6.7 Summary

In this chapter, we have presented the characteristics and limitations of sampling, quantization, and processing of dynamic variables external to a computer. Lab exercise 6 integrates all of these ideas, along with interrupt-based timing, to explore real-time signal processing of the type used in filtering or control. The main points are as follows:

• Physical variables are both continuous valued and continuous time. To be processed by the computer, these variables are converted to and from digital form by means of sensors, actuators, and conversion interfaces.
• Conversion of a physical variable to numerical representation involves two processes: sampling, the evaluation of a continuous signal at discrete points in time; and quantization, the assignment of the value of a continuous signal to one of a finite set of discrete states.
• An analog-to-digital converter (ADC) interface measures an analog input (AI) signal, within a finite voltage range, and expresses the value as a numerical value of finite resolution.
• The length of time required by an ADC to complete its conversion, called the aperture time, limits the measurement accuracy of a time-varying signal. A sample-and-hold circuit is one means of minimizing this error.
• A digital-to-analog converter (DAC) transforms a numerical value, encoding each of a finite number of input states, to an analog output (AO) voltage. The specific technology determines the conversion time and resolution.
• The sampled signal transform theorem establishes that the Fourier transform of a continuous function sampled at frequency \(f_s\) is equal to the superposition of the transform of the function shifted by all integer multiples of the sampling frequency.
• The Nyquist-Shannon sampling theorem establishes that, if a continuous-time signal contains no frequency components higher than \(f_c\), it is completely determined by sampling at a frequency \(f_s\) greater than \(2f_c\). Violations of this theorem may result in the aliasing of some frequency components.
• Digital signal processing and controls, based on periodically sampled dynamic signals, are commonly implemented by means of discrete-time dynamic systems in the form of difference equations.
• The Z-transform provides the connection between the difference equation and the transfer function of a discrete-time system.
• A continuous system, described by its transfer function, may be approximated by a corresponding discrete system using Tustin's method and efficiently implemented using a biquad cascade of difference equations.
• In all of our lab exercises, the myRIO ADC and DAC interfaces have \(\pm 10\) V ranges, with 12-bit resolution. Timing is controlled by means of periodic interrupts.