Digital Signal Processing Reference
In-Depth Information
u
(
t
t
0
)
1
0
t
0
t
Figure 2.17
Unit step function.
As previously stated, the unit step is often used to switch functions. An exam-
ple is given by
cos vt,
t 7 0
b
cos vtu(t) =
t 6 0
.
0,
The unit step allows us mathematically to switch this sinusoidal function on at
Another example is volts; this function is equal to 0 volts for and
to 12 volts for In this case, the unit step function is used to switch a 12-V source.
Another useful switching function is the unit rectangular pulse, rect(
t/T
),
which is
defined
as
t = 0.
v(t) = 12u(t)
t 6 0
t 7 0.
1,
-T/2 6 t 6 T/2
b
rect(t/T) =
.
0,
otherwise
This function is plotted in Figure 2.18(a). It can be expressed as three different func-
tions of unit step signals:
u(t + T/2) - u(t - T/2)
u(T/2 - t) - u(-T/2 - t).
u(t + T/2)u(T/2 - t)
c
rect(t/T) =
(2.35)
These functions are plotted in Figure 2.18(b), (c), and (d).
The time-shifted rectangular pulse function is given by
1,
t
0
- T/2 6 t 6 t
0
+ T/2
b
rect[(t - t
0
)/T] =
.
(2.36)
0,
otherwise
This function is plotted in Figure 2.19. Notice that in both (2.35) and (2.36) the rec-
tangular pulse has a duration of
T
seconds.
The unit rectangular pulse is useful in extracting part of a signal. For example,
the signal
x(t) = cos t
has a period
T
0
= 2p/v = 2p.
Consider a signal composed of
one period of this cosine function beginning at
t = 0,
and zero for all other time.
This signal can be expressed as
cos t,
06 t 6 2p
b
x(t) = (cos t)[u(t) - u(t - 2p)] =
.
0,
otherwise
The rectangular-pulse notation allows us to write
x(t) = cos t rect[(t - p)/2p].
