1. Budapest University of Technology &Econ. 1 Summer school on ADC & DAC June-July 2006 Testing of A/D Converters II. Improvement of Sine Fit István Kollár Budapest University of Technology and Economics Dept. of Measurement and Information Systems Budapest, Hungary

2. Budapest University of Technology &Econ. 2 Summer school on ADC & DAC June-July 2006 Outline ADC testing: standards, sine wave fitting Problem: improper fit of low-noise sine at low level numbers (<25) Cause: LS fit is used for deterministic error pattern Solution: elimination of “pathological” data Suggestion for modification of standard

3. Budapest University of Technology &Econ. 3 Summer school on ADC & DAC June-July 2006 Quantized Sine Wave

4. Budapest University of Technology &Econ. 4 Summer school on ADC & DAC June-July 2006 Probability Density Function

5. Budapest University of Technology &Econ. 5 Summer school on ADC & DAC June-July 2006 Properties LS is maximum likelihood if noise Gaussian, zero-mean and white Therefore it is (asymptotically) unbiased and has (asymptotically) minimum variance (effective) It is quite good also if noise is uniform, zero-mean and white NONE IS TRUE! ADC errors are, even in the ideal case: deterministic, interrelated if statistically treated correlated (interdependence of samples) non-Gaussian, non-uniform

6. Budapest University of Technology &Econ. 6 Summer school on ADC & DAC June-July 2006 Algorithm Minimize a)vs. , A, B, C, or b)vs. A, B, C the following sum (LS criterion):

7. Budapest University of Technology &Econ. 7 Summer school on ADC & DAC June-July 2006 Effective Number of Bits B: nominal number of bits RMS: root mean square of error q: nominal quantum size (LSB)

8. Budapest University of Technology &Econ. 8 Summer school on ADC & DAC June-July 2006 Probability Density Function

9. Budapest University of Technology &Econ. 9 Summer school on ADC & DAC June-July 2006 Fitting the Samples

10. Budapest University of Technology &Econ. 10 Summer school on ADC & DAC June-July 2006 ENOB as function of dc and A

11. Budapest University of Technology &Econ. 11 Summer school on ADC & DAC June-July 2006 Corrected ENOB

12. Budapest University of Technology &Econ. 12 Summer school on ADC & DAC June-July 2006 Corrected ENOB, Noisy Input

13. Budapest University of Technology &Econ. 13 Summer school on ADC & DAC June-July 2006 The Fitted Sine is Still Smaller than True One

14. Budapest University of Technology &Econ. 14 Summer school on ADC & DAC June-July 2006 Weight of the Considered Samples

15. Budapest University of Technology &Econ. 15 Summer school on ADC & DAC June-July 2006 The Fitted Sine is Smaller than True One

16. Budapest University of Technology &Econ. 16 Summer school on ADC & DAC June-July 2006 Re-weighting by the Reciprocal of the PDF

17. Budapest University of Technology &Econ. 17 Summer school on ADC & DAC June-July 2006 Proper Handling of Noise: Corrected Re-weighting by the Reciprocal of the PDF, with Elimination of Samples Around Zero

18. Budapest University of Technology &Econ. 18 Summer school on ADC & DAC June-July 2006 The “Best” Fitted Sine

19. Budapest University of Technology &Econ. 19 Summer school on ADC & DAC June-July 2006 Algorithm 1.Eliminate the ‘pathological’ bins, that is, samples which are at the maximum histogram bins or outside these, 2.Eliminate the ‘central’ samples which yield zero at the quantized output, 3.Make an estimation of the parameters of the sine wave, by an LS fit (3- or 4-parameter method), 4.Determine the corrected reciprocal weighting, 5.Make a WLS fit with these weights, 6.If the correction is important, return to 4.

20. Budapest University of Technology &Econ. 20 Summer school on ADC & DAC June-July 2006 Summary Error in sine fit (and also in the error sequence) for small noise and small bit number Robust elimination of the error source is possible (not affecting other calculations) It could be built into the standards