Graphics Programs Reference
In-Depth Information
t
2
∫
Ex
()
2
=
d
(13.3)
t
1
The signal
x
()
is said to be an energy signal if and only if it has finite
energy,
∞
∫
x
()
2
E
=
d
t
<
∞
(13.4)
∞
A signal
x
()
is said to be periodic with period
T
if and only if
x
()
xt nT
=
(
+
)
for all t
(13.5)
where
n
is an integer.
Example:
Classify each of the following signals as an energy signal, as a power signal,
or as neither. All signals are defined over the interval
(
∞
<<
t
∞
)
:
α
2
t
2
x
1
()
=
cos
t
+
cos
2
t
,
x
2
()
=
exp
(
)
.
Solution:
T
⁄
∫
2
1
---
)
2
P
x
1
=
(
cos
t
+
cos
2
t
d
t
=
1
⇒
power signal
T
⁄
2
Note that since the cosine function is periodic, the limit is not necessary
.
∞
∫
∞
∫
)
2
π
22α
1
---
π
---
e
α
2
t
2
2α
2
t
2
E
x
2
=
(
d
=
2
e
dt
=
2
--------------
=
⇒
energy signal
∞
0
Electrical systems can be linear or nonlinear. Furthermore, linear systems
may be divided into continuous or discrete. A system is linear if the input sig-
nal produces and produces ; then for some arbitrary
constants and the input signal produces the output
. A linear system is said to be shift invariant (or time invari-
ant) if a time shift at its input produces the same shift at its output. More pre-
cisely, if the input signal
x
1
()
y
1
()
x
2
()
y
2
()
a
1
a
2
a
1
x
1
()
a
2
x
2
+
()
a
1
y
1
()
a
2
y
2
+
()
x
()
produces
y
()
then the delayed signal
xt t
0
(
)
produces the output . The impulse response of a Linear Time Invariant
(LTI) system, , is defined to be the systemÓs output when the input is an
impulse (delta function).
yt t
0
(
)
h
()
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