# Dither: architecture, injection, identification and correction This is the entry point for the repository's Dither material. **A common PN generator does not imply a common injection node, estimator, or correction.** Parameter calibration, quantization randomization, DAC element selection and timing excitation are separate mechanisms. Random thermal noise and random aperture jitter are not deterministic parameters removed by these algorithms. Public Python entry points are `DitherGainTracker`, `apply_dither_gain` and `DitherHDCTracker`, imported from `adctoolbox` or `adctoolbox.calibration`. The first two cover the gain-only model; `DitherGainTracker` takes the stage's probe amplitude, initial gain, backend resolution and gain bounds explicitly. HDC covers the three-PN nominal-gain-2 model below and needs balanced random DAC role tags, or `estimate_dac_mismatch=False` for a stage without role swapping. Pass aligned observables, retain the tracker across contiguous blocks, and reset for independent captures. Plants and architecture controls remain examples; their simulation truth is not an estimator input. ## Shared excitation contract New comparisons use the HDC construction ``` p1,p2,p3 in {-1,+1}, separate persistent 31/29/23-bit LFSRs d[n] = (0.2/3)*(p1[n]+p2[n]+p3[n]) ``` The four levels are {-0.2,-1/15,1/15,0.2}; ideal probabilities are 1:3:3:1, variance is 0.2^2/3. PN streams are deterministic and finite-record correlations are not identically zero. The triple product p1*p2*p3 is a **digital correlation reference**, not an analog cubic operation. Injection peak is 0.2 in each model's declared node coordinates; this does not equate circuit voltage/power or ADC performance across architectures. Other examples use probes matched to their own estimators: exp_cal01 uses binary PN, exp_cal02/06/07 include a two-PN weighted four-level probe, and the randomization study in exp_cal02 uses continuous uniform dither. They do not implement this shared three-PN contract; changing their distributions without re-deriving their moments would invalidate the estimators. HDC's `--dither-peak` configures the plant, tracker and subtraction together; the default and all published acceptance results use 0.2. See the exposed parameter table in [HDC_FIXED.md](HDC_FIXED.md). A nonzero cubic denominator, adequate analog headroom and convergence are separate requirements. That guide also documents the offset/amplitude scan and large-to-small input continue/freeze controls in `hdc_small_signal.py`. ## Architecture and injection map | Architecture / objective | Physical node or action | Observable information | Identification / correction | |---|---|---|---| | Pipeline gain/cubic HDC | Stage DAC/reference charge before residue amplifier | Digitized residue, stage code, aligned PN | PN and triple-PN moments; gain and cubic approximate inverse | | Pipeline memory | Probe must traverse the path with memory | Current and delayed probe, residue | Lagged correlations; dynamic inverse with state | | Pipeline DAC mismatch | Random physical element/role selection | Selection tags, stage codes, residue | Code/element-conditioned moments; per-element or code correction | | SAR DAC weights | Offset charge at held-sample/CDAC comparator node, or redundant decision selection | Individual SAR decisions and known offset/selection | Paired conversion residual or bit-level correlations; weighted bit reconstruction | | TI timing | Common analog path **before** the aperture error, with known local probe slope | Channel labels, PN level/slope references, output | Timing-sensitive correlations; clock trim or digital resampling | | TI gain/offset | Probe through the relevant gain path; output moments | Channel labels and probe | Gain/offset statistics; channel scaling/offset correction | | Delta-sigma randomization | Input summing node or quantizer input; different transfer functions | Output stream and known probe | Dither decorrelates quantization; subtraction must match its transfer function | | Flash / generic static quantizer | Analog input or threshold network, depending on target | Output codes; threshold tags if explicitly modulated | Randomization alone does not estimate every comparator threshold | | Pipeline-SAR | Separate residue-amplifier and CDAC nodes | Stage residue plus SAR decision bits | Combine architecture-specific methods; no universal cubic correction | The map states physical requirements, not a claim that every row is a fully implemented ADC. A probe added downstream cannot characterize an upstream error it never experiences. A digital-only addition cannot excite an analog block. An unknown injection gain is confounded with the gain being measured. ## Evidence and scope **Published HDC:** [Panigada/Galton 2006](https://ispg.ucsd.edu/uploads/2017/05/2006-TCASI-A.-Panigada-Digital-Background-Correction-of-Harmonic-Distortion-in-Pipelined-ADCs.pdf) derives residue-amplifier polynomial calibration. The [2009 chip](https://ispg.ucsd.edu/uploads/2017/05/2009-JSSC-A.-Panigada-A-130-mW-100-MSps-Pipelined-ADC-With-69-dB-SNDR-Enabled-by-Digital-Harmonic-Distortion-Correction.pdf) implements gain/cubic HDC and separate DAC noise cancellation. Its small-signal measurements use coefficients trained at larger input then frozen. The [2013 follow-up](https://ispg.ucsd.edu/wordpress/wp-content/uploads/2017/05/2013-TCASI-N.-Rakuljic-Suppression-of-Quantization-Induced-Convergence-Error-in-Pipelined-ADCs-With-Harmonic-Distortion-Correction.pdf) addresses quantization-correlated estimation errors. Our three-PN example does not implement that follow-up's complete remedy. **Published SAR mechanisms:** [Chiu et al., SampTA 2013](https://www.eurasip.org/Proceedings/Ext/SampTA2013/papers/p544-chiu.pdf) describes offset double conversion (ODC), PN-based bit-weight calibration and redundancy dithering. ODC compares two conversions of the same held input with opposite offsets; a single SAR incurs twice the conversions. Our new SAR example adapts that residual equation to a varying three-PN offset and a declared sub-binary model. It is not a reproduction of the paper's silicon or its ICA/internal-redundancy algorithms. **Published TI placement:** the authors' entry for [ISSCC 2024, 22.4](https://icas.tsinghua-sz.edu.cn/WebPublications/PaperDetail/1477?unit=1002) describes a dithered reference clock injected at an input buffer and discusses path-mismatch limitations of other injection locations. Our linearly interpolated PN waveform is an analytic timing-observability model, not that circuit. No analog ramp generator or clock actuator is transistor-verified. **Delta-sigma foundation:** [Galton 2002](https://ispg.ucsd.edu/wordpress/wp-content/uploads/2017/05/2002-TMTT-I.-Galton-Delta-Sigma-Data-Conversion-in-Wireless-Transceivers.pdf) explains signal/noise transfer functions, granular quantization and overload limitations. Our first-order loop below is separately derived. A four-level PN sum does not satisfy every theorem for continuous random dither and does not guarantee removal of all idle tones. Source retrieval: 2026-11-02. Full HDC/SAR/Delta-sigma papers were inspected; the TI reference supports placement at author-abstract level only. ## Equations used in the new architecture experiments ### Pipeline: the probe must pass through the amplifier For an independent held residue u and d=A*sum(p), let F(v)=g*v+k*v^3. ``` before: r=F(u+d) -> E[(r-d)*p1*p2*p3]/(6*A^3) = k after: r=F(u)+d -> E[(r-d)*p1*p2*p3] = 0 ``` Thus identical PN values at different nodes carry different information. Balanced Cartesian enumeration removes sample variance for this boundary check; it is explicitly an analysis bench, not a background acquisition protocol. Additional controls include a fifth-order term and history-dependent nonlinearity: the fitted cubic coefficient then mixes unmodeled terms. Streaming convergence, fixed arithmetic and all four joint errors are already covered by [HDC_FIXED.md](HDC_FIXED.md), including DC/low-frequency overload. ### SAR: redundant decisions and paired residual The example uses 14 signed decisions and radix 1.85, not a binary 14-bit ADC. Weights nominally sum to one; four leading weights are unknown, the lower ten are held at their nominal values. Greedy decisions subtract +/-w_j from the current comparator residue. This assumes an ideal comparator apart from specified noise, known offset scale and sufficient architectural redundancy. For two conversions of the **same held input**, with injected offsets +d,-d: ``` Delta_b = b_plus-b_minus error = 2*d - sum(Delta_b[j]*w_hat[j]) w_hat[j] <- w_hat[j] + mu*Delta_b[j]*error / sum(Delta_b[0:4]^2) x_hat = sum((b_plus+b_minus)*w_hat_used)/2 ``` Only the first four weights update, after producing the current output. A zero denominator holds the update; weights have nominal +/-10% guards. Signed bit products and denominator values permit muxes/shifts and a small reciprocal table. This is floating arithmetic intended to expose the mechanism; only exp_cal08 has a fixed-point hardware budget. SVD/rank diagnostics are evaluation-only. The learner never sees the true input or true weights. Validation includes a new frequency with coefficients frozen and dither off, and an intentionally invalid pair using different held inputs. It also compares correct bit-weight reconstruction with inappropriate global cubic HDC on the same DAC-mismatch problem. A non-redundant SAR can lose information before digital correction; bit weighting cannot restore arbitrary gaps. ### TI: voltage PN needs a known time slope Let d_n be the same three-PN sum and take nonnegative skew 0<=tau<1 sample. A specified linear interpolation before the sampler gives ``` d(n+tau)=(1-tau)*d_n+tau*d_(n+1) y=g*[x(n+tau)+(1-tau)*d_n+tau*d_(n+1)] m0=E[y*d_n]/Var(d); m1=E[y*d_(n+1)]/Var(d) g_hat=m0+m1; tau_hat=m1/g_hat ``` This requires PN/input independence and the specified analog waveform, not just its four digital levels. A flat zero-order-held probe away from its edges has no skew information, as does a probe inserted after the sampler. Two matched failure controls use those placements. Streaming lookahead retains the same d_(n+1) at block boundaries. Evaluation applies estimated gain and a quantized ideal clock trim on an independent dither-off record. Real delay range, jitter, generator bandwidth and channel-path mismatch remain unmodeled. The linear ramp takes intermediate analog levels between the four digital endpoints while retaining peak 0.2. In the `after_sampler` control it also bypasses the modeled channel gain, so neither that gain nor skew is observable from its PN path. The next PN value is precomputed, not an unknown future input. ### Delta-sigma: subtract the correct transfer function Our first-order loop is ``` u[n]=state[n-1]+x[n]; y[n]=sign(u[n]+d[n]); state[n]=u[n]-y[n] e[n]=y[n]-(u[n]+d[n]) ``` Algebra gives `y=x+(1-z^-1)d+(1-z^-1)e` (with consistent initial state). Injection at the loop input instead gives `y=x+d+(1-z^-1)e`. Therefore quantizer-input dither requires subtracting **d[n]-d[n-1]**, not simply d[n]. This removes the explicit probe path, while e still depends on the randomized quantizer. It does not reconstruct the undithered bitstream or identify an analog error coefficient. Tests report in-band noise/spur levels for DC, in-band SNDR/SFDR for tones, and internal-state bounds; DC has no SNDR label. Subtraction is output postprocessing; the analog loop continues to feed back the unmodified quantizer output. ## Experiment and documentation navigation | Entry | Role and evidence | |---|---| | exp_cal01 | Binary-PN gain tracking, ramp/step and frozen foreground controls | | [DITHER_COVERAGE.md](DITHER_COVERAGE.md), exp_cal02 | Isolated mechanisms; distinct plants/probes | | [JOINT_DITHER.md](JOINT_DITHER.md), exp_cal06 | Derived joint inverse-cubic behavioral model | | [JOINT_CONVERGENCE.md](JOINT_CONVERGENCE.md), exp_cal07 | Statistical diagnosis; grouped matrix estimator is software-only | | [HDC_FIXED.md](HDC_FIXED.md), exp_cal08 | Published-form gain/cubic HDC, fixed arithmetic, extended memory/DAC model | | exp_cal09_pn_architectures | Shared-PN placement, SAR paired weight calibration, TI timing visibility, Delta-sigma subtraction | The examples are research/teaching models. `DitherGainTracker` and `DitherHDCTracker` are the public estimators; the frozen sine-fit baseline and the architecture-specific estimators remain example-only helpers. Full multi-stage silicon, general SAR weight calibration, nonlinear memory, arbitrary Delta-sigma topologies and complete TI PVT/jitter tracking are not implied by passing these tests. ## Reproduction and acceptance From the copied `11_calibration/` folder (or `python/src/adctoolbox/examples/11_calibration/` in a repository checkout), choose a new output directory: ```bash python exp_cal09_pn_architectures.py --output runs/pn-architectures ``` Numba is optional acceleration; output directories are never overwritten. The runner records arguments, source hashes, environment, PN distribution, per-case measurements and figures. Use `--quick` only for command/artifact smoke checks, not convergence qualification. Acceptance checks: no truth leakage into estimators; causal coefficient use; stream partition invariance; identical captures for paired method comparisons; probe peak/range accounting; independent-seed/held-out evaluation; expected failures retained; finite-record spectral metrics labeled by waveform/band; fixed-point claims limited to tested arithmetic; shipped examples run outside the repository. Dated experiment reports are snapshots, not user guides.