Databases Reference
In-Depth Information
This introductory account of complex signals gives us the chance to remake a very
important point. In engineering and applied science, measured signals are
real
.Cor-
respondingly, in all of our examples, the components
u
and
are
real
. It is only our
representation
x
that is complex. Thus one channel's worth of complex signal serves
to represent two channels' worth of real signals. There is no fundamental reason why
this would have to be done. We aim to make the point in this topic that the algebraic
economies, probabilistic computations, and geometrical insights that accrue to complex
representations justify their use. The examples of the next several sections give a preview
of the power of complex representations.
v
1.2
Simple harmonic oscillator and phasors
The damped harmonic oscillator models damped pendulums and second-order electrical
and mechanical systems. A measurement (of position or voltage) in such a system obeys
the second-order, homogeneous, linear differential equation
d
2
d
t
2
u
(
t
)
d
d
t
u
(
t
)
2
+
2
ξω
0
+
ω
0
u
(
t
)
=
0
.
(1.1)
The corresponding characteristic equation is
s
2
2
0
+
2
ξω
0
s
+
ω
=
0
.
(1.2)
ξ
≤
ξ<
1, the system is called
underda
mped
, a
nd
the quadratic equation (
1.2
) has two complex conjugate roots
s
1
=−
ξω
0
+
If the damping coefficient
satisfies 0
j
1
−
ξ
2
ω
0
s
1
. The real homogeneous response of the damped harmonic oscillator is then
and
s
2
=
A
e
−
ξω
0
t
cos(
1
A
e
−
j
θ
e
s
1
t
A
e
j
θ
e
s
1
t
A
e
j
θ
e
s
1
t
2
u
(
t
)
=
+
=
Re
{
}=
−
ξ
ω
0
t
+
θ
)
,
(1.3)
and
A
and
0.
The real response (
1.3
) is the sum of two complex modal responses, or the real part of
one of them. In anticipation of our continuing development, we might say that
A
e
j
θ
e
s
1
t
is a complex representation of the real signal
u
(
t
).
For the
undamped
system with damping coefficient
θ
may be determined from the initial values of
u
(
t
) and (d
/
d
t
)
u
(
t
)at
t
=
ξ
=
0, we have
s
1
=
j
ω
0
and the
solution is
A
e
j
θ
e
j
ω
0
t
u
(
t
)
=
Re
{
}=
A
cos(
ω
0
t
+
θ
)
.
(1.4)
In this case,
A
e
j
θ
e
j
ω
0
t
is the complex representation of the real signal
A
cos(
ω
0
t
+
θ
).
The complex signal in its polar form
A
e
j(
ω
0
t
+
θ
)
A
e
j
θ
e
j
ω
0
t
x
(
t
)
=
=
,
t
∈
IR
,
(1.5)
is called a
rotating phasor
.The
rotator
e
j
ω
0
t
rotates the
stationary phasor A
e
j
θ
at the
angular rate of
ω
0
radians per second. The rotating phasor is periodic with period 2
π/ω
0
,
thus overwriting itself every 2
π/ω
0
seconds. Euler's identity allows us to express the
rotating phasor in its Cartesian form as
x
(
t
)
=
A
cos(
ω
0
t
+
θ
)
+
j
A
sin(
ω
0
t
+
θ
)
.
(1.6)
Search WWH ::
Custom Search