Fixed-amplitude joint convergence study

Hardware scope: Keep this estimator as a software diagnostic reference. The target project requires a simple hardware implementation; the grouped covariance weighting, eigendecomposition and SVD have not been validated in fixed-point RTL, synthesis, transistor-level simulation or silicon. Floating- point spectral improvements are not hardware performance claims. Subsequent implementation work must start from an injection circuit and a bounded digital datapath, with correlation accumulators and scalar updates preferred where their identifiability and stability have actually been verified.

exp_cal07_joint_convergence.py diagnoses the gain/cubic fluctuations in the joint experiment (exp_cal06) without modifying it. This is an example, not a public API.

python 11_calibration/exp_cal07_joint_convergence.py --output output/joint_convergence

The default uses independent validation seeds 101, 202, 303; only seed 44 was used in the small exploratory comparison. Probe peak is fixed at 0.2, input peak at 0.8, block size at 32768. All methods process identical captures. The probe is the samplewise weighted sum (2*p31+p29)/3, not concatenated time segments, Gaussian noise, or a single binary PN. Its actual injected levels are {-0.2,-0.2/3,+0.2/3,+0.2} in the normalized w/2 coordinate. The separate binary PN s controls capacitor roles. Backend Gaussian noise with sigma 0.0002 is a distinct physical noise source, not the dither.

Experiments and controls

The five estimators are pooled moments with EMA weights 0.04 and 0.01, cumulative pooled moments, and code-stratified moments with weights 0.04 and 0.01. Each starts from nominal coefficients and updates after correcting the current block. Neither signal frequency, clean input, device truth nor the known evaluator change point enters an estimator. No estimator is reset at the step. The capacitor equation and acceptance guards are unchanged.

For each seed, both sine and multitone inputs run for 256 blocks, with either constant FINAL parameters or INITIAL-to-FINAL parameters at block 128. The constant case uses FINAL here to match the long-record bias controls; exp_cal06 uses INITIAL for its constant case. Do not compare their constant-case averages without accounting for that difference.

An additional 1024-block sine record compares the default joint plant with a 24-bit, zero-noise joint plant and a 24-bit, zero-noise plant with H=delta=0. Their input and PN streams match. Saved block moment matrices permit replay of averaging policies without regenerating signals. An oracle-only diagnostic fixes inverse cubic coefficient c3 to truth and fits c1,h; it is never used to report corrected-output SNDR/SFDR. QR reparameterization of the exactly identified equation checks whether numerical conditioning alone changes the solution.

Cartesian-product controls enumerate independent input/probe/switch/noise states, removing accidental finite PN/input correlation. They check exact identification, quantization error and independent symmetric backend-noise bias. The two-point noise integrates cubic noise moments exactly; it is a diagnostic distribution, while streaming experiments use Gaussian noise.

Why exp_cal06’s EMA averaging can fluctuate indefinitely

The update is an EMA of correlation matrices followed by a direct solve, not gradient descent. With block-independent equal-variance moments, the asymptotic variance-equivalent record length is

N_eff = block_size * (2-alpha)/alpha
alpha=0.04: 49 blocks = 1,605,632 samples
alpha=0.01: 199 blocks = 6,520,832 samples

These are idealized variance-equivalent counts, not exact independence claims for a finite PRBS stream. The first block has weight one, so startup behavior also matters. After a parameter step, old moments decay as (1-alpha)^k: at 128 blocks about 0.54% remains for 0.04, but 27.6% for 0.01. Cumulative averaging reduces stationary variance but keeps stale pre-step observations.

Preserve valid conditional moments instead of pooling them

The pooled estimator of exp_cal06 centers three instruments within each of nine observed (q[n],q[n-1]) pairs, then pools them into three equations. The grouped estimator retains a separate equation for each occupied group. With

z = [p, p^3-41*p/45, p_previous]
f = [r, r^3, -a*p_previous]
theta = [c1, c3, h] = [1/G, beta/G, H/G]
A_g = mean(1_group * centered(z) * f^T)
b_g = mean(1_group * centered(z) * a*p)

it fits up to 27 moments to three unknowns. Conditional exogeneity is valid in this plant because q is determined before the probe and independently of the capacitor-switch sequence. Business residue remains a nuisance, but different coarse-code pairs retain different cubic response sensitivities. Both schemes center within the same groups; the difference is retaining these moments separately, not adding new analog excitation or assuming a sine fit.

Weight matrices come from the within-group covariance of z*residual, using start-of-block coefficients for residuals, and are averaged at the same rate as A and b. Eigenvalue truncation removes empty/numerically unsupported directions. A whitened SVD solve avoids squaring the condition number. This is feasible generalized method of moments (GMM); the weights do not model all temporal correlation and are not claimed universally optimal. Finite records and estimated weights can still introduce bias and variation.

An invertible basis change to the three pooled equations cannot supply additional statistical information. The improvement being tested comes from additional valid conditional moments; whitening is numerical implementation.

Bias boundary

For independent symmetric additive backend noise e with variance sigma^2, E[(r+e)^3 | r]=r^3+3*sigma^2*r. With no quantization, this shifts the fitted inverse linear coefficient by -3*sigma^2*c3. It does not make the physical gain/cubic ratios exactly unbiased. The balanced controls quantify the shift and separate it from finite excitation variance. Unknown probe amplitude, wrong delay, clipping or a missing physical nonlinear term can create much larger structural errors; this study keeps the model assumptions of exp_cal06.

Outputs and interpretation

Per-block CSV files retain physical parameter errors, statuses, SNDR/SFDR, headroom and a causality check. Summary windows are the second quarter, first quarter after midpoint, and final quarter. Multitone spectral quantities are desired-tone to residual-power and largest desired-tone to spur ratios, not single-tone SNDR/SFDR. Parameter bands remain gain 1%, H/delta/beta 10%.

long_summary.csv reports the last quarter of each acquired-record length. Its diagnostic solves are post-update estimates without deployment bounds, whereas main-run parameters and spectra use accepted start-of-block estimates. Those overlapping windows are not independent Monte Carlo trials. Aggregate uncertainty should be evaluated across seeds, not by treating all time blocks as independent. Three seeds and two stationary input spectra are evidence for this example, not a universal convergence guarantee.

configuration.json saves source hashes and versions; audit.json records sample counts, clipping and causal application. Output directories must be new or empty. fixed_probe_convergence.png and sample_budget.png provide figures.

The fixed-seed measurements and remaining multitone limitations are in the archived convergence report linked from docs/experiments/dither-results.md in the repository.