Introduction to ADC testing I Definition of basic

Introduction to ADC testing I Definition of basic parameters Ján Šaliga Dept. of Electronics and Telecommunications Technical University of Kosice, Slovakia

Agenda n n n Introduction Deterministic and probabilistic models Basic static parameters Basic dynamic parameters Other parameters

A/D converter – A/D interface Reference and power sources x Signal conditioning S&H (optional) ADC Timing and control circuit Buffer

ADC parameters (characteristics & errors) n Static (quasistatic) parameters – derived from transfer characteristic n n n Dynamic parameters – characterize a behavior of ADC at time-varying signals n n Point (gain, gain error, offset, missing code, . . . ) Function (transfer characteristic, INL, DNL, . . . ) SINAD, ENOB, SNR, SFDR, THD, IMD, . . . ADC parameter testing requires extraordinaire accuracy n E. g. : 12 -bit ADC: detetermination of transition level with uncertainty < 1% →uncertainty of measurement < 1/(100*4096) ~ 0, 00025%=2, 5 ppm of ADC FS

Accuracy versus precision

ADC transfer characteristic Gain (slope) error Input T[k] - transition level (threshold code k of code k), W[k]= T[k]- T[k-1] – code bin 111 width Nonlinearity Ideal and real straight lines 110 N – nominal resolution (number of bits) of ADC -4 Real ADC -3 -2 -1 101 Missing code 100 0 Ideal ADC 1 011 2 3 4 Input analogue value x(t) [Vfs/Q] Error in monotonicity 010 Offset error 001 000 Vfs - full scale range Vfs = Vref(2 N-1)/(2 N)

Gain and offset + their errors n Fitting the straight line: n n n End points straight line - connecting the two end code transition or code midstep values Least-square fit straight line according a least -square fitting algorithm Minimum-maximum straight line - the line which leads to the most positive and the most negative deviations from the ideal straight line

ADC transfer characteristic Deterministic model Output code k 111 Stochastic model Output code k 101 Input analogue value x(t) 110 101 100 -4 -3 -2 -1 00111 010 001 000 2 3 1 2 Deterministic definition Channel profile 4 Input analogue value x(t) P(k |x) [Vfs/2 N] 1 k =0, 1, . . . , 2 N-1 0 Conditional probability 1 1, 5 2 Stochastic definition Input analogue value x(t) [Vfs/2 N]

DNL and INL n n Differential non-linearity Integral non-linearity DNL [ k ] = W [ k ] - Q nom INL [k ]= Q nom T [k ] - Tnom [k ] Qnom

Dynamic parameters I n n Bandwidth (BW) - the band of frequencies of input signal that the ADC under test is intended to digitize with nominal constant gain. It is also designated as the Halfpower Bandwidth, i. e. , the frequency range over which the ADC maintains a dynamic gain level of at least � 3 d. B with respect to the maximum level. Gain flatness error ( G(f)) - the difference between the gain of the ADC at a given frequency in the ADC bandwidth, and its gain at a specified reference frequency, expressed as a percentage of the gain at the reference frequency. The reference frequency is typically the frequency where the bandwidth of ADC presents the maximum gain. For DC-coupled ADCs the reference frequency is usually fref = 0.

Quantisation noise and errors n n Caused by rounding in quantisation process (and ADC non-linearity) Power of quantisation noise for ideal ADC (s 2 eq, h 2 rms) n n n Is it dependent/independent on input signal? Is the value Q 2/12 correct? Distribution? Answer: see the simulation

ADC noise and distortion n ADC output random noise – random signal: n n n Quantisation noise - uniform Noise generated in input analogue circuits - Gaussian Noise caused by sampling frequency jitter and aperture uncertainty (Kobayashi) Spurious – unwanted deterministic spectral components uncorrelated with input signal (e. g. 50 Hz) Total noise – any deviation between the output signal (converted to input units) and the input signal, except deviations caused by linear time invariant system response (gain and phase shift), harmonics of the fundamental up to the frequency fm, or a DC level shift. Distortion – new unwanted deterministic spectral components correlated with input signal

Noise floor n determines the lowest input signal power level which is reliably detectable at the ADC output, i. e. , it limits the ultimate ADC sensitivity to the weak input signals, since any signal whose amplitude is below the noise floor (SNR < 0 d. B) will become difficult to recover.

Dynamic parameters II Signal to noise and distortion ratio n SINAD: for a pure sinewave input of specified amplitude and frequency, the ratio of the rms amplitude of the ADC output fundamental tone to the rms amplitude of the output noise, where noise is defined as to include not only random errors but also non-linear distortion and the effects of sampling time errors, i. e. , the sum of all nonfundamental spectral components in the range from DC (excluded) up to half the sampling frequency (fs/2).

Dynamic parameters III SNR n Signal to noise ratio (SNR) - harmonic signal power (rms) to broadband noise power ratio excluding DC, fundamental, and harmonics

Dynamic parameters IV THD, THD+noise, IMD n THD+noise = 1/SINAD n Intermodulation distortion (IMD) - for an input signal composed of two or more pure sinewaves, the distortion due to output components at frequencies resulting from the sum and difference of all possible integer multiples of the input frequency tones.

Dynamic parameters V Effective Number of Bits n n n Effective Number of Bits (Nef, ENOB) - for a sinusoidal input signal, Nef is defined as: where hrms is the rms total noise including harmonic distortion and seq the ideal rms quantisation noise for a sinusoidal input. (SINADd. BFS = SINADd. B - 20 log(SFSR)) SFSR – signal to full scale ratio Nef can be interpreted as follows: if the actual noise is attributed only to the quantisation process, the ADC under test can be considered as equivalent to an ideal Nef-bit ADC insofar as they produce the same rms noise level.

Dynamic parameters VI SFDR n n Spurious-free dynamic range (SFDR) - expresses the range, in d. B, of input signals lying between the averaged amplitude of the ADC's output fundamental tone, fi, to the averaged amplitude of the highest frequency harmonic or spurious spectral component observed over the full Nyquist band, for a pure sinewave input of specified amplitude and frequency, i. e. , max{|Y(fh)| , |Y(fsp)|}: where: Yavm is the averaged spectrum of the ADC output, fi is the input signal frequency, fh and fsp are the frequencies of the set of harmonic and spurious spectral components.

Dynamic parameters VII Experimental demonstration n Measurement setup (run generator first and then demonstration) Sound out AI 1 (DUT) USB NI USB 6009 ADC: 12 bits, 10 k. Hz, differential Software (Lab. VIEW): 1. Sinewave generator = Sound card 2. Control: AI 1 = DUT (FS, record) Data processing and visualisation

Other parameters n n Various electrical parameters, e. g. input impedance, power requirements, grounding, … Time parameters, e. g. clock frequency, conversion time, sampling frequency, … Digital output: data coding, levels (logic), serial/parallel, error bit rate, … …

Introduction to ADC testing II Basic standardized test methods

Agenda n n n Standardization Static test method Histogram test Dynamic test with data processing in time domain Dynamic test with data processing in spectral domain

Standardization n n IEEE Std. 1057 - 1994, "IEEE Standard for Digitizing Waveform Recorders", IEEE Std. 1241 - 2000, "IEEE Standard for Terminology and Test Methods for Analog-to-Digital Converters European project DYNAD – SMT 4 -CT 98 -2214, „Methods and draft standards for the DYNamic characterisation of Analogue to Digital converters“ http: //www. fe. up. pt/~hsm/dynad IEC Standard 62008 “Performance characteristics and calibration methods for digital data acquisition systems and relevant software” Additional and related standards: n n n IEEE Standard on Transition and Pulse Waveforms, Std-181 -2003 (IEC 60469 -1, -2) IEEE and IEC standards for DAQ and ADM – in preparation IEC 60748 - covers only static ADC and DAC operations … Detail overview of standards and standardisation – see the lecture of Pasquale Arpaia: A/D and D/A Standards, CD from SS on DAQ 2005 Standard comparison: Sergio Rapuano: Figures of Merit for Analog-to-Digital Converters: Analytic Comparison of International Standards , In Proc. of IMTC 2006, Sorrento, Italy, pp. 134 -139

ADC static test Standardized method

ADC static test - basic ideas n n Yields ADC transfer characteristic Static point and function parameters can be derived and calculated: n n Gain, offset, FS, DNL, INL, … Based on the stochastic model of ADC Simple test setup – DC voltmeter is the only accurate instrument Time consuming – each T[k] is determined individually. The total time: 2 N x longer than determination of one T [k]

Static test setup (IEEE 1057)

ADC static test - algorithm n n n Start with the code k = 1 Find an input voltage level for which the probability of codes lower than k in the record is slightly higher than 0. 5 – the voltage is below T[k]. Find a bit higher voltage (the usual step is a quarter of Q) for which the probability of codes lower than k is slightly lower than 0. 5 – the voltage is above T[k] Fit these two point by line and calculate the voltage for which the probability of codes smaller than k is 0. 5 – this is the transition level of code k – the voltage equal to T[k] Repeat the procedure for all k = 1, 2, …. , 2 N-1 – the complete transfer characteristic will be measured out

Uncertainty in the static test n n The uncertainty can be reduced by increasing the number of acquired samples (M). The table shows the measurement precision for a confidence level of 99, 87%. Number of acquired samples ( M) Transition level measurement precision (% of noise standard deviation) 64 256 1024 4096 45% 23% 12% 6%

The main disadvantage of the static testing n The test is long time consuming: n n Let’s test 16 bit ADC with sampling frequency 10 k. Hz, testing step is Q/4, additive noise: s=1 LSB, required precision: better than 10%. The chosen record length: 2000 samples Measurement on one level takes 2000 x 0. 1 ms = 0. 2 s Total required time: 0. 2 s x 2(16+4)= 58. 2 hours!!!

Static test Experimental demonstration n Measurement setup (run demonstration) AI 0 (DUT) USB AI 1 (Voltmeter) NI USB 6009 ADC: 12 bits, 10 k. Hz, differential DAC: 12 bit, static, RSE Software (Lab. VIEW) controls: 1. AO 0 = DC test voltage 2. AIO = DUT - FS, record 3. AI 1– virtual DC voltmeter with averaging 1: 10 AO 0 (DC source) 4. Statistical data processing and visualisation

Alternative static method with feedback - IEEE 1241

Alternative static method with feedback - IEEE 1241

Some experimental results I NI USB 6008 (12 bits, 10 k. Hz, 10000 s/T)

Some experimental results II NI USB 6008/9 (10000 s/T) Difference of two following measurements Switching monitor during the measurement

Histogram (statistical) test Standardized method

Histogram (statistical) test Basic ideas I n n Goal: to determine ADC transfer characteristic (the same as in static test method) The calibrating signal is a time invariant repetitive signal covering the ADC full scale n n n The stream of ADC output codes is recorded Histogram is built from the record The relative count of hits in code bin k in the histogram in comparison to the calibrating signal probability density function (or counts for code bin k in cumulative histogram in relation to signal probability distribution function) gives information about the code bin width (or code transition levels)

Histogram (statistical) test Basic ideas II n n n The best shape would be ramp or triangular signal. Why? Problem? The basic recommended signal by all standards: sinewave. Why? To achieve a required accuracy a relative long record (or records) is required Faster than the static test Requirement: an accurate generator with an extremely high accuracy (low distortion, high linearity, high spectral purity)

Histogram (statistical) test General test setup

Ramp signal (IEEE 1241) T[k]=C+G. HC[k-1]/S for k=1, 2, . . , (2 N- 2) G is a gain factor, C is an offset factor, The code bins 0 and 2 N-1 are usually excluded from data processing (why? )

Sinewave signal (All standards) – theoretical background I n n n Signal: Density of probability: Distribution of probability:

Sinewave signal (All standards) – theoretical background I n Ideal theoretical histogram: n DNL: n Transition levels:

Sinewave signal (All standards) – theoretical background II n n n Problem in praxis: what are the sinewave parameters – A, C →Hid[k]? Various ways of estimation, e. g Dynad: Incorrect estimation →error in gain and offset

Sinewave signal Test conditions I n n The total record must contain exactly an integer number J of sinewave cycles R partial records can be used instead of one long record Total recorded number M of samples must be relatively prime with J, i. e. they have no common factor Then the sampling and sinewave frequency are:

Sinewave signal Test conditions II n The number of samples (M) to acquire in the histogram test, depends on: n n n The noise level in the measurement system, The required tolerance (B is measured in LSBs) and confidence level (a) and the M is different if DNL (quantization interval) or INL (transition levels) it to be determined. The specification of tolerance for an individual transition level or code bin width, or for the worst case in all range.

Sinewave signal Test conditions III n The equation generally used to determine the number of records to acquire is: J=1 for INL, J=2 for DNL, s is the standard deviation of noise level in volt for the INL determination and the smaller of the values of s and Q/1, 1 for the DNL determination.

Sinewave signal Simulation n Simulation = (see the simulation): n n n Form of histogram for various test signals Error caused by limited number of samples Error caused non-coherent sampling Error caused by noise in input signal Error caused by higher harmonics …

Histogram test Experimental demonstration n Measurement setup (run generator first and then demonstration) Sound out 1: 2 AI 1 (DUT) USB NI USB 6009 ADC: 12 bits, 10 k. Hz, differential Software (Lab. VIEW): Sinewave generator = Sound card AI 1 control = DUT - FS, record Data processing and visualisation

Results of experimental tests Comparison generators (USB 6009) Stanford DS 360 (20 -bits, 100 mil. samples) Agilent 33220 A (14 -bits, 100 mil. samples)

Histogram (statistical) test Some non-standardized methods

Non standardized histogram tests Basic ideas n Reasons: n n n To use signals that are closer to real signal digitized by ADC in common applications To use signal that can be simply generated with required precision Common signals: n n n Gaussian noise Exponential signal Uniform noise, small sinewave or triangular with DC steps, …

Non standardized histogram test Gaussian noise I Martins, R. C. , Serra, A. C. : ADC Characterisation by using Histogram Test stimulated by Gaussian Noise. Theory and experimental results, Measurement, Elsevier Science B. V. , n n vol. 27, n. 4, pp. 291 -300, June 2000 The noise is centred within ADC input range and overlap the whole ADC range Problem generate the noise with really precise Gaussian distribution – convenient methods for low resolution ADCs and very high and very low frequencies where it is difficult to generate sinewave with required purity

Non standardized histogram test Gaussian noise II Holub J. , Komárek M. , Macháček J. , Vedral J. : STEP-GAUSS STOCHASTIC TESTING METHOD APPLICATION FOR TRANSPORTABLE REFERENCE ADC DEVICE, Proc. 8 th IWADC 2003, n n Perugia, Italy, pp. 223 -226 Gaussian noise with a small standard deviation is moved within the ADC input range by adding a DC voltage (mean) in small steps so that the results will be the same as using uniform noise overlapping the whole ADC full scale Discussion: is really possible in praxis to fulfil the requirement of the limit with finite DC steps with acceptable precision?

Non standardized histogram test Small amplitude sinewave or triangular with a DC component Michaeli L. , Serra A. C. , . . : In: IEEE transactions on instrumentation and measurement, Measurement, proc. of IMTC, IMEKO – IWADC Idea: multistep test with fractional histograms (and INLs) acquired at small signal (sinewave, triangular) covering only a few tens/hundreds of codes shifted within ADC FS by known DC voltage Advantage: the quality of test signal may be much worse than those of signal covering the whole FS of ADC Disadvantage: connecting the partial histograms to build the final histogram

Non standardized histogram test Exponential signal Holcer R. , Michaeli L. , Šaliga J. : DNL ADC testing by the exponential shaped voltage, In: IEEE transactions on instrumentation and measurement, Vol. 52, no. 3 (2003), pp. 946 -949. Šaliga J. , Holcer R. , Michaeli L. : Noise sensitivity of the exponential histogram ADC test, In: Measurement, Vol. 39, no. 3 (2006), pp. 238 -244 n n n We will continue with a new Ph. D. Student next year Exponential signal is simple to generate – native signal in electronic circuit Problem: distortion by other exponential with different time constant and keeping the final value of the signal known and constant.

Non standardized histogram test Small signals with a DC component n Measurement setup (run generator first and then demonstration) Sound out 1: 2 1: 10 AI 0 (DUT) USB AO 0 (DC shift) Software (Lab. VIEW): Arbitrary generator = Sound card DC shift = AO 0 NI USB 6009 ADC: 12 bits, 10 k. Hz, differential AI 0 = DUT (FS, record) Data processing and visualisation

Histogram test Conclusions n Histogram versus static test: histogram test gives usually better – more reliable results because: n n n Faster = the test conditions are “constant” and measurement of any T [k] is distributed and repeated in time over the all testing time Disadvantage: an precise generator is needed Non standardised test procedures can bring simplifying in test setup and decrease the requirements on instrumentation precision.

ADC dynamic testing

Dynamic test Introduction n Goal: n n Determination of various dynamic ADC parameters such as SINAD, ENOB, SNR, THD, IMD SFDR, … Two ways of data processing: n n n Time domain – directly SINAD, ENOB Spectral domain (DFT test): SINAD, ENOB, SNR, THD, IMD SFDR, … No way can be generally supposed to be the best one

Dynamic test General test setup

Dynamic test Requirements n n Coherent sampling – the same as for sinewave histogram test - the precise coherence is not necessary Minimal size of record: Record can consist of a few partial records Sinewave must cover the ADC input range as much as possible (more than 90 – 95%) but must not overload it.

Dynamic test Data processing in time domain

Dynamic test Data processing in time domain I n n n See the following lectures by prof. Kollár and prof. Händel Basic idea: to calculated the noise in the record (residuals) as the deference between the input signal – sinewave (analogue samples) and the record (digitized samples). Knowing the noise the SINAD and ENOB can be calculated according the definitions

Dynamic test Data processing in time domain II n n Difficult task and question: the input signal must be precisely know – how to do it? Common solution: recovering the input signal from the record by a fitting method (LMS) n n n Three-parameter fit (A, C, f) Four-parameter fit (A, C, f, f) Question: is the recovered fitted signal really the origin input signal? !

Dynamic test Three-parameter fit I n Simple calculation = system of linear system of 3 equations is to be solved

Dynamic test Three-parameter fit II n In matrix form:

Dynamic test Three-parameter fit III Necessary condition: The input (and sampling) frequency must be precisely known!!! If not – incorrect results SINAD, … n SEE THE SIMULATION

Dynamic test Four-parameter fit I n n n Unknown parameters: A, C, f, f Difficult calculation = system of non-linear system of 4 equations is to be solved The system can be solved only by iteration process

Dynamic test Four-parameter fit II Let the first estimation is Repeat calculation:

Dynamic test Four-parameter fit III n n Problem with convergence – one global minimum and a few local minima If the first estimation is incorrect the iteration converges to the fault minimum n n One of best estimations is the estimation from spectrum within the interval (J-s, J+s): See the simulation

Dynamic test Data processing in spectral domain – DFT test

Dynamic test Data processing in spectral domain I n n n The same test setup, requirements and the first step as for Data processing in time domain The DFT spectrum is calculated from the record Using the definitions (see the beginning part of this lecture) the unknown ADC parameters can be estimated

Dynamic test Data processing in spectral domain II n n Common problem in praxis: incoherent sampling – leakage effect in the record spectrum Solution: applying a window function (Hanning, 7 term Blackman-Harris, …) to suppress the leakage effect and then correction of results according the window parameters (see the general theory of windowing in DSP) n n n Introduced in detail in DYNAD Rule: the higher the ADC resolution is, the lower the side-lobes of the window have to be. Nevertheless, lowering the side-lobes results in increasing the main lobe width Calculation is much more complex

Dynamic test Data processing in spectral domain III n Spectrum calculation: n Error in coherency: Processing gain Equivalent Noise Bandwidth

Dynamic test Data processing in spectral domain IV n Changes in formulas: example 1: Noise floor:

Dynamic test Data processing in spectral domain V n Changes in formulas: example 2: SINAD

Dynamic test Conclusions n n n No method of data processing can be suppose to be absolutely the best Processing in time domain is less sensitive on coherency but the 4 -parameter fit can be problematic Processing in frequency domain gives directly much more parameters but it is very sensitive on coherency

The final conclusions n n ADC testing is not a simple task Extremely difficult task: to test ADC with high resolution (more than 20 bits) Methods are in the process = a challenge for you Another challenge: test procedures for special ADC, e. g. band-pass for direct digitalization and demodulation of high frequency signals, etc.

Thank you for your attention