Open any modern audio DAC chip — ESS Sabre, AKM Velvet Sound, the Cirrus part inside your iPhone — and trace the signal path. The input is 24-bit PCM at 48 kHz. The output is analog. Somewhere between those two points, almost every commercial design squeezes the data through a 1-bit converter switching at several megahertz. That is not a typo. The final stage of a 24-bit, 144 dB-dynamic-range DAC is a single bit toggling between two voltages.
This is sigma-delta modulation, and once you understand why it works, you also understand DSD, SACD, why 1-bit DACs sound clean, and the strange identity between dither noise shaping and converter design.
The problem with 1-bit quantization
A 1-bit converter has two output states. By the standard formula SQNR ≈ 6.02·N + 1.76 dB, that gives you about 7.78 dB of signal-to-quantization-noise ratio at the Nyquist rate. Useless for audio.
But that formula assumes you sample at the Nyquist rate and that quantization noise is uniformly spread across the band from DC to fs/2. Both assumptions can be broken.
Trick one: oversampling
Sample faster than you need to, and total quantization noise power stays roughly constant — but it spreads over a wider frequency band. The portion that lands inside your audio band of interest (say, 0–24 kHz) gets smaller. Oversampling ratio (OSR) is the multiplier above Nyquist.
Each doubling of OSR buys you 3 dB inside the audio band. To reach 100 dB SQNR with a 1-bit converter using oversampling alone, you would need an OSR of roughly 2.5 billion — a sampling rate around 5 THz. Not viable.
Oversampling on its own is not the answer. It is half the answer.
Trick two: noise shaping
Now the clever part. Wrap the quantizer in a feedback loop with an integrator. The structure looks like this: subtract the output from the input, integrate the difference, then quantize. Feed the quantized result back to the subtractor.
In the z-domain, this gives you two transfer functions. The signal transfer function (STF) is approximately 1 — your audio passes through unchanged. The noise transfer function (NTF) becomes 1 − z⁻¹ for a first-order modulator, which is a high-pass filter with a zero at DC. The quantization noise that the 1-bit decision adds gets pushed up the spectrum, away from your audio band, while the signal sits unaffected at the bottom.
Each integrator you add raises the order. The NTF becomes (1 − z⁻¹)ᴸ for order L. The slopes:
- 0th order (oversampling only): 3 dB per OSR doubling
- 1st order: 9 dB per OSR doubling
- 2nd order: 15 dB per OSR doubling
- 3rd order: 21 dB per OSR doubling
A 3rd-order modulator at OSR of 64 (typical for audio DACs running at 64·48 kHz = 3.072 MHz) easily clears 120 dB SQNR inside the audio band. The noise hasn't disappeared — there is enormous quantization noise above 20 kHz. A simple analog low-pass filter at the DAC output removes it before it reaches your speakers.
Why higher order is not free
You cannot just keep stacking integrators. Above 2nd order with a 1-bit quantizer, the loop becomes conditionally stable. Large input transients can drive the integrators into states from which the loop never recovers — the modulator latches up, output saturates, and the analog signal turns into a buzzing carrier.
Lee's rule gives a practical bound: keep ‖H‖∞ < 2 for the NTF and you have a reasonable safety margin with a binary quantizer. Tighter bounds buy you stability but flatten the noise-shaping slope, costing in-band SNR.
This is why high-end audio converters do not just crank the order. They use MASH architectures (Multi-stAge noise SHaping) — cascade two or three independently stable lower-order modulators, then digitally combine their outputs so that the quantization error of the first stage is cancelled by the second. You get the steep noise shaping of a high-order modulator without the instability. Every modern audio DAC that handles real-world signals reliably uses some variant of this idea: ESS calls it Hyperstream, AKM has its OSR-Doubler architecture, Cirrus and TI have their own takes.
DSD is just the raw 1-bit output
Direct Stream Digital — the format SACD ships in — is a sigma-delta bitstream with no decimation. DSD64 is 1-bit at 64·44.1 kHz = 2.8224 MHz. DSD128 doubles that. DSD256 doubles again.
This means: a DSD file is essentially the unprocessed output of a sigma-delta ADC. Playback can be remarkably simple — feed the bitstream into a low-pass filter and you have analog audio. No PCM-to-DSD conversion stage, no extra modulator, no extra noise shaping.
The catch: any signal processing in the DSD domain is hard. Mixing, EQ, level changes — all of these typically require converting back to PCM, processing, and re-modulating. SACD mastering pipelines work around this by doing as much as possible in PCM and modulating only at the very end.
The connection to dithering
The noise-shaping math is identical to the technique we covered earlier in the dithering article. When you noise-shape a 16-bit dither, you are running a 1st- or 2nd-order delta-sigma loop in software, with the quantizer being the truncation from 24 to 16 bits. The same 1 − z⁻¹ filter is doing the same job: pushing quantization noise above 4 kHz where the ear is less sensitive.
The difference is purely scale. A 16-bit shaped dither runs at 48 kHz with maybe 5th-order shaping. A DAC modulator runs at 3 MHz or higher with 3rd-order shaping plus MASH.
Why everything converged on this design
R-2R ladder DACs — discrete resistor networks where each bit corresponds to a current source — can in principle hit any resolution if the resistors are precise enough. They were the standard until the 1990s. Their problem is matching: a 24-bit R-2R DAC needs resistors matched to one part in 16 million. That is brutal.
A 1-bit DAC has exactly two output states and one switching threshold. As long as those two voltages are stable, the converter is perfectly linear by construction. There is no monotonicity problem, no missing codes, no DNL issues. Combine that with 100+ dB of in-band SNR from noise shaping, and the engineering tradeoff is obvious: take the simplest possible DAC and let the math do the rest.
The DAC industry made that bet thirty years ago. Today, nearly every audio chip you use — phone, laptop, audio interface, hi-fi DAC — runs the signal through a sigma-delta modulator before it reaches air. The 24 bits at the input are real. The bit at the output is also real. The clever part is what happens in between.
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