# Correction of ADC errors by iterative method with dithering using dither with uniform distribution.

1. INTRODUCTIONSelf-correction functions become important property of measuring channels of modern equipment. Very often analog-to-digital converter (ADC) integrated within monolithic micro controller is used for signal level measurements. So ADC characteristics determine the overall metrology properties of the channel. The improvement of them necessitates increasing the precision of such ADC. It is not difficult to make correction of offset and gain error of ADC. But correction of nonlinearities is problematic especially if they vary in time.

Methods for automatic correction of ADC have been employed as discussed below. Proposed correction is focused on ADC nonlinearities (Michaeli, 2001). Additive iterative method (AIM) is suitable for nonlinear error correction in the case of analog MT. But the ideal characteristic of ADC is fundamentally nonlinear reflecting quantisation error. Quantisation error limits efficiency of AIM, therefore in designed measurement system AIM is combined with dithering. Implementation of non-subtractive dithering (ND) is easier than implementation of the subtractive form. ND has been chosen for this application, where price is important aspect.

The proposed combination of both correction methods was already analysed in our previous publication (Kamensky & Kovac, 2009) for dithering noise with Gaussian distribution. Now we bring the results for uniformly distributed noise.

2. CORRECTION METHOD PRINCIPLE

One of ways for a gain of measurement results accuracy is the use of so-called structural-algorithmic methods (Muravyov, 2000), when the measurement errors are diminished with help of auxiliary means. In our case designed measurement system with AIM consists only of single-chip microcomputer (with some external components) and it was described in our previous publication. Generally for the AIM three new block should be added to MT and then four main blocks of the system are distinguished: MT--in our system is represented by ADC; block of processing (BP)--CPU (processor) with memory; inverse element (IE)--Pulse Width Modulation (PWM) with RC-filter; switch--multiplexer.

The correction is performed iteratively in several steps using signal from IE for suppression of measurement error. The BP controls the whole process. If condition of convergence is satisfied then the iterative process is convergent and it tends to value given by characteristics of IE. If it is not possible to have an ideal IE, error of IE [DELTA][h.sub.IE] will be present in the measurement result after correction. Deeper investigation of IE properties will help to obtain negligible error [DELTA][h.sub.IE].

2.1 Inverse element operation

Transfer characteristic of IE determines the best resp. theoretically reachable accuracy of measurement output after correction. Inverse element for ADC is digital-to-analog converter (DAC) and it has been built by means of pulse width modulation output of microprocessor. PWM circuits are naturally precise but to get the mean of PWM output corresponding to precise DAC result, low-pass (LP) filter should be added. Simple RC-filter has been used. Frequency characteristic of the filter is influenced by time constant [[tau].sub.RC], which has to be large to get small amplitude of signal from IE. But big [[tau].sub.RC] slows down measurement process because after every step the process should wait until settling of filter output.

To speed up the process we proposed to use combination of analog and digital filter. This technique suppresses also impact of quantization on signal from IE. Output of the analog RC-filter oscillates in the range of several LSB. As digital filtering sampling and averaging of N samples in each step of the iterative correction is used. The best way is to use synchronous sampling here but there could be no possibility to synchronize ADC and PWM circuits. If choosing appropriate number of samples N, special case of quasi-synchronous sampling could be achieved (e.g. N=20,39,59,79 in our system). Properties of IE have been analyzed using mathematical model of IE error (error caused by non-synchronous sampling) and simulations. Expecting error of measurement after correction near to 0,01 % of scale, the values [[tau].sub.RC]=0,1 s and N=59 have been chosen.

3. UNIFORM DITHER AND AVERAGING

Quantization error limits measurement accuracy of an ADC corrected with AIM. The way to overcome this limitation is averaging of samples employing noise (dither) intentionally added to measured signal. Such a process of resolution improvement is called nonsubtractive dithering (ND), as the noise is not subtracted from the signal after quantization.

There are several types of dither (Carbone & Petri, 1994) and usually uniform noise leads to best results, if peak-to-peak value [D.sub.d] of noise equal to one quantization step q could be provided, [D.sub.d] = q. But it is often assumed in the theory, that the number of processed samples N is big and only deterministic part of error determines achievable accuracy. In the application for microcomputers like this, N is small and therefore dispersion of measurement output should be included in model of error. Analyses of second moment of error have been presented in (Wannamaker et al., 2000). It could be shown that uniform noise gives still good results and mathematical model of the whole error after averaging will be presented next.

If only deterministic part of error of ideal quantizer is investigated, mean error is the right parameter. To model error after averaging of small number of samples N the mean-square error (MSE) was chosen as suitable parameter of dithering and averaging performance rating for finding an optimal noise dispersion. MSE is theoretically evaluated as mean for one whole quantization step according to (Skartlien & Oyehaug, 2005), where Gaussian noise was analyzed. Following these ideas we got theoretical model of MSE for uniform noise

[mu].sup.2.sub.a] ([D.sub.d], N) = [q.sup.2] + [D.sup.2.sub.d]/12N + (1 - 1/N) [q.sup.2/2 [[pi].sup.2] [sinc.sup.2] ([pi][D.sub.d]/q) (1)

This formula embodies influence of both the mean error and dispersion of measurement results. The second part of (1) describes the deterministic part of error which is zero for [D.sub.d] = l.q (l is positive integer) and negligible for big [D.sub.d]. But the first part reflects dispersion of error and impact of added noise is negative here. The first part is meaningful for small number N of averaged samples, when optimal dispersion of dither is dependent from N. For N=59 according to theoretical relation from (1) the optimal value of [[sigma].sub.d] ([[sigma].sub.d.sup.2] = [D.sub.d.sup.2]/12) is 0,2813q.

4. EXPERIMENTAL RESULTS AND DISCUSSION

Experiments were performed with designed measurement system, where AIM and ND with averaging of N=59 samples was implemented. The mean error curve obtained from 20 measurement results is depicted in the Fig.1. The iterative method used for averaged samples corrects error considerably under the 1 LSB level. In our case appropriate dither helps to suppress nonlinearities involved by quantization. Black curves correspond to quasi-optimal standard deviation of dither.

To evaluate also dispersion of results within error analysis, RMSE (Root MSE) resp. [[mu].sub.a] curves have been depicted in the Fig.2. As presented in previous section, implemented method can significantly correct gain error and offset. Main contribution of designed correction is in correction of nonlinear error component. Therefore linear error part had been subtracted from measurement results before evaluation of RMSE. Using only natural noise averaging could improve accuracy. According to Fig.2 dither with standard deviation close to 0.025 % (0.25 LSB) has enhanced accuracy, but integral nonlinearity (INL) of ADC has caused notable shift of experimental curve against the theory. AIM corrects INL and therefore it shifts the curve closer to theoretical values. Optimal dither dispersion is lower than theoretical optimum because natural noise is present in the input signal. According to Fig.2 quasi-optimal standard deviation of dither has been found [[sigma].sub.d] = 0.02279 % (0.2333q), for which the obtained mean error is presented in the Fig. 1.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

5. CONCLUSION

In our experimental measuring system combination of two methods has been implemented for ADC error correction. Additive iterative method (AIM) has automatically corrected integral nonlinearity and it is based on precisely designed inverse element. Non-subtractive dithering with averaging has enabled correction under level of 1 LSB of used 10-bit ADC. For uniform dither theoretical dependence of root mean square error (RMSE) upon standard deviation of added noise has been proved through measurements in the whole range. Quasi-optimal value of dither standard deviation has been found. The RMSE has been reduced from 0.036 % to 0.0076 %. Evaluated in ENOB accuracy improvement from 9.64 bit to 11.90 bit has been obtained, which excels previous results with Gaussian dither (11.73 bit). The designed technique requires minimum additional hardware components (only passive LP filter) so it may be used in each microcontroller based measuring channel for which the high speed is not the necessity.

6. ACKNOWLEDGEMENTS

Work presented in this paper was supported by the Slovak Ministry of Education under grant No. 2003SP200280802 and by the Slovak Grant Agency VEGA under grant No. 1/0551/09.

7. REFERENCES

Carbone, P. & Petri, D. (1994). Effect of Additive Dither on the Resolution of Ideal Quantizers, IEEE Transactions on Instrumentation and Measurement, Vol. 43, No. 3 (June 1994), pp 389-396, ISSN: 0018-9456

Kamensky, M. & Kovac, K. (2009). Iterative Method and Dithering with Averaging Used for Correction of ADC Error, Measurement Science Review, Vol. 9, No. 4 (September 2009), pp 86-88, ISSN 1335-8871

Michaeli, L. (2001). Modeling of Analog-to-Digital Interfaces, (in Slovak), Mercury, ISBN 80-968550-1-8, FEI TU Kosice (Slovak Republic)

Muravyov, S. V. (2000). Model of procedure for measurement result correction, Proceedings of the XVI IMEKO World Congress, Hofburg, Vienna, September 2000, published on CD, Vienna (Austria)

Skartlien, R. & Oyehaug, L. (2005). Quantization error and resolution in ensemble averaged data with noise, IEEE Trans. on Instrumentation and Measurement, Vol. 54, No. 3, (June 2005) pp 1303-1312, ISSN: 0018-9456

Wannamaker, R. A.; Lipshitz, S. P.; Vanderkooy, J. & Wright, J. N. (2000). A Theory of Nonsubtractive Dither, IEEE Trans. on Instrumentation and Measurement, Vol. 48, No. 2 (February 2000), pp 499-516, ISSN: 1053-587X

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Title Annotation: | analog-to-digital converter |
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Author: | Kamensky, Miroslav; Kovac, Karol |

Publication: | Annals of DAAAM & Proceedings |

Article Type: | Report |

Geographic Code: | 4EUAU |

Date: | Jan 1, 2009 |

Words: | 1677 |

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