3.1.1 Capacity as a function of correct PPM symbol detection

The capacity of the communications system in Fig. 1 is the maximum mutual information between the input and output,

$$C \triangleq \max_{p({\bf X})} I({\bf U}; {\bf\hat U}) = \max_{p({\bf X})} H({\bf\hat U}) - H({\bf\hat U}\vert {\bf U}),$$

where $H({\bf\hat U})$ is the entropy of $\bf\hat U$, $H({\bf\hat U}\vert {\bf U})$ is the conditional entropy of $\bf\hat U$ given $\bf U$, and $I({\bf U}; {\bf\hat U})$ is the mutual information between $\bf U$ and $\bf\hat U$. Since the encoder and decoder are deterministic, invertible functions, the capacity of the system reduces in the usual way to

$$C = \max_{p({\bf X})} I({\bf X}; {\bf Z}) = \max_{p({\bf X})} H({\bf Z}) - H({\bf Z}\vert {\bf X}).$$

The channel ${\bf X} \rightarrow {\bf Z}$ is an $M$-ary symmetric channel (repeated $n$ times), whose capacity depends on the probability of correct uncoded symbol detection $p \triangleq$   Pr$(X_i = Z_i)$. Under the assumptions of perfect timing and negligible inter-slot interference, the $M-1$ possible incorrect decisions are equally likely, and each incorrect $M$-PPM symbol has probability $q = (1-p)/(M-1).$ The capacity of the $M$-ary symmetric channel is given by [Ash65]

$$C = \log_2 M + p\log_2 p + (M-1)q\log_2 q$$ (1)

Thus, to compute the capacity of the optical channel, we need only determine $p$. Note that the analysis thus far has not depended on the particular type of detector used, only that the detector operates in a memoryless fashion.