What are Digital Filters and Why Are They Required In Today's Audio DACS? by Resonessence Labs Technical Staff
In any digital audio system a digital filter is needed to remove what are called aliases of the signal. These aliases are not imperfections in the design in any sense; they are a mathematical artifact of the act of sampling a continuous signal into a series of digital numbers taken at different points in time.
Aliases are not present in the ‘older’ analog recording formats: tape and vinyl records capture continuous signals, and do not create these artifacts.
Perhaps your first introduction to aliasing due to a finite sample rate, would have been when you watched cowboy movies in the sixties and seventies: sometimes the wheels of the wagon trains would appear to be going the wrong way, or even slowing and reversing in direction despite the wagon clearly continuing to move.
This was due to the camera used to make the movie: it was sampling the scene at 24 frames per second, but the wheel spokes were moving much faster than that. They were captured by the camera having moved more than one-spoke in revolution, and the camera therefore generated artifacts in the video playback that showed the wheels moving at the wrong rate. This effect will occur every time something is represented in a non-continuous fashion. The first time physicists discovered this phenomena was when they looked at vibrations in crystals. Something very odd was happening as the frequency of the vibration increased: the energy was coming out in the wrong place!
It took some very clever physicists to realize that the crystal was made up of discrete atoms, all the same distance apart in the crystal, and the vibrations (called phonons) passing though the crystal, were being sampled by the atoms all a similar distance apart. So, because these pieces of crystal were made of a finite number of atoms all the same distance apart, when the phonon frequency was such that it moved more than one cycle in the distance between atoms (equivalent to the wagon wheel moving more that one spoke-distance between frames) the phonon frequency was changed – it was wrong.
Leon Brillouin, a French Physicist was amongst the first to figure out what was going on and what are called “Brillouin Zones” define how a crystal creates phonon aliases. He figured this out in the decade of the 1920′s.
Our problem in the audio world is much simpler than Brillouin’s, because it is only in one dimension, and engineers are used to thinking about the Brillouin zones as just certain frequencies that cannot be exceeded before there is “a problem”.
The frequency where problems start to occur is at half the equivalent sampling rate. So, for example, in digital music recorded on a CD, the studio has sampled the signal at 44.1kS/s, and what physicists would call the first Brillouin zone, ends at half this rate: at 22.05Khz. Engineers just call this the half sample rate, or sometimes the Nyquist limit, (after Harry Nyquist who recognized that there was a limit to the rate at which information could travel down a telegraph line in 1928, and later defined the maximum frequency that could be encoded into discrete samples).
If we ask the studio to encode a sound of 30Khz into the CD, 30Khz will not come out when we playback. Rather, 14.1Khz will come out. You can perhaps see where the 14.1Khz comes from: it is the difference between 30Khz we applied and the 44.1Khz we used to sample the signal.
Nothing is wrong, nothing is faulty, in this scenario: each element is operating at mathematical perfection, it is just that a signal of 30Khz cannot be captured into a series of samples taken at 44.1Khz because it exceeds half the sample rate – it exceeds 22.05Khz.
How can we cope with this? What if the music content has a cymbal sound with more than 22.05Khz in it?
The answer assumed by the clever engineers at Phillips and Sony who first came up with the CD, was to argue as follows: since the human ear cannot hear above about 20Khz, let us make an analog filter that removes all above 20Khz, then there can be no problem, 20Khz is just below 22.05Khz and no aliases will be created, since there is no signal above 20Khz.
You may ask why did they not simply increase the sample rate to say 100Khz and so the first problem would not occur until 50Khz? The answer is that they could not do that because that would have more than doubled the number of samples on the CD, and the CD had to play for at least 45mins so that it could capture one whole vinyl album. In other words, there were commercial considerations that dictated that the sample rate be as slow as possible. Not good for us audiophiles, and it has taken us years to break this constraint: now we can finally get 24 bit 192Khz sampled music without compromise.
"The problem they have is that any signal above 22.05Khz will alias, some engineers use the term “fold back”, into the audio domain and so there has to be a filter, an analog filter, that removes all the sounds above 22.05Khz (in fact they choose 20Khz) to prevent this problem."
But let’s return to what Phillips and Sony had to do in the 1970′s to make CDs viable. The problem they have is that any signal above 22.05Khz will alias, some engineers use the term “fold back”, into the audio domain and so there has to be a filter, an analog filter, that removes all the sounds above 22.05Khz (in fact they choose 20Khz) to prevent this problem.
This is not trivial: they are asking the analog designer to make a filter that lets through 20Khz, but blocks off all signals above 22.05Khz! Any analog designer would tell you this is far from trivial: the 22.05Khz is much too close to the 20Khz. “Can you not give me a break and say let through 20Khz and block off say 50Khz?” the analog designer would say, to which the company has to reply, “No if you can’t do this, we can’t fit an album on a CD, and who would buy that?”