0.3 Analog or Discrete? 9of a different type than the reference. Such would be the case, for instance, if the plant output is a
temperature while the reference signal is a voltage.
0.3 Analog or Discrete?
Infinitesimal calculus, or just plaincalculus, deals with functions of one or more continuously changing
variables. Based on the representation of these functions, the concepts ofderivativeandintegralare
developed to measure the rate of change of functions and the areas under the graphs of these
functions, or their volumes. Differential equations are then introduced to characterize dynamic
systems.
Finite calculus, on the other hand, deals with sequences. Thus, derivatives and integrals are replaced
by differences and summations, while differential equations are replaced by difference equations.
Finite calculus makes possible the computations of calculus by means of a combination of digital
computers and numerical methods—thus, finite calculus becomes the more concrete mathematics.^1
Numerical methods applied to sequences permit us to approximate derivatives, integrals, and the
solution of differential equations.
In engineering, as in many areas of science, the inputs and outputs of electrical, mechanical, chemical,
and biological processes are measured as functions of time with amplitudes expressed in terms of
voltage, current, torque, pressure, etc. These functions are calledanalog or continuous-time signals, and
to process them with a computer they must be converted into binary sequences—or a string of ones
and zeros that is understood by the computer. Such a conversion is done in a way as to preserve as
much as possible the information contained in the original signal. Once in binary form, signals can
be processed using algorithms (coded procedures understood by computers and designed to obtain
certain desired information from the signals or to change them) in a computer or in a dedicated piece
of hardware.
In a digital computer, differentiation and integration can be done only approximately, and the solu-
tion of differential equations requires a discretization process as we will illustrate later in this chapter.
Not all signals are functions of a continuous parameter—there exist inherently discrete-time signals
that can be represented as sequences, converted into binary form, and processed by computers. For
these signals the finite calculus is the natural way of representing and processing them.
Analog or continuous-time signalsare converted into binary sequences by means of an ADC, which, as we will
see, compresses the data by converting the continuous-time signal into a discrete-time signal or a sequence
of samples, each sample being represented by a string of ones and zeros giving a binary signal. Both time and
signal amplitude are made discrete in this process. Likewise, digital signals can be transformed into analog
signals by means of a DAC that uses the reverse process of the ADC. These converters are commercially
available, and it is important to learn how they work so that digital representation of analog signals is obtained(^1) The use ofconcrete, rather than abstract, mathematics was coined by Graham, Knuth, and Patashnik inConcrete Mathematics: A
Foundation for Computer Science[26]. Professor Donald Knuth from Stanford University is the the inventor of the Tex and Metafont
typesetting systems that are the precursors of Latex, the document layout system in which the original manuscript of this book was
done.