Supervised ADC post-correction: meaning, mathematics and evidence#

The short explanation#

During calibration, supply measured ADC outputs and an aligned reference. The polynomial or MLP learns a digital output-to-reference mapping, not individual capacitor errors, bit weights or amplifier gains. During normal operation, feed new ADC outputs through the frozen mapping to obtain corrected samples. No reference, fitting, backpropagation or parameter update is needed then. The MLP implementation adds a learned residual to a frozen affine baseline.

Calibration: ADC output/history + aligned reference -> fit -> saved model
Operation:   ADC output -> history buffer -> frozen model -> downstream DSP

This is foreground digital post-correction, after ADC conversion and usually before spectral analysis, measurement or demodulation. It does not change the analog conversion circuit. A background algorithm that identifies physical gain or bit weights is a different estimator, even if it also corrects digital data.

For history=3, the inputs are [y[n], y[n-1], y[n-2], y[n-3]]: four features with three preceding samples. This does not mean a third-degree polynomial or a third-order physical ADC. Polynomial degree and memory depth are separate.

What is the reference?#

Reference source	Requirements and limits
Independent reference ADC or instrument	Characterize gain, offset, distortion, bandwidth and relative sampling delay; compare aligned samples. A slower reference can label aligned instants if excitation remains adequate.
Characterized calibration source	The waveform amplitude, phase/timebase and source errors must be known to the required accuracy. Generator settings alone are not an accuracy certificate.
Sine fit to the DUT’s own output	A useful spectral-shape target under a sine assumption, but it retains the output’s fundamental gain/phase and fitted DC. It does not independently establish absolute input accuracy.

exp_cal03 and exp_cal05 use an independent 16-bit monitor followed by a known-frequency least-squares projection. For one tone,

\[r[n]=A\sin(\omega n)+B\cos(\omega n)+C.\]

Only the frequency is supplied; A, B and C come from monitor observations. For two tones, both sine/cosine pairs and one DC column are fitted. Projection reduces random monitor noise under this signal assumption; systematic reference errors remain. Arbitrary business signals cannot be labeled by a sine fit. Known-frequency sine fitting and its quantization-bias limitations are discussed in Kollár’s 2004 paper, full text.

The simulator’s clean input is used only by the evaluator. Our monitor assumes unity gain, synchronized sampling and no bandwidth or nonlinear error; it is a behavioral reference, not measured reference hardware. The configured gain-bias control explicitly demonstrates that monitor gain error enters the labels.

Polynomial: solve coefficients, then evaluate#

Let y be ADC output, r the aligned reference, and z the input features standardized using training-only statistics:

\[z_k[n]=\frac{y[n-k]-\mu_k}{\sigma_k}.\]

The implementation replaces a numerically constant feature’s scale with 1. With M features and total degree d, include the intercept and every monomial:

\[\phi_{\alpha}(\mathbf z)=\prod_{k=0}^{M-1}z_k^{\alpha_k}, \qquad \sum_k\alpha_k\le d.\]

For two features and degree two, the row is [1, z0, z1, z0*z0, z0*z1, z1*z1]. Four features and degree five need 126 coefficients; one feature and degree five need only six.

Stack rows into Phi and solve

\[c^*=\arg\min_c\left[\frac1N\|\Phi c-r\|^2+ \lambda\sum_{j>0}c_j^2\right].\]

This is nonlinear in the inputs but linear in the unknown coefficients. _ridge solves an augmented least-squares system, with Phi/r divided by sqrt(N) and a sqrt(lambda) diagonal penalty whose intercept entry is zero. It does not explicitly invert the normal-equation matrix. The API default ridge is 1e-8; the foreground benchmarks fix it at 1e-10.

In normal operation, compute the same standardized features and monomials:

\[\hat x[n]=\phi(\mathbf z[n])^T c^*.\]

There is no iterative inversion per sample. Coefficients describe the inverse correction, not the forward circuit. For example, if y=gx+kx^3, its weakly nonlinear inverse begins as x=y/g-k*y^3/g^4. Its cubic coefficient is not k. Bit-weight calibration instead fits offset + sum(weight_i * decision_i) to raw bit/segment decisions. Those are different features from powers of final code.

MLP: learn a residual, then add it to the baseline#

One hidden unit performs h = tanh(w*z + beta). A single input is multiplied by a trainable weight, shifted by a trainable bias, then passed through tanh. Multiple units learn different nonlinear responses and the output combines them. This is a feedforward MLP, not a recurrent network with additional hidden state. See Deep Learning, Chapter 6 for the general network definition and section 6.5 for backpropagation.

The actual implementation first fits and freezes an affine baseline:

\[x_{\mathrm{base}}[n]=a_{\mathrm{bias}}+\sum_{k=0}^{M-1}a_kz_k[n].\]

It uses training data and ridge=1e-10. Then construct

\[e[n]=r[n]-x_{\mathrm{base}}[n],\quad s=\max(\operatorname{std}(e),10^{-12}),\quad t[n]=e[n]/s.\]

The network predicts this normalized residual using H hidden units:

\[h_j[n]=\tanh\left(\sum_kW_{kj}z_k[n]+\beta_j\right),\qquad g[n]=\sum_jv_jh_j[n]+\gamma.\]

Symbol	Meaning
y[n], r[n]	Observable DUT output and aligned training reference
z_k, mu_k, sigma_k	Standardized feature and its training mean/scale
a_bias, a_k	Affine intercept and feature weights; fitted before network training
x_base[n]	Baseline prediction, not a hidden-layer bias
e[n], s, t[n]	Reference-minus-baseline residual, its scale and normalized target
W_kj, beta_j	Input-to-hidden weight and hidden-unit bias
h_j[n]	Hidden-unit response, between -1 and 1
v_j, gamma	Hidden-to-output weight and output bias
g[n]	Predicted normalized residual; its weighted sum is not bounded to [-1,1]

Train W, beta, v and gamma to minimize

\[L=\frac1N\sum_n(g[n]-t[n])^2+ \lambda\left(\sum_{k,j}W_{kj}^2+\sum_jv_j^2\right).\]

Biases are not penalized. The MLP’s default lambda is 1e-6 and applies to the normalized residual loss; it is not equivalent to the polynomial’s ridge prior. For one example without the penalty, let epsilon=g-t. The chain rule gives

\[\frac{\partial L_n}{\partial v_j}=2\epsilon h_j,\qquad \frac{\partial L_n}{\partial W_{kj}}=2\epsilon v_j(1-h_j^2)z_k.\]

The implemented gradients average these contributions over the batch and add the weight-penalty derivatives. SciPy’s L-BFGS-B optimizer uses the loss and analytic gradient to update the parameters. The budget counts optimizer iterations, not mini-batch epochs. The selected model minimizes unpenalized reference MSE on separate validation records among observed iterates. Reaching the iteration limit is not optimizer convergence or proof of overfitting.

At inference the baseline, residual scale and network weights are frozen:

\[\boxed{\hat x[n]=x_{\mathrm{base}}[n]+s\,g[n].}\]

For example, baseline=0.503 V, s=0.002 V and g=-1.5 give 0.500 V. This is an illustrative calculation, not a measured result. With four inputs and 16 hidden units there are 97 network parameters plus five baseline parameters; normalization statistics are stored separately.

Applicability and operating conditions#

Error / architecture context	Appropriate interpretation
Gain and offset	An affine model is usually sufficient.
Smooth, invertible static distortion in an ADC or its front end	Start with a static polynomial; MLP is an alternative learned mapping.
Repeatable short memory, such as acquisition/settling or reference-path effects	History models may help if features and excitation capture the dynamics. Our plant tests the repository’s previous coarse-quantized-output feedback, including its coarse/fine quantization; it does not validate every settling mechanism.
SAR capacitor mismatch or Pipeline stage errors visible in decision bits	Physical bit-weight/stage-code methods may be preferable. Final-code models do not identify those internal parameters.
Sharp code-dependent INL	A global low-degree polynomial may underfit; consider characterized LUT/bit-weight methods. Not validated by this smooth plant.
Time-interleaved mismatch	Channel identity, alignment or separate channel mappings are generally needed. The default four-history example is not a validated TI calibration.
Long memory, nonminimum-phase dynamics, nonmonotonicity or overload	A short causal inverse may be inadequate or unavailable; lost information has no general inverse.
Random thermal noise or aperture jitter	No per-sample deterministic correction is guaranteed. Signal-prior denoising is a different claim.
PVT drift or changing sampling rate/bandwidth	Frozen models may become stale; revalidate/recalibrate. No online adaptation is implemented here.

Training requires a dedicated reference capture and separate validation/test records. Normal operation requires identical feature order, units, timing and sampling conditions, but no reference. Keep history state only across contiguous blocks; reset it for independent captures. Corrected output is in reference units. The generic APIs can accept other feature matrices, but this does not validate every ADC architecture or make unobservable errors identifiable.

The capture helper now calls the original signal-generator APIs: static u=x+.015*x^2+.04*x^3, optionally apply_memory_effect(u, memory_strength=.04), noise, then a clipped 14-bit quantizer. The memory API combines coarse/fine quantization with delayed coarse-code feedback; it is not the removed custom linear FIR. Details and boundary handling are in FOREGROUND_WORKFLOWS.md. The static transfer is monotonic, but the memory model has code discontinuities. Reusing repository code does not establish physical fidelity or eliminate model selection bias. In particular, low-order smooth distortion favors polynomial approximation. Quantization and overload are not undone by a general inverse; the real-valued output is an estimate. Independent circuit/measurement evidence and device-parameter sweeps remain outside this behavioral comparison.

Single-tone training is especially restrictive for four history features. Ideal zero-DC sine samples satisfy x0*x2 + x2*x2 - x1*x1 - x1*x3 = 0, obtained from two sine recurrences. Distortion/noise can make the numerical design full rank without providing robust joint excitation. Inspect design_rank and design_condition_number (unregularized standardized design; None for numerical rank deficiency), and test independent waveform families. Marginal range flags are not joint-coverage guarantees. Even good conditioning is not a generalization certificate. Static distortion does not automatically require history features.

Physical evidence: what the papers do and do not establish#

All sources below were read as full text. They establish related methods on real converter data, not silicon validation of this repository’s exact models.

Original source	Physical evidence	Difference from this code
Vallant et al., 2011, A Linearization Strategy for Undersampling ADCs, sections III-V	LTC2208, 16-bit, 120 MS/s measurements; up to 25 dB SFDR improvement using dynamic nonlinear correction. Data processed in MATLAB; hardware implementation listed as future work.	Constrained Volterra/Hammerstein structure and dynamic-phase FIR design, not our full four-input fifth-degree ridge fit.
Bhanushali et al., Circuits-Informed Machine Learning Technique for Blind Open-Loop Digital Calibration of SAR ADC, author manuscript, pp. 1-3	28 nm, 12-bit, 84 MS/s main ADC and 10.5 MS/s reference ADC; shallow tanh error learning with history/derivative features. Near-Nyquist SNDR/SFDR improvements reported as 30.5/38.6 dB.	41 features and 40 hidden neurons, separately calibrated reference, different training design. Network power is a digital-synthesis estimate, not measured full-calibrator silicon power. No support for calling our 4-by-16 baseline/residual combination a reproduction.

The current NumPy/SciPy floating-point implementation has no measured ADC data, fixed-point error budget, FPGA throughput result or on-chip area/power claim. The literature’s spectral improvements do not transfer to our model, nor do they establish absolute accuracy under an uncharacterized reference channel.

Use packaged FOREGROUND_WORKFLOWS.md for runnable experiments, reference-bias controls, frozen-model replay and the optional mixed-excitation MLP comparison. Their numerical results are behavioral experiments with stated budgets.

See foreground workflows for commands, and Calibration API for signatures.