Dynamic Testing of A/D Converters Using Sinusoid Signals

Theoretical Background

Dynamic parameters of an analog-digital converter, such as signal-to-noise ratio or total harmonic distortion, are important in the characterization or comparison of such converters. Some of these parameters can be found using time-domain methods. Among these methods, curve fitting is widely used in testing. During the measurement, a high-quality sinusoid signal is applied to the input of the device under test (DUT). The output samples are saved into a y (y[n]) vector. A sinewave is fitted to the measurement results, which consists of a series of numbers. This fit is basically an estimate of the unknown input signal. Calculating the difference between the measured values and the fitted sinewave, some consequences may be drawn about the dynamic parameters of the converter. Based on this difference (the so-called residual vector), the rms value of the error can be calculated, and from that the effective number of bits (ENOB) in the converter and the signal-to-noise and distortion ratio (SINAD) can be determined.

Let us assume that the input signal applied to the input of the converter is given in the form

where A, f_i, Φ and C are the amplitude, the frequency, the phase and the constant offset of the signal, respectively. Let's take M samples from this signal, at f_s sampling rate. A sinewave signal should be fitted to the column vector y[n], n=1..M containing the output samples. There are two different cases: the frequency of the input signal is known or unknown. In the first case there are three parameters to determine (A, Φ, C), while in the second, four (A, Φ, C, f_i). Let us assume that the frequency is known, thus, discuss only the three-parameters fitting in the following. (Note that from the theoretical and research viewpoint the four-parameters fitting is much more interesting - see the publications related to this topic.)

The fit with the least squares error can be found by minimizing the following cost function

where , the parameter-vector containing the unknown variables is p=[Acos(Φ), -Asin(Φ), C]^T. Creating the following matrix

the cost function given in (2) can be formulated as . At the minimum value of the cost function the parameter-vector becomes:

Note that if the sampling is coherent, i.e., the number of periods of the incoming signal is integer, then the solution of the parameter-vector can be found in closed form:

which (ignoring the sign of the second equation) is equal to the calculation of the real and imaginary components of the discrete Fourier-transform formula at a given input frequency and the real component at DC.

By using the estimated parameter-vector, the estimators of the amplitude, phase and offset can be easily calculated (, if and if , and finally ), thus the estimated input signal can also be calculated (). Subtracting this signal from the measured samples, the error vector (the residual) can be calculated.

The SIgnal-to-Noise-And-Distortion ratio (SINAD) is defined as the ratio of the input signal's power and the power of the residual vector, containing the noise and the harmonics:

i.e., the variance of the residual vector (the difference between the vector of the fitted sinewave and the vector of the measured samples). Since the SINAD parameter compares the noise power to the incoming signal's power, it is important to drive the input of the converter rail-to-rail, but not to overdrive.

Knowing the error, the effective number of bits can be also calculated, where the rms value of the measured error is compared to that of the ideal quantization noise:

where N is the number of bits of the converter, and Q is the nominal code-width of the converter.

Figure 1 shows the steps described above in the case of an ideal quantizer (MATLAB simulation). The original sinewave is quantized, then a sinewave is fitted to the quantized samples, then the residual vector is calculated by subtracting the fitted sinewave and the input signal. In this ideal case the residual vector contains only the quantization error.

Fig. 1. The steps of sine fitting

Results

During the first part of the research, a MATLAB-implementation of the sinewave-fit routines defined in the standard IEEE-STD-1241 has been released. The program can be downloaded from the URL http://www.mit.bme.hu/projects/adctest/. Figure 2 shows the input and the result window of the program.

Fig. 2. The input and the result windows of the ADC test program

Among the theoretical results, the parameters of the sine wave fit algorithm, which were not exactly specified in the standard have been defined (e.g., exact stop condition of the iteration as a function of the number of bits of the converter). Another current research direction is the study of non-ideal effects. (One topic: how big the error caused by the fact that the quantization error of a sine wave is not gaussian. In this case the least-squares fit is not the best estimator. Another topic: how big the error caused by the fact that the histogram of the sinewave is not uniform, i.e., at the top and the bottom the converter's errors are weighted. What can be done to minimize this error, etc.)

Related Publications:

Further publications on A/D converters can be downloaded from here.

Useful links:

• Program to demonstrate the three- and four-parameter sine wave fitting algorithms

 Further information: János Márkus