Dither mechanisms and reproducible experiments#

Start with DITHER_GUIDE.md for the architecture/injection map and the shared three-PN experiments. This document covers the isolated-mechanism models and their different probe distributions. Published-form fixed-point HDC is documented separately in HDC_FIXED.md.

exp_cal02_dither_coverage.py separates randomization from parameter identification. It extends the gain-only exp_cal01 example with linear memory, capacitor-role modulation and nonlinear moment identification. These are software behavioral examples, not public calibration APIs.

For all four errors in one conversion path, see JOINT_DITHER.md and exp_cal06_joint_dither.py. That joint model uses a 1.5-bit capacitor stage (nominal gain 2) and a newly derived joint estimator; it is not a direct concatenation of the isolated gain-4 helpers.

Run and outputs#

From a copied example tree:

python 11_calibration/exp_cal02_dither_coverage.py --output output/dither_coverage

Defaults: 4 mechanisms x 2 waveforms x 2 scenarios x 3 seeds = 48 runs. Each run has 256 x 32768 normal-operation samples (83.88608 ms at a behavioral 100 MS/s), preceded by 16 x 32768 dedicated foreground samples (5.24288 ms). The static case retains initial parameters; the step case changes them at 41.94304 ms. Changes are deliberate stress tests, not measured PVT rates.

The output includes configuration and source hashes, per-block CSVs, individual/aggregate JSON summaries, aggregate CSV, two PNG figures, 72 randomization measurements and a JSON file of failure controls. The latter 72 measurements are 3 seeds x 3 amplitudes x 2 upstream-nonlinearity cases x 4 dither modes. Existing nonempty output directories are rejected.

Short command-level check (not enough data to qualify convergence):

python 11_calibration/exp_cal02_dither_coverage.py --blocks 16 --block-size 1024 --foreground-blocks 1 --seeds 11 --mechanisms memory --waveforms sine --scenarios step --output output/dither_smoke

Experimental contract#

  • The plant receives the true input and device parameters. Background trackers receive only observable subcodes/backend data, the known probe/switch sequence and a rail flag derived from output codes. True overload counts are evaluator diagnostics and never drive updates.

  • The current block uses the estimate available at its start; update prepares the next block. No best-SFDR selection or truth-based stopping is used.

  • PN, analog history, inverse-filter history, noise streams and waveform phase persist across blocks. A seed changes the LFSR initial state and noise.

  • Real LFSRs are used: a 31-bit maximal-length sequence (period 2^31-1), plus a 29-bit sequence for the second bit of the four-level nonlinear probe. Finite windows are not exactly independent of the deterministic input.

  • Sine input: peak 0.8, frequency nearest odd bin to 0.137 fs. Two-tone input: peaks 0.46 and 0.34, nearest odd bins to 0.083 fs and 0.217 fs. The latter metric is total desired-tone / noise+distortion, not single-tone SNDR/SFDR.

  • Backend range is [-1, 1]; Gaussian backend-input noise RMS is 0.0002. This mechanism-isolation experiment has no input sampling noise, jitter, comparator offset or reference dynamics. Its noise floor is consequently different from exp_cal01 and the five-stage research reconstruction.

  • All main methods share the same probe-on captures. A paired probe-off plant uses identical analog-noise draws. In the capacitor case, “probe off” means fixed capacitor roles (p=+1), not zero mismatch.

  • The oracle uses true coefficients only for evaluation. It corrects the imperfect capture; it does not remove its noise or quantization error.

  • Reported windows average per-block dB values, then average across seeds. No waveform fitting is performed before error-SNR evaluation. FFT ratios exclude DC; error-SNR includes absolute gain/phase/DC error against truth.

  • Different mechanisms use different plants. Comparing their absolute dB values is not an algorithm ranking or a comparison with commercial chips.

1. Subtractive dither: randomization, no learning#

A monotone 14-bit quantizer has thresholds

t[j] = -1 + j*LSB + 2*LSB*sin(2*pi*j/64), LSB=2/2^14.
d ~ Uniform(-32*LSB, +32*LSB).
y = Q(x+d)-d.

Four modes are compared: no dither, unsubtracted dither, ideal subtraction, and a 1% error between actual analog injection and digital subtraction. Three input peaks (0.01, 0.1, 0.8) are used. A second plant includes an upstream cubic term x+0.03*x^3 BEFORE injection. Dither cannot undo that upstream transfer function in this model.

The periodic INL is chosen to expose the mechanism; it is not an extracted LTC2208 or AD9208 transfer curve. Dither may spread spurs into a noise floor, improving SFDR without improving integrated SNDR. This experiment is not parameter calibration, despite its relationship to commercial dither features.

2. Gain and linear memory#

The seven-level coarse quantizer is identical to exp_cal01; q is its DAC level. The backend has 12 bits. Let v=x-q+a*p, with a=0.0625.

r[n] = G*v[n] + H*v[n-1] + backend_noise, followed by quantization.
P[n] = [p[n], p[n-1]].
theta_observed = inv(Cov(P,P))*Cov(P,r)/a = [G,H], approximately.
theta_next = (1-0.08)*theta_used + 0.08*theta_observed.
v_hat[n] = r[n]/G_hat - (H_hat/G_hat)*v_hat[n-1].
y[n] = q[n] + v_hat[n] - a*p[n].

Solving with the measured PN covariance handles finite-window cross-correlation between the two probe columns. It does not eliminate correlation with unknown user input. The gain-only control uses only the current PN and fixes H=0.

Initial/final gain is 3.88/4.08. The gain case has H=0 throughout; the memory case has H=0.12/0.24. Inversion is stable for these values; general dynamic paths can have nonminimum-phase zeros or noise amplification and need a different correction design. The model is a two-tap equivalent residue path, not the full dielectric-relaxation/shared-opamp model of Keane et al.

Foreground identification uses a dedicated probe-off sine and fits r = -G*q - H*q_previous + A*sin(w*n)+B*cos(w*n)+DC. Frequency is known; amplitude, phase and DC are fitted. True x is not a label. The first startup sample is excluded from this foreground regression.

3. Capacitor mismatch: random role switching and code-weighted correlation#

Two physical capacitors are normalized to C1=1+delta, C2=1-delta. PRBS p chooses Cf=1-delta*p, Cs=1+delta*p. The 1.5-bit coarse code q is -1, 0 or +1, with thresholds -0.25/+0.25. The backend has 13 bits. Ideal charge redistribution gives

r = ((Cf+Cs)*x-Cs*q)/Cf = (2*x-(1+delta*p)*q)/(1-delta*p).
y_raw = (q+r)/2.
x = y_raw + delta*p*(q-y_raw), before backend noise/quantization.

To identify delta, use an observable instrument z=(p-E[p|q])*q:

E[z*x] = 0 under independent switching.
delta = -E[z*y_raw] / E[z*p*(q-y_raw)].

Both moments use exponential averaging with factor 0.08. Mean p is computed within each visited coarse code in the current block. This removes constant within-code nuisance terms without accessing the true input. The initial/final delta is 0.006/0.012; this is one capacitor-pair mismatch parameter, not a Monte Carlo sigma for a multi-element DAC.

The unweighted control replaces z with centered p. For symmetric input its denominator tends to zero, so it is held by a rank/excitation guard. Merely averaging longer cannot create the missing information. If only q=0 is visited, the selected code-weighted instrument also has no excitation and is held.

This example follows the published principle of capacitor switching plus correlation, but differs from Sun/Lee/Ham’s CFCS selection logic. We use exact capacitor ratios and a derived observable moment equation, not their first-order approximation. It is not a reproduction of their 15-stage ADC.

Foreground identification retains switching during a dedicated sine and fits y_raw+delta*p*(q-y_raw) = A*sin(w*n)+B*cos(w*n)+DC, then freezes delta.

4. Nonlinear response: multiple probe levels and inverse moments#

The coarse stage/backend match the gain model. The amplifier has a monotone compressive characteristic implicitly defined by

r + beta*r^3 = G*v, v=x-q+a*p.
v = c1*r+c3*r^3, c1=1/G, c3=beta/G.

The plant solves the forward equation with Newton iterations before adding backend noise and quantization. This model deliberately has an exact cubic inverse. An arbitrary amplifier does not; model mismatch remains future work.

Use four nominally equiprobable levels p in {-1,-1/3,1/3,1}, a=0.1. Define features phi=[r,r^3] and instruments Z=[p,p^3-(41/45)*p]. The coefficient 41/45 equals E[p^4]/E[p^2] for this distribution. Independence of p from x-q gives two moment conditions:

Cov(Z,phi) * [c1,c3] = Cov(Z,a*p).
y = q+c1*r+c3*r^3-a*p.

The matrix and RHS are exponentially averaged with factor 0.04. Instrument rank, matrix condition and monotonicity/broad parameter bounds are checked. Noise/quantization make these moments approximate; no claim of statistically unbiased finite-sample estimates is made.

Initial G=3.88, beta=0.16; final G=4.08, beta=0.24. The scalar control uses only p and r, and can fit only an effective gain. A binary p makes the two selected odd instruments collinear and is explicitly rejected. This does NOT prove that every nonlinear calibration must use four levels: binary schemes can add windowing, other observables or structural assumptions.

This is our derived educational inverse-moment estimator. The cited parallel correlation paper instead uses two asymmetric dither windows and a different estimator; no exact reproduction is claimed. Foreground identification fits q+c1*r+c3*r^3 = A*sin(w*n)+B*cos(w*n)+DC with probe off, then freezes c1/c3.

Controls and coverage boundaries#

diagnostics.json includes a one-sample reference delay error, unknown probe amplitude, deliberately input-correlated probes, an adversarial short-period input, binary-probe rank loss, insufficient capacitor-code coverage and physical backend clipping. A guard can stop an invalid update; it cannot undo clipping.

The main runs isolate four mechanisms rather than combining all of them in one ADC. They do not cover complete multi-stage joint identification, dynamic reference settling, nonlinear memory, aperture jitter, metastability, fixed-point implementation or chip area/power. exp_cal01 remains the example for continuous gain ramp tracking and probe-amplitude sweeps. Band-limited non-subtractive dither, DEM/DNC and a full dual-dither circuit are documented in the literature review but not numerically reproduced here.

Primary references and implementation relationship#

Source retrieval date: 2026-11-01. The date is not the publication date.