A. Capacity of the Soft Decision Optical Channel

This report considered the capacity of the ``hard decision'' PPM optical channel, in which the decoder operates on PPM symbol decisions. However, there are practical ways to provide the decoder with additional information that can improve performance. When additional information is available, the capacity of the channel will increase (or least, will not decrease). Preliminary work on this has been started [DDP98].

There are several forms the additional information could take. It could be a reliability metric associated with each PPM symbol decision, indicating the conditional probability that the symbol decision is correct given the values of all the slot statistics. Or, the information could consist of the $l$ most likely PPM symbols and each of their reliabilities. Decoders can incorporate the additional information into a decoding algorithm that performs better as a result. Ultimately, the symbol detector could be removed entirely and the decoder could operate on all $M$ soft statistics directly. This approach has been taken in [Ham98b] and has shown improvement over Reed-Solomon coding in the cases considered there. This option is often within practical limits. In situations where full slot statistics are impractical, it is still useful to quantify the capacity one is giving up by not being able to use such an approach.

The capacity of the communications system when the symbol demodulator (see Fig. 1) is removed is at least as high as the channel that contains the symbol demodulator. This is a simple consequence of the data processing theorem. Using Fig. 1 with the demodulator removed, let $X \in \{0,\ldots, M-1\}$ denote the $M$-PPM symbol sent, and let ${\bf Y} = (Y_1, \ldots, Y_M)$ be the vector of slot statistics, $Y_i \in {\mathbb{R}}$. The capacity of the modified communications system is

$$C = \max_{p(X)} \sum_{j=0}^{M-1} \int_{\bf Y} p(X=j) p({\bf Y}\vert X... ...Y} \vert X=j) \over \sum_{l=0}^{M-1} p(X=l) p({\bf Y}\vert X=l)\right) d{\bf Y}$$

$$= \int_{\bf Y} p({\bf Y}\vert X=j) \log_2\left(p({\bf Y} \vert X=j) \over {1\over M} \sum_{l=0}^{M-1} p({\bf Y}\vert X=l)\right) d{\bf Y}.$$ (16)

This can difficult to compute for APD statistics and typical PPM orders such as $M=256$.

As a practical matter, codes are not yet available that can take advantage of the additional soft information within the individual slot statistics. RS codes cannot use soft information, except to the extent that they can define the erasure probabilities. Recent work on turbo-coded PPM has shown promise[Ham98b], but code rates needed for the optical channel have not been studied yet. Numerical analysis could indicate whether the difference in capacity is worth the extra effort needed to retain the soft slot statistics.