In this appendix, we will proof some important results for the geometric proof of the channel capacity of the additive white Gaussian noise channel.
Let n = [n1, n2,…, nD] be the noise vector composed of D independent terms. The probability density of each of these terms is Gaussian:
The probability density of the vector n is the product of the D probability densities p(ni):
Let be the norm of n. Since the variance of ni is = σ2, the mean of r2 is given by:
And the variance of is:
For high values of D, using the limit central theorem, we can show that the variance of r2 is equal to 2σ4D. Consequently, when D tends to infinite, the norm r is concentrated around