Quibbles and Bits

Sigma Delta Modulators Part II

In my last column, I wrote about how a SDM might be used to faithfully reconstruct an incoming signal, even if its output is constrained to an apparently hopelessly reduced bit depth.  We can do this if we can ensure that the Signal Transfer Function (STF) and Noise Transfer Function (NTF) have appropriate characteristics.  Today I will expand on how we can make that work.  I had originally planned to conclude with some personal remarks on the DSD-vs-PCM debate, but I will leave those for a complete post in the next issue.

We want the NTF and STF to be frequency dependent, but the Sigma stage I described is clearly not frequency dependent, and neither is the Delta stage, so it’s not at all apparent how that can come to pass.  Therefore we need to introduce some frequency dependence into the picture.  The way we do that is to augment the Sigma stage with a filter called the “Loop Filter”.  This is usually a low-pass filter, and its incorporation means that the NTF and STF now become frequency dependent.  If the “Transfer Function” of the Loop Filter (a deeply technical way of expressing its frequency response) is H(z), then the frequency response of the STF will be also given by H(z), and that of the NTF will be given by 1- H(z).  This is the basis of the simple Linear Model of SDM design.

Don’t worry if none of that means anything to you, because addressing these design issues takes us right to the bleeding edge of today’s digital audio technology.  The best approach to understanding the design of an SDM remains this Linear Model, where we treat the Quantization error introduced at the Quantizer stage as a noise source.  However, any model’s accuracy is – self-evidently – limited by its “limiting assumptions”.  In this case the limiting assumption is that the Quantization error can be adequately represented by a noise source.  According to the Linear 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, a noise source is not as ideal a substitute for the Quantization error as we had hoped.  Unfortunately, though, we don’t yet have a better candidate available.

In the absence of a good guiding model, SDM designers stick to an empirical methodology based on trial and error.  The most successful approach is based on increasing the “order” of the modulator.  The simple SDM I described last time 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 NTF 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.  In effect, we are forcing the Quantizer into hard clipping.  In a SDM, because of the feedback, the result of such an overload will reverberate within the output of the SDM for a very considerable time.  Overloads become a progressively higher problem as the “order” of the SDM is increased, and ultimately impose an upper limit on the “order” of practical SDMs.

A second problem is that high order digital filters can themselves be unstable, not so much because of any inherent instability, but because of truncation and other precision errors in the processing and execution of the filter.  Proper filter design tools can identify and optimize for these errors, but can rarely 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.

A 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 frustratingly difficult to detect – sometimes 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, offers a choice of SDM structures, having orders 6, 8, and 10.  Each SDM structure produces a DSD output file with subtly different sonic signatures, 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 is not sufficient to guarantee that one of them is correct!

These and other limitations mean that the best SDMs deliver a dynamic range of about ~120dB – better than the ~98dB of 16–bit PCM, but less than the ~146dB capacity of 24–bit PCM.  Distortion is also an issue, albeit usually a minor one.  Overall, the noise and distortion performance of the very best SDMs is more or less equivalent to those of the very best analog circuits currently used in preamplifiers and amplifiers.  In principle we ought to be able to achieve better than that, and in future we very well might, but right now you could argue that what we are getting is as good as we need.

Interestingly enough, one of those limitations can be readily made to go away.  The problem of overloads can be largely 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 simple 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 design 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, 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; the skill – and golden ears – of the designer tend to become the critical differentiating factors.
The conclusion of Richard’s examination of Sigma-Delta Modulators will appear in Copper #17–-Ed.