PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

14 Practical MATLAB® Applications for Engineers

R.1.30 The analog unit parabolic function pK(t) u(t) is defi ned by

ptut K

tt K

t

K

K

() ()!

,,

1

023

00

for

for





and ...











R.1.31 The discrete unit parabolic function pK(n) u(n) is defi ned by

pnun K

nn K

n

K

K

()()!

,,

1

023

00

for

for



and ...











R.1.32 Observe that

a. The unit ramp presents a sharp 45°^ corner at t = 0.

b. The unit parabolic function presents a smooth behavior at t = 0.

c. The unit step presents a discontinuity at t = 0.

R.1.33 The step, ramp, and parabolic functions are related by derivatives as follows:

a. (d/dt)[r(t)] = u(t)

b. __d

dt

[p 2 (t)] = r(t)u(t)

c. (d/dt)[pa(t)] = pa− 1 (t)

Observe that the fi rst relation makes sense for all t ≠ 0 , since at t = 0 a discontinu-

ity occurs, whereas the second and third relations hold for all t.

R.1.34 Note that, in general, the product f(t) times u(t) [f (t)u(t)] defi nes the composite func-
tion given by

ftut

ft t

t

() ()

()



for  0

00 for







R.1.35 A wide class of engineering systems employ sinusoidal and exponential* signals as
inputs. A real exponential analog signal is in general given by

f(t) = Aebt^

where e = 2.7183 (Neperian constant) and A and b are in most cases real constants.
Observe that for f(t) = Aebt,
a. f(t) is a decaying exponential function for b < 0.
b. f(t) is a growing exponential function for b > 0.
The coeffi cient b as exponent is referred to as the damping coeffi cient or constant.
In electric circuit theory, the damping constant is frequently given by b = 1/τ, where
τ is referred as the time constant of the network (see Chapter 2).
Note that the exponential function f(t) = Aebt repeats itself when differentiated
or integrated with respect to time, and constitutes the homogeneous solution of

*^ Recall that sinusoids are complex exponentials (Euler), see Chapter 4 of Practical MATLAB® Basics for
Engineers.