Digital Signal Processing Reference
In-Depth Information
V
T
t,
v(t) =
0 6 t 6 T.
We can multiply this pulse by the unit rectangular pulse to force
v(t)
to be zero for all other
values of time, with the final result
V
T
t[u(t) - u(t - T)] =
V
T
t rect
T
2
v(t) =
B¢
t -
≤
n
T
R
.
(2.49)
Recall from (2.35) that the rectangular pulse can also be expressed as other functions of unit
step signals.
■
The equation for a sawtooth waveform
EXAMPLE 2.14
This example is a continuation of Example 2.13. The results of that example are used to write
the equation of the periodic waveform of Figure 2.29. This waveform, called a
sawtooth wave
because of its appearance, is used to sweep an electron beam repeatedly across the face of the
CRT. From Example 2.13, the equation of the sawtooth pulse for
0 6 t 6 T
is
V
T
t[u(t) - u(t - T)] =
V
T
t rect
T
2
[eq(2.49)]
v
1
(t) =
B¢
t -
≤
n
T
R
.
Note that this pulse has been denoted as Hence, as in the case of the half-wave rectified
signal of Example 2.9 and (2.37), the pulse from
v
1
(t).
T 6 t 6 2T
is
v
1
(t - T):
V
T
[t - T][u(t - T) - u(t - 2T)]
v
1
(t - T) =
V
T
[t - T] rect
3T
2
B¢
≤
R
=
t -
n
T
.
The interested reader can plot this function to show its correctness. In a similar manner, the
pulse from
kT 6 t 6 (k + 1)T
is
v
1
(t - kT):
V
T
[t - kT][u(t - kT) - u(t - kT - T)].
v
1
(t - kT) =
(2.50)
This equation applies for
k
either positive or negative. Hence, the equation for the sawtooth
wave of Figure 2.29 is
v
(
t
)
V
Figure 2.29
T
0
T
2
T
t
Sawtooth waveform.
