## Using Histogram Techniques to Measure A/D Converter Noiseby Steve Ruscak and Larry Singer
This article describes a technique for using histograms to determine the rms noise of an A/D converter, referred to the input. It can complement the popular qualitative approach of evaluating A/D converter performance by applying a dc analog voltage and observing "code flicker" at the A/D's digital output.
- Connect a "clean" dc source to the analog input of the A/D converter (and a "clean" reference, if an external one is used).
- Record the results of a large number of conversions, depending on the expected noise. One-to-two million conversions are usually more than adequate for a low-noise A/D.
- Sort the conversions into code "hits." The core of the histogram is essentially a bank of counters (or bins); there is one bin corresponding to each possible digital output of the A/D converter. After each conversion, the digital code at the A/D output is determined (by hardware or software) and the corresponding counter is incremented. If an ideal (noiseless) A/D were connected to a dc voltage, the histogram of
*n*conversions would indicate*n*code 'hits' in the bin corresponding to the digital value of the dc input. All other bins would equal zero.
While the ideal A/D only produces code hits in a single bin, an actual A/D will produce additional codes outside the main bin for any dc input value, due to the presence of noise. The number of codes that fall outside the main bin is the key to measuring the converter's rms noise. Avoidance of an additional effect at certain codes due to differential nonlinearity will be discussed later.
Figure 1. Quantization of a probability distribution.
What are our important assumptions? First, that the model for the A/D is an ideal quantizer with a Gaussian noise source added to the input of the device. The measurement will be somewhat erroneous if the dominant noise source is non-Gaussian; for example, digital feedthrough tends to be code and signal dependent. Second, the architecture of the device affects the measurements; but the technique is valid for the majority of high-speed, pipelined converters available from Analog Devices.
Figure 2 is a plot of a PDF for A/D input noise in which corresponds to the least significant bit (LSB) of the converter under test. The dc input to the A/D is adjusted to center the PDF on the midpoint of code
Figure 2. A Gaussian distribution scaled in LSBs, about the mean. For a symmetrical Gaussian PDF, the equation(1),
expresses the probability of getting a code outside the main bin for that PDF. The terms inside the integral are simply the expression for a Gaussian distribution with unit area and standard deviation, . Integrate from -infinity to -0.5 LSB to find the area under one tail, and from +0.5 LSB to +infinity for the other. By symmetry, one can simply multiply the area under one tail by 2.
To ultimately determine the A/D's noise, equation (1) must be solved for , with
with equal to unity. For each point,
The procedure's simplicity can be demonstrated with an example. Suppose that the fraction of codes outside the main bin equals 0.0027. Using Figure 3b, a magnified version of 3a, find 0.0027 on the y-axis [these values can also be found from a standard normal curve, or Figures 3a and 3b relate rms noise and code histogram values. Do a histogram, find the fraction of codes that occur outside the main bin, look up that number on the y-axis of 3a or 3b, and read off the x-axis value. Dividing 0.5 LSBs by this number gives the rms noise in LSBs.
Figure 3. Plots of F vs. X for equation (2)
Determining the noise of the converter is useful for making a comparison, either to a specified requirement or to another A/D converter of the same, or different type. Figure 4a shows a typical histogram generated from the AD872A 12-bit, 10-MSPS A/D converter. All the codes appear in either the main bin or the two adjacent bins, roughly symmetrically disposed. From the histogram, the ratio of the 34,729 code hits outside the main bin to the 652,790 total number of codes counted is 0.053. From Figure 3a, the x-value corresponding to 0.053 on the y-axis is approximately 1.9 . From equation (3), 1.9 = 0.5 LSBs, and the rms noise () equals 0.26 LSBs rms.
For comparison, Figure 4b shows a histogram of the AD871, a 12-bit, 5-MSPS high-speed A/D converter. Using the same procedure, the hit ratio is 2,581/1,638,400, about 0.0016, and the input-referred noise of the AD871 is thus 0.5/3.17, or 0.16 LSBs rms. With low-noise A/Ds, like the AD871, the test should be run long enough to acquire enough code hits outside the main bin in order to obtain a credible noise measurement.
(1) Drake, Alvin W., Fundamentals of Applied Probability Theory. McGraw-Hill Book Company, 1967 |