1. FE8113 ”High Speed Data Converters”

2. Part 2: Digital background calibration

3. Paper 9: Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”, IEEE Transactions on Circuits and Systems-I: Regular Papers, Vol. 51, No. 1, January 2004, pp 38-46Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters

4. This week’s funny picture:

5. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters Outline: We present an adaptive digital technique to calibrate pipelined analog-to-digital converters. The new approach infers component errors from conversion results and applies digital postprocessing to correct those errors. With the help of a low, but accurate ADC, the proposed code-domain adaptive FIR filter is sufficient to remove the effect of component errors.

6. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters Introduction  Pipeline ADC High throughput and high resolution Accuracy limited to about 10 bits without calibration.  Front-end T/H bandwith limitation, capacitor mismatch, finite opamp gain, charge injection...  Digital calibration techniques Analog signal paths disturbed during calibration Foreground calibration suffers from lack of tracking capability ”Accuracy boot-strapping”  Calibration is performed stage-by-stage from LSB to MSB  Sequental operation cumbersome to implement, adds digital and analog overhead  This calibration technique Treats errors analogous to distortion in communication channels Errors from all stages removed simultaneously The analog signal path is completely intact Corrects dominant memory-less linear errors  Capacitor mismatch, finite opamp-gain, switch-induced offset errors

7. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters Nonlinear channel model  Dynamic and static errors cause nonlinearity Finite bandwidth and slew-rate introduce memory effects  Distortion Capacitor mismatch, finite opamp-gain and charge injection introduces static errors  INL, DNL, offset, gain-error  ADC-model: Decision vector:  D=[D 1 D 2.....D N ] T  D K is decision from k’th stage Weighting vector (1.5bit):  W=[2 N-1 2 N-2....2 0 ] T Ideal ADC decision is thus:  Linear transformation of the code vector

8. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters Nonlinear channel model:  A real ADC exhibits nonlinearity and memory effect. How do we compensate?  Volterra filtering: Of the digital output W T D  Correct decision can not be recovered when a multiple-to-one mapping from V in to W T D occurs (e.g. Nonmonotonic codes)  High-order Volterra filter is very complex.  Non-linear transform: The correct decision can be obtained by a non-linear transform of D  Analog errors modelled as a code- domain non-linear channel.  Still a form of Volterra-filter but the operand is vector D in lieu of the scalar W T D  Assume D as a sufficient statistic for D in =V in /V r, the quantized version of Vin, when quantization error is ignored.

9. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters Code space and sufficient statistics:  Redundancy and error correction: ”The final decision is not made until the last stage resolves where circuit nonidealities are negligible due to large residue gain accumulated.”  {D K } is a binary decomposition of D in. {D K } fully span the code space of D in if no missing codes occur. {D K } consequently represents a set of sufficient statistics for D in.  I.e. D in can be fully recovered from {D K }, when quantization and circuit noise are ignored.  Not susceptible to nonmonotonicity as the Volterra filtering of W T D. In the rest, we focus on the dominant, memoryless linear errors.  Capacitor mismatch, finite opamp-gain and switch-induced offsets.

10. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters The LE approach:  Stage I-O transfer function:  V r : Reference voltage  V os : Input-ref. opamp offset  C x : Virtual gnd parasitic cap.  d=0,1 or 2.  Rewrite:  Define D i =V i /V r, D o =V o /V r, D os =V os /V r for digital representation:

11. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters The LE approach:  Rewrite to simplify:  Applied to each stage in sequence, recursively, we get:  We can rewrite this to a sum of three terms:  This is a FIR-filter. The first term (A) is the weighted sum of {D K } The second term is quantization error The third term is total input-referred offset.

12. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters The LE approach:  In reality, tap values are not known due to mismatch, finite gain and offset errors. Adaptive techniques to obtain them on the fly  Steepest gradient descent adaptive digital background calibration scheme The output vector D is decimated and applied to an adaptive digital filter. A parallel slow-but-accurate ADC is used to obtain D in The ADF tap values are updated using a least-mean-square algorithm driven by the error signal and the update is performed at the speed of the slow ADC.

13. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters Simulations  10-bit prototype  Adaption runs at the speed of the ADC (to decrease simulation time)  Slow-ADC has 16b output, pipeline has 12b output  nLMS-algorithm to update filter taps Step-size µ=0.1 (0<µ<2 for stability).

14. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters Simulation results

15. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters Simulation results

16. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters Simulation results  Steady-state MSE: Memoryless linear errors can be fully removed when the quantization noise is negligible. Simulations confirm this, steady-state MSE is close to 12 bit.  Noise enhancement: LE suffers from a noise-enhancement problem when the noise spectrum is not flat.  Missing codes cause this and greatly enhances noise.  Increase the word length of the raw code, adding more stages.  Singularity: Equations for least-squares formulation: Solve for F subject to least-squares criteria, there Y is a vector of N+1 input samples. If X is non-singular:  A set of N+1 noiseless observations of D in and D should yield enough information to determine N+1 unknown tap values.  The input signal does not have to exercise the whole input range for calibration to work.

17. Y.Chiu et.al: “Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters”Least Mean Square Adaptive Digital Background Calibration of Pipelined Analog-to-Digital Converters Simulation results  Singularity: If V in is confined to a range where D 1 (n) does not toggle for all N+1 samples, the D-matrix will be singular. Solution still exists:  LMS or recursive least-squares (RLS) algorithm will still work.  However, it will find only a local optimum for the filter coefficients within this range.  Non-linear errors are not treated Will need non-linear, i.e. Volterra, filtering. Very complex. Conclusions:  All-digital, adaptive, data-driven algorithm has been presented.  Based on theory for equalization of communications channels.  Corrects linear pipeline ADC errors Capacitor mismatch, finite opamp-gain, input-referred offsets

18. Discussion...