Deducing capacity theorem [Under
Information theory > Shannon's paper]
Let us combine our discussions from last two articles to deduce Channel capacity assuming white thermal noise and certain average power for both transmitted signals and noise.
From article on Rate in terms of noise entropy,
When transmitted signal and noise are independent and received signal is sum of transmitted signal and noise, rate of transmission is:
R = H(y) — H(n)
If we would like to know the capacity of this (noisy) channel, we will have maximise the rate of transmission i.e.
C = R_{max} = ( H(y) — H(n) )_{max}
n being independent of x, adjustable parameter in above equation is H(y). So,
C = H(y)_{max} — H(n)
From article on Continuous symbols,
We saw that taking N as average noise power, maximum possible entropy of noise power is:
H(n) = W log(2πeN)
As seen earlier (mathematically), for achieving maximum entropy, signal should form Gaussian distribution. To achieve maximum H(y), we can control x so that y forms Gaussian distribution corresponding to addition of average noise power N and average transmitted power (say P). The maximum entropy achieve would be:
H(y)_{max} = W log(2πe (P+N) )
Putting above values in capacity equation, we get (theorem 16, page 43):
C = W log(2πe (P+N) ) — W log(2πeN)
That is:
References: A Mathematical Theory of Communication by Claude E. Shannon, An Introduction to Information Theory by John R. Pierce.
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