How a delta-sigma DAC works: oversampling and noise shaping

From samples to a fast, coarse pulse stream

A 16-bit sample describes one of 65,536 levels, and a converter that built each level directly would need parts matched to a tiny fraction of a percent. A delta-sigma DAC avoids that. It first raises the sample rate many times over with a digital interpolation filter, then hands the result to a modulator that can only output one bit, or a handful of levels, but does so millions of times a second.

No single output value is accurate. The average of many is. To make a level a quarter of the way up from the bottom with a one-bit output, the modulator spends about a quarter of its time high and the rest low, and the pattern shifts continuously as the music changes. Precision in amplitude has been traded for speed, which cheap silicon supplies easily.

The name comes from the loop inside the modulator. It measures the difference, or delta, between each pulse it sends and the signal it should be sending, sums those differences over time, which is the sigma, and uses that running total to choose the next pulse. An error is never left alone. It is carried forward and corrected by the pulses that follow.

Noise shaping and the filter after it

Describing a smooth waveform with so few levels leaves a large rounding error. Spread evenly across all frequencies, it would be loud hiss. Noise shaping is how the modulator's loop arranges that error: very little of it stays in the audio band, and most is pushed to ultrasonic frequencies far above hearing. Higher-order modulators push harder, and designers have to keep them stable while they do.

The result is a very low noise floor from 20 Hz to 20 kHz and a great deal of noise above it. An analog low-pass filter at the output removes that ultrasonic noise, so the filter is part of the converter. A poorly filtered output can send the noise on into an amplifier.

Two more refinements matter in current chips. Modulators with several levels suffer from small mismatches between their output elements, so the chip keeps rotating which elements it uses, and the mismatch averages out into noise. A small amount of dither also keeps a quiet or silent input from settling into repeating patterns, called idle tones, that could otherwise fall in the audio band.

Why nearly every DAC chip works this way

A one-bit output can only be high or low, and any two points make a straight line, so the conversion is inherently linear. Multi-level designs get close to that with the element rotation above. Nothing in the chip needs the precision a resistor ladder does, which makes delta-sigma cheap, small and consistent from one unit to the next. It is what you'll find in phones, laptops, dongles and most desktop DACs.

Measured performance followed. The SMSL PS200, an $89 DAC on an ESS ES9039Q2M chip, is rated at 0.0001% THD+N (−118 dB) and 123 dB of dynamic range. The SMSL PS100, built on the older ES9023, is rated at 0.005% (−85 dB) and 96 dB. Two DACs from one maker, on an older and a newer ESS chip, sit more than 30 dB apart on distortion.

Same chip, different DAC

A delta-sigma chip leaves several decisions to whoever builds the DAC around it: the digital filter setting, the clock, the analog stage that follows, and the power supply that feeds it all. Those choices are why DACs on the same chip can measure and sound different.

The PS200's review is titled on that point, that DAC chips are not everything. Its figures are excellent and it sounded fine without being exceptional, and the review heard the PS100 as generally similar, which it put down to the two sharing a similar output stage. The SMSL RAW-MDA1 uses two of the same ES9039Q2M chips, one per channel, with six OPA1612A op-amps after them, and its review describes it as warmer than the typical ESS box, with a forward midrange. The Henry Audio DA 256 is a €1,600 DAC built on a Texas Instruments PCM5142, with no upsampling ahead of it, the chip's own filter chosen by listening, and the effort spent on power and clocking.

Filter settings are the part you can change yourself. Many delta-sigma DACs offer sharp or slow roll-off, and linear or minimum phase. A sharp filter keeps the treble flat to 20 kHz and removes more ultrasonic images; a slow one rolls off earlier and rings less around a transient, and a minimum-phase filter moves the ringing after the transient. The measured differences are clear. Whether they are audible on music at normal levels is disputed, so compare settings blind before trusting a preference.

Delta-sigma, DSD and the R-2R alternative

A delta-sigma modulator's one-bit output is, in effect, what the DSD format stores: a recording kept as the fast pulse stream instead of as PCM samples. Most delta-sigma chips accept DSD directly, and the PS200 plays it up to DSD512.

The main alternative is the R-2R ladder, which builds each sample's voltage directly from a network of matched resistors. It needs precision parts where delta-sigma needs heavy filtering, and that page covers how the ladder works and what its matching costs. Neither architecture decides the result by itself. The DAC guide ranks both kinds side by side by score.