# Communication in presence of noise: Why is the number of distinguishable amplitudes K*sqrt((S+N)/N)

In section VII: The Capacity of a channel in the presence of white thermal noise in his 1949 paper Communication in the presence of noise, C.E.Shannon says, that for a signal with average power P, the total number of reasonably distinguishable amplitudes in the presence of white noise with average power N, is given by

$$K\,\sqrt{\frac{P+N}{N}}$$

where

K is a small constant in the neighborhood of unity depending on how the phrase reasonably well is interpreted.

I wonder where above formula comes from and if it has any deeper theoretical background, since for me it seems just like a simple metric: The Rx signal's amplitude has a standard deviation of

$$\sigma_\text{Rx}=\sqrt{P+N},$$

the noises amplitude standard deviation is

$$\sigma_\text{Tx} = \sqrt{N},$$ so above formula gives the factor by which the Rx's signals amplitude is larger than the noise amplitude that corrupts that signal on average. But why not compare the Tx signal's amplitudes to the noise amplitudes?

• Probaly this is rather a question for dsp.SE? – Andy Ef Apr 9 at 10:33
• What is Shannon's criteria for "reasonably distinguishable amplitudes"? This sounds like "bit error probability" metric. Also, his paper appeared before the work on "maximum likelihood detection" theories. Has this part of Shannon's philosophy (not to criticize him, but concepts do evolve) been superseded? – analogsystemsrf Apr 9 at 11:30
• @analogsystemsrf They haven't. – MBaz Apr 9 at 20:00