Amplitude Quantization

Transcription

1 Amplitude Quantization Amplitude quantization Quantization noise Static ADC performance measures Offset Gain INL DNL ADC Testing Code boundary servo Histogram testing EECS Lecture : Amplitude Quantization B. Boser Ideal Quantizer Quantization step (= LSB) N = Bits Full-scale input range: -. ( N -.) Characteristics [] - 8 Quantization error: bounded by / + / for inputs within full-scale range Quantization error [LSB] ADC Input Voltage [/ ] EECS Lecture : Amplitude Quantization B. Boser

2 Quantization Error PDF Uniformly distributed from / + / provided that Busy input Amplitude is many LSBs No overload Not Gaussian! Pdf / Zero mean Variance e + / = / e de = Spectral density white if the joint pdf of the input at different sample times is smooth Ref: W. R. Bennett, Spectra of quantized signals, Bell Syst. Tech. J., vol., pp. -, July / - / error B. Widrow, A study of rough amplitude quantization by means of Nyquist sampling theory, IRE Trans. Circuit Theory, vol. CT-, pp. -, 9. EECS Lecture : Amplitude Quantization B. Boser Signal-to-Quantization Noise Ratio Since if some conditions are met (!) the quantization error is random, it is often referred to as noise In this case, we can define a peak signal-to-quantization noise ratio, SQNR, for sinusoidal inputs: N SQNR = =. =.N +. db N Actual converters do not quite achieve this performance due to other errors, including Electronic noise Deviations from the ideal quantization levels e.g. N SQNR 8 db db 98 db db EECS Lecture : Amplitude Quantization B. Boser

3 Static ADC Errors Deviations of characteristic from ideal staircase Offset Gain error Differential Nonlinearity, DNL Integral Nonlinearity, INL EECS Lecture : Amplitude Quantization B. Boser Offset and Gain Error Characteristics [] Full-scale error Offset error - 8 ADC Input Voltage [LSB] EECS Lecture : Amplitude Quantization B. Boser

4 Differential Nonlinearity Characteristics [] DNL = deviation of bin width from 8 -. LSB DNL error +. LSB DNL error ADC Input Voltage [/ ] EECS Lecture : Amplitude Quantization B. Boser Differential Nonlinearity Characteristics [] Characteristics [] 8 8 Missing code (+./- LSB DNL) Non-monotonic (> LSB DNL) ADC Input Voltage [/ ] ADC Input Voltage [/ ] EECS Lecture : Amplitude Quantization B. Boser 8

5 Large DNL Errors Characteristics [] A converter with DNL larger than LSB could be equivalent an ideal ADC with bit less resolution At right: alternating DNL /+ LSB Quantization error [LSB] ADC Input Voltage [/ ] EECS Lecture : Amplitude Quantization B. Boser 9 Integral Nonlinearity Characteristics [] INL = deviation of actual center of bin from its ideal location A straight line through the endpoints is usually used as reference, i.e. offset and gain errors are ignored in INL calculation -. LSB INL Note that INL errors can be much larger than DNL errors and vice-versa - 8 ADC Input Voltage [/ ] EECS Lecture : Amplitude Quantization B. Boser

6 Monotonicity Monotonicity guaranteed if INL =. LSB The best fit straight line is taken as the reference for determining the INL. This implies DNL = LSB Note: these conditions are sufficient but not necessary for monotonicity Ref: R. J. van de Plassche, Integrated Analog-to-Digital and Digital-to-Analog Converters, Kluwer Academic Publishers, 99. EECS Lecture : Amplitude Quantization B. Boser DAC DNL and INL Ref: Understanding Data Converters, Texas Instruments Application Report SLAA, Mixed-Signal Products, 99. EECS Lecture : Amplitude Quantization B. Boser

7 DNL and INL Testing Code boundary servo Code density (histogram) testing EECS Lecture : Amplitude Quantization B. Boser Code Boundary Servo i Code Input A A<B Digital Comp. B A B i C R C ADC Input V REF ADC f S ADC Output EECS Lecture : Amplitude Quantization B. Boser

8 Code Boundary Servo i and i are small, and C is large, so the ADC analog input moves a small fraction of an LSB each sampling period For a code input of, the ADC analog input settles to the code boundary shown ADC Digital Output V REF ADC Analog Input V REF EECS Lecture : Amplitude Quantization B. Boser Code Input Code Boundary Servo A A<B Digital Comp. B A B i C R C i Good DVM V REF f S ADC ADC Output EECS Lecture : Amplitude Quantization B. Boser

9 Code Boundary Servo A very good digital voltmeter (DVM) measures the analog input voltage corresponding to the desired code boundary DVMs have some interesting properties They can have very high resolutions (8½ decimal digit meters are inexpensive) To achieve stable readings, DVMs average voltage measurements over multiple Hz ac line cycles to filter out pickup in the measurement loop Hz pickup in typical measurement loops is ~mv EECS Lecture : Amplitude Quantization B. Boser Code Boundary Servo A high-accuracy (as opposed to highresolution) DVM is unnecessary The same meter can measure the ADC s voltage reference A high-accuracy ADC reference voltage is likewise unnecessary V REF must be stable to within a fraction of an LSB for the duration of the INL test EECS Lecture : Amplitude Quantization B. Boser 8

10 Code Boundary Servo ADCs of all kinds are notorious for kicking back high-frequency, signal-dependent glitches to their analog inputs R Good DVM V REF f S ADC A magnified view of an analog input glitch follows C EECS Lecture : Amplitude Quantization B. Boser 9 Code Boundary Servo Just before the input is sampled and conversion starts, the analog input is pretty quiet As the converter begins to quantize the signal, it kicks back charge analog input start of conversion /f S time EECS Lecture : Amplitude Quantization B. Boser

11 Code Boundary Servo The difference between what the ADC measures and what the DVM measures is not ADC INL, it s error in the INL measurement analog input DVM measures the average input including the glitch How do we control this error? ADC converts this voltage /f S time EECS Lecture : Amplitude Quantization B. Boser Code Boundary Servo A large C fixes this Good DVM At the expense of longer measurement time (the DVM is slow anyway) R V REF ADC f S C EECS Lecture : Amplitude Quantization B. Boser

12 Histogram Testing Code boundary measurements are slow Long testing time May miss dynamic errors Histogram testing Quantize input with known pdf (e.g. ramp or sinusoid) Derive INL and DNL from deviation of measured pdf from expected result EECS Lecture : Amplitude Quantization B. Boser Histogram Test Setup DNL is usually measured via a code frequency of occurrence histogram Hardware looks like this: V REF Ramp V REF ADC PC EECS Lecture : Amplitude Quantization B. Boser

13 Measuring DNL Error Ramp speed is adjusted to provide an average of outputs of each ADC code (for / LSB resolution) Ramps can be quite slow for high resolution ADCs: ( codes)( conversions/code) conversions/sec =. sec EECS Lecture : Amplitude Quantization B. Boser Histogram of Ideal Bit ADC Characteristics [] 8 Counts 8-8 ADC Input Voltage [/ ] ADC output code EECS Lecture : Amplitude Quantization B. Boser

14 Histogram of Sample Bit ADC Characteristics [] 8 -. LSB DNL +. LSB INL Counts 8 +. LSB DNL - 8 ADC Input Voltage [/ ] ADC output code EECS Lecture : Amplitude Quantization B. Boser DNL from Histogram Remove over-range bins ( and ) Counts, End bins removed 8 ADC output code EECS Lecture : Amplitude Quantization B. Boser 8

15 DNL from Histogram. Scale:. divide by average count. subtract (ideal bins have exactly the average count, which, after normalization, is ) DNL = Counts / Mean(Counts) Result is DNL ADC output code EECS Lecture : Amplitude Quantization B. Boser 9 INL from Histogram The DNL tells us width of all bins (DNL + ) We can use it to reconstruct the exact converter characteristic (having measured only the histogram) by simply adding up all bin-width Reconstructed Characteristic The INL is the deviation from a straight line through the end points - 8 ADC Input Voltage EECS Lecture : Amplitude Quantization B. Boser

16 DNL and INL of Sample ADC Characteristics [] -. LSB DNL DNL [in LSB].. -. DNL and INL of Bit converter (from histogram testing) avg=-.9e-, std.dev=., range=.8 +. LSB INL +. LSB DNL INL [in LSB] - bin.. avg=., std.dev=., range= ADC Input Voltage [/ ] - bin EECS Lecture : Amplitude Quantization B. Boser Sinusoidal Inputs Ramps are limited to slow inputs and may miss dynamic effects Solution: use sinusoidal test signal Raw Histogram of ADC Output Problem: ideal histogram is not flat but has bath-tub shape EECS Lecture : Amplitude Quantization B. Boser

17 Sinusoidal Inputs After correction for sinusoidal pdf. x -. Linearized Histogram.8... EECS Lecture : Amplitude Quantization B. Boser DNL and INL. DNL = +. / - LSB, missing codes (DNL<-.9) DNL [LSB]. -. Ref: [] M. V. Bossche, J. Schoukens, and J. Renneboog, Dynamic Testing and Diagnostics of Converters, IEEE Transactions on Circuits and Systems, vol. CAS-, no. 8, Aug. 98. [] IEEE Standard INL [LSB] - code.. -. INL = +. / -.9 LSB - code EECS Lecture : Amplitude Quantization B. Boser

18 DNL/INL Code function [dnl,inl] = dnl_inl_sin(y); %DNL_INL_SIN % dnl and inl ADC output % input y contains the ADC output % vector obtained from quantizing a % sinusoid % Boris Murmann, Aug % Bernhard Boser, Sept % histogram boundaries minbin=min(y); maxbin=max(y); % histogram h = hist(y, minbin:maxbin); % cumulative histogram ch = cumsum(h); % transition levels T = -cos(pi*ch/sum(h)); % linearized histogram hlin = T(:end) - T(:end-); % truncate at least first and last % bin, more if input did not clip ADC trunc=; hlin_trunc = hlin(+trunc:end-trunc); % calculate lsb size and dnl lsb= sum(hlin_trunc) / (length(hlin_trunc)); dnl= [ hlin_trunc/lsb-]; misscodes = length(find(dnl<-.9)); % calculate inl inl= cumsum(dnl); EECS Lecture : Amplitude Quantization B. Boser DNL/INL Code Test % converter model B = ; % bits range = ^(B-) - ; % thresholds () th = -range:range; % ideal thresholds th() = th()+.; % error DNL [LSB]. -. DNL = +. / -. LSB, missing codes (DNL<-.9) fs = e; fx = 9e + pi; % try fs/! C = round( * ^B / (fs / fx)); code.8 INL = +. / -. LSB t = :/fs:c/fx; x = (range+) * sin(*pi*fx.*t); y = adc(x, th) - ^(B-); hist(y, min(y):max(y)); INL [LSB] code dnl_inl_sin(y); EECS Lecture : Amplitude Quantization B. Boser

19 Limitations of Histogram Testing The histogram (as any ADC test, of course) characterizes one particular converter. Test many devices to get valid statistics. Histogram testing assumes monotonicity. E.g. code flips will not be detected. Dynamic sparkle codes produce only minor DNL/INL errors. E.g.,,,,,,, look at ADC output to detect. Noise not detected or improves DNL. E.g. 9, 9, 9,, 9, 9, 9,, 9,,,, Ref: B. Ginetti and P. Jespers, Reliability of Code Density Test for High Resolution ADCs, Electron. Lett., vol., pp. -, Nov. 99. EECS Lecture : Amplitude Quantization B. Boser