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Channel capacity

Channel capacity ofDMC is defincd as ihc largcst mutual information /(A/B)that can be transmitted over the channel in one use, niaximi/ed over all input probability assignments P_A.

C = max /(A/B)

Pa

It is worth to observe that, whereas /(A/B)is a function of both the channel and the input probability assignment. C is a function only of the channel assignmcnt! It is obvious that information can be transmitted at capacity, simply by using the appropriate input probability assignment.

For the Symmetric Channel, misinformation:

//(B/A) = p(b_f la_i) log i p(bj /a,),

>=i

index i is arbitrary, from 1 to D. as cach row of P_AB contains same set of probabilitics. Capacity can be achicvcd by using inputs of cqual probabilities, p(a,) = 1 /D: i = 1 „...Z). For such channel:

//(B)_mtt=log₂£

and thcn.

F.

C_sc = log, E + Y,P&j ^,ai) ^,0S: P(^bj = l or - orD.

/=!

For the Binary Symmetric Channel:

H( B/ A) = -p log, p - (1 - p)log, (1 - p)

and

Cbsc ^{= ,+} P^2 P + (l~ P)\og₂(\-p)

It is worth to observe, that for such channel, if p = 0 or p = 1,

then, C_BSC = 1, capacity reaches the maximum value for the BSC, such channel is the noiseless one,

if p- 1/2,

then C_BSC = 0, such channel is practically useless.

For the given models of input sourcc (probability assignment) and BSC (bit crror probability). ratę of channel utilization can be calculatcd: