Digital Signal Processing Reference
In-Depth Information
x
[
n
]
y
[
n
]
l
g
r
,
y
i
d
.
,
©
,
L
s
Figure 5.1
A single-input, single-output discrete-time system (black-box view).
Linearity is expressed by the equivalence
α
]
=
α
x
1
]
+
β
x
2
]
x
1
[
n
]+
β
x
2
[
n
[
n
[
n
(5.1)
for any two sequences
x
1
[
n
]
and
x
2
[
n
]
and any two scalars
α
,
β
∈
.Time-
invariance is expressed by
]=
x
]
⇐⇒
x
]
=
y
[
n
[
n
[
n
−
n
0
y
[
n
−
n
0
]
(5.2)
Linearity and time-invariance are very reasonable and “natural” require-
ments for a signal processing system. Imagine a recording system: linear-
ity implies that a signal obtained by recording a violin and a piano playing
together is the same as the sum of the signals obtained recording the vio-
lin and the piano separately (but in the same recording room). Multi-track
recordings in music production are an application of this concept. Time in-
variance basically means that the system's behavior is independent of the
time the system is turned on. Again, to use a musical example, this means
that a given digital recording played back by a digital player will sound the
same, regardless of when it is played.
Yet, simple as these properties, linearity and time-invariance taken to-
gether have an incredibly powerful consequence on a system's behavior. In-
deed, a linear time-invariant system turns out to be
completely
character-
ized by its response to the input
x
[
n
]=
δ
[
n
]
.Thesequence
h
[
n
]=
{
δ
[
n
]
}
[
]
is called the
impulse response
of the system and
h
is all we need to know
to determine the system's output for
any
other input sequence. To see this,
we know that for any sequence we can always write the canonical orthonor-
mal expansion (i.e. the reproducing formula in (2.18))
n
∞
x
[
n
]=
x
[
k
]
δ
[
n
−
k
]
k
=
−∞
and therefore, if we let
δ
[
n
]
=
h
[
n
]
, we can apply (5.1) and (5.2) to ob-
tain
∞
]=
x
]
=
y
[
n
[
n
x
[
k
]
h
[
n
−
k
]
(5.3)
k
=
−∞
