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A lower bound for the decoder error probability (P sub E (u)) of a linear maximum distance separable (MDS) code is derived by counting the dominant types of decoding words around code words. It is shown that the lower bound derived is similar in form, and close numerically, to the upper bound derived.
A highly efficient error correcting code has been selected by NASA as a CCSDS standard: the 16 symbol error correcting Reed Solomon code. A VLSI implementation of this decoder is described in this paper. A total of 4 full custom VLSI chips are needed that correct data in real time at an sustained rate of up to 80 Mbits/second.
Visual Object Oriented Unification System (VOUS) computer program provides facility for development of other, high-level software. Does not replace, but rather extends, preexisting software tools for development of other software. Provides comprehensive, graphical, interactive medium for all phases in development of computer code from early exploration of concepts, through detailed coding-and-error-checking process, to final reporting of finished code and compilation of instruction manual for its use. Simplifies and partly automates programmer's task.
This is a compilation of NELS (NASA Electronic Library System) Optimization progress/problem, interim, and final reports for all phases. The NELS database was examined, particularly in the memory, disk contention, and CPU, to discover bottlenecks. Methods to increase the speed of NELS code were investigated. The tasks included restructuring the existing code to interact with others more effectively. An error reporting code to help detect and remove bugs in the NELS was added. Report writing tools were recommended to integrate with the ASV3 system. The Oracle database management system and tools were to be installed on a Sun workstation, intended for demonstration purposes.
Laser communication and ranging experiments were successfully conducted from the satellite laser ranging (SLR) station at NASA Goddard Space Flight Center (GSFC) to the Lunar Reconnaissance Orbiter (LRO) in lunar orbit. The experiments used 4096-ary pulse position modulation (PPM) for the laser pulses during one-way LRO Laser Ranging (LR) operations. Reed-Solomon forward error correction codes were used to correct the PPM symbol errors due to atmosphere turbulence and pointing jitter. The signal fading was measured and the results were compared to the model.
An upper bound on the average probability of error for maximum-likelihood decoding of the ensemble of random L-branch binary trellis codes of rate R = 1/n with distinction between memory length and tail length is given. It is shown that the bound is independent of the length L of the information sequence if the memory length exceeds the tail length by a specified amount that depends on L. Sequential decoding simulations using the stack algorithm were conducted to test the dependence of the undetected error probability on tail length and memory length, and the results corroborated the theory.
Addressed in this paper is the dependence of telecommunication channel, forward error correcting coding and source data compression coding on integrated circuit technology. Emphasis is placed on real time high speed Reed Solomon (RS) decoding using full custom VLSI technology. Performance curves of NASA's standard channel coder and a proposed standard lossless data compression coder are presented.
This slide presentation reviews the progress that the workgroup on Low-Density Parity-Check (LDPC) for space link coding. The workgroup is tasked with developing and recommending new error correcting codes for near-Earth, Lunar, and deep space applications. Included in the presentation is a summary of the technical progress of the workgroup. Charts that show the LDPC decoder sensitivity to symbol scaling errors are reviewed, as well as a chart showing the performance of several frame synchronizer algorithms compared to that of some good codes and LDPC decoder tests at ESTL. Also reviewed is a study on Coding, Modulation, and Link Protocol (CMLP), and the recommended codes. A design for the Pseudo-Randomizer with LDPC Decoder and CRC is also reviewed. A chart that summarizes the three proposed coding systems is also presented.
A robust error-control coding system is presented. This system is a cascaded FEC (forward error control) scheme supported by parity retransmissions for further error correction in the erroneous data words. The error performance and throughput efficiency of the system are analyzed. Two specific examples of the error-control system are studied. The first example does not use an inner code, and the outer code, which is not interleaved, is a shortened code of the NASA standard RS code over GF(28). The second example, as proposed for NASA, uses the same shortened RS code as the base outer code C2, except that it is interleaved to a depth of 2. It is shown that both examples provide high reliability and throughput efficiency even for high channel bit-error rates in the range of 0.01.
Digital Flight Control System (DFCS) software was used as a test case for assertion testing. The assertions were written and embedded in the code, then errors were inserted (seeded) one at a time and the code executed. Results indicate that assertion testing is an effective and efficient method of detecting errors in flight software. Most errors are eliminate at an earlier stage in the development than before.
In the late seventies and early eighties a technique known as Trellis Coded Modulation (TCM) was developed for providing spectrally efficient error correction coding. Instead of adding redundant information in the form of parity bits, redundancy is added at the modulation stage thereby increasing bandwidth efficiency. A digital communications system can be designed to use bandwidth-efficient multilevel/phase modulation such as Amplitude Shift Keying (ASK), Phase Shift Keying (PSK), Differential Phase Shift Keying (DPSK) or Quadrature Amplitude Modulation (QAM). Performance gain can be achieved by increasing the number of signals over the corresponding uncoded system to compensate for the redundancy introduced by the code. A considerable amount of research and development has been devoted toward developing good TCM codes for severely bandlimited applications. More recently, the use of TCM for satellite and deep space communications applications has received increased attention. This report describes the general approach of using a concatenated coding scheme that features TCM and RS coding. Results have indicated that substantial (6-10 dB) performance gains can be achieved with this approach with comparatively little bandwidth expansion. Since all of the bandwidth expansion is due to the RS code we see that TCM based concatenated coding results in roughly 10-50% bandwidth expansion compared to 70-150% expansion for similar concatenated scheme which use convolution code. We stress that combined coding and modulation optimization is important for achieving performance gains while maintaining spectral efficiency.
A code trellis is a graphical representation of a code, block or convolutional, in which every path represents a codeword (or a code sequence for a convolutional code). This representation makes it possible to implement Maximum Likelihood Decoding (MLD) of a code with reduced decoding complexity. The most well known trellis-based MLD algorithm is the Viterbi algorithm. The trellis representation was first introduced and used for convolutional codes [23]. This representation, together with the Viterbi decoding algorithm, has resulted in a wide range of applications of convolutional codes for error control in digital communications over the last two decades. There are two major reasons for this inactive period of research in this area. First, most coding theorists at that time believed that block codes did not have simple trellis structure like convolutional codes and maximum likelihood decoding of linear block codes using the Viterbi algorithm was practically impossible, except for very short block codes. Second, since almost all of the linear block codes are constructed algebraically or based on finite geometries, it was the belief of many coding theorists that algebraic decoding was the only way to decode these codes. These two reasons seriously hindered the development of efficient soft-decision decoding methods for linear block codes and their applications to error control in digital communications. This led to a general belief that block codes are inferior to convolutional codes and hence, that they were not useful. Chapter 2 gives a brief review of linear block codes. The goal is to provide the essential background material for the development of trellis structure and trellis-based decoding algorithms for linear block codes in the later chapters. Chapters 3 through 6 present the fundamental concepts, finite-state machine model, state space formulation, basic structural properties, state labeling, construction procedures, complexity, minimality, and sectionalization of trellises. Chapter 7 discusses trellis decomposition and subtrellises for low-weight codewords. Chapter 8 first presents well known methods for constructing long powerful codes from short component codes or component codes of smaller dimensions, and then provides methods for constructing their trellises which include Shannon and Cartesian product techniques. Chapter 9 deals with convolutional codes, puncturing, zero-tail termination and tail-biting.Chapters 10 through 13 present various trellis-based decoding algorithms, old and new. Chapter 10 first discusses the application of the well known Viterbi decoding algorithm to linear block codes, optimum sectionalization of a code trellis to minimize computation complexity, and design issues for IC (integrated circuit) implementation of a Viterbi decoder. Then it presents a new decoding algorithm for convolutional codes, named Differential Trellis Decoding (DTD) algorithm. Chapter 12 presents a suboptimum reliability-based iterative decoding algorithm with a low-weight trellis search for the most likely codeword. This decoding algorithm provides a good trade-off between error performance and decoding complexity. All the decoding algorithms presented in Chapters 10 through 12 are devised to minimize word error probability. Chapter 13 presents decoding algorithms that minimize bit error probability and provide the corresponding soft (reliability) information at the output of the decoder. Decoding algorithms presented are the MAP (maximum a posteriori probability) decoding algorithm and the Soft-Output Viterbi Algorithm (SOVA) algorithm. Finally, the minimization of bit error probability in trellis-based MLD is discussed.
The control system of NASA's Orbital Maneuvering Vehicle (OMV) will employ range/range-rate radar, a forward command link, and a compressed video return link. The video data is compressed by sampling every sixth frame of data; a rate of 5 frames/sec is adequate for the OMV docking speeds. Further axial compression is obtained, albeit at the expense of spatial resolution, by averaging adjacent pixels. The remaining compression is achieved on the basis of differential pulse-code modulation and Huffman run-length encoding. A concatenated error-correction coding system is used to protect the compressed video data stream from channel errors.
Using the finite field transform and continued fractions, a simplified algorithm for decoding Reed-Solomon (R-S) codes is developed to correct erasures caused by other codes as well as errors over the finite field GF (q(m), where q is a prime and m is an integer. Such an R-S decoder can be faster and simpler than a decoder that uses more conventional methods.
The performance of a concatenated coding scheme for error control in ARQ systems is analyzed for both random-noise and burst-noise channels. In particular, the probability of undetected error and the system throughput are calculated. In this scheme, the inner code is used for both error correction and error detection, and the outer code is used for error detection only. Interleaving/deinterleaving is assumed within the outer code. A retransmission is requested if either the inner code or the outer code detects the presence of errors. Various coding examples are considered. The results show that concatenated coding can provide extremely high system reliability (i.e., low probability of undetected error) and high system throughput.
This research project was set up to study various kinds of coding techniques for error control in satellite and space communications for NASA Goddard Space Flight Center. During the project period, researchers investigated the following areas: (1) decoding of Reed-Solomon codes in terms of dual basis; (2) concatenated and cascaded error control coding schemes for satellite and space communications; (3) use of hybrid coding schemes (error correction and detection incorporated with retransmission) to improve system reliability and throughput in satellite communications; (4) good codes for simultaneous error correction and error detection, and (5) error control techniques for ring and star networks.
A detailed discussion of the use of channel coding for error correction, privacy/secrecy, channel separation, and synchronization is presented. Channel coding, in one form or another, is an established and common element in data systems. No analysis and design of a major new system would fail to consider ways in which channel coding could make the system more effective. The presence of channel coding on TDRS, Shuttle, the Advanced Communication Technology Satellite Program system, the JSC-proposed Space Operations Center, and the proposed 30/20 GHz Satellite Communication System strongly support the requirement for the utilization of coding for the communications channel. The designers of the space station data system have to consider the use of channel coding.