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A recent Audio Asylum discussion on true 32-bit resolution involved the Sabre DAC and, amongst others, Thorsten Loesch of Abbingdon Music Research and finally the Sabre's own designer Dustin. I've reproduced it here for posterity in the context of our review.


Thorsten Loesch: I think we need to define what we mean by 32-bit DAC if we want to see if the claims are true or not. We can interpret it in many ways. 1) A 32-bit DAC is one that shows 32 bits of equivalent analogue resolution, i.e. 183dB dynamic range measured the traditional analogue way. Not only can't we measure such a DAC should it actually exist, no such thing exists nor is it possible outside a laboratory system suspended in liquid nitrogen and maybe not even then. 2) A 32-bit DAC is one that uses 32 individual binary weighted bit switches to theoretically be capable of producing 2^32 discrete steps though its analogue dynamic range is less than the postulated 183dB (or 192dB as some may say). No such thing exists either—yet—but it is at least theoretically possible to make such a device. 3) A 32-bit DAC is a DAC that accepts a 32-bit wide data word and outputs whatever real resolution it is capable of. In other words it is a marketing number without any appreciable meaning.


It should be added that recent specifications for computer-based audio call for systems that are able to handle 32-bit words simply because this is how computers like to work. They want 8/16/32-bit words to work with, not 24 bits. This is the reason for 32-bit DACs becoming more common now. This has no meaning other than that a DAC should accept a 32-bit word for compatibility reasons, not that it actually does anything meaningful with those whole 32 bits. In fact, several '32-bit' DACs simply take the 32-bit data and dither it down to 24 bits which are then applied to another manufacturer's 24-bit DAC core.


In 2010 all but one DAC targeted at audio employ a concept that used to be called hybrid DAC. This means these DACs combine several bits worth of multi-bit core with a delta-sigma modulator (aka one-bit DAC). The combination is used to achieve the total resolution with noise shaping from the true resolution of the DAC (the number of levels which the DAC can directly represent in the analog domain plus the additional resolution attained from noise shaping). The Sabre has 2^6 or 64 so-called unitary weighted or thermometer code bit switches. These are able to represent directly 2^6 or 64 individual levels.


Dustin: This is true on a per-pin basis. Each pin has 64 unitary weighted DACs connected to it. So in the stereo config many people use, that's 8 pins for 512 DACs per channel.

Thorsten: Further, the Sabre uses asynchronous sample rate conversion on all input data and converts into a clock rate of 40MHz.


Dustin: Again true but not the whole story. You can use the ASRC if you like - or not by simply clocking the XIN pin synchronously (at an integer multiple) to the BCLK. Then the ASRC drops itself out, reverting in this case to a more conventional method as the other DACs I'm aware of do.


Thorsten: If we assume for ease of calculation a 50KHz data sample rate (close enough to the 44.1KHz used on CD), we can represent as many as 40MHz:50KHz or 800 individual levels using classic pulse width or pulse density modulation. For ease of calculation I will round up to 1024 levels, which is equivalent to 10-bit resolution. This means that the raw resolution built into the ESS DAC is around 16 bit for single speed (44.1/48KHz) data, 15 bit for double speed (88.2/96KHz) and 14 bit for quad speed (176.4/192KHz) data.


Dustin: Using the ratio of oversampling of the FS to the Xin clock (the 40MHz in this case) and deriving the amount of extra bits from that assumes a moving average type filter, meaning it just takes the average over a certain period of time. This is somewhat approximate to 1st-order shaping. So the assumption that the 1024 x upsampling can only give you 10 more bits of resolution is true only if you use 1st-order shaping to your quantization noise. (1st-order is 6dB/octave, there are 10 octaves in 1024 x so that means 60dB, i.e. the 10 bits.) Now imagine you shape with a filter that does 12dB/octave (2nd order). Now that's 120dB or 20 bits for 1024 x oversampling. The Sabre DAC has a 5th-order modulator meaning it does 30dB/octave, however the 5 poles in the noise transfer function don't all kick in at the same point so it won't do the 30dB/octave * 10 octaves = 300dB. It does however have a digital noise floor of -200dB up to about 200kHz when clocked at 40MHz. This noise is impossible to reach in the analog domain since that is the amount of noise a 1-ohm resistor generates at room temperature, approximately.

Thorsten: I have to say that's appreciably more real raw resolution than most DACs on the market offer. The ESS Sabre DAC can actually represent CD data in the analogue domain with no or very little noise shaping. For reference, a highly regarded 32-bit DAC by another manufacturer uses a 32-level (5-bit) multi-bit section and 128 x oversampling at all data rates (7 bits), thus meaning the actual core of the DAC is able to provide only 12-bit real resolution without noise shaping.


It means the ESS DAC relies on appreciably less noise shaping to represent the full needed resolution than most (or at this time perhaps all?) others using the same principle - though it is less than what's attainable using a true multi-bit DAC. In fact it is barely able to match the mid 1980's TDA1541 in terms of real (non-noise shaped) resolution. For reference, if we combine analogue resolution (24 binary weighted bits) and the possibility to run at 8 x oversampling (3 bits), the Burr Brown PCM 1704 (the last true multi-bit audio DAC in production) allows us in effect 27 bits of analogue levels.


Dustin: Noise shaping took me a long time to wrap my head around but I think I have it figured out. At first I thought that if I have 1024 pulses which I can set at either 1 or 0, then surely I can get no better than 1/1024 resolution from this system. This is only true if you use the type of filtering mentioned above, i.e. a 1024-tap long rectangular shaped filter. This makes a SINC filter that has a 1st-order roll-off to the peaks of the lobes which is why I say above it's approximately a 1st-order roll off.


But the part left out of any text book I have read on DS (or SD) modulators is that the post filter after the DAC actually weights the pulses into it in time by its own impulse response. This is the 'magic' trick to how higher orders of modulation can actually give better than 1/1024 resolution with only 1024 pulses. However, the better than 1/1024 is only valid for a certain period of time. That's the tradeoff. This time is set to 5us or 200kHz. I think we can all agree that's sufficient for audio purposes. It's up to the post-DAC filter in the analog domain to suppress the out-of-band noise so it cannot affect things down the line like a preamp or a amplifier.
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