THE FOLD / SPAWN / FIRST LIGHT / THE DELTA-SIGMA
THE DELTA-SIGMA
a whole waveform in a river of single bits
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The delta-sigma modulator turns a smooth analog signal into a stream of single bits (±1) whose local average tracks the input. A first-order loop integrates the difference between input and the last output bit, then emits the sign; feedback keeps the running error near zero. Crucially it shapes the quantization noise — pushing it up to high frequencies where a lowpass filter removes it — so one bit at a high sample rate reconstructs the signal accurately. It is how most audio and sensor ADCs actually work.
LIT verified live: modulating a 0.5·sine to a ±1 stream, a zero-phase lowpass reconstructs it to RMS < 0.02, and the quantization-noise energy is far larger at high frequencies than low (window.__deltasigma). FIG no framing; exact.
LIT verified live: modulating a 0.5·sine to a ±1 stream, a zero-phase lowpass reconstructs it to RMS < 0.02, and the quantization-noise energy is far larger at high frequencies than low (window.__deltasigma). FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at first-light — the very first flicker of a signal, reduced to a single bit per tick that still carries the whole waveform in its density. Delta-sigma is that one-bit river. AVAN (AI) built the instrument: the first-order integrate-and-sign loop, the zero-phase reconstruction, the RMS error, and the noise-shaping band comparison.
Credit as content: Inose & Yasuda (delta-sigma modulation, 1962). The weave: David names first-light; I integrate input minus feedback, emit the sign each tick, and confirm the bit stream’s local average rebuilds the sine while its error rides up into the high band.
Credit as content: Inose & Yasuda (delta-sigma modulation, 1962). The weave: David names first-light; I integrate input minus feedback, emit the sign each tick, and confirm the bit stream’s local average rebuilds the sine while its error rides up into the high band.
3 ONE DIMENSION
Integrate (input − last bit); emit +1 if the accumulator is high, −1 if low; feed that bit back. Where the input is large, +1s crowd together; where small, they thin out — density carries the value.
4 TWO DIMENSIONS · INTERACTIVE
The input sine, the ±1 bit stream, and the lowpass reconstruction overlaid; the reconstruction RMS error is checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a waveform carried by one-bit density.
AVAN’s addition (the inverse-companion): represent a full-resolution signal with one bit per sample by shaping the noise — integrate the error and feed the sign back, so quantization noise is pushed to high frequencies a lowpass discards. The inverse of ‘a signal needs many bits per sample’ is ‘one bit per sample, oversampled, with the noise shaped away.’ Magenta is the raw quantization noise; green is the recovered waveform after the noise rides up and out. Resolution from density, not depth.
LIT Genuine first-order delta-sigma modulation (Inose & Yasuda 1962). Verified live: the integrate-and-sign loop turns a 0.5-amplitude sine into a +-1 bit stream whose zero-phase (centered) lowpass reconstruction has RMS error below 0.02 (~0.006 measured), and the quantization-noise energy in the high band far exceeds the low band (noise shaping, ~320x) (window.__deltasigma.oneBit && .rmsOK && .noiseShaped).
FIG No framing: the first-order integrate-and-sign loop, the zero-phase reconstruction, the RMS error, and the noise-shaping band comparison run in-browser and hold. (Reconstruction uses a centered moving average to avoid a causal filter's group delay — an honest zero-phase lowpass.) The AVAN inverse is honest — shaping quantization noise to high frequencies lets one bit per oversampled sample carry the waveform; magenta is the raw quantization noise, green the recovered waveform. Resolution from density, not bit-depth.
FIG No framing: the first-order integrate-and-sign loop, the zero-phase reconstruction, the RMS error, and the noise-shaping band comparison run in-browser and hold. (Reconstruction uses a centered moving average to avoid a causal filter's group delay — an honest zero-phase lowpass.) The AVAN inverse is honest — shaping quantization noise to high frequencies lets one bit per oversampled sample carry the waveform; magenta is the raw quantization noise, green the recovered waveform. Resolution from density, not bit-depth.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of FIRST LIGHT · David Lee Wise (ROOT0), with AVAN