# Digital Signal Processing, Discrete-Time Control, and Numercial Analysis: All About the Same

Digital Signal Processing (DSP) was, a few decades ago, a relatively obscure field associated with the mathematics of numerical analysis. As computers have become cheaper and faster, it is now commonplace in cell phones, motor drives, medical instruments, and many other places, often mixed in with analog circuits. It is so common that it has moved into the mainstream of what an electronics engineer needs to know nowadays. Yet the emergence of DSP has happened well within the career lifetime of most engineers, and it has required that we all learn the subject to some extent, or else hope to get around it.

One of the early confusion factors in learning DSP is the rather different emphases placed upon it in DSP books. If you learn DSP from a book like the Prentice-Hall classic, Digital Signal Processing , by Oppenheim and Schafer (or from Alan Oppenheim’s excellent MIT video course, which follows his book to a great extent), or even more so, from another Prentice-Hall book, Digital Filters , by R.W. Hamming (the “Hamming window” Hamming), you will get a filter-oriented view of DSP. If you learn it by reading digital control books, you will acquire a control-oriented view. And more basically, if you learn it by reading numerical analysis books of mathematics, you might wonder how it is related to electronics (or engineering). What then is the best approach to an efficient acquisition of the subject?

I know of no single DSP book that explains this at the outset so that the reader knows of the different emphases and styles of presentation of what is essentially the same subject-matter. Consequently, each learner of DSP must go through the somewhat bewildering experience of finding this out and of making the correspondences between the different styles. For instance, in numerical-methods mathematics are different methods of integration, yet in control DSP, essentially the same topic is covered by expressing these different methods in different domains of analysis (s , w , z ) as poles, and emphasizing how they show feedback loop stability. In filter DSP the emphasis is instead on steady-state frequency response and how sharp the filter rolls off to provide frequency band separation. Numerical methods tend to relate difference and differential equations more readily while the differing engineering DSP styles can roughly be categorized as time- (control) and frequency-domain (filter) presentations of DSP.

For communications and pure signal processing, such as FFTs, Oppenheim and Schafer’s book offers the optimal approach. For microcomputer real-time control, Franklin and Powell’s Digital Control (Addison-Wesley, 1980) might be better. For control DSP, my favorite is an older book – the best control theory book I have found – by Roberto Saucedo and Earl Schiring: Introduction to Continuous and Digital Control Systems , published by Macmillan in 1968. More recent books have shown an increasingly refined presentation of DSP for control, as digital or discrete-time control, such as Phillips and Harbor, Feedback Control Systems, Prentice-Hall, 1988.

DSP is categorized in control theory as discrete-time control. The quantity represented in discrete time either has discrete (digital) value representation (fully digital) or the range of values is continuous (analog). Sample-and-hold circuits, for instance, produce discrete-time samples with continuous (analog) values while DACs input and output discrete values in both time and quantity. In dual-slope DVMs and power electronics with switching converters, time is discrete in measurement or switching cycles while voltages and currents remain continuous. Whether a quantity has discrete or continuous values, the sampling behavior in time applies to both.

For circuit design, the best of the three ways to learn the same topics – numerical-analysis math, DSP, or discrete-time control – depends on whether you work largely in the frequency or time domain. In both cases, having a numerical analysis book or two is a recommended supplement to either DSP books for frequency-domain work or control theory books for time-domain work. Although this differentiation is not perfect, filter implementation is typically emphasized in DSP books while sampling circuits and feedback stability with sampling in the loop is the domain of control theory. Sampling, as in sample and hold or track and hold circuits, causes continuous (analog) waveforms to be made into discrete-time, continuous-value waveforms. The discrete-time aspect can be either an impulse train or piecewise-continuous (stair-stepped) waveforms – usually the latter. Sampling oscilloscopes have front-end samplers that sample at an instant in time to result in a series of dots on a graph of the sampled waveform. More commonly, the instant in time that the sample occurs is the active edge of a clocking waveform that captures a digital value into an A/D converter digital hold circuit, which is a register or latch. A/D converters (ADCs) perform the inverse function of D/A converters (DACs). Both convert between an analog ratio of an analog quantity, vX to full-scale reference quantity, VR , as vX /VR . The other side of the converter represents this ratio as a digital quantity, wX to the full-scale value for N bits of representation, as wX /2N . Then

ADCs and DACs are sample-and-hold (S&H) functions. The sampling occurs on a digital clock edge and the hold is performed by a digital register. Most analog S&Hs also have a digital sampling waveform output in that sampling occurs during an active level and holding is performed by a capacitor as a voltage. (An inductor can hold a current as an electrical dual for holding an analog value and is found most commonly in peak-current loop converter circuits, not in instrumentation.) In actual circuits, the sample and hold functions are found together though it is conceptually useful to separate them and place them in cascade.

In analog design, op-amps are common for amplification and also integration. An open-loop op-amp is essentially an integrator. Amplification is easier in the digital domain as multiplication by a constant. Multipliers are substantial as analog circuits but rather simple though time-consuming in μCs. However, analog circuits are typically faster. A good design lets the analog circuits and μC or digital circuits each do what they do best.

In closing, discrete and continuous waveforms are intermixed in circuits nowadays. By understanding sampling theory, in whatever form it can be learned, and some of its applications, the behavior and even the design of these mixed-signal circuits can be understood.

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