A system HH is said to be linear if it satisfies two important conditions. The first, additivity, states for every pair of signals x,yx,y that H(x+y)=H(x)+H(y)H(x+y)=H(x)+H(y). The second, homogeneity of degree one, states for every signal xx and scalar aa we have H(ax)=aH(x)H(ax)=aH(x). It is clear that these conditions can be combined together into a single condition for linearity. Thus, a system is said to be linear if for every signals x,yx,y and scalars a,ba,b we have that
H
(
a
x
+
b
y
)
=
a
H
(
x
)
+
b
H
(
y
)
.
H
(
a
x
+
b
y
)
=
a
H
(
x
)
+
b
H
(
y
)
.
(1)Linearity is a particularly important property of systems as it allows us to leverage the powerful tools of linear algebra, such as bases, eigenvectors, and eigenvalues, in their study.
A system HH is said to be time invariant if a time shift of an input produces the corresponding shifted output. In other, more precise words, the system HH commutes with the time shift operator STST for every T∈ZT∈Z. That is,
S
T
H
=
H
S
T
.
S
T
H
=
H
S
T
.
(2)Time invariance is desirable because it eases computation while mirroring our intuition that, all else equal, physical systems should react the same to identical inputs at different times.
When a system exhibits both of these important properties it opens. As will be explained and proven in subsequent modules, computation of the system output for a given input becomes a simple matter of convolving the input with the system's impulse response signal. Also proven later, the fact that complex exponential are eigenvectors of linear time invariant systems will encourage the use of frequency domain tools such as the various Fouier transforms and associated transfer functions, to describe the behavior of linear time invariant systems.
Consider the system HH in which
H
(
f
(
n
)
)
=
2
f
(
n
)
H
(
f
(
n
)
)
=
2
f
(
n
)
(3)for all signals ff.
Given any two signals f,gf,g and scalars a,ba,b
H
(
a
f
(
n
)
+
b
g
(
n
)
)
)
=
2
(
a
f
(
n
)
+
b
g
(
n
)
)
=
a
2
f
(
n
)
+
b
2
g
(
n
)
=
a
H
(
f
(
n
)
)
+
b
H
(
g
(
n
)
)
H
(
a
f
(
n
)
+
b
g
(
n
)
)
)
=
2
(
a
f
(
n
)
+
b
g
(
n
)
)
=
a
2
f
(
n
)
+
b
2
g
(
n
)
=
a
H
(
f
(
n
)
)
+
b
H
(
g
(
n
)
)
(4)for all integers nn. Thus, HH is a linear system.
For all integers TT and signals ff,
S
T
(
H
(
f
(
n
)
)
)
=
S
T
(
2
f
(
n
)
)
=
2
f
(
n
-
T
)
=
H
(
f
(
n
-
T
)
)
=
H
(
S
T
(
f
(
n
)
)
)
S
T
(
H
(
f
(
n
)
)
)
=
S
T
(
2
f
(
n
)
)
=
2
f
(
n
-
T
)
=
H
(
f
(
n
-
T
)
)
=
H
(
S
T
(
f
(
n
)
)
)
(5)for all integers nn. Thus, HH is a time invariant system.
Therefore, HH is a linear time invariant system.
"My introduction to signal processing course at Rice University."