PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

10 Practical MATLAB® Applications for Engineers

The discrete shifted impulse function δ(n − m)m − 1 ... m ... m + 1n is defi ned as

()nm

nm

nm





1

0

for

for







R.1.18 Let us go back to the analog world. The analog unit step function denoted by u(t) is
illustrated in Figure 1.9.
Analytically, the analog unit step is defi ned by

ut

t

t

()

10 

00

for

for 







The step is related to the impulse by the following relations:

du t

dt

t

()

()

or in general

d

dt

[(ut too)]

(t t)

()tdt ut
 ()





∫

or in general

ut t t d

t

t

t

o t

t

() ( )o



 

∫
 







1

0

0

0

for

for

The derivative of the unit step constitutes a break with the traditional differen-
tial and integral calculus. This new approach to the class of functions called sin-
gular functions is referred to as generalized or distributional calculus (mentioned
in R.1.4).

R.1.19 The analog unit step u(t) can be implemented by a switch connected to a voltage
source of 1 V that closes instantaneously at t = 0 , illustrated in Figure 1.10.

u(t)

1

t

0

FIGURE 1.9
Plot of the step function u(t).