Digital Signal Processing Reference
In-Depth Information
2.1.3.5 Linear Systems
In system theory, it is often convenient to introduce some classes of possi-
ble operators. A very relevant distinction is established between linear and
nonlinear systems.
Linear systems
are those whose defining
S
[·]
operator
obeys the following
superposition principle
:
S
[
k
1
x
1
+
k
2
x
2
]=
k
1
S
[
x
1
]+
k
2
S
[
x
2
]
(2.19)
The idea of superposition can be explained in simple terms: the response
to a linear combination of input stimuli is the linear combination of the indi-
vidual responses. Conversely, a nonlinear system is simply one that does not
obey this principle.
2.1.3.6 Time-Invariant Systems
Another important feature is time-invariance. A system is said to be time-
invariant when its input-output mapping does not vary with time. When the
contrary holds, the system is said to be time-variant. Since this characteristic
makes the system easier to be dealt with in mathematical terms, most models
of practical systems are, with different degrees of fidelity, time-invariant.
2.1.3.7 Linear Time-Invariant Systems
A very special class of systems is that formed by those that are both lin-
ear and time-invariant (linear time-invariant, LTI). These systems obey the
superposition principle and have an input-output mapping that does not
vary with time. The combination of these desirable properties gives rise to
the following mathematical result.
Suppose that
x
are, respectively, the input and the output of a
continuous-time LTI SISO system. In such case,
(
t
)
and
y
(
t
)
∞
y
(
t
)
=
h
(
t
)
∗
x
(
t
)
=
h
(
τ
)
x
(
t
−
τ
)
d
τ
(2.20)
−∞
where
h
(
t
)
is the system impulse response, which is the system output when
x
(
t
)
is equal to the Dirac delta function δ
(
t
)
. The symbol
∗
denotes that the
output
y
(
n
)
is the result of the convolution of
x
(
t
)
with
h
(
t
)
.
are, respectively, the input and the output
of a discrete-time LTI SISO system, it holds that
Analogously, if
x
(
n
)
and
y
(
n
)
∞
y
(
n
)
=
h
(
n
)
∗
x
(
n
)
=
h
(
k
)
x
(
n
−
k
)
(2.21)
k
=−∞
