Maximum-entropy probability distributions under Lp-norm constraints

Continuous probability density functions and discrete probability mass functions are tabulated which maximize the differential entropy or absolute entropy, respectively, among all probability distributions with a given L sub p norm (i.e., a given pth absolute moment when p is a finite integer) and unconstrained or constrained value set. Expressions for the maximum entropy are evaluated as functions of the L sub p norm. The most interesting results are obtained and plotted for unconstrained (real valued) continuous random variables and for integer valued discrete random variables. The maximum entropy expressions are obtained in closed form for unconstrained continuous random variables, and in this case there is a simple straight line relationship between the maximum differential entropy and the logarithm of the L sub p norm. Corresponding expressions for arbitrary discrete and constrained continuous random variables are given parametrically; closed form expressions are available only for special cases. However, simpler alternative bounds on the maximum entropy of integer valued discrete random variables are obtained by applying the differential entropy results to continuous random variables which approximate the integer valued random variables in a natural manner. All the results are presented in an integrated framework that includes continuous and discrete random variables, constraints on the permissible value set, and all possible values of p. Understanding such as this is useful in evaluating the performance of data compression schemes

Exact closed-form expressions for the performance of the split-symbol moments estimator of signal-to-noise ratio

Previously, the performance of the split-symbol moments estimator (SSME) of signal-to-noise ratio (SNR) has been evaluated by means of approximate expressions for the estimator mean and variance. These are asymptotic formulas in the sense that they become accurate as the number of estimator samples gets large. Here, exact closed-form expressions are obtained for the same quantities. These expressions confirm the accuracy of the previously derived asymptotic results, and, unlike the asymptotic formulas, they are useful even when the number of samples is small. It is also shown that the conventional split-symbol estimator can be trivially scaled to form a signal-to-noise ratio estimator which is precisely unbiased (as long as the estimate is based on more than two split-symbols)

Validity of the two-level model for Viterbi decoder gap-cycle performance

A two-level model has previously been proposed for approximating the performance of a Viterbi decoder which encounters data received with periodically varying signal-to-noise ratio. Such cyclically gapped data is obtained from the Very Large Array (VLA), either operating as a stand-alone system or arrayed with Goldstone. This approximate model predicts that the decoder error rate will vary periodically between two discrete levels with the same period as the gap cycle. It further predicts that the length of the gapped portion of the decoder error cycle for a constraint length K decoder will be about K-1 bits shorter than the actual duration of the gap. The two-level model for Viterbi decoder performance with gapped data is subjected to detailed validation tests. Curves showing the cyclical behavior of the decoder error burst statistics are compared with the simple square-wave cycles predicted by the model. The validity of the model depends on a parameter often considered irrelevant in the analysis of Viterbi decoder performance, the overall scaling of the received signal or the decoder's branch-metrics. Three scaling alternatives are examined: optimum branch-metric scaling and constant branch-metric scaling combined with either constant noise-level scaling or constant signal-level scaling. The simulated decoder error cycle curves roughly verify the accuracy of the two-level model for both the case of optimum branch-metric scaling and the case of constant branch-metric scaling combined with constant noise-level scaling. However, the model is not accurate for the case of constant branch-metric scaling combined with constant signal-level scaling

VLA telemetry performance with concatenated coding for Voyager at Neptune

Current plans for supporting the Voyager encounter at Neptune include the arraying of the Deep Space Network (DSN) antennas at Goldstone, California, with the National Radio Astronomy Observatory's Very Large Array (VLA) in New Mexico. Not designed as a communications antenna, the VLA signal transmission facility suffers a disadvantage in that the received signal is subjected to a gap or blackout period of approximately 1.6 msec once every 5/96 sec control cycle. Previous analyses showed that the VLA data gaps could cause disastrous performance degradation in a VLA stand-alone system and modest degradation when the VLA is arrayed equally with Goldstone. New analysis indicates that the earlier predictions for concatenated code performance were overly pessimistic for most combinations of system parameters, including those of Voyager-VLA. The periodicity of the VLA gap cycle tends to guarantee that all Reed-Solomon codewords will receive an average share of erroneous symbols from the gaps. However, large deterministic fluctuations in the number of gapped symbols from codeword to codeword may occur for certain combinations of code parameters, gap cycle parameters, and data rates. Several mechanisms for causing these fluctuations are identified and analyzed. Even though graceful degradation is predicted for the Voyager-VLA parameters, catastrophic degradation greater than 2 dB can occur for a VLA stand-alone system at certain non-Voyager data rates inside the range of the actual Voyager rates. Thus, it is imperative that all of the Voyager-VLA parameters be very accurately known and precisely controlled

Frame synchronization methods based on channel symbol measurements

The current DSN frame synchronization procedure is based on monitoring the decoded bit stream for the appearance of a sync marker sequence that is transmitted once every data frame. The possibility of obtaining frame synchronization by processing the raw received channel symbols rather than the decoded bits is explored. Performance results are derived for three channel symbol sync methods, and these are compared with results for decoded bit sync methods reported elsewhere. It is shown that each class of methods has advantages or disadvantages under different assumptions on the frame length, the global acquisition strategy, and the desired measure of acquisition timeliness. It is shown that the sync statistics based on decoded bits are superior to the statistics based on channel symbols, if the desired operating region utilizes a probability of miss many orders of magnitude higher than the probability of false alarm. This operating point is applicable for very large frame lengths and minimal frame-to-frame verification strategy. On the other hand, the statistics based on channel symbols are superior if the desired operating point has a miss probability only a few orders of magnitude greater than the false alarm probability. This happens for small frames or when frame-to-frame verifications are required

The theoretical limits of source and channel coding

The theoretical relationship among signal power, distortion, and bandwidth for several source and channel models is presented. The work is intended as a reference for the evaluation of the performance of specific data compression algorithms

Performance of Galileo's concatenated codes with nonideal interleaving

The Galileo spacecraft employs concatenated coding schemes with Reed-Solomon interleaving depth 2. The bit error rate (BER) performance of Galileo's concatenated codes, assuming different interleaving depths (including infinite interleaving depth) are compared. It is observed that Galileo's depth 2 interleaving, when used with the experimental (15, 1/4) code, requires about 0.4 to 0.5 dB additional signal-to-noise ratio to achieve the same BER performance as the concatenated code with ideal interleaving. When used with the standard (7, 1/2) code, depth 2 interleaving requires about 0.2 dB more signal-to-noise ratio than ideal interleaving

Wiring Viterbi decoders (splitting deBruijn graphs)

A new Viterbi decoder, capable of decoding convolutional codes with constraint lengths up to 15, is under development for the Deep Space Network (DSN). A key feature of this decoder is a two-level partitioning of the Viterbi state diagram into identical subgraphs. The larger subgraphs correspond to circuit boards, while the smaller subgraphs correspond to Very Large Scale Integration (VLSI) chips. The full decoder is built from identical boards, which in turn are built from identical chips. The resulting system is modular and hierarchical. The decoder is easy to implement, test, and repair because it uses a single VLSI chip design and a single board design. The partitioning is completely general in the sense that an appropriate number of boards or chips may be wired together to implement a Viterbi decoder of any size greater than or equal to the size of the module

Processing and Transmission of Information

Contains reports on one research project.National Aeronautics and Space Administration (Grant NGL 22-009-013

Uncorrectable sequences and telecommand

The purpose of a tail sequence for command link transmission units is to fail to decode, so that the command decoder will begin searching for the start of the next unit. A tail sequence used by several missions and recommended for this purpose by the Consultative Committee on Space Data Standards is analyzed. A single channel error can cause the sequence to decode. An alternative sequence requiring at least two channel errors before it can possibly decode is presented. (No sequence requiring more than two channel errors before it can possibly decode exists for this code.