# HDC memory and capacitor-mismatch model basis Scope: the joint HDC model of `exp_cal08`, the role-swapping capacitor models of `exp_cal02` and `exp_cal06`, the waveform-error and SAR weight models already in ADCToolbox, and the papers these models rely on. The companion [switch-phase and memory-state derivation](hdc-physical-state-and-phases.md) adds a plate connection table, finite-reset and finite-open-loop-gain equations, and the conditions under which the PN path observes memory. **There is no basis for replacing these models wholesale with the existing ADCToolbox models. The capacitor models differ mainly because the circuits differ; the current HDC memory model is mathematically self-consistent but has weaker physical support, so its target state and approximation conditions must be stated.** The existing ADCToolbox models also include simplified behavioral models and are not a physical ground truth by default. ## 1. Criteria and conclusions Evidence is weighed in this order: circuit mechanism, state and node, mathematical equation, range of approximation, and only then code reuse. That a calibrator can invert a formula does not show that the formula comes from the target circuit; that a formula differs from the existing library does not show that it is unreasonable. | Model | Current evidence | Scope | Status | |---|---|---|---| | HDC one-tap residue memory | Explicit linear FIR; corresponds to the first-order expansion of a specific recursive residue path | Equivalent model when one tap dominates and PN and signal pass through the same dynamic path | Kept as a restricted behavioral model; not yet a general settling-error or dielectric-absorption model | | ADCToolbox previous-MSB leakage | Explicit code-dependent additive error; no derivation from device parameters to the coefficient | Equivalent coupling caused by the previous coarse decision | Kept; not a direct replacement for residue memory | | ADCToolbox incomplete sampling | Recursion with a tracking time and a time constant | Single-state settling with a target voltage roughly constant within the tracking window | For this mechanism its physical parameters are more directly interpretable; not equivalent to the reset memory of a whole MDAC | | Role-swapping sampling/feedback capacitors | Ideal charge conservation directly yields the rational expression | 1.5-bit stage whose two capacitors swap roles | Clear structural basis, kept; the selection logic is not an item-by-item reproduction of the paper's CFCS | | Current HDC fixed feedback capacitor with switched DAC elements | The current equation follows from the stated ideal charge-transfer convention | Fixed sampling/feedback ratio, one differential mismatch mode | Kept; does not generalize to all MDAC/CDAC mismatch | | ADCToolbox SAR capacitor-weight mismatch | Independent weight perturbations scaled by unit-capacitor area, applied in successive approximation | SAR behavioral model with static effective weights | Better suited to SAR; not a complete switched-capacitor circuit solver | ## 2. What the papers support ### 2.1 HDC 2009 The Panigada/Galton model targets a memoryless, weakly nonlinear residue amplifier; the chip calibrates DAC mismatch separately with DNC. The circuit uses independent switched-capacitor DAC networks and resets the amplifier to reduce memory effects. The paper supports the feasibility of HDC and of independent DAC networks, **but does not directly validate the one-tap memory, the two-element correlator or the combined parameter set added here**. See Sections II-B, III-C and IV and Fig. 16. [Paper](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) ### 2.2 Keane/Hurst/Lewis 2006 Starting from dielectric relaxation, incomplete reset and parasitic coupling through a shared op-amp, the paper derives a state equation that contains the previous stage output and the previous DAC code. Its unified form is \[ X_{i+1}(z)=\frac{G_i}{1-\gamma_i z^{-1}} \left[X_i(z)-(1-\delta_{m,i}z^{-1})K_iD_i(z)\right]. \] The paper's memory parameter is written here as \(\delta_m\) to avoid confusion with the capacitor mismatch \(\delta\) of this repository. The paper's two-tap FIR \(M_i(z)\) is an equivalent correction relation and cannot be used directly as a forward stage model. See Eqs. (14), (24), (32) and (65)–(73). [DOI](https://doi.org/10.1109/TCSI.2005.858760), [author preprint mirror](https://citeseerx.ist.psu.edu/document?doi=54e99e923feb71e2119dcd82b1cf57876baeec3d&repid=rep1&type=pdf) ### 2.3 Sun/Lee/Ham 2008 CFCS changes the feedback-capacitor selection to obtain different transfer curves, then extracts the mismatch by correlation. Its capacitor selection depends on the coarse decision and the CFCS mode, and the 1.5-bit case must also handle crossing transfer curves. The role-swapping models share the charge-conservation and random-switching principles but use a different selection logic and estimator; the current fixed-feedback model is even further from a direct reproduction of CFCS. See Figs. 1–5 and Sections II–IV; the paper is validated by behavioral simulation. [Paper](https://people.seas.harvard.edu/~donhee/SLH_TCAS2.pdf) The equivalences, error bounds and comparisons below are derived from the code; they are not validation results that the papers give for this repository. ## 3. Memory models: mathematics and physics ### 3.1 What system the current HDC model is In [hdc_fixed.py](../../python/src/adctoolbox/examples/11_calibration/hdc_fixed.py): \[ v_n=x_n-\frac{(1+\delta s_n)q_n}{2}+d_n, \quad t_n=v_n+h v_{n-1},\quad h=H/G, \] \[ r_n=g t_n+k t_n^3,\qquad g=G/2, \] Backend noise and quantization follow. The analog state stores **the residue plus PN before the previous nonlinearity, \(v_{n-1}\)**, not the previous actual amplifier output and not the previous coarse code. For \(k=0\) the forward dynamic transfer is \(g(1+h z^{-1})\). After gain and nonlinearity are removed, the corrector uses \[ \hat v_n=\hat t_n-\hat h\hat v_{n-1} \] to implement the inverse system \(1/(1+h z^{-1})\). This inverse is stable for \(|h|<1\), but a stable inverse does not mean that the forward model has been validated against a circuit. The model describes residual charge that acts as a one-tap linear coupling to the previous residue, with older history negligible. The code gives no quantitative mapping from a target circuit's capacitor states, reset phases and time constants to \(H/G\); the companion derivation works out such a mapping for one candidate connection, and there the stored state is not \(v_{n-1}\). The model therefore cannot yet be identified as an exact model of a particular device mechanism. ### 3.2 When it approximates the paper's recursive model Writing the paper's unified model in the time domain, without nonlinearity: \[ R_n=G(X_n-KD_n)+\gamma R_{n-1} +G\delta_m K D_{n-1}. \] Only when \(\delta_m=0\), with \(V_n=X_n-KD_n\), and when the PN also passes through this residue path, does this give \[ \frac{R_n}{G}=V_n+\gamma V_{n-1} +\gamma^2 V_{n-2}+\cdots. \] The current one-tap model can be the first-order truncation of this expansion with \(h=\gamma\). If \(\delta_m\ne0\), an independent delayed DAC-code term appears, which in general cannot be absorbed into the same \(hV_{n-1}\). The comparison also explains the FIR/IIR difference: the inverse of the recursive forward system \(1/(1-\gamma z^{-1})\) is the FIR \(1-\gamma z^{-1}\), whereas the inverse of the current forward FIR \(1+h z^{-1}\) is recursive. That both use two taps and both correlate delayed PN does not make them the same. For the first-order pole system assumed above, on the unit circle \[ \left|\frac{1}{1-hz^{-1}}-(1+hz^{-1})\right| \le\frac{|h|^2}{1-|h|}. \] | Parameters | \(h=H/G\) | Truncated second-tap coefficient \(h^2\) | Bound on the transfer difference | |---|---:|---:|---:| | INITIAL | 0.03093 | 0.0009565 | 0.0009871 | | FINAL | 0.05882 | 0.0034602 | 0.0036765 | Multiplying the bound by an amplitude bound of the input sequence gives the corresponding time-domain truncation-error bound; it is neither an SNDR prediction nor a measured circuit error. The FINAL one-tap coefficient is about 5.9%, so explaining it as a truncated weak pole requires evaluating the later taps against the target precision; calling them second order is not enough to neglect them. ### 3.3 The order of memory and nonlinearity is also a model assumption The current model chooses \(r=f(v+h v_{-1})\). Expanding the cubic term: \[ r=g v+ghv_{-1}+kv^3+3khv^2v_{-1} +3kh^2vv_{-1}^2+kh^3v_{-1}^3. \] This contains nonlinear memory cross terms, but all their coefficients are tied to the same \(k,h\). It equals neither \(f(v)+h f(v_{-1})\) nor a nonlinear settling process with independent states. The cascade order corresponds to a circuit only if the linear history coupling occurs before the static nonlinearity. The HDC cubic formula itself cannot establish this order. ### 3.4 The two existing ADCToolbox models are not automatically more realistic `apply_memory_effect` in [nonidealities.py](../../python/src/adctoolbox/siggen/nonidealities.py) actually computes \[ m_n=\lfloor16x_n\rfloor/16,\qquad \ell_n=\lfloor4096(x_n-m_n)\rfloor/4096, \] \[ y_n=m_n+\ell_n+\lambda m_{n-1}. \] It is an additive leakage that depends on the previous coarse code and suits constructing that kind of code-dependent error; the coefficient \(\lambda\) is not derived from switches, parasitic capacitance or a specific charge-transfer process. The implementation also uses `np.roll`, so the first sample refers to the last one. That is a periodic batch boundary, not the initialization of a circuit with continuous state. `apply_incomplete_sampling` computes, in DC-removed coordinates, \[ \tau_n=\tau_0(1+c x_n^2),\qquad a_n=e^{-T_{track}/\tau_n},\qquad y_n=(1-a_n)x_n+a_ny_{n-1}. \] If the target voltage within each tracking window is approximately constant, this is the discrete form of a single-time-constant settling equation; the quadratic voltage dependence of \(\tau\) is an additional behavioral assumption. Its state is the previous output, and its physical parameters are easier to interpret than an arbitrary one-tap coefficient, but it does not contain the full two-phase MDAC reset/amplify state or the DAC charge. The code also has no explicit current limit or maximum-slope constraint, so the word `slew-rate` in its docstring does not make it a hard slewing model. **Assessment: for single-node settling, the existing recursive model fits the problem better; for previous-coarse-code coupling, the existing coarse-code model fits better; for one-tap residue coupling, the current model is the more direct definition. The choice needs a circuit mechanism; it cannot be decided by which model the existing calibrator removes more easily.** ## 4. Capacitor-mismatch models: mathematics and physics ### 4.1 Two-capacitor role swapping has a direct charge basis `CapacitorPlant` in [dither_mechanisms.py](../../python/src/adctoolbox/examples/11_calibration/dither_mechanisms.py) and [joint_dither_model.py](../../python/src/adctoolbox/examples/11_calibration/joint_dither_model.py) use \[ C_f=1-\delta s,\qquad C_s=1+\delta s. \] If both capacitors sample the input and then one becomes the feedback capacitor while the other switches to the coarse reference level, the ideal charge relation is \[ C_fR=(C_f+C_s)x-C_s q, \quad R=\frac{2x-(1+\delta s)q}{1-\delta s}. \] This is not a polynomial assembled for the corrector: the denominator comes from the feedback capacitor, and the mismatch changes both the signal gain and the DAC weight. Defining \(y=(q+R)/2\) and solving the equation directly gives \[ x=y+\delta s(q-y). \] The exact inverse is a property of the ideal charge model and is not a circular argument. The model is limited to an ideal op-amp, complete charge transfer, fixed capacitor and reference values, and no parasitic or dynamic errors; random role selection is also not the complete CFCS logic of the Sun paper. The paper's ratio mismatch cannot be compared directly with the symmetric parameter used here. If in some state \(C_s=1+\delta\) and \(C_f=1-\delta\), then \[ \Delta=C_s/C_f-1=2\delta/(1-\delta). \] ### 4.2 The fixed-feedback model also has a conditional charge interpretation [HDC_FIXED.md](../../python/src/adctoolbox/examples/11_calibration/HDC_FIXED.md) uses the capacitor convention \[ C_f=15C_u,\quad C_x=30C_u,\quad C_{D,s}=15(1+\delta s)C_u,\quad C_{p,j}=2C_u. \] If the input sampling charge, the reference charge of the selected DAC element and the three PN charges are transferred, with initial charges reset, then \[ C_fB=C_xx-C_{D,s}q+\sum_j C_{p,j}p_j, \] \[ B=2x-(1+\delta s)q+\frac{2}{15}\sum_jp_j =2(u+d). \] Because \(C_f\) is fixed, \(\delta\) does not enter the denominator; this difference from the role-swapping model has a structural cause. After ideal inversion, \(y=x-\delta s q/2\) directly gives the target term of the current correlation estimate and digital compensation. These capacitor ratios have a charge-level interpretation, but the implementation has no switch-phase diagram that fixes the connection, reset path and load of the unselected elements; the companion derivation proposes one. The evidence level is therefore an ideal charge macromodel, not a verified switched-capacitor implementation. If the actual selection changes the feedback factor, load or settling time, gain, memory and PN amplitude can also depend on \(s\); the current model does not describe these couplings. ### 4.3 Degrees of freedom left out by one differential mismatch parameter Any pair of positive capacitors can be written as \[ C_+=\bar C(1+\delta),\quad C_-=\bar C(1-\delta), \quad \delta=(C_+-C_-)/(C_++C_-). \] so the symmetric form itself is reasonable. The model, however, also sets \(\bar C/C_f\), \(C_x/C_f\) and the injection-capacitor ratios to nominal values, so only the **differential mismatch mode** between the two DAC elements is active. For example, if \(\bar C/C_f=1+\epsilon_D\), the reconstruction also contains \(-\epsilon_D q/2\) and products with the differential mode. This coarse-code term generally cannot be removed by the overall gain or by the single \(\delta\). Sampling, feedback and PN-injection capacitor errors can also change the input ratio and the identifiable probe scale separately. The model therefore cannot claim to cover independent random mismatch of every capacitor, but studying one chosen mismatch mode is reasonable in itself. Whether to add these degrees of freedom depends on the target capacitor network and error budget. ### 4.4 Strengths and limits of the ADCToolbox SAR model `sar_apply_cap_mismatch` in [sar.py](../../python/src/adctoolbox/models/sar.py) uses \[ w'_j=w_j(1+\epsilon_j),\qquad \epsilon_j\sim\mathcal N(0,\sigma_u^2/N_j), \] and `sar_convert` then uses the true weights in successive approximation. This shows how SAR weights change decisions more directly than adding an arbitrary code error to the final output, and it suits statistical mismatch experiments with many weights. The function returns independently perturbed weights that are not renormalized, however. For an ideal charge-redistribution array scaled by its total capacitance, with \(w_j=C_j/C_T\), the first-order change is \[ \frac{\Delta w_j}{w_j}\approx \frac{\Delta C_j}{C_j}-\frac{\Delta C_T}{C_T}. \] The shared denominator correlates the weights; whether the total capacitance includes dummy, parasitic or split-bridge capacitors depends on the structure. This is a derivation about actual capacitor ratios, not a claim that the library's weight model is necessarily defective: it can fold the common ratio into the gain and model effective weights directly. It also does not solve switch phases, reference dynamics or parasitic networks. **Assessment: the ADCToolbox SAR model suits static SAR weight problems better, and the two pipeline models here suit their own capacitor networks better. None of the three can replace another unconditionally.** ## 5. What correlation identification additionally requires A reasonable model and identifiable parameters are two independent conditions. For the current model, \(w_n=v_n+h v_{n-1}\) after ideal inversion. For \[ E[w_np_{1,n-1}]/A=h \] to hold, the corresponding cross moments of the delayed PN with the other inputs and probe terms must vanish, and the PN must actually excite this memory state. If the PN is injected after the memory source, or the memory comes mainly from a coarse code on another capacitor, the correlation no longer represents this physical parameter directly. Similarly, for \(y=x-\delta s q/2\), \[ \hat\delta=-2\frac{E[s q y]}{E[q^2]} =\delta-2\frac{E[s q x]}{E[q^2]}. \] so \(E[sqx]=0\) and \(E[q^2]>0\) are required. In the implementation the coarse decision precedes the independent element selection, which matches this structural assumption; if the actual switch selection affects sampling or the coarse decision, the condition must be checked again. Finite PN windows, quantization correlation and approximate inversion still cause estimation errors; the existing correlation experiments diagnose them numerically and do not validate a device. ## 6. What is supported and what still needs evidence The current code and results can be kept, stated precisely as follows: **for a cascade of a single differential DAC mismatch with a fixed feedback capacitor, one-tap residue memory and cubic nonlinearity, the fixed-point numerical behavior of correlation calibration has been tested.** Still to be supported: 1. Which capacitor or amplifier node the memory state comes from, and why it stores \(v_{n-1}\) rather than the actual output or the DAC code. 2. Whether the target circuit's reset/settling equations can be truncated to one tap over the present \(h\) range and precision requirement, and whether the PN passes through the same state. 3. The phase and connection table of the fixed-feedback scheme, especially how the unselected DAC element and the three injection capacitors connect and reset. 4. To claim general capacitor mismatch, whether errors of the mean DAC ratio, the sampling/feedback ratio and the PN ratio are calibrated separately; they cannot all be assumed to be part of \(\delta\). The [companion derivation](hdc-physical-state-and-phases.md) answers items 1 and 3 for its candidate connection: the stored states are capacitor voltages and the previous output, and the one-tap model is a conditional first-order approximation. Items 2 and 4 remain open for a specific target circuit. With that evidence, choose by mechanism. If the target state is previous-output feedback from incomplete reset, its state equation decides whether the plant should become recursive; if it really is one-tap residue coupling, the current model can stay. For a swapped feedback capacitor use the role-swapping equation; for an independent DAC with a fixed feedback capacitor, keep the current charge form. This requires neither larger experiments nor replacing the existing library functions first. The next threshold is the physical state and phase relations, not better-looking calibration results. ## 7. Checks - The code was compared with the paper equations and circuit diagrams; the relevant HDC and CFCS figure pages were checked against local PDF renderings. The Keane author-site PDF returned HTTP 403, so its equations were checked against the readable author preprint mirror; figures that could not be read are not used as evidence. - Exact rational arithmetic checks 216 charge-relation/inversion combinations, covering positive, negative and zero mismatch, two selections, three coarse codes and four PN sums. This verifies algebraic consistency only, not a physical implementation or calibration performance. - From the repository root, `python docs/experiments/check_hdc_model_basis_20261102.py --output ` repeats these checks; it does not overwrite an existing record. - The bound for truncating a first-order pole to a one-tap FIR is analytic; it is not an ADC measurement or a convergence experiment. - The [audit record](hdc-model-basis-audit-20261102.json) stores the computed results and the SHA-256 of the five model sources examined.