FE8113 ”High Speed Data Converters”

Part 2: Digital background calibration

Papers 3 and 4: J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”, IEEE Transactions on Circuits and Systems I: Regular Papers : Accepted for future publication, Volume PP, Issue 99, 2005 pp 1-15 J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”, IEEE Journal of Solid-State Circuits, Vol. 36, No. 10, October 2001, pp Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital ConvertersAn 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration

This week’s funny picture: ”Nervous moment” by G.I.Joe

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Outline: Memory errors can occur in the stages of a pipeline ADC due to several effects. This paper describes the sources of such memory effects and the impact they have on ADC performance. Then, two new algorithms for digital calibration of memory effects are proposed.

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Introduction to the pipeline ADC and notation Stage i output: Stage transfer function Total ADC output Everything ideal

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Sources of memory effects Capacitor dielectric absorption/relaxation effect: 1) Charge cap to V C =V init 2) At t=0, discharge by short until t=t 0 3) Allow the capacitor to float until t=t f Charge gather back from dielectric to plates Ideal pipleline SC-stage Normalized values

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Sources of memory effects - Add model for cap dielectric absorption/relaxation memory effect 1) At t=(k-0.5)T: C in and C f charged to V DASC (kT-T/2) and V i+1 (kT-T/2) 2) During φ 1 : C in and C f connected to V i and ground, voltage => V i (t) 3) At t=kT: Switch opens, capacitor terminals float during φ 2 => Relaxation voltage V Cin =γV DACS (kT-T/2) and γV i +1(kT-T/2) at C i and C f during φ 2 - Additional charge on C in transferred to C f - Net additional charge on C f : C f V Cf +C in V Cin - Set up linear charge transfer equation for residue output - Normalized:

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Sources of memory effects - Incomplete stage reset effects - T is chosen long enough to allow settling to the required accuracy of the conv - Incomplete settling in φ 2 => errors in interstage gain G i and DASC gain K i - Can be corrected using conventional calibration techniques - Incomplete settling in φ 1 causes memory-effects - Can not be corrected using conventional calibration techniques - Linear settling model - Phase φ 1 from t=kT-T/2 to t=kT - γ in and γ f are settling time constants of C f and C in - At the end of φ 1 - Assuming V cf (kT+T/2)=V i+1 (kT+T/2), V Cin (kT+T/2)=V DASC (kT+T/2)

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Sources of memory effects - Incomplete stage reset effects - Assuming V Cf (kT+T/2)=V i+1 (kT+T/2), V Cin (kT+T/2)=V DASC (kT+T/2) - Changed coefficient for V i (kT) - G i =(1-γ f )+((1-γ f )C in /C f - The gammas, and hence the gain is dependant on previous output and DASC input.

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Sources of memory effects - Opamp sharing effects - Subsequent stages work in opposite phases and opamp is only needed in the amplification phase => two and two stages share one opamp. Memory effects due to input parasitic capacitance and limited gain - Voltage at the end of amplifying phase: V m =-V out /a - At the end of φ 2 : V out =V i+1 (kT-T/2) => V m =-V i+1 (kT-T/2)/a - φ 1 : C p injects C p V m at summing node of second stage generating V out =V i+2 (kT) - Transfer function of stage 2 during φ 1 - Normalized form

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Sources of memory effects - Opamp sharing effects - φ 2 : C p injects C p V m at summing node of first stage generating V out =V i+1 (kT+T/2) - Transfer function of stage 1 during φ 2 - Normalized form - Output of stage 1 depends on previous output of stage 2

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Sources of memory effects - Opamp sharing effects - Only odd stages have memory effect - Even stages have an interstage gain error - Model for opamp sharing memory effects - Can be constrained within one stage - Equivalent model for ADC transfer function given when:

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Linearity error due to memory effects First: summarize the memory effects described in stage transfer function γ i and δ i correspond to memory effects γ f and γ in resp. Can often be assumed equal for mentioned memory effects

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Linearity error due to memory effects - Now, we can find nonlinearity due to memory effects by tracing recursively from the final stage output (X N (z)) to the output signal of the input SHA (X 0 (z)). - First terms is linear filtering, second term is non-linear error, containing a weighted sum of past stage decisions. - Past decisions are correlated with the input.

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Simulation of memory error effect on INL in otherwise ideal ADC 12b LSB-level, 95% of full scale sinus in a) γi and δi = for first stage only b) γi and δi = two first stages c) γi and δi = all stages Memory error dominated by second stage

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Background calibration for memory effects A) LMS-method using parallel ADC

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters LMS-method using parallel ADC - Parallel reference ADC operating at 1/M of main ADC speed - Ref. ADC is low-power, low-speed high linearity (e.g. delta-sigma) - Error-signal e[n] is an estimate of main-ADC nonlinearity - Used in a negative feedback loop that adjusts the calibration to minimize the mean-square value of the error - Digital calibration block performs: - Where are estimates of the ideal weights w i LMS-algorithms gives:

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters LMS-method using parallel ADC - Output with memory effect rewritten: - Since Q is unknown, appropriate expression to correct for mem.errors: - To find the estimation coefficients for w i,k we apply the LMS-algorithm - µ i,k is a small number chosen as compromise between tracking speed and steady-state variation

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters LMS-method using parallel ADC - In direct form, i+1 coefficients are needed for stage i. - Total number of coefficients for N-stage ADC: - Since later stages contribute very little, only L first stages are calibrated - Removing quantization terms similar to the previous example yields:

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters LMS-method using parallel ADC - Now the number of coefficients is reduced to: - Drawback: Converged value of coefficients will be dependent on input PDF. - Good calibration relies on the input varying over many possible levels to provide different decisions for all d i [n] and hence estimation of all w i - Input signal waveform with uniform distribution over full signal range is preferable. - Convergence time is fastest when input signal is correlated - Input signal waveform with discrete distribution is preferable.

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Background calibration for memory effects B) Calibration using DAC dithering and stage redundancy

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Calibration using DAC dithering and stage redundancy Interstage gain estimated by randomly dithering DASC input Stage input estimate: Estimate of m i =1/G i Estimate of error in coefficient: Scale of dither signal determines amount of information about errors in present in Limits tracking speed

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Calibration using DAC dithering and stage redundancy - Extend model to include interstage error and memory effect. - Signal flow - Generally γ i ≈δ i : - If opamp-sharing, δ i =0 and:

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Calibration using DAC dithering and stage redundancy - Stage transfer function: - In time-domain: - Then, calibration expression becomes:

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Calibration using DAC dithering and stage redundancy - After some algebra, we get: - Where: - If the error signal is uncorrelated and spectrally white, we get: - And the coefficient update equation is given by:

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Calibration using DAC dithering and stage redundancy In words: - The ADC error due to memory effect are proportional to the previous value of the stage residue. - Random dithering of DASC causes the mean value of the stage residue to change by K i r i [n] - Adjusting the coefficient as given forces the stage input estimate to be uncorrelated with the previous stage residue, overcoming the memory effect. - The different dither signals ri[n] must each be uncorrelated with each other.

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters Simulation results: - 6-stage pipeline ADC - 2.5b stages with 6 ADSC levels - G i =4 - Final stage is 4b flash - When simulating dither calibrated ADC, each stage has 7 ADSC levels and 16 DASC levels to accomodate overhead needed for dither signal - Memory effect with γ i =δ i =0.005 in both cases. - Mean interstage gain error of -1% also introduced - Calibration of first five stages

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters

J.P.Keane et.al: “Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters”Digital Background Calibration for Memory Effects in Pipelined Analog-to-Digital Converters - LMS algorithm gives best results - LMS algorithm has fastest tracking - Dither algorithm is independent of input signal statistics - Dither algorithm does not require highly linear refrerence DAC.

J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration Outline: An 8-bit 80Msample/s pipelined ADC uses monolothic background calibration to reduce the nonlinearity caused by interstage gain errors

J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration Introduction and review - Stage with k-bit ADSC, k-bit DASC and SHA with gain G. - Internal resolution greater than output resolution to introduce redundancy - Overcome effects of comparator and SHA offset. - Accuracy limitations stem from linearity of DASC and gain accuracy of SHA.

J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration Introduction and review - Conventional SHA - With infinite gain and perfect capacitor matching G=2 - Gain error due to mismatch, limited gain and incomplete settling - Proposed algorithm corrects all these.

J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration Interstage Gain-Error Compensation - Interstage gain-error transformed to ADC gain error by moving G E to output of DASC - Effect is eliminated in model if V R1 =V R2 /G E - Proposed algorithm calibrates gain error by adjusting DASC reference voltages during the normal operation of the ADC

J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration Interstage Gain-Error Compensation - RNG produces random, while calib signal N[i]=±1 for all I - Converted with 1-bit DAC and added to stage input - Multiplied by G D1 and quantized at the back end ADC 2 - DAC 1 reference V n digitized by slow-but-accurate ADC - Result is multiplied with N[i] and subtracted from ADC 2 output - Producing error signal ε i - Error signal ε i is multiplied with N[i], scaled by µ and accumulated. - Accumulator output control DAC 2 to set reference voltage V R1

J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration Interstage Gain-Error Compensation - Accumulator input averages to N 2 - The N·Vin product part averages to zero since they are uncorrelated - If slow-but-accurate ADC and ADC 2 are ideal: - To set the average accumulator input to zero, the loop adjusts V R1_average until ε c_average = 0, which gives:

J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration Interstage Gain-Error Compensation - In practice, random fluctuation occur around V R1_average - Can be made arbitrarily small by reducing step-size µ - Also assures stability of the loop - Injection of calibration signal must not saturate ADC 2 - V n ≈ V R2 /4 chosen along with G D1 ≈ 1 - Amplitude is half the correction range of ADC 1, leaving headroom for comparator offset etc

J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration Effect of non-linearity in the back-end ADC - Back-end measure the calibration signal injected into first stage - Comparator offset - Digital redundancy and correction - Input-referred offset - Has little effect on loop convergence as it is whitened by N[i] - Gain-error in first backend interstage amplifier - G 2 ≠2

J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration Effect of non-linearity in the back-end ADC - Gain error in first backend interstage amplifier. - Case 1: V res [i] has a value such that adding cal.sig. does not change D 2 - The calibration signal appears entirely in V G2 - Change in V G2 : N[i]·1/2·LSB·G 2 - Case 2: V res [i] has a value so that adding cal.sig. changes D 2 one code - The noise appears in combination of D 2 and G 2 - Change in V G2 : -N[i]·1/2·LSB·G 2 - Combining the two cases equal cancels out average error caused by G 2 - Given by shaded regions in figure - Calibration is only done when the signal is in these regions.

J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration Prototype implementation - 0.5µm CMOS - 1.5b stages - Calibration of two first stages - V R1 adjusted with respect to V R2 - Delta sigma slow-but-accurate-ADC - 7b DAC for reference voltage control - µ= pF sampling cap, 54dB open-loop gain, 0.1% settling

J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration

J.Ming, S.Lewis: “An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration”An 8-bit 80-Msample/s Pipelined Analog-to-Digital Converter With Background Calibration