<s>
In	O
control	O
theory	O
,	O
a	O
time-invariant	B-Algorithm
(	O
TI	O
)	O
system	O
has	O
a	O
time-dependent	O
system	O
function	O
that	O
is	O
not	O
a	O
direct	O
function	O
of	O
time	O
.	O
</s>
<s>
If	O
this	O
function	O
depends	O
only	O
indirectly	O
on	O
the	O
time-domain	O
(	O
via	O
the	O
input	O
function	O
,	O
for	O
example	O
)	O
,	O
then	O
that	O
is	O
a	O
system	O
that	O
would	O
be	O
considered	O
time-invariant	B-Algorithm
.	O
</s>
<s>
Mathematically	O
speaking	O
,	O
"	O
time-invariance	B-Algorithm
"	O
of	O
a	O
system	O
is	O
the	O
following	O
property	O
:	O
</s>
<s>
Given	O
a	O
system	O
with	O
a	O
time-dependent	O
output	O
function	O
,	O
and	O
a	O
time-dependent	O
input	O
function	O
,	O
the	O
system	O
will	O
be	O
considered	O
time-invariant	B-Algorithm
if	O
a	O
time-delay	O
on	O
the	O
input	O
directly	O
equates	O
to	O
a	O
time-delay	O
of	O
the	O
output	O
function	O
.	O
</s>
<s>
If	O
a	O
system	O
is	O
time-invariant	B-Algorithm
then	O
the	O
system	O
block	O
commutes	O
with	O
an	O
arbitrary	O
delay	O
.	O
</s>
<s>
If	O
a	O
time-invariant	B-Algorithm
system	I-Algorithm
is	O
also	O
linear	O
,	O
it	O
is	O
the	O
subject	O
of	O
linear	O
time-invariant	B-Algorithm
theory	O
(	O
linear	O
time-invariant	B-Algorithm
)	O
with	O
direct	O
applications	O
in	O
NMR	O
spectroscopy	O
,	O
seismology	O
,	O
circuits	O
,	O
signal	O
processing	O
,	O
control	O
theory	O
,	O
and	O
other	O
technical	O
areas	O
.	O
</s>
<s>
Nonlinear	O
time-invariant	B-Algorithm
systems	I-Algorithm
lack	O
a	O
comprehensive	O
,	O
governing	O
theory	O
.	O
</s>
<s>
Discrete	O
time-invariant	B-Algorithm
systems	I-Algorithm
are	O
known	O
as	O
shift-invariant	O
systems	O
.	O
</s>
<s>
Systems	O
which	O
lack	O
the	O
time-invariant	B-Algorithm
property	O
are	O
studied	O
as	O
time-variant	O
systems	O
.	O
</s>
<s>
To	O
demonstrate	O
how	O
to	O
determine	O
if	O
a	O
system	O
is	O
time-invariant	B-Algorithm
,	O
consider	O
the	O
two	O
systems	O
:	O
</s>
<s>
Since	O
the	O
System	O
Function	O
for	O
system	O
A	O
explicitly	O
depends	O
on	O
t	O
outside	O
of	O
,	O
it	O
is	O
not	O
time-invariant	B-Algorithm
because	O
the	O
time-dependence	O
is	O
not	O
explicitly	O
a	O
function	O
of	O
the	O
input	O
function	O
.	O
</s>
<s>
This	O
makes	O
system	O
B	O
time-invariant	B-Algorithm
.	O
</s>
<s>
Clearly	O
,	O
therefore	O
the	O
system	O
is	O
not	O
time-invariant	B-Algorithm
.	O
</s>
<s>
Clearly	O
,	O
therefore	O
the	O
system	O
is	O
time-invariant	B-Algorithm
.	O
</s>
<s>
For	O
time-invariant	B-Algorithm
systems	I-Algorithm
,	O
the	O
system	O
properties	O
remain	O
constant	O
with	O
time	O
,	O
</s>
<s>
in	O
general	O
,	O
so	O
it	O
is	O
not	O
time-invariant	B-Algorithm
,	O
</s>
<s>
so	O
it	O
is	O
time-invariant	B-Algorithm
.	O
</s>
<s>
We	O
can	O
denote	O
the	O
shift	B-Algorithm
operator	I-Algorithm
by	O
where	O
is	O
the	O
amount	O
by	O
which	O
a	O
vector	O
's	O
index	O
set	O
should	O
be	O
shifted	O
.	O
</s>
<s>
This	O
system	O
is	O
time-invariant	B-Algorithm
if	O
it	O
commutes	O
with	O
the	O
shift	B-Algorithm
operator	I-Algorithm
,	O
i.e.	O
,	O
</s>
<s>
then	O
it	O
is	O
time-invariant	B-Algorithm
if	O
we	O
can	O
apply	O
the	O
system	O
operator	O
on	O
followed	O
by	O
the	O
shift	B-Algorithm
operator	I-Algorithm
,	O
or	O
we	O
can	O
apply	O
the	O
shift	B-Algorithm
operator	I-Algorithm
followed	O
by	O
the	O
system	O
operator	O
,	O
with	O
the	O
two	O
computations	O
yielding	O
equivalent	O
results	O
.	O
</s>
