Statistics behind error analysis of ADC system
This video walks through the statistics behind error analysis. Specifically, it covers the statistical implications of the typical and maximum data sheet specifications and how they can be used to determine a worst case analysis and a statistical analysis.
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Hello and welcome to the TI Precision Lab discussing the statistics behind error analysis. Specifically, in this section, we will learn the statistical implications of the typical and maximum data sheet specifications. Furthermore, we consider the difference between a worst case analysis and a statistical analysis looking at typical values.
Let's start with a basic error analysis. To better understand the concept of calculating total offset error in a system. This slide shows the example circuit that we will use for this error analysis. It is a high side current shunt monitor that will monitor currents from 50 milliamps to 20 amps. Let's find the offset error seen by the data converter considering all of the elements in the signal chain.
The current shunt amplifier u1 has a gain of 20. So the output offset is increased by 20. The buffer, U2, has a gain of 1. So the output offset from u1 directly adds to the offset of u2 and the ADC u3. a. Common approach is to take the maximum limit for each of the three devices and add them directly to find the worst case offset. However, this calculation assumes that all three devices have a worst case offset at the same time.
What is the statistical likelihood that all these worst cases lineup at the same time? We will take a look at this in the next few slides. This slide shows the statistical background behind the typical and maximum specifications in a datasheet. For a zero centered specification, the typical value is the absolute value of one standard deviation plus the mean of the distribution.
Often, the mean value is near zero. So for the purposes of this discussion, we will assume that the mean is zero. In this example, the typical offset is plus or minus 1 millivolt, which corresponds to plus or minus one standard deviation. The maximum offset is a tested parameter, so any device that exceeds the maximum limit is discarded and will not be shipped to customers.
Thus, the distribution is actually a truncated Gaussian distribution, as there is no population above the maximum or below the minimum. The maximum limit is set on the tail of the distribution to minimize yield loss during manufacturing. Typically, the maximum is set above three standard deviations.
In this example, you can see that the maximum was set to four standard deviations. Note that 68.27% of the population is inside the typical limits. From a statistical perspective, how likely is it that we find a device at the maximum limit? For this example, what is the probability that we get a device at exactly 4 millivolts?
Since probability is equal to the area under the curve, and the area under a single point is zero, technically, the probability is zero. This isn't very helpful. So let's consider the probability that we are near the maximum limit. In this example, the probability that we are between 2 millivolts and 4 millivolts is 2.272%. This makes sense when you look at the measured histogram, as you really can't see any bends in this region.
We will use the statistical information in the next slide to find the compound and probability that all three offsets from our example circuit are near the worst case value. Here, we show the Gaussian distribution for all three components based on the datasheet specifications. What is the probability that all three devices are near the worst case?
You can see that for each device, the probability of being near worst case is about 2%. Since the three distributions are random and uncorrelated, the compound probability that all three events occur simultaneously is the product of the three probabilities. Stepping through the math, you can see the probability that all the devices are near worst case is 0.0011%. You can imagine that as the number of components in the system increases, the probability that they are all at the worst case value is very small.
So directly adding the worst case offset for each device isn't the best approach for understanding a system's total error. In the next slide, we will look at an approach that gives a better statistical understanding of the errors. This slide shows the distributions for all three devices in the signal chain. Rather than adding the maximum values, we will add the three distributions.
The standard deviation of uncorrelated Gaussian distributions can be combined as the square root sum of the squares. In this example, all the distributions are referred to the input of the data converter. So the offset from u1 is multiplied by the gain of 20 we noted earlier. The standard deviation for the total combined offset distribution is plus or minus 1.887 millivolts.
Let's take a closer look at the combined distribution. The final system distribution has a standard deviation of plus or minus 1.887 millivolts. This is analogous to a typical plus or minus 1.887 millivolts for the system. But what is the maximum offset error for our system? The maximum can be set according to the risk tolerance in your system specification.
The table shows what percentage of a population will be inside a limit set according to the number of standard deviations. For example, if the system specification maximum is set to plus or minus 3 standard deviations, the 99.73% of the population will be inside that limit and 0.37% of the population will fall outside of the limit.
Depending on the requirement, a more conservative limit can be set. It is important to realize that most specifications like offset have other factors that can impact the total error. Be careful not to adjust the statistical limits beyond the worst case limit we started with. Remember the actual distribution of the device is a truncated Gaussian distribution, so the combined distribution will also be truncated at the worst case we calculated earlier. That concludes this video. Thank you for watching. Please try the quiz to check your understanding of this video's content.
This video is part of a series
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Precision labs series: Analog-to-digital converters (ADCs)
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