Digital Communications 1

Appendix AProof of the Channel Capacity of the Additive White Gaussian Noise Channel

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,…, n_D] be the noise vector composed of D independent terms. The probability density of each of these terms is Gaussian:

[A.1]

The probability density of the vector n is the product of the D probability densities p(n_i):

[A.2]

Let be the norm of n. Since the variance of n_i is = σ², the mean of r² is given by:

[A.3]

And the variance of is:

[A.4]

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