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Channel capacity is the limit of the transmission rate which can be delivered to the destination with arbitrary small error.

For the Gaussian channel case, the channel capacity is given as


C = \log_2 ( 1 + \mathrm{SNR})

which shows that the channel capacity of the Gaussian channel is determined by SNR.

We can derive the capacity of Gaussian channel by assuming the receive signal is represented as


y = x + n

where x, y, n are all Gaussian signal with the power spectral densities are P+N,P,N respectively. If the length of the input and out signal sequence \{x\}, \{y\}, \{n\} is infinitely long, the velocity of the sphere of infinite multi-dimensional sequence of \{y\} is 1+P/N times bigger than the velocity of the sphere of infinite multi-dimensional sequence of \{n\}. Hence, we predicts that the number of maximally possible transmission codeword is \log_2 ( 1 + \mathrm{SNR})
where \mathrm{SNR} = P/N.

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