Everything about Binary Symmetric Channel totally explained
A
binary symmetric channel (or BSC) is a common
communications channel model used in
coding theory and
information theory. In this model, a transmitter wishes to send a
bit (a zero or a one), and the receiver receives a bit. It is assumed that the bit is
usually transmitted correctly, but that it'll be "flipped" with a small
probability (the "crossover probability"). This channel is used frequently in information theory because it's one of the simplest channels to analyze.
Description
The BSC is a
binary channel; that is, it can transmit only one of two symbols (usually called 0 and 1). (A non-binary channel would be capable of transmitting more than 2 symbols, possibly even an infinite number of choices) The transmission isn't perfect, and occasionally the receiver gets the wrong bit.
This channel is often used by theorists because it's one of the simplest
noisy channels to analyze. Many problems in
communication theory can be
reduced to a BSC. On the other hand, being able to transmit effectively over the BSC can give rise to solutions for more complicated channels.
Definition
A
binary symmetric channel with crossover probability p is a channel with binary input and binary output and probability of error
p; that is, if
X is the transmitted
random variable and
Y the received variable, then the channel is characterized by the
conditional probabilities » Pr(
Y = 0 |
X = 0) = 1-
p
Pr(
Y = 0 |
X = 1) =
p » Pr(
Y = 1 |
X = 0 ) =
p
Pr(
Y = 1 |
X = 1 ) = 1-
p
It is assumed that 0 ≤
p ≤ 1/2. If
p>1/2, then the receiver can swap the output (interpret 1 when it sees 0, and visa versa) and obtain an equivalent channel with crossover probability 1-
p ≤ 1/2.
Capacity of the BSC
The
capacity of the channel is 1 - H(
p), where H(
p) is the
binary entropy function.
The converse can be shown by a
sphere packing argument. Given a codeword, there are roughly 2
n H(p) typical output sequences. There are 2
n total possible outputs, and the input chooses from a
codebook of size 2
nR. Therefore, the receiver would choose to
partition the space into "spheres" with 2
n / 2
nR = 2
n(1-R) potential outputs each. If
R> 1 - H(
p), then the spheres will be packed too tightly
asymptotically and the receiver won't be able to identify the correct codeword with vanishing probability.
Further Information
Get more info on 'Binary Symmetric Channel'.
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