Digital Signal Processing Reference
In-Depth Information
where This signal, is equal to the half-wave rectified signal for
and is zero elsewhere. Thus, the half-wave rectified signal can be expressed as a
sum of shifted signals,
T
0
= 2p/v
0
.
v
1
(t),
0
F
t
F
T
0
v(t)
= v
1
(t) + v
1
(t - T
0
) + v
1
(t - 2T
0
) +
Á
=
q
k= 0
v
1
(t - kT
0
),
(2.37)
since is delayed by one period, is delayed by two periods,
and so on. If the half-wave rectified signal is specified as periodic for all time, the lower limit
in (2.37) is changed to negative infinity. As indicated in this example, expressing a
periodic signal as a mathematical function often requires the summation of an infinity of
terms.
v
1
(t - T
0
)
v
1
(t)
v
1
(t - 2T
0
)
v
1
(t)
■
Engineers have found great use for , even though this is not a real number
and cannot appear in nature. Electrical engineering analysis and design utilizes
j
ex-
tensively. In the same manner, engineers have found great use for the
unit impulse
function,
even though this function cannot appear in nature. In fact, the impulse
function is not a mathematical function in the usual sense [2]. The unit impulse
function is also called the
Dirac delta function.
The impulse function was introduced
by Nobel Prize winning physicist Paul Dirac.
j =
2
-1
d(t),
Paul Adrien Maurice Dirac
(born August 8, 1902, in Bristol, Gloucester-
shire, England; died October 20, 1984 ): A theoretical physicist who was one
of the founders of quantum mechanics and quantum electrodynamics, Dirac
introduced his “delta function” in the 1920s to simplify the mathematical
treatment of actions that are highly localized, or “spiked,” in time or space.
Dirac is most famous for his 1928 relativistic quantum theory of the electron
and his prediction of the existence of antiparticles. Dirac was awarded the
Nobel Prize for Physics in 1933.
Reference:
“Dirac, P. A. M.” Encyclopedia Britannica. 2007. Encyclopedia Britannica
Online. 5 Jan. 2007,
http://www.britannica.com/eb/article-9030591
To introduce the impulse function, we begin with the integral of the unit step
function; this integral yields the unit ramp function
t
t
t
`
f(t) =
L
u(t-t
0
)dt =
L
dt = t
= [t - t
0
]u(t - t
0
),
(2.38)
0
t
0
t
0
where , by definition, is the
unit ramp function
. In (2.38), the factor
in the result is necessary, since the value of the integral is zero for
The unit step function and the unit ramp function are illustrated in Figure 2.22.
[t - t
0
]u(t - t
0
)
u(t - t
0
)
t 6 t
0
.
