Analogue-to-digital converter resolution, accuracy, and error sources

Analogue and Digital Converter

This is an electronic device that helps in the conversion of continuous signals to discrete or isolated digital numbers. When an analogue voltage or current is fed into the device as an input, it converts it into a digital number relative to the voltage or current magnitude. There are a number of terminologies related to ADC, which include resolution, accuracy, response type, sampling rate, aliasing, dither, oversampling, relative speed and precision, and the sliding scale principle, however, just a few of them will be considered in this study.

The resolution of an ADC refers to an indication of the isolated values that in can generate over the range of analogue values. Since the electronic storage of these values is in binary form the resolution is normally expressed in bits and the available discrete values being a power of two. For instance, an ADC whose resolution is 6 bits has the capability of encoding an analog input to one in 64 varied levels, given that 26=64. It is also possible to define resolution electrically and give the expression in volts. The voltage resolution of an ADC is found by dividing the overall voltage measurement range by the number of isolated intervals. The formula is written as:

Where:

Q= resolution in volts/step i.e. (volts/output codes-1)

EFSR= full scale voltage range which is given by VRefHi -- VRefLow

M= ADC's resolution in bits

N= number of intervals= 2M -- 1 (Knoll 1989)

Consider an example given by Knoll (1989) where the Full scale measurement range is 0 to 7 volts, the ADC resolution will be 3 bits which means 8 quantization level sie. 23. When this is given in terms of ADC voltage resolution it equals 7V/7 steps which give 1V/step.

The ADC is not exempted from errors that are encountered by other instruments and has errors that have a number of sources which brings about the question of accuracy. These errors are categorized as quantization error, non-linearity error and aperture error. Quantization error is caused by the finite resolution of the ADC and cannot be avoided in any ADC while non-linearity error occurs due to the physical imperfections of the ADC which leads to a deviation between the output and the input from a linear function. The third error is caused by a clock jitter and is usually exposed when digitizing a signal that is time variant. The non-linearity error can be toned down by calibration or eve averted by testing. In most ADCs the range of input values that map to every output value are linearly related to that output value and are referred to as linear ADCs.

The speed and precision of an ADC varies depending on the type of the ADC with the Wilkinson ADCs being considered the best since they exhibit the best differential non-linearity. ADCs are usually represented using a symbol; the conventional electrical symbol used is as below (schematic).

Monte Carlo simulation

By the Monte Carlo method, the Gaussian message ?s (t) of bandwidth Ws rad/s and rms frequency deviation ?s rad/s is simulated by a sum of Ns sine waves

(1)

Where is the peak frequency deviation of the nth tone, is the fundamental modulation frequency so that Ns

a = Ws, and is a random phase distributed uniformly over (-

). Analogously, the Gaussian message ?i (t) is simulated by (2)

Where, and Ni

a = Wi are, respectively, the peak frequency deviation of the nth tone, the rms frequency deviation and the bandwidth of ?i (t), and is a random phase distributed uniformly over (-

). If the numbers Ns and Ni of tones simulating the Gaussian messages are large enough, the statistics of (1) and (2) approach that of Gaussian noise.

Omitting the details, the signals s2(t) and i2(t)at the output of the bandpass filter are derived using the Fourier method by processing the spectral components of s1(t) and i1(t)at the frequencies s ± n a and i ± n a respectively, by the filter transfer function. The amplitude modulation and the phase modulation of the filtered FM signals are calculated with arbitrary accuracy depending on the number of spectral components considered. The distorted frequency modulation ?s (t) of the filtered desired signal is obtained analytically as the derivative of ?s (t) with respect to t. The colored Gaussian noise processes nc (t) and ns (t) are simulated using the FFT-based filtering as follows. Uniform deviates are generated by a routine based on three linear congruential generators. Normal deviates are obtained from uniform deviates using the Box-Muller method and the resultant Gaussian processes are filtered in the frequency domain by the transfer function HLP (i

A fast recurrent algorithm

3

With linear prediction

4

Computational efficiency of the Monte Carlo simulation is enhanced by implementing the time-consuming Fourier method using a low sampling rate. The fast recurrent algorithm based on the above equations (3) -- (4) allows a high sampling rate with the amplitude and phase modulation of both filtered FM signals as well as the distorted frequency modulation of the desired signal cubic spline interpolated. (Hasan, 1991, p. 2-3).