Arbitrarily varying channel
An arbitrarily varying channel (AVC) is a communication channel model used in coding theory, and was first introduced by Blackwell, Breiman, and Thomasian. This particular channel has unknown parameters that can change over time and these changes may not have a uniform pattern during the transmission of a codeword. uses of this channel can be described using a stochastic matrix
, where
is the input alphabet,
is the output alphabet, and
is the probability over a given set of states
, that the transmitted input
leads to the received output
. The state
in set
can vary arbitrarily at each time unit
. This channel was developed as an alternative to Shannon's Binary Symmetric Channel (BSC), where the entire nature of the channel is known, to be more realistic to actual network channel situations.
Capacities and associated proofs
Capacity of deterministic AVCs
An AVC's capacity can vary depending on the certain parameters.
is an achievable rate for a deterministic AVC code if it is larger than
, and if for every positive
and
, and very large
, length-
block codes exist that satisfy the following equations:
and
, where
is the highest value in
and where
is the average probability of error for a state sequence
. The largest rate
represents the capacity of the AVC, denoted by
.
As you can see, the only useful situations are when the capacity of the AVC is greater than , because then the channel can transmit a guaranteed amount of data
without errors. So we start out with a theorem that shows when
is positive in an AVC and the theorems discussed afterward will narrow down the range of
for different circumstances.
Before stating Theorem 1, a few definitions need to be addressed:
- An AVC is symmetric if
for every
, where
,
, and
is a channel function
.
-
,
, and
are all random variables in sets
,
, and
respectively.
-
is equal to the probability that the random variable
is equal to
.
-
is equal to the probability that the random variable
is equal to
.
-
is the combined probability mass function (pmf) of
,
, and
.
is defined formally as
.
-
is the entropy of
.
-
is equal to the average probability that
will be a certain value based on all the values
could possibly be equal to.
-
is the mutual information of
and
, and is equal to
.
-
, where the minimum is over all random variables
such that
,
, and
are distributed in the form of
.
Theorem 1: if and only if the AVC is not symmetric. If
, then
.
Proof of 1st part for symmetry: If we can prove that is positive when the AVC is not symmetric, and then prove that
, we will be able to prove Theorem 1. Assume
were equal to
. From the definition of
, this would make
and
independent random variables, for some
, because this would mean that neither random variable's entropy would rely on the other random variable's value. By using equation
, (and remembering
,) we can get,
since
and
are independent random variables,
for some
because only
depends on
now
because
So now we have a probability distribution on that is independent of
. So now the definition of a symmetric AVC can be rewritten as follows:
since
and
are both functions based on
, they have been replaced with functions based on
and
only. As you can see, both sides are now equal to the
we calculated earlier, so the AVC is indeed symmetric when
is equal to
. Therefore,
can only be positive if the AVC is not symmetric.
Proof of second part for capacity: See the paper "The capacity of the arbitrarily varying channel revisited: positivity, constraints," referenced below for full proof.
Capacity of AVCs with input and state constraints
The next theorem will deal with the capacity for AVCs with input and/or state constraints. These constraints help to decrease the very large range of possibilities for transmission and error on an AVC, making it a bit easier to see how the AVC behaves.
Before we go on to Theorem 2, we need to define a few definitions and lemmas:
For such AVCs, there exists:
- - An input constraint
based on the equation
, where
and
.
- - A state constraint
, based on the equation
, where
and
.
- -
- -
is very similar to
equation mentioned previously,
, but now any state
or
in the equation must follow the
state restriction.
Assume is a given non-negative-valued function on
and
is a given non-negative-valued function on
and that the minimum values for both is
. In the literature I have read on this subject, the exact definitions of both
and
(for one variable
,) is never described formally. The usefulness of the input constraint
and the state constraint
will be based on these equations.
For AVCs with input and/or state constraints, the rate is now limited to codewords of format
that satisfy
, and now the state
is limited to all states that satisfy
. The largest rate is still considered the capacity of the AVC, and is now denoted as
.
Lemma 1: Any codes where is greater than
cannot be considered "good" codes, because those kinds of codes have a maximum average probability of error greater than or equal to
, where
is the maximum value of
. This isn't a good maximum average error probability because it is fairly large,
is close to
, and the other part of the equation will be very small since the
value is squared, and
is set to be larger than
. Therefore, it would be very unlikely to receive a codeword without error. This is why the
condition is present in Theorem 2.
Theorem 2: Given a positive and arbitrarily small
,
,
, for any block length
and for any type
with conditions
and
, and where
, there exists a code with codewords
, each of type
, that satisfy the following equations:
,
, and where positive
and
depend only on
,
,
, and the given AVC.
Proof of Theorem 2: See the paper "The capacity of the arbitrarily varying channel revisited: positivity, constraints," referenced below for full proof.
Capacity of randomized AVCs
The next theorem will be for AVCs with randomized code. For such AVCs the code is a random variable with values from a family of length-n block codes, and these codes are not allowed to depend/rely on the actual value of the codeword. These codes have the same maximum and average error probability value for any channel because of its random nature. These types of codes also help to make certain properties of the AVC more clear.
Before we go on to Theorem 3, we need to define a couple important terms first:
is very similar to the
equation mentioned previously,
, but now the pmf
is added to the equation, making the minimum of
based a new form of
, where
replaces
.
Theorem 3: The capacity for randomized codes of the AVC is .
Proof of Theorem 3: See paper "The Capacities of Certain Channel Classes Under Random Coding" referenced below for full proof.
See also
- Binary symmetric channel
- Binary erasure channel
- Z-channel (information theory)
- Channel model
- Information theory
- Coding theory
References
- Ahlswede, Rudolf and Blinovsky, Vladimir, "Classical Capacity of Classical-Quantum Arbitrarily Varying Channels," http://ieeexplore.ieee.org.gate.lib.buffalo.edu/stamp/stamp.jsp?tp=&arnumber=4069128
- Blackwell, David, Breiman, Leo, and Thomasian, A. J., "The Capacities of Certain Channel Classes Under Random Coding," http://www.jstor.org/stable/2237566
- Csiszar, I. and Narayan, P., "Arbitrarily varying channels with constrained inputs and states," http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=2598&isnumber=154
- Csiszar, I. and Narayan, P., "Capacity and Decoding Rules for Classes of Arbitrarily Varying Channels," http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=32153&isnumber=139
- Csiszar, I. and Narayan, P., "The capacity of the arbitrarily varying channel revisited: positivity, constraints," http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=2627&isnumber=155
- Lapidoth, A. and Narayan, P., "Reliable communication under channel uncertainty," http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=720535&isnumber=15554