082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

2.3 Basic concepts of functions 67
Technical computing languages often have a range of built-in functions for producing
waveforms. Sometimes specialist functions are provided in a separate softwarepackage.InMATLAB
® ,thesesoftwarepackagesareknownastoolboxes.The
signalprocessingtoolboxhasafunctionforgeneratingsaw-toothwaves.Thiscanbe
accessed by typing, forexample:
t=(-2*pi:0.1:2*pi);
plot(t, sawtooth(t));
This will plot two periods of a saw-tooth wave. The first line generates a set of time
values −2π ≤t < 2πinavectorformwithaspacingof0.1betweeneachpoint.The
secondlineplotst againsttheresultofpassingthevectort tothesawtoothfunction.
The sawtooth function always produces a wave with a period of 2π. It highlights
the need to read the manual pages carefully before using a function to understand
how itwill behave.
Engineeringapplication2.3
Squarewaveform
Periodic functions may be piecewise continuous. Consider the functiong(t) defined
by
{
1 0t<1
g(t) = period = 2
0 1t<2
Thefunctiong(t)isperiodicwithperiod2.Agraphofg(t)isshowninFigure2.16.
This function is commonly referred to as a square waveform by engineers. In Figure
2.16 the open and closed end-points have been shown for mathematical correctness.Note,however,thatengineerstendtoomitthesewhensketchingfunctionswith
discontinuities and usually they use a vertical line to show the discontinuity. This
reflectsthefactthatnopracticalwaveformcaneverchangeitslevelinstantaneously:
evenveryfastrisingwaveformsstillhaveafiniterisetime.Thefunctionhasdiscontinuities
att = ...,−3,−2,−1,0,1,2,3,4,5,....
g(t)
1
–5 –4 –3 –2 –1 0
1 2 3 4 5 t
Figure2.16
The functiong(t)is both piecewise continuous and periodic.
The square waveform is often used in electronic engineering, particularly in digital
electronic systems. One example is the clock signal that is generated to ensure that
all of the digital electronic circuits switch around the same time and so remain in
synchronisation.