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# Universal Decoder for PPM of any Order

- Friday, 26 February 2010

### Complexity can be reduced and flexibility increased, at small cost in performance.

A recently developed algorithm for demodulation and decoding of a pulse-position-modulation (PPM) signal is suitable as a basis for designing a single hardware decoding apparatus to be capable of handling any PPM order. Hence, this algorithm offers advantages of greater flexibility and lower cost, in comparison with prior such algorithms, which necessitate the use of a distinct hardware implementation for each PPM order. In addition, in comparison with the prior algorithms, the present algorithm entails less complexity in decoding at large orders.

An unavoidably lengthy presentation of background information, including definitions of terms, is prerequisite to a meaningful summary of this development. As an aid to understanding, the figure illustrates the relevant processes of coding, modulation, propagation, demodulation, and decoding. An *M*-ary PPM signal has M time slots per symbol period. A pulse (signifying 1) is transmitted during one of the time slots; no pulse (signifying 0) is transmitted during the other time slots.

The information intended to be conveyed from the transmitting end to the receiving end of a radio or optical communication channel is a *K*-bit vector **u**. This
vector is encoded by an (*N*,*K*) binary error-correcting code, producing an *N*-bit vector **a**. In turn, the vector a is subdivided into blocks of *m* = log_{2}(*M*) bits and each such block is mapped to an *M*-ary PPM symbol. The resultant coding/modulation scheme can be regarded as equivalent to a nonlinear binary code. The binary vector of PPM symbols, **x** is transmitted over a Poisson channel, such that there is obtained, at the receiver, a Poisson-distributed photon count characterized by a mean background count *n _{b}* during no-pulse time slots and a mean signal-plus-background count of

*n*+

_{s}*n*during a pulse time slot.

_{b}In the receiver, demodulation of the signal is effected in an iterative soft decoding process that involves consideration of relationships among photon counts and conditional likelihoods of *m*-bit vectors of coded bits. Inasmuch as the likelihoods of all the *m*-bit vectors of coded bits mapping to the same PPM symbol are correlated, the best performance is obtained when the joint *m*-bit conditional likelihoods are utilized. Unfortunately, the complexity of decoding, measured in the number of operations per bit, grows exponentially with m, and can thus become prohibitively expensive for large PPM orders. For a system required to handle multiple PPM orders, the cost is even higher because it is necessary to have separate decoding hardware for each order. This concludes the prerequisite background information.

In the present algorithm, the decoding process as described above is modified by, among other things, introduction of an *l*-bit marginalizer subalgorithm. The term “*l*-bit marginalizer” signifies that instead of *m*-bit conditional likelihoods, the decoder computes *l*-bit conditional likelihoods, where *l* is fixed. Fixing *l*, regardless of the value of *m*, makes it possible to use a single hardware implementation for any PPM order. One could minimize the decoding complexity and obtain an especially simple design by fixing *l* at 1, but this would entail some loss of performance. An intermediate solution is to fix l at some value, greater than 1, that may be less than or greater than *m*. This solution makes it possible to obtain the desired flexibility to handle any PPM order while compromising between complexity and loss of performance.

*This work was done by Bruce E. Moision of Caltech for NASA’s Jet Propulsion Laboratory. For more information, contact This email address is being protected from spambots. You need JavaScript enabled to view it.
. NPO-46013*

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