Measure an ADC's Transition Noise

Today’s low-noise, 12-bit ADCs don’t produce enough noise for you to use traditional RMS noise measurement techniques. You do, however, have options. In this article, I will describe how you can use a modified traditional statistical technique to calculate an ADC’s RMS noise. Next month, part two of the article will show you a technique that uses a modified servo-loop circuit and some statistics to improve measurement speed and accuracy while reducing the calculations.

An ADC’s RMS noise adds to the device’s input voltage (V_in) to produce a band of output-code transitions centered around an ideal transition voltage, V_trans. The RMS noise is Gaussian in nature and causes the ADC to produce more than one digital output code for a fixed V_in. Figure 1 shows the nature of transitions of three output codes.

Figure 1 . Real transitions between two adjacent ADC output codes increase as Vin approaches the transition voltage Vtrans. Halfway between two transition voltages, the ADC's output may be stable.

For some ADCs, you can measure RMS noise by observing an ADC’s output codes. In this traditional RMS noise-measurement technique, you select a fixed value for V_in and count the occurrences of all the codes that the ADC produces. Then, you plot a histogram of counts per code and calculate the mean output code and its standard deviation (s ), where s is a measure of the device’s RMS noise.

Several Output Codes
Low-noise, 12-bit ADCs, however, produce no more than two codes for any value of V_in. For some voltage ranges, these devices produce just one output code, which means you can’t get a sufficient distribution of codes to get meaningful results. Despite the limited number of output codes, you can still calculate the RMS noise.

If you tried to pick a value for V_in the way you do to make noise measurements with 16-bit devices, you might land in a stable range and get just one output code, which tells you nothing. You can get the necessary information from the output codes’ "frequencies of occurrence" around the transition voltage and then calculate RMS noise.

Figure 2 shows that a given value of V_in produces complementary probabilities (such as 90:10 and so on) for code_x and for code_x+1. When V_in = V_trans, the ADC produces the two codes in equal numbers. Thus, at V_trans, the probability for code_x equals that for code_x+1.

Figure 2.. The Gaussian nature of an output code's frequency of occurence shows that if the ADC produces just two codes for a given input, codex and codex+1 will occur with 50% probability when Vin=Vtrans.

For other than 50:50 probabilities, you will need a difference voltage, DV, which moves the value of V_in away from V_trans. The magnitude of DV for a given probability ratio, say 90:10, depends on the amount of RMS noise around V_trans. The greater the RMS noise, the larger DV you will need to achieve that 90:10 probablility ratio of code_x to code_x+1.

To make the measurements from which you calculate the RMS noise, start by choosing an ADC output code (code_x). Generally, you should pick a codex near the high end of the ADC’s range of codes, because most ADCs show higher noise when V_in nears full scale. A 12-bit ADC produces digital output codes from 000h to FFFh, so you might pick FF0h as code_x.

Find Transition Voltage

To calculate the RMS noise (s ), you first find V_trans for code FF0h. Adjust V_in until you find equal occurrences for FF0h and FF1h. Next, select a value for D V and measure a code’s frequency of occurrence, which becomes that code’s probability of occurrence in calculating s . Adjust D V and measure the occurrences until you find that FF0h occurs about 10% of the time. Generate a histogram of the output codes and take the frequency of occurrence of FF0h.

The equation a) below shows the relationship between the frequency of occurrence at D V, which is P_x+1(D V), and the value of s given that F = D V / s.

The math is easier than it looks. To calculate s , you need a value for F . Unfortunately, you can’t calculate F directly, but you can find it using a table of normal distributions and equation b) below:

Find the value of FF from the equation, and then get the corresponding value for F from Table 1. Once you know F , you can find s , the standard deviation of V_trans.

For example, if you observed a 10% occurrence of code_x+1, then p_x+1 as a function of DV equals 0.1. So, F(F ) = 0.4. From Table 1, the corresponding F for F( F ) = 0.4 is approximately 1.28. Now that you know the value of F , you can calculate s , the device’s RMS noise.

The above process is tedious, but you can use a servo-loop circuit to generate V_in. You set the frequency of occurrence for the output codes by selecting circuit components. With that fixed frequency of occurrence, F(F ) and, therefore, F become constant—you eliminate the substitution into the above equations. TMW

Frank Ohnhauser designs integrated circuits for motor control.
E-mail: ohnhauser_frank@burr-brown.com.