Channel model and notations

We consider the simple additive white Gaussian noise (AWGN) channel. In his 1959 paper [1], Claude E. Shannon showed that an optimal code is built by uniformly placing codewords on an n-dimensional sphere. An upper and a lower bound for the word error rate performance $P_{ew}$ of such a spherical code have been established by Shannon on an AWGN channel [1] for finite n. The main code parameters are its length n and its information rate $R$ . The length n is the number of real dimensions. The information rate $R$ is expressed in bits per real dimension. The spherical code is an ensemble of $2^{nR}$ points uniformly placed on a sphere in $\mathbb{R}^n$ .
A quick review of Shannon results and its generalization to a Rayleigh fading channel can be found in [7]. For $n \ge 100$ , the upper and lower bounds of $P_{ew}$ are superimposed. Hence, an accurate approximation for $P_{ew}$ is its lower bound $Q(\theta_0)$ , the probability of a codeword being moved outside its cone of half-angle $\theta_0$ .
Before you read [1] and [7], let me summarize all the numerical evaluations by two formulas. The first one is used to find $\theta_0$ from n and $R$ , the second one to evaluate $Q(\theta_0)$ , where $G = \frac{1}{2} \Big [ \sqrt{\frac{2E_s}{N_0}} \cos \theta + \sqrt{\frac{2E_s}{N_0} \cos^2\theta + 4} \Big ]$ . The two standard signal-to-noise ratios are related by $E_s/N_0=R \times E_b/N_0$ .
 
For the cone half-angle, please use

$$2^{nR} \approx \frac{\sqrt{2\pi n} \sin \theta_0 \cos \theta_0}{\sin^n \theta_0}$$ (1)

For the word error rate $P_{ew}$ , please use

$$Q(\theta_0) \approx \frac{1}{\sqrt{n\pi}} \frac{1}{\sqrt{1+G^2} \... ...heta_0 \Big) \Big]^n}{\sqrt{\frac{2E_s}{N_0}}G \sin^2 \theta_0 - \cos \theta_0}$$ (2)

The approximations are highly accurate in the above expressions. There is no need to compute the exact integrals involving the solid angle and the error probability [1].
In the next section, I give a C program implementation for $Q(\theta)$ . Sections 3 and 4 illustrate some numerical examples.