Mario Richard
One of the broadest and longest-running topics of scientific discussion, we find it in Pythagorean discussions as well as in the correspondence between Einstein and Bohr, refers to the matching of theory and practice. Neither of the two representative faces of reality, theory and experience, are at odds with each other, they complement each other allowing the theorist a better description of experience and the experimenter a deeper understanding of the theoretical hypothesis. Robert Watts has been working for more than 30 years to reconcile the two sides of the coin that bears the inscription DAC on its coinage, asking himself why some audio performances are so good and tying his judgment to what digital signal processing theory dictates to him, in this case for the development of his fantastic achievements. For many years he has concentrated his attentions on the rendering of transients in audio reproduction, attributing to these a large part of the transport of information of frequency, timbre, scene representation, etc… In simple terms we are talking about the correctness of the harmonic composition of a sound signal starting from its attack until the exhaustion of notes and the temporal correctness with which it develops. Or if we want to see the glass half empty we are talking about harmonic distortions and temporal or phase distortions.
What does theory say about the sampling of a signal and its reconstruction? Among the many things it points out to us there are 2 particularly important ones (actually one of them comes to you transversely since it is more general and concerns signal processing) related to the properties of the signal: having a maximum frequency, being periodic. Now what dictates the maximum frequency, the highest tone, of an audio signal? Certainly the ability to produce it by an instrument, the ability to capture and store it by microphone and recording equipment, our ability to hear it. To limit us to CD playback for example, this limit has been set at 22050 kHz, which is perfectly sufficient to our hearing but perhaps not fully representative of the original signal that can produce even higher harmonics. And it is precisely from harmonics that the second mentioned property arises, that is, the one that requires the signal to be periodic, repetitive in time, in order to be represented as a sum of several elementary signals, the harmonics precisely. This then allows the engineer, the physicist, the mathematician, the musician, the audiophile, whoever you want to move into the frequency dimension and navigate it at will. For most signals periodicity is not an issue, what distinguishes a note from a noise is precisely the repetitiveness over time of the “sound form” let me tell you. But what about Mahler’s mallet, for example, on the finale of the Sixth Symphony? What note is it? Here again a solution has been found, and in particular it all fits mathematically if the period to be considered in this case is……infinity, that is, the signal repeats after an infinite time for another infinite time.
Mr.Watts' approach
Watts wrestled with all this information and requirements and with the performance ultimately judged when listening to his implementations; hence his approach and implementation metology emerged. In particular, I am referring to what in Chord products goes by the name of WTA or rather Watts Transient Aligned Filter. Robert Watts, the designer precisely, argues that by relying on listening tests one cannot be satisfied with implementations derived from requirements dictated by theory alone, in particular he focuses on the sampling rate of the signal values presented to the Digital/Analog converter. For the designer, the correct reconstruction of the transient is crucial, and to do this the succession of data to be converted cannot be less than the temporal resolution we are able to perceive, which is 4 microseconds (!). Simply put, if our abilities allow us to distinguish something that varies in such a short time, the sound signal must be able to do so as well. What then about the sampling theory that, for example, for CD requires 22-microsecond intervals? It remains intact because it relates to the assumptions I described earlier, but if, on the other hand, the assumptions espoused Watts are different then something more needs to be done. To get below the 4-microsecond wall with the clock multiples we are used to seeing, we have to multiply the standard 44.1 kHz by 8, for example, and arrive at 352 kHz (DXD ring a bell? It is no coincidence that more and more recordings are being made at this sample rate), so we arrive at 2.8 microseconds. Having established, then, that our DAC in design needs to be presented with a sequence of samples at the given frequency, it is necessary to figure out where to get all this data since, for example, from a CD we get 8 times less value. The solution is again given by mathematics and is called interpolation. There are algorithms and mathematical functions that do this with very but very low errors, in fig.1 is depicted one of the main functions of interpolations, such that they are inaudible
The function shown as an example means that by juxtaposing 2 such functions whose peak assumes the value of the known samples, from the corresponding sum–better envelope–of all their side lobes (the tentacles surrounding the peak) it is possible to derive the intermediate values between the two known samples. Of course the example I have given is elementary and first approach but it serves to make it clear that the tool the designer now needs is a machine that performs calculations, lots of calculations perhaps elementary but in an extremely fast and safe way, there can be no indecision or error otherwise the result would be heard immediately. Watts realizes interpolation with algorithms that use digital filters composed of hundreds of thousands of values, and to do so he has chosen not to rely on computer processors (CPUs) but rather on components with which it is possible to implement programmable digital architectures but which are extremely fast and, above all, consume very little power. The componentry in question is called Field Programmable Gate Array (FPGA), and there are different types of them with more or less performance. They vary in size from less than 1 cm on a side to a few cm and contain tens of thousands of “elementary logic bricks” inside them; by connecting these logic bricks together appropriately, it is possible to make them perform mathematical or logical functions very quickly. Thus, they are programmable elements, but unlike classical computer CPUs, one does not program the instructions to be executed sequentially but programs an actual digital architecture that will perform the various tasks required. Today FPGAs contain within them not only logic elements but also entire processors, typically of the ARM family, the communication facilities such as serial or LAN ports, memories etc… These elements have now invaded our lives, we find them in almost all consumer electronics in our homes, and their commercial value goes from a few euros to several thousand a piece. Chord chose to use FPGAs from Xilinx. Almost every year they are at the Xilinx headquarters in San Jose in California’s silicon valley. Xilinx has 4,000 employees scattered around the world and turns over $2 billion a year. It is so important to world electronics that the street where it resides is called “All Programmable drive”! Just think that until last year it was in Logic drive….
In fig.2 you can see what they call the wall of fame, which is all the patents they have. There are thousands! They are bringing the resolution on the silicon-the distance between one logic brick and another-from the 28 nm of what was used here to the 5 nm of the new families. This represents the possibility of more integration, thus more bricks in the same area, but most importantly less consumption. Not only that, DAC and ADC will also already be found integrated inside. The new FPGA families make what is now called a System On Chip (SOC) or almost a computer motherboard on a single component. Simply put, the future designer can almost write the algorithm to find it then implemented directly as an ad hoc digital architecture. This is the path Chord has chosen with Robert Watts: the experience and skill of a designer directly into the soul of silicon.
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