∈ A
T
, F
∈ B
B
T
.
(9.2)
If the transformations are clear from context then we simply say that the chan
nel is stationary.
Intuitively, a right shift of an output event yields the same
probability as the left shift of an input event. The different shifts are required
because in general only
T
A
x
and not
T

1
A
x
exists since the shift may not be
invertible and in general only
T

1
B
F
and not
T
B
F
exists for the same reason. If
the shifts are invertible, e.g., the processes are twosided, then the definition is
equivalent to
ν
T
A
x
(
T
B
F
) =
ν
T

1
A
x
(
T

1
B
F
) =
ν
x
(
F
)
,
all
x
∈
A
T
, F
∈ B
B
T
(9.3)
that is, shifting the input sequence and output event in the same direction does
not change the probability.
The fundamental importance of the stationarity of a channel is contained in
the following lemma.
Lemma 9.3.1
If a source
[
A, μ
]
, stationary with respect to
T
A
, is connected
to channel
[
A, ν, B
]
, stationary with respect to
T
A
and
T
B
, then the resulting
hookup
μν
is also stationary (with respect to the cartesian product shift
T
=
T
A
×
B
=
T
A
×
T
B
defined by
T
(
x, y
) = (
T
A
x, T
B
y
)
).
Proof:
We have that
μν
(
T

1
F
) =
Z
dμ
(
x
)
ν
x
((
T

1
F
)
x
)
.
Now
(
T

1
F
)
x
=
{
y
:
T
(
x, y
)
∈
F
}
=
{
y
: (
T
A
x, T
B
y
)
∈
F
}
=
{
y
:
T
B
y
∈
F
T
A
x
}
=
T

1
B
F
T
A
x
9.3.
STATIONARITY PROPERTIES OF CHANNELS
165
and hence
μν
(
T

1
F
) =
Z
dμ
(
x
)
ν
x
(
T

1
B
F
T
A
x
)
.
Since the channel is stationary, however, this becomes
μν
(
T

1
F
) =
Z
dμ
(
x
)
ν
T
A
x
(
F
T
A
x
) =
Z
dμT

1
A
(
x
)
ν
x
(
F
x
)
,
where we have used the change of variables formula.
Since
μ
is stationary,
however, the right hand side is
Z
dμ
(
x
)
ν
x
(
F
)
,
which proves the lemma.
2
Suppose next that we are told that a hookup
μν
is stationary. Does it then
follow that the source
μ
and channel
ν
are necessarily stationary? The source
must be since
μ
(
T

1
A
F
) =
μν
((
T
A
×
T
B
)

1
(
F
×
B
T
)) =
μν
(
F
×
B
T
) =
μ
(
F
)
.
The channel need not be stationary, however, since, for example, the stationarity
could be violated on a set of
μ
measure 0 without affecting the proof of the
above lemma. This suggests a somewhat weaker notion of stationarity which is
more directly related to the stationarity of the hookup. We say that a channel
[
A, ν, B
] is
stationary with respect to a source
[
A, μ
] if
μν
is stationary. We also
state that a channel is stationary
μ
a.e. if it satisfies (9.2) for all
x
in a set of
μ
probability one.
If a channel is stationary
μ
a.e.
and
μ
is stationary, then
the channel is also stationary with respect to
μ
. Clearly a stationary channel
is stationary with respect to all stationary sources.
The reason for this more
general view is that we wish to extend the definition of stationary channels to
asymptotically mean stationary channels. The general definition extends; the
classical definition of stationary channels does not.
Observe that the various definitions of stationarity of channels immediately
extend to block shifts since they hold for any shifts defined on the input and
output sequence spaces, e.g., a channel stationary with respect to
T
N
A
and
T
K
B
could be a reasonable model for a channel or code that puts out
K
symbols
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 Spring '10
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 Electrical Engineering, Information Theory, Probability theory, Ergodic theory, Ergodic Properties, Don Ornstein