Analog input to digital output correlation using piecewise regression on a Multi Chip Module

Transcription

1 Analog input digital output correlation using piecewise regression on a Multi Chip Module Farrin Hockett ECE 557 Engineering Data Analysis and Modeling. Fall, Portland State University Abstract This paper demonstrates the calculation of a least squares linear regression model correlate a low resolution digital pulse width a non-measureable analog reference voltage on multi-chip modules (MCM). Because of a large quantization error in the digital pulse output of the MCM, the model was optimized for one tail of the distribution. This tail represented reference voltages outside of the normal distribution and was the specific area of interest for the model. The model was then extrapolated over the entire input range and tested against the entire data set of more than 600 MCMs. The model was shown be a conservative fit with all data points within the 95% confidence intervals. The model was further tested with a sample of data from modules specifically selected from the upper distribution tail of a larger population. Again, with one exception, all data points fell within the 95% confidence interval. This analysis provides the ability predict, with statistical confidence, the worst case voltage of an analog parameter on MCMs that can only be measured indirectly with a low resolution digital signal. The significance of these results are ensure the value of this analog voltage without implementing expensive test fixturing, test instruments and manufacturing inefficiencies. Index Terms Piecewise Linear Regression, Least Squares Analysis, Confidence Interval. I. INTRODUCTION An internal analog reference voltage (Vref) of a multichip module (MCM) cannot be measured directly without manual measurements or a large investment in test fixturing and aumated equipment. This paper uses a statistical confidence interval determine the worst case value of the Vref voltage based on a low resolution, analog--digital parameter that can be measured with existing equipment. The challenge in establishing the confidence interval is the low resolution of the digital pulse width generated by the module. The pulse width is quantized in 5 µs increments representing approximately 4mV resolution of the Vref voltage. Even with averaging multiple samples of the analog -digital measurements, the corresponding Vref voltage is still quantized due the constant analog DC voltage being measured. There are numerous articles on linear regression and optimizing the accuracy of analog--digital converters. An improved method that was demonstrated be more accurate than linear regression was shown for calibrating digital--analog converters in continuous time []. There were also several methods used using piecewise linear regression [2,3] and data weighted averaging. [4] The technique used in this paper is similar the piecewise approach in that it optimizes the linear regression for a range of the input that is of most interest. The model in this paper differs from the piecewise model in that it extrapolates this optimized equation over the entire set of data. The benefit of establishing the Vref digital pulse width correlation is multi-thousands of dollars savings avoid the investment of cusmized test fixturing, test instruments, engineering development and manufacturing inefficiencies. II. METHODOLOGY A. Evaluation of parameters be analyzed The MCM uses Vref as a reference voltage in converting analog voltages a digital value and transmits the digital value as a pulse width with 5 µs resolution. One analog--digital parameter was selected as the correlation variable as it was determined be the most sensitive the reference voltage. This parameter is measured as a pulse width and is referred in this report as PW. By design, a linear correlation exists between the Vref voltage and PW. However, errors are introduced in the linear relationship due A-D converter offsets, A-D converter resolution, noise in the supplied analog value, pulse width measurement error and repeatability in the test system. An assumption is made that these errors average out a zero offset but contribute the variation that must be accounted for by a statistical confidence interval. This variation is expected create a symmetrical distribution of the measured PW parameter. Any asymmetry or tail in the distribution would be suspected be caused by the Vref voltage being outside the normal distribution. The PW distribution was collected on over 0,000 MCMs and

2 2 shown in Fig.. As expected, the majority of MCMs are clustered around one value at approximately 73 µs. The same distribution is shown in Fig. 2 with the Y-axis expanded show quantities less than 00. This expanded view shows slight peaks at ± 5 µs around the major peak of 73 µs. The distributed values between the 5 µs quantization steps are due averaging multiple PW measurements from the MCM. Qty of MCMs PW Fig.. Hisgram of the PW parameter for more than 0,000 MCMs. Note, the majority of MCMs measure a value around 730 µs with minor peaks at approximately 5 µs above and below the peak. Qty of MCMs with associated PW value PW Distribution for 3 Months PW Distribution for 3 months (Exploded View for Qty < 00) L S MM PW Fig. 2. Expanded Y-axis view of the hisgram in Fig. showing the quantities < 00. Note the 0,000 peak at a PW of 73 is cuff. The local peak at 760 µs and the outliers between 750 µs and 790 µs are in the region of interest. 4 C7 C 3 of measurements in the upper distribution. This tail is only apparent on the upper side of the distribution and suggests outlier MCMs with Vref values outside the normal distribution. B. Data Collection & Analysis To determine the Vref PW correlation, a data set was created by measuring both parameters on 604 MCMs. These samples were an exclusive population of MCMs built during a specified period of time and assumed represent the entire population. The following analysis was performed: A hisgram of the Vref parameter was graphed show the distribution. A scatter plot of the Vref PW relationship was created with a least squares linear regression model created using the entire data set. The least squares linear model was optimized for the upper tail of the PW distribution. The upper tail optimized model was applied for the entire data set and compared using 95% and 99% confidence intervals. A test sample was created from MCMs in the upper tail population of the PW distribution and compared against the confidence intervals. Lastly, the worst case range of Vref was determined from the corresponding PW value based on the confidence intervals. III. RESULTS Figure 3 shows the Vref distribution of 604 MCMs. The distribution has a large peak at approximately.000v and a one-sided tail on the low side of the distribution. The one sided tail is similar a mirrored image of the one sided tail observed in the PW distribution in Fig. 2 and supports the assumption that the cause of the one-sided tail in Fig. 2 is due the vref voltage being outside of the normal distribution. A significant observation in Fig. 2 is the peak at 76 µs (2 quantization steps above the major peak) and a scattering 250 Vref Hisgram for qty 604 MCMs s 50

3 3 tail of the PW distribution, a new linear regression model was created using only samples with Vref < 7V. Fig. 5 shows the XY scatter plot of these data samples and the optimized linear regression model. R 2 :0.809 R :0.793 R:0.900 a Approximation Out put y 5 Fig. 3. Hisgram distribution of Vref for 604 MCMs. Note, the tail on the low side of the distribution approximately matches a mirror image of the distribution of the outlier PW values in Fig. 2. An XY scatter plot of Vref versus PW and the Least Squares linear regression line is shown in Fig. 4. The linear model shown in Fig. 4 appears be biased in the positive X direction and is a poor fit for the three data points with a PW value > 760 µs. Vref (V) n= e e E E E-0 Scatter Plot of 604 Samples Data Samples Best Fit with 604 samples y = E-3 * x PW (us) Fig. 5. Scatter plot of Vref vs PW and the best fit linear regression line for the data samples with Vref < 7V. The best-fit line calculated in Fig. 5 is for the area of interest where PW is significantly higher than the majority of MCMs. Visually, this model appears be a good fit for this sample of data. The linear model created from the tail of the distribution was tested against the entire data set by graphing the model and confidence intervals of 95% and 99% with the data. This is shown in Figure 6 and all data samples fit within the 95% confidence intervals. Additionally, a test sample group was created by measuring Vref on MCMs selected from the upper tail of the PW distribution from a large sample of modules. The XY data is shown in Fig. 6 as indicated by a red x. One sample of 25 is outside the 99% confidence interval. All other samples are within the 95% confidence interval. Fig. 4. Vref versus PW scatter plot of 604 MCMs. The best fit line is based on the Least Squares regression analysis weighting all data points equally. In comparing the Vref distribution in Fig. 3, there is a major concentration of MCMs with Vref at.000v. This also corresponds the majority of MCMs having a PW value of 73 µs per Fig.. This concentration of values at 73 µs/.000v is due the low resolution of the analog-digital converter and causes the least squares linear regression model be heavily weighted at this data point. This causes an overall bias the model as shown in Fig 4. Because our specific area of interest is around the upper

4 4 O ut pu ty Confidence Intervals for y Model 95% CI 99% CI Outliers the resolution of the PW parameter. With 5 µs resolution of the MCM pulse width, most of the PW data points measured exactly 73 µs. It was not possible for a linear regression model calculate an accurate least squares fit without artificially providing a weighting the XY region of heavy sample concentration. The bias of the least squares linear regression model was overcome by ignoring the heavily concentrated data points and using a piecewise linear regression model with the data samples in the outlier range of the distribution. This essentially gave a weighting value of 0 all data points in the heavily sampled region and provided a high weighting value the data points in the tail of the distribution. This technique proved fit the data using extrapolation and also fit the test samples for the tail of the distribution. Fig % and 99% confidence intervals are overlaid on a scatter plot of all data samples with the best fit linear regression line calculated from data samples with Vref < 7V. The confidence intervals were calculated using MATLAB. Also shown are outlier samples (red x) measured from a larger population. The minimum value of Vref, for a given PW can be determined from the confidence intervals on the graph in Fig. 6. This graph is enlarged showing only the data from the tail of the PW distribution in Appendix provide better resolution for determining the Vref range. At a PW value of 750 µs, the 95% confidence interval for Vref is between approximately 0V 7V. Therefore, the model predicts a minimum Vref voltage of 0V with a 95% confidence for a PW maximum limit of 750 µs. This PW value corresponds a natural break in the distribution as shown in Fig. 2 and also shown in the scatter plots both in Fig. 6 and Appendix. IV. DISCUSSION The Least Squares linear regression created a biased model when all samples from over 600 data points were used. There are two facrs that contribute the bias of the model: The first facr that contributed strongly the biased model is a high concentration of data points with XY values at a PW of 73 µs and Vref of.000v. Over 90% of the population falls within a narrow distribution for both Vref and PW. The least squares linear regression technique weighs all data points equally, and provides a model that minimizes the mean squared error of all data points. This minimizes the error for the majority of data points and causes the linear model equation pass close that XY data point. The second facr that contributed the biased model is V. CONCLUSIONS AND SUMMARY By extrapolating a piecewise linear regression model from a sampling distribution that is approximately uniform, the bias from a heavily sampled distribution can be compensated. The 95% and 99% confidence intervals, as shown in Appendix, can be used set the PW value a corresponding range of Vref. If the variation, or lerance of the expected Vref value is acceptable, the expense of measuring Vref directly can be eliminated. It may be determined that using the PW value is inadequate test the Vref parameter due the lerance of the 95% interval being o large. The confidence interval may be able be reduced by further narrowing the piecewise linear regression model a narrower data set near the expected PW limit of 750 µs. REFERENCES [] Keller, M.; Manoli, Y.; Gerfers, F., A calibration method for current steering digital analog converters in continuous time multi-bit sigma delta modulars, 2004 IEEE International Symposium on Circuits and Systems, 2004, PPI Vol. [2] Mourot, G.; Maquin, D., Parameter estimation of switching piecewise linear system, 42 nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), 2003, pt. 6, p Vol. 6 [3] Abdelmalek, Nabih N., Piecewise linear least-squares approximation of planar curves, International Journal of Systems Science, v 2, N 7, Jul, 990, p [4] Baird, R.T., Improved Delta Sigma DAC linearity using data weighted averaging, 995 IEEE Symposium on Circuits and Systems, 995, pt., p3-6 vol.

5 5 APPENDIX Appendix. Least Squares best-fit Linear regression model calculated from data points with Vref <7 V. With the exception of one outlier data point, all points are bounded by the 95% confidence interval. Note, at a confidence interval of 95%, a minimum Vref of 0V can be assured with a maximum PW limit of 750 µs..05 Confidence Intervals for y.0 Model 95% CI 99% CI Outliers.005 Output y