that we can “hear” these frequencies clearly.  But all of those systems, analog and digital, also have an associated phase response.

Some types of phase response are quite trivial.  For example, if the phase response is linear, which means that the phase is linear with frequency, this means simply that the signal has been delayed by a fixed amount of time.  More generally if we look at the phase response plot (phase vs frequency), the slope of the line at any frequency tells us how much that frequency is delayed by.  Clearly, if the slope is linear, all frequencies will be delayed by the same amount, and the effect will be a fixed delay applied to the entire signal.  However, if the slope is anything other than linear, it means that different delays apply to each frequency and the result will be a degree of waveform distortion as discussed regarding the square wave.

So, we have clear ideas about errors in the magnitude of the frequency response.  We classify these as dips, humps, roll-offs, etc, in the frequency response, and we have expectations as to how we expect these defects to sound, plus a reasonably well-cultivated language with which to describe those sounds.  But we are still trying to develop an equivalent understanding of phase responses.

One development I don’t like is to focus on the impulse response, and to ascribe features of the impulse response to corresponding qualities in the output waveform.  So, for example, pre-ringing in the impulse response is imagined to give rise to “pre-ringing” in the output waveform, which is presumed to be a BAD THING.  This loses sight of a simple truth.  If you mathematically analyze a pure perfect square wave and remove all of its components above a certain frequency, what you get is pre-ringing before each step, and post-ringing after it.  We’re not talking about a filter here, we’re talking about what the waveform inherently looks like if its high frequency components were absent, which they need to be if we are going to encode it digitally.

You might argue that a perfect phase response would be a zero-phase response, where there is no phase error whatsoever at each and every frequency.  Such characteristics cannot be achieved at all in the analog domain, but in the digital domain there are various ways of accomplishing it.  However, it can be shown mathematically that all zero-phase filters must have a symmetrical impulse response.  In other words, whatever post-ring your filter has, it will have the exact same pre-ring before the impulse.  This, by the way, is another way of describing what happened to the pure perfect square wave.

Another impulse response characteristic that gets a lot of favourable press is the Minimum Phase filter.  This is a misleading title because, although it does mathematically minimize the net phase error, it lacks a theoretical basis upon which to suppose a monotonic relationship exists between the accumulated net phase error and an observed deterioration in the sound quality.  For example, linear phase filters exhibiting no waveform distortion can in principle have significant different fixed delays, with corresponding significant differences in their net phase error, yet with no difference whatsoever in the fidelity of their output signals.  On the other hand, Minimum Phase filters do concentrate the filter’s “energy” as much as possible into the “early” part of its impulse response, which can mean that it is more mathematically “efficient”, which may make for either a better-designed filter, or a more accurate implementation of the filter’s design (sorry for the “air quotes”, but this is a topic that could take up a whole post of its own).

One thing I must be clear on is that this discussion is purely a technical one.  I discuss the technical properties of phase and impulse responses, but I don’t hold up a hand and claim that one thing is better than the other.  Someone may state an opinion that such-and-such a filter sounds better than so-and-so’s filter because it has a “better” impulse response.  I might agree or disagree with the opinion regarding which filter sounds best, but I will argue against attributing the finding to certain properties of the impulse response without a good model to account for why the properties advocated should be beneficial.  As regards the impulse responses no such “good” model yet exists (that I know of).

Where I do stand from a philosophical standpoint is that I like zero-phase responses and linear phase responses because these contribute no waveform distortion at the output.  For that reason, we are, here at BitPerfect, developing a zero-phase DSP engine that, if successful, we will be able to apply quite broadly.  We will try it out first in our DSD Master DSD-to-PCM conversion engine, where I am convinced that it will provide PCM conversions that are, finally, indistinguishable from the DSD originals.  If listening tests prove us out, we will release it.  From there it will migrate to SRC, where I believe it will deliver an SRC solution superior to the industry-leading Izotope product (which is too expensive for us to use cost-effectively).  Finally, it will appear in our new design for a seriously good graphical equalizer package that is in early-stage development, with possible application to room-correction technology.

Audio Files for Audiophiles

A few years back I purchased a Windows App called dBpoweramp.  It met my needs for a while.  Upon installation, I learned that the App supports a huge number of different music file formats.  Today, that list reads:  AIFF, ALAC, CDA, FLAC, MP3, WAV, AC3, AAC, SND, DFF, DSF, MID, APE, MPP, OGG, OPUS, WVC, W64, WMA, OFR, RA, SHN, SPX, TTA, plus a number of variants.  Who knew there were so many audio formats?  I for one have never heard of most of these.  Counting through them, I have only ever used eight of ’em, and of the rest I have only ever come across three.  Well, good for dBpoweramp!  I can sleep comfortably knowing that if I ever want to convert a TTA file to OFR I probably have just the the tool for the job.  Today I use iTunes, and it only supports a handful of those file formats.  What's the story with that, I wonder?

Music file formats arise to fill a need, and each and every one of those file formats I mentioned represents a need which went unmet at the time the format was devised.  Actually, I even invented an audio file format of my own, way back in 1979.  In my lab at work I had a Commodore Pet computer which was attached to an X-Y graphic printer.  I used the Pet to control a laser test apparatus and had the printer output the results graphically.  As the printer’s two stepper motors (one for each axis) drove the pen holder across the paper, the tone of each motor would sound a certain note.  By having the printer draw out a certain pattern I could get it to play “God Save the Queen”.  Not very imaginative, I agree, but it was quite a party trick in its day.  I then wrote a program that would allow you to compose a tune which you could then play on the printer.  Finally, I devised a simple format with which to store those instructions in a file which the Commodore Pet saved on its audio-cassette tape drive.  I could conceivably claim to have developed one of the world’s first audio file formats!  Looking back, the Zeitgeist was quite delicious - a computer audio file stored in digital form on an analog audio cassette tape.

But back to the myriad file formats supported by dBpoweramp.  Each one has a purpose, and I suppose not all of those involve the distribution of music for commercial or recreational purposes.  For what it’s worth, the developers of iTunes could have arranged for it to support all of these weird and wonderful file formats too, but they didn’t.  In some cases there are good technical reasons why they would elect not to support a particular file type.  In others it is a matter of choice.  Some of those formats are Audio-Video formats, and iTunes is, after all, a multi-media platform.  But for the purposes of this post I am going to constrain the discussion to audio-only playback.

Not just the developers of iTunes, but every developer who writes an audio playback App has to decide for themselves which of those (and, perhaps others too) file formats their App is going to support.  I am going to break these formats down into four camps - Uncompressed, Lossless Compressed, Lossy Compressed, and DSD.  Lets look at each one, and discuss how they handle the audio data.

The simplest audio file formats contain raw uncompressed audio data.  The actual audio data itself is written straight into the file.  It is not manipulated or massaged in any way.  The advantage of doing it this way is that the audio data can be both written and read with the minimum of fuss.  The two most commonly used examples of this type of file format are AIFF (released by Apple in 1987) and WAV (released by Microsoft in 1991).  iTunes will happily load either file type.

Back in those days the file size of a AIFF or WAV file was utterly prohibitive.  A five-minute track ripped from a CD would be require a file size of 53MB which represented something like three times the capacity of a good-sized hard disk drive at that time.  Clearly, if computers were going to be able to handle digital audio something needed to be done to reduce the file size.  To address this problem, during the early 1990’s the Fraunhofer Institute in Germany developed what we now call the MP3 file format.  What this does is, effectively, to figure out which parts of an audio signal are the least audible and throw them away.  By throwing away more and more of the audio signal the file size can be reduce rather dramatically.  This approach is referred to as Lossy Compression, because it compresses the file size but loses data (and therefore sound quality) along the way.

The first MP3 codec was released in 1995.  In 1997 Apple introduced their own version of MP3 called AAC.  Structurally, AAC is very similar to MP3 but has some significant differences aimed at improving the subjective audio quality.  However, each format requires a separate codec to be able to read it.

By the turn of the millennium, the confluence of the ubiquitous MP3 codec and the ready availability of hard discs with capacities exceeding 100MB had ushered in the age of computer audio.  As always, there was a fringe element who still preferred the improved sound quality of uncompressed WAV and AIFF files, but who were still troubled by the enormous file sizes.  Programs like PKZip proved that ordinary computer files could be compressed to a smaller file size and subsequently regenerated in their exact original form.  However, PKZip did not do a very good job of reducing the file size of audio files.  A dedicated lossless compressor was needed, one specifically optimized around the characteristics of audio data.  In 2001 the first FLAC format specification was released.  The FLAC codec could produce compressed files that are approximately 50% of the size of the original WAV or AIFF file.  Later, in 2004, Apple responded with their own lossless compression format ALAC (or Apple Lossless).

Meanwhile, in 1999, Sony and Philips tried and failed to launch the SACD format as a successor to the ubiquitous CD.  SACD uses a radically different form of audio encoding called DSD.  Ultimately, the SACD launch flopped, although the format has never actually gone away, and the DSD format acquired its own band of loyal followers.  The developers of SACD each developed a file format that could handle DSD data - the DFF format developed by Philips, and the DSF format developed by Sony.  By 2011, DSD enthusiasts had demonstrated the ability to manage DFF and DSF files on their computers, and to transmit DSD data to a DAC, and the first DSD-compatible DACs trickled onto the market.  Consumer-level DSD recording equipment is also now available, and produces output files in either DSF or DFF format.

Today, although other file formats do persist, the computer audio market has more or less settled down to four format types, with two competing format offerings for each type.  AIFF (Apple) and WAV (everybody else) for uncompressed audio; ALAC (Apple) and FLAC (everybody else) for lossless compression; and AAC (Apple) and MP3 (everybody else) for lossy compression.  DSF and DFF continue to duke it out in the DSD world.  Note that, except for DSD which Apple does not support in any form, the formats have shaken down into pairs of Apple and everybody else.  Why is this?

Frankly, there is absolutely no reason why any software player should not be able to support all of these file formats.  The process of reading (or writing) any of them is quite straightforward.  Yet, Apple originally refused to support WAV and MP3 formats in its iTunes software and iPod players, instead requiring users to use its own AIFF and AAC formats.  In fact, to this day Apple products continue to refuse to support FLAC files, instead requiring its customers to use ALAC.  From a functionality viewpoint none of this really matters.  ALAC and FLAC can be seamlessly transformed from one to the other and back again using high quality free software (as can AIFF and WAV, AAC and MP3).  But this is not what customers want.  So why is it that Apple takes this unhelpful stance?

The reason is simple.  From a business perspective, Apple’s entire iTunes ecosystem exists not to provide you with a platform on which to manage and play your music, but as a platform to sell you the music that you listen to.  Apple’s business model is for you to buy your music from them rather than from anybody else.  Therefore when you buy music from the iTunes Store it comes in AAC format only and not in MP3 or FLAC.  But if you buy your music virtually anywhere else, it only comes in the MP3 and FLAC formats.  Virtually nobody outside of Apple is interested in selling AAC or ALAC files.

The situation is even more bizarre when it comes to lossless compressed audio.  Apple isn’t actually selling any ALAC files on its iTunes Store!  You really have to wonder what their thinking is.  Do they consider that they are motivating me to buy lossy AAC files from Apple instead of lossless FLAC files from someone else?  Really?  Hey, maybe they’re right - maybe that’s exactly what we do.  Consumers are a pretty dumb species after all.  It has also been suggested that Apple is scared of becoming targets of a patent troll if they start offering FLAC support, but that seems to be an even more feeble explanation.  Google have been supporting FLAC in Android for some time now, and have not attracted any trolls’ attention that we know of.  In any case, as far as I know, nobody has ever identified any significant patents FLAC might possibly be infringing, given that it is all open-source.  But given the size of Big Apple (even bigger than Big Google!), they would certainly make for a tasty target.

Interestingly, back in the early 2000's, with the overwhelming consumer embrace of MP3, Apple realized very early on that if they were going to continue refusing to support MP3 they could risk losing out on the whole mobile music opportunity to one of the competing platforms such as Rio, Zune, Nomad/Zen and others.  Deciding to support MP3 was a key tactical business decision that took the air out of their competitors’ sails and ultimately paved the way for the total dominance of iPod and iTunes.  Today, despite the overwhelming consumer embrace of FLAC, there is no such pressure on Apple to encourage them towards supporting FLAC.

At one time there was an App called Fluke which allowed users to import FLAC files into iTunes.  Unfortunately, that loophole relied on a 32-bit OS kernel, and as a result Fluke no longer works with OS X 10.7 (Lion) and up.  Just to be clear, there are absolutely no technical reasons whatsoever that prevent Apple from supporting FLAC files.  It would be a trivial move for them to make, if they wanted to.  Their refusal to support FLAC is entirely a tactical decision on their part.

The situation with DSD is significantly different.  OS X and iOS are both fundamentally incapable of supporting DSD.  It would require significant changes to the way their audio subsystems work in order for that to happen, and, being honest, I see some fundamental issues that they would face if they ever considered doing that.  Consequently, I don’t see DSD being supported by Apple in any form for the foreseeable future.  The way the audio industry has got around that is with the DoP data transmission format.  This dresses up native DSD data so that it looks like PCM, which OS X can then be fooled into sending to your DAC, but it means that any Mac Apps which support DSD would have to be extremely careful how they went about it.  BitPerfect, for example, will do that for you, but iTunes won’t.  This is different from the situation with FLAC files.  Whereas iTunes would have no problems reading a FLAC file if Apple chose to let it, it would have absolutely no idea what to make of a DSD file.  You might as well ask it to load an Excel spreadsheet.

In order for BitPerfect to manage DSD playback, we have created what we call the Hybrid-DSD file format.  Hybrid-DSD files are ALAC files that iTunes recognizes, and can import and play normally.  However they also contain the native DSD audio data as a sort of “trojan horse” payload.  If iTunes plays a Hybrid-DSD file it plays the ordinary ALAC content.  But if BitPerfect plays the file it plays the DSD content.  We really like that system.  Other software players have instead adopted the idea of a “proxy” file.  This is a similar thing, but instead of containing ordinary ALAC music plus the DSD payload, they contain no music and include information that enables the playback software to locate the original DSF or DFF file.  Some may like the proxy file format, indeed some may prefer it, but we don’t, and this isn’t the place to discuss that.

It has often been suggested that BitPerfect could adopt a mechanism similar to either the Hybrid-DSD file or the proxy file to import FLAC files into iTunes.  And yes, we could do that.  But frankly, why bother converting from FLAC to "Hybrid-FLAC", when it is even easier to transcode FLAC files to ALAC using a free App such as XLD.  It is simple and effective, and the ALAC files can just as easily be transcoded back into FLAC form if needed, with zero loss of fidelity.

The final topic I want to cover in this post is Digital Rights Management (DRM).  This is a method by which the audio content in the file is encrypted in such a way as to prevent someone who does not “own” an audio file from playing it.  In other words, it is an anti-piracy technique.  Files containing DRM are pretty much indistinguishable from files that do not contain it, and most audio file formats support the inclusion of DRM (I am given to understand that FLAC does not, but I am not 100% sure).  For example, Apple included DRM in almost all of the music downloads sold on iTunes between 2004 and 2009.   

DRM is something that tends to get forced on the distributors (i.e. the iTunes Store) by content providers (i.e. the record labels), and is a major inconvenience for absolutely everybody involved in the playback chain.  Between 2004 and 2009 Apple had grown to hold sufficient clout that they could dictate to the content providers their intention to discontinue supporting DRM.  Today, DRM is a non-factor, although the new Apple Music service, plus TIDAL, and other streaming-based services which offer off-line storage, rely on their own versions of it.  The advance and retreat of DRM is an interesting barometer of who has the upper hand at any time in the music business between the distributors and the content providers.

Sigma-Delta Modulators - Part II.

Yesterday, we saw how a SDM can be used to faithfully reconstruct an incoming signal, even if the output is constrained to an apparently hopelessly reduced bit depth.  We do this by ensuring that the Signal Transfer Function (STF) and Noise Transfer Function (NTF) have appropriate characteristics.  This, of course, is a lot harder to achieve that you might have concluded from the expansive tone of yesterday’s post, which we concluded with the open question of how to design an appropriate loop filter.

Addressing those issues remains at the bleeding edge of today’s digital audio technology.  The best approach to understanding the design of an SDM remains the “Linear Model” I alluded to yesterday, where we treat the quantization error introduced at the quantizer stage as a noise source.  This model ought to be as accurate as its limiting assumption, which is that the quantization error is well represented by a noise source.  Unfortunately, the results don’t appear to bear that out.  According to this model, relatively simple SDMs should exhibit stunningly good performance, where in reality they do not.  In fact they fall very substantially short of the mark.  Clearly, the noise source is not as good a substitute for the quantization error as we thought.  Furthermore the reasons why are not clear, and we don’t have a better candidate available.

In the absence of a good guiding model, SDM designers stick to an empirical methodology based on the well-known “suck-it-and-see” approach.  The most successful approach is based on increasing the “order” of the modulator.  The simple SDM I described yesterday has a single Sigma stage, and is called a “first order” SDM.  If we simply add a second Sigma stage we get a “second order” SDM.  We can add as many Sigma stages as we like, and however many we add, that’s the “order” of the SDM.  The higher the “order” of the SDM, the better its performance ought to be.  I make that sound so much easier than it actually is, particularly when it comes to the task of fine-tuning the SDM’s noise-shaping (or the NTF if you like) performance.

In practice, real-world SDM designs run into problems.  Lots of them.  First of these is overloads.  If the signal fed into the quantizer overloads the quantizer then the SDM will go unstable.  This is the same as any PCM representation - if the signal level is too high, then the PCM format, due to its fixed bit depth, will not have an available level with which to represent the signal, and something has to give (typically, a simple PCM encoder will allow the signal to hard clip).  In a SDM, because a Sigma modulator is in fact a very simple IIR filter, the result of such an overload will reverberate within the output of the SDM for a very considerable time.

The second problem is that high-order digital filters can themselves be rather unstable, not so much because of any inherent instability, but generally because of CPU truncation errors in the processing and execution of the filter.  Proper filter design tools can identify and optimize for these errors, but can never make them go away entirely.  Unstable filters can cause all sorts of problems in SDMs, from the addition of noise and distortion to total malfunction.

The third problem is that SDMs are found to have any number of unexpected error or fault loops in which they can find themselves trapped, which are not yet adequately explained or predicted by any theoretical treatment.  These include phenomena known as “limit cycles”, “birdies”, “idle tones” and others.  They can be astonishingly difficult to detect, or even to describe, let alone to design around.

Real-world high performance SDMs for DSD applications are typically between 5th and 10th order.  Below 5th order the performance is inadequate, and above 10th order they are rarely sufficiently stable.  The professional audio product Weiss Saracon, for example, contains a choice of loop filters in its SDM, having orders 6, 8, and 10.  Each loop filter produces a DSD output file with subtly different sonic characteristics, differences which many well-tuned audiophile ears can reliably detect.  And, as with religion, the fact that there are several of them from which to choose doesn’t guarantee that one of them is correct!

Interestingly enough, one of those limitations can be readily made to go away.  The problem of overloads can be entirely eliminated by using a multi-bit quantizer.  This approach is used in almost all commercial ADCs which use an analog SDM in the input stage, configured to provide a 3-bit to 5-bit intermediate result.  This intermediate result is then converted in the digital domain to the desired output format, whether PCM or DSD.  Likewise, almost all commercial DACs employ a digital SDM in the input stage, configured to provide a 3-bit to 5-bit intermediate result which is then converted to analog using a 3-bit to 5-bit R-2R ladder.  SDMs are therefore deeply involved at both ends of the PCM audio chain, though they mostly don’t use the 1-bit bit depth of DSD (or, for that matter, its 2.8MHz sample rate).  When you listen to PCM, you cannot escape the fact that you are listening to SDMs.

The key takeaway from the study of SDMs is that while their performance can indeed be extremely good, the current state-of-the-art does not permit us to quantify that performance on an a priori basis to a high degree of accuracy.  Instead, SDMs must be evaluated phenomenologically.  In other words we must carefully measure their characteristics - linearity, distortion, noise, dynamic range, phase response, etc.  In this regard, SDMs are very much like analog electronic devices such as amplifiers.  We can bring a lot of design intelligence to bear, but at the end of the day those designs cannot tell us all we need to know about their performance, and the skill of the designer (not to mention the keen ear of the person making the final voicing decisions) becomes the critical differentiating factor.

At this point I promised to conclude by touching on some of the differences between DSD and PCM formats.  Much has been written about this, and it can tend to confuse and obfuscate.  Frankly, I'm not so sure this will help much.  On one hand, with a PCM data stream, the specific purpose of every single bit in the context of the encoded signal is clear and unambiguous.  Each bit is a known part of a digital word, and each word stipulates the exact magnitude of the encoded signal at a known instant in time.  The format responds to random access, by which I mean that if we want to know the exact magnitude of the encoded signal at some stipulated moment in time, we can go right in there and grab it.  Of course, when we say “exact” we understand that to be limited by the bit depth of the PCM word.

The situation with SDM bitstreams is slightly different, and I will illustrate this with the extreme example of a DSD 1-bit bitstream.  On one level, we can see the DSD bitstream as being exactly identical to what I have just described.  Each bit is a known part of a digital word, except that in this case the single bit comprises the entire word!  This word then represents the exact magnitude of the encoded signal at a known instant in time - but this time to a resolution of only 1-bit.  That is because the DSD bitstream has encoded not only the signal, but also the heavy dose of shaped noise that we have been describing in noxious detail.  That noise gets in the way of our ability to interpret an individual word in the light of the original encoded signal.  By examining one word in isolation we cannot determine how much of it is signal and how much is noise.

If we want to extract the original signal from the DSD bitstream, we must pass the entire bitstream through a filter which will eliminate the noise.  And because we have already stipulated that the SDM is capable of encoding the original signal with a very high degree of fidelity, it stands to reason that we will require a bit depth much greater than 1-bit to store the result of doing so.  In effect, by passing the DSD bitstream through a low-pass filter, we end up converting it to PCM.  This is how DSD-to-PCM conversion is done.  You simply pass it through a low-pass filter.  The quality of the resultant PCM representation can be very close to a perfect copy of the original signal component in the DSD file.  It will be limited only by the accuracy of the low-pass filter used.

When we started developing our product DSD Master, we realized very quickly that the choice of filter was the most critically important factor in getting the best possible DSD-to-PCM conversions.  A better choice of filter gave rise to a better-sounding conversion.  FYI, we continue to work on better and improved filters for our DSD Master product, and for our next release we will be introducing a new class of filter that we believe will make virtually perfect PCM conversions!

Unlike SDMs, digital filters are very well understood.  There is virtually no significant aspect of a digital filter’s performance which has not been successfully analyzed to the Nth degree.  The filter’s amplitude and phase responses are fundamentally known.  We can stipulate with certainty the extent to which computer rounding errors are going impact the filter’s real-world performance, and take measures to get around that if necessary.  In other words, if we know what is in the filter’s input signal, then we know exactly, and I mean EXACTLY, what is going to be in the filter’s output signal.  SDMs, as we have seen above, are not like that.

What does that mean for the DSD-vs-PCM argument?

I really don’t know the answer to that!  On one hand, I am convinced that before too long I will be able to make conversions from DSD to PCM which are virtually perfect, at least to the extent that any PCM representation can be perfect.  On the other hand, I am equally convinced that conversions from PCM to DSD are less perfect, and that SDM technology still has some major advances to be made.  Here at BitPerfect we are working on Look-Ahead SDMs (which, being pedantic, are not strictly speaking SDMs at all) which have the potential to take some small steps forward.  The problem is, they require phenomenal computing power, so our LA-SDM remains very much a lab curiosity.  My feeling is that when each is performed in accordance with the current state-of-the-art, PCM-to-DSD conversion lags DSD-to-PCM conversion in the ultimate quality stakes.

So why - and I’ve said this before - do I still have a lingering preference for DSD over PCM?  I have come to the following conclusion.

DSD is primarily listened to by audio enthusiasts.  The market for DSD comprises people who like music, but still want to hear it well recorded.  It is still a small market, and it is served almost exclusively by specialist providers why are happy to put in the time, expense, and inconvenience required to generate quality product for that market.  People like Cookie Marenco at Blue Coast Records, Jared Sacks at Channel Classics, Morten Lindberg at 2L, Todd Garfinkel at MA Recordings, Gus Skinas at Super Audio Centre and many others, focus on delivering to consumers truly exceptional recordings of uncompromised quality.  DSD, for those people, drives three things, aside from the fact that some of them have their own firmly-established preference for DSD.

First, because of the issues described at length above, tools do not exist to do even the simplest of studio work in the DSD domain.  Even panning and fading require conversion to an intermediate PCM format.  Forget added reverb, pitch correction, and any number of studio tricks of the Pro-Tools ilk.  Recording to DSD forces recordists to strip everything down to its basics, and capture the music in the simplest and most natural manner possible.  That alone usually results in significant increases in the sort of qualities that appeal to audiophiles.

Second, when remastering old recordings for re-release on SACD, or even for digital download as DSD files, mastering engineers will typically pay a lot more attention to details than would normally be the case for a CD release.  Gone will be the demands for compression (or loudness).  The mastering engineer will get the opportunity to dust off that old preamp he always wanted to use, or those old tube amplifiers that he only brings out when the twenty-something suits from the label are not prowling around.  Try Dire Straits’ classic “Brothers In Arms”, which sounds a million times better when specially remastered for SACD (I love the Japanese SHM-SACD remastering) than it ever did on any CD, even though the master tape was famously recorded in 16-bit PCM and mixed down to DAT.  Go figure.

Third, unless you are using one of the few remaining ancient Sonoma DSD recording desks, if you are recording to DSD you will be using some of the latest and highest-spec studio equipment.  That’s where the DSD options are all positioned.  You will be using top-of-the-line mics, mic preamps, ADCs, cables, etc.  As with most things in life, you tend to get what you pay for, and if you are using the best equipment your chances of laying down the best recording can only improve.

So I like DSD, I continue to look out for it, and it continues to sound dramatically better than the vast majority of PCM audio that comes my way.  Is that due to some fundamental advantages of the DSD format, or is it that PCM offers a million new and exciting ways to shoot a recording in the foot?  I’ll leave to others decide.

Sigma-Delta Modulators - Part I.

I have mentioned SDMs many times in the past.  These are, in effect, complex filter structures that are used to produce DSD and other bitstreams.  I know I talk about DSD a lot, and I also know that digital audio is way more about PCM that it it ever is - or ever will be - about DSD.  But, as I have already written, SDMs are in fact core to both ADCs and DACs and therefore also, I think, to resolving (or maybe just understanding) the debate concerning the relationship of DSD to PCM.  So I thought I would devote a post to an attempt to explain what SDMs are, how they work, and what their limitations are.  This will be doubly taxing, because I am far from being any sort of expert, and this is a deeply technical subject.  Finally, I will attempt to place the results of my ramblings in the context of the PCM-vs-DSD debate, with perhaps a surprising result.

The words Sigma and Delta refer to two Greek letters, Σ and Δ, which are used by convention in mathematics to denote addition (Σ) and subtraction (Δ).  Negative feedback, where the output signal is subtracted from the input signal, is a form of Delta modulation.  Similarly, in an unstable amplifier, where the output signal is inadvertently added to the input signal causing it to increase uncontrollably, this is a form of Sigma modulation.  Sigma Delta Modulators work by combining those functions into a single complex structure.  I use the term ‘structure’ intentionally, because SDMs can be implemented both in the analog domain (where they would be referred to as circuits) and in the digital domain (where they would be referred to as algorithms).  In this context, analog and digital refer only to the inputs of the SDM.  An SDM’s output is always digital.  For the remainder of this post I will refer only to digital SDMs, mainly because it is easier to describe.  But you should read it all as being equally applicable to the analog case.

At the core of an SDM lies the basic concept of a negative feedback loop.  This is where you take the output of the SDM and subtract it from its input.  We’ll call that the Delta stage.  If the output of the SDM is identical to its input, then the output of this Delta stage will always be zero.  Between the Delta Stage and the SDM output is a Sigma stage.  A Sigma stage works by maintaining an accumulated value to which it adds every input value it receives.  This accumulated value then becomes its output and therefore also the output of the SDM itself.  Therefore, so long as the output of the SDM remains identical to its input, the output of the Delta stage will always be zero, and consequently will continue to add zero to the accumulated output of the Sigma stage which will therefore also remain unchanged.  This is what we call the “steady-state case”.

But music is not steady-state.  It is always changing.  Let’s look at what happens when the input to the SDM increases slightly.  This results in a small difference between the input and the output of the SDM.  This difference appears at the output of the SDM’s Delta stage, and, consequently, at the input of it’s Sigma stage.  This causes the output of the Sigma stage to increase slightly.  The output of the Sigma stage is also the output of the SDM, and so the SDM’s output also increases slightly.  Now, the output of the SDM is once more identical to its input.  The same argument can be followed for a small decrease in the input to the SDM.  The SDM as described here is basically a structure whose output follows its input.  Which makes it a singularly useless construct.

So now we will modify the SDM described above in order to make it useful.  What we will do is to place a Quantizer between the output of the Sigma stage and the output of the SDM, so that the output of the SDM is now the quantized output of the Sigma stage.  This apparently minor change will have dramatic implications - for a start, this is what gives it its digital-only output.  To illustrate this, we will take it to its logical extreme.  Although we can choose to quantize the output to any bit depth we like, we will elect to quantize it to 1-bit, which means the output can only take on one of two values.  We’ll call these +1 and -1 although we will represent them digitally using the binary digits 1 and 0.  One result of this is that now the input and output values of the SDM will always be different, and the output of the Delta stage will never be zero.  The SDM is still trying to do the same job, which is to try to make the output signal as close as possible to the input signal.  However, since the output signal is now constrained to taking on only the values +1 or -1 it would appear that the SDM is going to flounder.

At this point, mathematics takes over, and it no longer becomes practical to reduce what I am going to describe to simple illustrative concepts.  I hope you will bear with me.

In order to understand what the SDM is actually doing, we need to make some sort of model.  In other words we’ll need a set of equations which describe the SDM’s behaviour.  By solving those equations we can then gain an understanding of what the SDM is and is not capable of doing.  There is a problem, though.  The quantizer introduces a non-linear element.  If we know what the input value to the quantizer is, we can determine precisely what the output value will be.  However, the opposite is not true.  If we know the output of the quantizer, we cannot deduce what the input value was that resulted in that output value.  The way we treat problems such as this is to consider the quantizer instead as a noise source.  We consider that we are instead adding noise (i.e. random values) to the output of the Sigma stage, such that the output values of the SDM end up being either +1 or -1.

The next thing we do is to observe that one thing we have said about how the SDM works is not entirely correct.  We said that at the input to the Delta stage we take the SDM’s input and subtract from it the SDM’s output.  In fact what we subtract is the SDM’s output at the previous time step.  This is very important, because it means that we can use this one-step delay to express the SDM’s behaviour in terms of a digital transfer function, using theories developed to understand how filters work.  I have mentioned such matters before in my previous posts on “Pole Dancing”.  Transfer functions allow you to calculate the structure’s frequency response, and when we apply this approach to the SDM we come up with two equations which we call the Signal Transfer Function (STF) and the Noise Transfer Function (NTF).  These are two very useful properties.

The STF tells us how much of the signal applied to the input of the SDM makes it through and appears in the output, whereas the NTF tells us how much of the quantization noise generated by the quantizer makes it to the the SDM’s output.  Both of these properties are strongly inter-related, and are strongly frequency dependent.  Generally, we would like to see STF~1 at low frequencies.  By contrast, we would like to have NTF~0 at the low frequencies but transition to NTF~1 at the high frequencies.  What exactly does all that gobbledygook mean?

The important thing is that at low frequencies we want the combination of STF~1 and NTF~0.  This means that at these low frequencies the output of the SDM contains all of the signal and none of the quantization noise.  However, at high frequencies we would like the opposite to be true, so that the output of the SDM contains none of the signal and all of the quantization noise.  If we can arrange it such that those so-called “low frequencies” actually comprises the audio frequency band, then our SDM can be capable of encoding that music signal with surprising precision even though the format has a bit depth of only 1-bit.  Analysis of the STF and NTF enables us to figure out how high the sample rate rate must be in order for the full 20kHz+ of the audio frequency bandwidth to fit into the “low frequency” part of the STF/NTF spectrum where sufficiently good performance can be obtained.  The answer, not surprisingly, is what drives DSD to use a sample rate of 2.8MHz.

A simpler way for the performance potential of this SDM to be viewed is to consider only the quantization noise.  This is nothing more than the difference between what the ideal (not quantized) output signal would look like and what the actual (quantized) output signal actually does looks like.  If those differences could be stripped off, then what we would end up with is the ideal output signal in all its glory.  What the NTF of the SDM has done is to arrange for all of those differences to be concentrated into a certain band of high frequencies which are quite separate from the audio frequency band containing the ideal output.  By the simple expedient of applying a suitable low-pass filter, we can filter them out completely, and thereby faithfully reconstruct the ideal output signal.

Unfortunately, the simplistic SDM I have just described is not quite up to the task I set for it.  The NTF is not good enough to meet our requirements.  In reality, there is a final step in the design of the SDM where we need to be able to fine tune the STF and NTF to acquire the characteristics needed to make a high-performance SDM.  What we do is to replace the Sigma modulator with a filter, which is generally termed the Loop Filter.  The transfer function of the loop filter then determines the actual STF and NTF of the final SDM.  Designing the SDM then becomes the task of designing the loop filter.  This is a big challenge.

In Part II I will discuss some of the limitations and challenges of SDM design, and conclude by attempting to place my observations in the context of the ongoing PCM-vs-DSD debate.