Digital Signal Processing Reference
In-Depth Information
Basic characteristics of step function s (t) and į-impulse
Signals carry information by changing temporally. There are two "extreme" signal
forms of great theoretical and practical relevance already dealt with in chapter 6.
The step function s(t)
The step function
s(t)
s tt
(
−
)
s t
()
0
1
1
t
t
t
0
1
t
>
0
-
1
0
tt
tt
>
<
-
°
−=
0
st
()
=
bzw.
st t
(
)
®
®
°
0
0
t
<
0
¯
¯
0
st
( ) is not defined for
t
=
0 or
t
=
t
.
0
Graphic interpretation
()
The
The -impulse
The
δ
δ
-impulse
δ
δ
(t
)
t
ignal alteration.
Mathematically the "infity fast" changes causes big problems. With
the following definitions capable of graphically demonstration
δ
-impulse represents the most extreme form of a s
ε
→
0
1
ε
t
t
t
0
0
0
t
≠
0
0
0
t
≠
0
-
-
()
(
)
δ
t
=
respectively
δ
t
−
t
=
®
®
0
¯
∞=
t
0
¯
∞=
tt
0
ε
ε
³
()
³
(
)
δ
tdt
=
1 respectively
δ
t t dt
−
=
1
0
ε
−
ε
−
ε
ε
→
0 and the plane - the integral - remains constant
=
1
The following definition is mathematically reasonable and practically useful which deals with a
continuous function respectively a signal ( )
xt
∞
∞
()
(
)
³
³
xt
( )
δ
t dt
=
x
(0) bzw.
xt
( )
δ
t t dt
−
=
xt
(
)
0
0
−∞
−∞
Here, the signal is sampled at the time point
t
=
0 or
t
. If the signal is sampled
0
continuously at all time points
τ
(see Illustration 37), it is composed (synthesi
zed) entirely
of weighted
δ
-impulses. Every continuous signal can be modeled this way.
∞
(
)
³
x t
()
=
x
( )
τδ
t
−
τ
d
τ
−∞
Illustration 314:
Step function, į- impulse and continuous signals as weighted į- impulse sequence
Despite their extreme step locations - these singularities tend to represent a certain difficulty in mathe-
matics - s(t) and į(t) play an important role in the examination of linear systems. The latter equation,
introduced by Norbert WIENER, shows a demonstration which to a certain extent rivals the FOURIER-
demonstration of the synthesis of continuous signals. Both demonstrations are of equal importance. While
the FOURIER- form connects more to the continuous side, the WIENER- form gives access to statistical
characteristics and structures of signals. It was only this statistical point of view that led to the current
information theory and cybernetics of SHANNON.
