Digital Signal Processing Reference
In-Depth Information
x
(
t
)
m
0
m
1
m
2
t
0
t
1
t
2
t
Figure 2.26
Signal.
Example 2.11 illustrates the construction of waveforms composed only of
straight-line segments. Recall the general equation of a straight line:
y - y
0
= m[x - x
0
] .
(2.44)
In this equation,
y
is the ordinate (vertical axis),
x
is the abscissa (horizontal axis),
m
is the slope and is equal to
dy/dx
, and is any point on the line.
A technique is now developed for writing the equations for functions com-
posed of straight-line segments. An example is given in Figure 2.26. The slopes of
the segments are denoted as
(x
0
, y
0
)
m
i
.
The signal is zero for
t 6 t
0
.
For
t 6 t
1
,
the equa-
tion of the signal, denoted as
x
0
(t),
is given by
x
0
(t) = m
0
[t - t
0
]u(t - t
0
);
t 6 t
1
,
(2.45)
where for
To write the equation of the signal for
x
0
(t) = x(t)
t 6 t
1
.
t < t
2
,
first we set the slope to zero by
subtracting the slope
m
0
:
x
1
(t) = x
0
(t) - m
0
[t - t
1
]u(t - t
1
).
(2.46)
Next we add the term required to give the slope
m
1
,
x
2
(t) = x
1
(t) + m
1
[t - t
1
]u(t - t
1
),
(2.47)
with
x
2
(t) = x(t)
for
t 6 t
2
.
Then, from the last three equations, for
t 6 t
2
,
x
2
(t)
= m
0
[t - t
0
]u(t - t
0
) - m
0
[t - t
1
]u(t - t
1
) + m
1
[t - t
1
]u(t - t
1
)
= m
0
[t - t
0
]u(t - t
0
) + [m
1
- m
0
][t - t
1
]u(t - t
1
);
t 6 t
2
.
(2.48)
This result is general. When the slope of a signal changes, a ramp function is added
at that point, with the slope of this ramp function equal to the new slope minus the
previous slope At any point that a step occurs in the signal, a step func-
tion is added. An example using this procedure is now given.
(m
1
- m
0
).
Equations for straight-line-segments signal
EXAMPLE 2.12
The equation for the signal in Figure 2.27 will be written. The slope of the signal changes
from 0 to 3 for a change in slope of 3, beginning at
t =-2.
The slope changes from 3 to
-3
at
t =-1,
for a change in slope of
-6.
At
t = 1,
the slope becomes 0 for a change in slope of 3.
