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A design for a single chip implementation of a Reed-Solomon encoder is presented. The architecture that leads to this single VLSI chip design makes use of a bit serial finite field multiplication algorithm.
A VLSI architecture and layout for a convolutional encoder is described. This architecture allows a single chip implementation of an encoder that is capable of handling many different convolutional codes including all the convolutional codes that are presently used for deep space missions.
This paper describes a parallel processing computer consisting of a 16-bit microcomputer as a master processor which controls and coordinates the activities of 8086/8087 VLSI chip set slave processors working in parallel. The hardware is inexpensive and can be flexibly configured and programmed to perform various functions. This makes it a useful research tool for the development of, and experimentation with parallel mathematical algorithms. Application of the hardware to computational tasks involved in the finite element analysis method is demonstrated by the generation and assembly of beam finite element stiffness matrices. A number of possible schemes for the implementation of N-elements on N- or n-processors (N is greater than n) are described, and the speedup factors of their time consumption are determined as a function of the number of available parallel processors.
Fermat number transformation convolutes two digital data sequences. Very-large-scale integration (VLSI) applications, such as image and radar signal processing, X-ray reconstruction, and spectrum shaping, linear convolution of two digital data sequences of arbitrary lenghts accomplished using Fermat number transform (ENT).
A pipeline structure of a transform decoder similar to a systolic array was developed to decode Reed-Solomon (RS) codes. An important ingredient of this design is a modified Euclidean algorithm for computing the error locator polynomial. The computation of inverse field elements is completely avoided in this modification of Euclid's algorithm. The new decoder is regular and simple, and naturally suitable for VLSI implementation.
The specification and verification work is described in detail, and some of the problems and issues to be resolved in their application to Very Large Scale Integration VLSI systems are examined. The hierarchical design methodologies enable a system architect or design team to decompose a complex design into a formal hierarchy of levels of abstraction. The first step inprogram verification is tree formation. The next step after tree formation is the generation from the trees of the verification conditions themselves. The approach taken here is similar in spirit to the corresponding step in program verification but requires modeling of the semantics of circuit elements rather than program statements. The last step is that of proving the verification conditions using a mechanical theorem-prover.
Decoder for digital communications provides high-speed, pipelined ReedSolomon (RS) error-correction decoding of data streams. Principal new feature of proposed decoder is modification of Euclid greatest-common-divisor algorithm to avoid need for time-consuming computations of inverse of certain Galois-field quantities. Decoder architecture suitable for implementation on very-large-scale integrated (VLSI) chips with negative-channel metaloxide/silicon circuitry.
Proposed Reed-Solomon (RS) decoder assembled from very-large-scale integrated-circuit (VLSI) building blocks. Decoder exploits recursive forms in RS decoding algorithms. RS codes capable of correcting random or burst errors in telemetry and other data-communication signals. Because of small size and low power consumption, advantageous to employ several such decoders in parallel-processing scheme to increase decoding speed.
Fast Viterbi decoder with fully parallel, pipeline architecture implemented on two VLSI NMOS chips. Decoder used with constraint-length-7, rate-1/2, convolutional error-correcting code widely used by NASA for deepspace telemetry data. With this (7,1/2) code, bit stream contains 2 bits per original data bit, and information about 1 data bit distributed over 7 pairs of bits. Design principles of decoder also applicable to Viterbi codes of other lengths and rates.
Feedback shift registers are efficient periodic binary sequence generators. Polynomials of degree r over a Galois field characteristic 2(GF(2)) characterize the behavior of shift registers with linear logic feedback. The algorithmic determination of the trinomial of lowest degree, when it exists, that contains a given irreducible polynomial over GF(2) as a factor is presented. This corresponds to embedding the behavior of an r-stage shift register with linear logic feedback into that of an n-stage shift register with a single two-input modulo 2 summer (i.e., Exclusive-OR gate) in its feedback. This leads to Very Large Scale Integrated (VLSI) circuit architecture of maximal regularity (i.e., identical cells) with intercell communications serialized to a maximal degree.
The integer complex multiplier and adder over the direct sum of two copies of a finite field is specialized to the direct sum of the rings of integers modulo Fermat numbers. Such multiplications and additions can be used in the implementation of a discrete Fourier transform (DFT) of a sequence of complex numbers. The advantage of the present approach is that the number of multiplications needed for the DFT can be reduced substantially over the previous approach. The architectural designs using this approach are regular, simple, expandable and, therefore, naturally suitable for VLSI implementation.
A computer aided design (CAD) system that is being used at the Jet Propulsion Laboratory for the design of custom and semicustom very large scale integrated (VLSI) chips is described. The system consists of a Digital Equipment Corporation VAX computer with the UNIX operating system and a collection of software tools for the layout, simulation, and verification of microcircuits. Most of these tools were written by the academic community and are, therefore, available to JPL at little or no cost. Some small pieces of software have been written in-house in order to make all the tools interact with each other with a minimal amount of effort on the part of the designer.
Realization of a bit-serial multiplication algorithm for the encoding of Reed-Solomon (RS) codes on a single VLSI chip using NMOS technology is demonstrated to be feasible. A dual basis (255, 223) over a Galois field is used. The conventional RS encoder for long codes often requires look-up tables to perform the multiplication of two field elements. Berlekamp's algorithm requires only shifting and exclusive-OR operations.
A pipeline structure of a transform decoder similar to a systolic array was developed to decode Reed-Solomon (RS) codes. An important ingredient of this design is a modified Euclidean algorithm for computing the error locator polynomial. The computation of inverse field elements is completely avoided in this modification of Euclid's algorithm. The new decoder is regular and simple, and naturally suitable for VLSI implementation.
A recursive algorithm using the error-trellis decoding technique is developed to decode convolutional codes (CCs). An example, illustrating the very large scale integration (VLSI) architecture of such a decode, is given for a dual-K CC. It is demonstrated that such a decoder can be realized readily on a single chip with metal-nitride-oxide-semiconductor technology.
A Very Large Scale Integration (VLSI) architecture and layout for an 8-bit finite field multiplier is described. The algorithm used in this design was developed by Massey and Omura. A normal basis representation of finite field elements is used to reduce the multiplication complexity. It is shown that a drastic improvement was achieved in this design. This multiplier will be used intensively in the implementation of an 8-bit Reed-Solomon decoder and in many other related projects.
One of the most time consuming facets of custom Very Large Scale Integration (VLSI) design is obtaining hardcopy plots of the mask geometries of cells and chips. The traditional method of generating these plots is to use a multicolor pen plotter. Pen plotters are inherently slow and the plotting speed increases linearly with the number of edges that must be plotted. A moderate custom chip design at the Jet Propulsion Laboratory (JPL) now consists of more than 200,000 such edges and can take as much as eight hours to plot using a pen plotter. Software is described that was written at JPL to produce similar plots using a laser printer. It is shown that, for rather small layouts, the laser printer can provide nearly instantaneous turnaround. For moderate to large chip designs, the laser printer provides a factor of five or more improvement is speed over pen plotting.
A conventional prime factor discrete Fourier transform (DFT) algorithm is used to realize a discrete Fourier-like transform on the finite field, GF(q sub n). A pipeline structure is used to implement this prime factor DFT over GF(q sub n). This algorithm is developed to compute cyclic convolutions of complex numbers and to decode Reed-Solomon codes. Such a pipeline fast prime factor DFT algorithm over GF(q sub n) is regular, simple, expandable, and naturally suitable for VLSI implementation. An example illustrating the pipeline aspect of a 30-point transform over GF(q sub n) is presented.