Figure 1. To calculate ENOB of an ADC, digitize a sine wave and apply a curve-fitting program in your PC.
© 2008 Brad Brannon, Analog Devices, Inc.
Introduction
When selecting an analog to digital converter (ADC), you must examine several specifications. One spec to examine is effective number of bits (ENOB). ENOB takes many ADC errors (such as integral nonlinearity, differential nonlinearity, and total harmonic distortion) and conveniently reports them as one specification, providing an overall picture of an ADC’s performance.
There are two methods for calculating ENOB. One lets you convert an ADC’s signal to noise ratio (SNR) directly to effective bits. Another method relies on curve fitting, and that’s the method I’ll discuss. This method is also useful for evaluating data-acquisition boards and digital scopes.
ENOB represents an ADC's dynamic performance expressed in number of bits. ENOB is related to the number of output bits, but the two specs differ. An ADC’s output bits describe its theoretical performance, while the effective number of bits represents the device’s actual performance.
Many parameters affect ENOB. Some change with frequency, while others don’t. Integral nonlinearity (INL) and differential nonlinearity (DNL) directly change the ADC’s quantization effects but don’t change with frequency. That limits their usefulness as parameters for evaluating ADCs.
One parameter that does change with frequency is harmonic distortion. As input frequency increases, the ADC’s internal amplifiers or comparators begin to slew rate limit the ADC’s performance. Harmonic distortion rises, and ENOB falls.
Another important factor in ENOB is signal amplitude. If an ADC has a bad DNL error near it’s fullscale, then increasing the input signal’s amplitude so the signal traverses this bad code will cause the ADC’s rms error to increase, thus decreasing ENOB.
Figure 1. To calculate ENOB of an ADC, digitize a sine wave and apply a curve-fitting program in your PC.
How to Calculate ENOB
To perform the ENOB calculation, you need to drive the ADC under test with a pure sine wave. Figure 1 shows a typical setup you can use to measure the ENOB of an ADC. Because a signal generator’s output may not be pure enough to test the ADC accurately, you should add a narrow band pass filter to remove extraneous harmonics and noise.
This filter’s pass band should be between 5% and 10% of the test signal’s frequency. For example, if your test signal is 10 MHz, the filter’s pass band should be between 500 kHz and 1 MHz, with a center frequency of 10 MHz.
Collect the unit under test (UUT) digital output in a high-speed data capture memory (such as the Digital Converter FIFO Evaluation Kit available from ADI). If you are evaluating a data acquisition card, do not disable the car’s front end filters; consider the filters to be part of the overall system. After collecting the data, transfer them to a computer for storage and analysis.
Curve Fitting
You can calculate ENOB with a sine wave curve fit program that I have written (see my website www.converter-radio.com for a copy). My program extracts a best-fit sinusoid signal (it reconstructs the original signal based on the data samples from the ADC) using a least mean square fitting technique. The technique is similar to linear regression except that it is based on a sinusoid rather than a straight line. The rms error function is written as:
In this equation, yn represents the sampled signal data with a record from 1 to M samples. The sine and cosine function represent the modeling signal, and the C is the signal offset.
To determine the solution to the best-fit sinewave, this function is differentiated by the four variables: amplitude, offset, frequency and phase. These equations are then minimized and simultaneously solved. The unit sample rate is assumed know for the tests. The result is the rms error between the input signal and the fitted sinusoidal signal.
After the program
calculates the rms error, it can compare the results to the ideal quantization
error of the ADC. The ideal quantization error is the error that would result
from a perfect ADC with the same number of bits as the ADC under test. The rms
error of an ideal 12-bit ADC is 
,
where Q is the size of the ADC’s least significant bit (1/4096 for a 12 bit
ADC).
The logarithmic ratio between the actual rms error and the ideal error gives the error in terms of bits. This is subtracted from the actual number of bits to provide ENOB:
While complex in nature, this method is easy to implement in software. The method, however, is recursive and requires initial values. You need to accurately “guess” all but one of the variables for the equation to work. Providing the initial values for four variables to within a few percent can be difficult.
To simplify the process, you can use a three-term sine wave curve fit. The three-term equation assumes know values for sample rate and analog input frequency. This is a reasonable assumption for test environment because you can use a synthesized signal generator to generate an input signal of know frequency.
If you don’t know the analog input frequency, though, you can use a spectrum analyzer to measure the frequency of your test signal. You can also perform an FFT on your test signal after digitizing it but before applying the curve fitting algorithm-just be sure the number of samples you take is a power of two. The three-term curve fit makes initial assumptions about the amplitude, offset or phase of the input signal.
The three-term method provides a poor estimate of the actual effective bits, but it does provide an excellent estimate of the signal parameters, usually within 1% of their actual values. You can calculate the ENOB by first performing a three term since wave curve fit and then using the updated estimates of frequency, phase, amplitude and offset as initial values for the four term solution.
ENOB Has Limits
Despite being a convenient spec for comparing ADCs, ENOB does have several drawbacks. It provides no measure of gain or offset error and does not provide information of gain flatness over frequency.
Although ENOB will show degraded performance over frequency, the measurement made at each frequency become normalized in amplitude and do not directly show information on width. Therefore, you should not rely on ENOB as the sole criterion for evaluation ADCs.
For Further Reading
“Dynamic Performance Testing of A to D Converters,” Hewlett-Packard Product Note 5180A-2, Hewlett-Packard, Santa Clara, CA, January 1989.
IEEE Std. 1057, Standard for Digitizing Waveform Recorders, IEEE, New York, NY.
Kester, Walt, “High Speed Design Seminar,” Analog Devices, Norwood, MA, 1990.
Peetz, Bruce E., Arthur Muto, and J. Martin Neil, “Measuring Waveform Recorder Performance,” Hewlett-Packard Journal, Palo Alto, CA, Vol. 33, No. 11, November 1982.