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A decoding method for a (23,12) Golay code is extended to the important 1/2-rate (24,12) Golay code so that three errors can be corrected and four errors can be detected. It is shown that the method can be extended to any decoding method which can correct three errors in the (23,12) Golay code.
The Arroyo C++ libraries, written by Caltech post-doc student Matthew Britton, have the ability to simulate optical systems and atmospheric signal interference. This program was chosen for use in an end-to-end simulation model of a laser communication system because it is freely distributed and has the ability to be controlled by a remote system or "smart agent." Proposed operation of this program by a smart agent has been demonstrated, and the results show it to be a suitable simulation tool. Deformable mirrors, as a part of modern adaptive optics systems, may contain thousands of tiny, independently controlled actuators used to modify the shape of the mirror. Each actuator is connected to two wires, creating a cumbersome and expensive device. Recently, an alternative actuation method that uses gas-filled tubes known as Golay cells has been explored. Golay cells, operated by infrared lasers instead of electricity, would replace the actuator system thereby creating a more compact deformable mirror. The operation of Golay cells and their ability to move a deformable mirror in excess of the required 20 microns has been demonstrated. Experimentation has shown them to be extremely sensitive to pressure and temperature, making them ideal for use in a controlled environment.
A simplified procedure is developed to decode the three possible erors in a (23,12) Golay codeword. A computer simulation shows that this algorithm is modular, regular and naturally suitable for both Very Large Scale Integration (VLSI) and software implementation. An extension of this new decoding procedure is used also to decode the 1/2-rate (24,12) Golay code, thereby correcting three and detecting four errors.
The (24,12;8) extended Golay Code can be generated as a 6 x 4 binary matrix from the (15,11;3) BCH-Hamming Code, represented as a 5 x 3 matrix, by adding a row and a column, both of odd or even parity. The odd-parity case provides the additional 12th dimension. Furthermore, any three columns and five rows of the 6 x 4 Golay form a BCH-Hamming (15,11;3) Code. Similarly a (80,58;8) code can be generated as a 10 x 8 binary matrix from the (63,57;3) BCH-Hamming Code represented as a 9 x 7 matrix by adding a row and a column both of odd and even parity. Furthermore, any seven columns along with the top nine rows is a BCH-Hamming (53,57;3) Code. A (80,40;16) 10 x 8 matrix binary code with weight structure identical to the extended (80,40;16) Quadratic Residue Code is generated from a (63,39;7) binary cyclic code represented as a 9 x 7 matrix, by adding a row and a column, both of odd or even parity.
Digital computer components which perform two dimensional array logic operations (Tse logic) on binary data arrays are described. The properties of Golay transforms which make them useful in image processing are reviewed, and several architectures for Golay transform processors are presented with emphasis on the skeletonizing algorithm. Conventional logic control units developed for the Golay transform processors are described. One is a unique microprogrammable control unit that uses a microprocessor to control the Tse computer. The remaining control units are based on programmable logic arrays. Performance criteria are established and utilized to compare the various Golay transform machines developed. A critique of Tse logic is presented, and recommendations for additional research are included.
The development of an improved Golay cell is reported. This new sensor is constructed entirely from micromachined silicon components. A silicon oxynitride (SiO(x)N(y)) membrane is deflected by the thermal expansion of a small volume of trapped gas. To detect the motion of the membrane, an electron tunneling transducer is used. This sensor detects electrons which tunnel through the classically forbidden barrier between a tip and a surface; the electron current is exponentially dependent on the separation between the tip and the surface. The sensitivity of tunneling transducers constructed was typically better than 10(exp -3) A/square root of Hz. Through use of the electron tunneling transducer, the scaling laws which have prevented the miniaturization of the Golay cell are avoided. This detector potentially offers low cost fabrication, compatibility with silicon readout electronics, and operation without cooling. Most importantly, this detector may offer better sensitivity than any other uncooled infrared sensor, with the exception of the original Golay cell.
A technique proposed by Chase (1972) is used to evaluate the performance of several fairly long binary block codes on a wideband additive Gaussian channel. Considerations leading to the use of Chase's technique are discussed. Chase's concepts are first applied to the most powerful practical class of binary codes, the BCH codes with Berlekamp's (1972) decoding algorithm. Chase's algorithm is then described along with proposed selection of candidate codes. Results are presented of applying Chase's algorithm to four binary codes: (23, 12) Golay code, (32, 16) second-order Reed-Muller code, (63, 36) 5-error correcting BCH code, and (95, 39) 9-error correcting shortened BCH code. It is concluded that there are many block codes of length not exceeding 100 with extremely attractive maximum likelihood decoding performance on a Gaussian channel. BCH codes decoded via Berlekamp's binary decoding algorithm and Chase's idea are close to being practical competitors to short-constraint length convolutional codes with Viterbi decoding.
Convolutional codes of any rate and any constraint length give rise to a sequence of quasi-cyclic codes. Conversely, any quasi-cyclic code may be convolutionally encoded. Among the quasi-cyclic codes are the quadratic residue codes, Reed-Solomon codes and optimal BCH codes. The constraint length K for the convolutional encoding of many of these codes (Golay, (48, 24) QR, etc.) turns out to be surprisingly small. Thus using the soft decoding techniques for convolutional decoding, a new maximum likelihood decoding algorithm for many block codes is established. Conversely an optimal quasi-cyclic code will yield a convolutional encoding with optimal local properties and therefore with good infinite convolutional coding properties.
As new analytical instruments and techniques emerge with increased dimensionality, a corresponding need is seen for data processing logic which can appropriately address the data. Two-dimensional measurements reveal enhanced unknown mixture analysis capability as a result of the greater spectral information content over two one-dimensional methods taken separately. It is noted that two-dimensional convolute integers are merely an extension of the work by Savitzky and Golay (1964). It is shown that these low-pass, high-pass and band-pass digital filters are truly two-dimensional and that they can be applied in a manner identical with their one-dimensional counterpart, that is, a weighted nearest-neighbor, moving average with zero phase shifting, convoluted integer (universal number) weighting coefficients.
Sophisticated telecommunications techniques are described that were used to acquire images of Uranus, its 14 moons and ten narrow rings darker than coal. The images, equal in quality to those transmitted from Saturn several years earlier despite the signal being weaker by 6 dB due to the increased distance, were received from Voyager 2 during its January 24, 1986 flyby of Uranus. Solutions to the problem of the weakening signal were found in modifications to Voyager's image processing system and NASA's ground tracking network. In April 1985, Voyager's prime flight data computer was reconfigured to accept only nonimaging science data, and its backup, only imaging data; the latter was reprogrammed to determine only arithmetic differences between adjacent pixel intensities rather than absolute intensities. By image compression, equivalent imaging information could be sent at lower bit rates. Instead of Golay coding, Reed-Solomon onboard encoding was used. These techniques gained the equivalent of 4-dB in imaging yield. Additional improvements were gained by using earth station antennas in pairs (the Parkes radio telescope and the Canberra ground station antenna). Moves under way to prepare for the Voyager encounter with Neptune in 1989 are described (using additional antennas and arrays, scaling up the Deep Space Network antennas from 64 m to 70 m, etc.) to assure almost Saturn-equivalent pictures despite a further 3.5-dB drop in signal strength.
A general construction for finite-state (FS) codes is applied to some well-known block codes. Subcodes of the (24,12) Golay code are used to generate two optimal FS codes with d sub free = 12 and 16. A partition of the (16,8) Nordstrom-Robinson code yields a d sub free = 10 FS code. Simulation results are shown and decoding algorithms are briefly discussed.
New codes generated and analyzed. Report discusses application of general construction for finite-state (FS) codes to known block codes. Decoding algorithms and results of simulations discussed briefly. First, general construction applied to (24,12) Golay code. Next, general construction applied to (16,8) Nordstrom-Robinson code. Results presented as plots of bit-error probability versus ratio of bit energy to noise energy.
The pneumatic infrared detector, originally developed by Golay in the late 1940s, uses the thermal expansion of one cm(exp 3) of xenon at room temperature to detect the heat deposited by infrared radiation. This detector was limited by thermal fluctuations within a 10 Hz bandwidth, but suffered from long thermal time constants and a fragile structure. Nevertheless, it represents the most sensitive room temperature detector currently available in the long wavelength infrared (LWIR). Fabrication of this type of detector on smaller scales has been limited by the lack of a suitably sensitive transducer. Researchers designed a detector based on this principle, but which is constructed entirely from micromachined silicon, and uses a vacuum tunneling transducer to detect the expansion of the trapped gas. Because this detector is fabricated using micromachining techniques, miniaturization and integration into one and two-dimensional arrays is feasible. The extreme sensitivity of vacuum tunneling to changes in electrode separation will allow a prototype of this detector to operate in the limit of thermal fluctuations over a 10 kHz bandwidth. A calculation of the predicted response and noise of the prototype is presented with the general formalism of thermal detectors. At present, most of the components of the prototype have been fabricated and tested independently. In particular, a characterization of the micromachined electron tunneling transducer has been carried out. The measured noise in the tunnel current is within a decade of the limit imposed by shot noise, and well below the requirements for the operation of an infrared detector with the predicted sensitivity. Assembly and characterization of the prototype infrared detector will be carried out promptly.
The paper describes the design, fabrication, and characteristics of a novel infrared detector based on the principle of Golay's (1947) pneumatic infrared detector, which uses the expansion of a gas to detect infrared radiation. The present detector is constructed entirely from micromachined silicon and uses an electron tunneling displacement transducer for the detection of gas expansion. The sensitivity of the new detector is competitive with the best commercial pyroelectric sensors and can be readily improved by an order of magnitude through the use of an optimized transducer.
A method for determining the detector electrooptical transfer function (DEOTF) at different discrete frequencies simultaneously is presented. It involves simulation of the detector with a waveform of unknown frequency composition, such as a square wave or impulse function. The DEOTF is calculated as the ratio of the discrete Fourier transform of the detector output to the transform of the input waveform. This technique was successfully applied to Golay cell and bolometer detectors and can be used for other linear detector systems.
Pneumatic/thermal infrared detectors based partly on Golay-cell concept, but smaller and less fragile. Include containers filled with air or other gas trapped behind diaphragms. Infrared radiation heats sensors, causing gas to expand. Resulting deflections of diaphragms measured by displacement sensors based on principle of electron-tunneling transducers of scanning tunneling microscopes. Exceed sensitivity of all other miniature, uncooled infrared sensors presently available. Expected to include low consumption of power, broadband sensitivity, room-temperature operation, and invulnerability to ionizing radiation.
There exist almost complete convolutional encodings of self-dual codes, i.e., block codes of rate 1/2 with weights w, w = 0 mod 4. The codes are of length 8m with the convolutional portion of length 8m-2 and the nonsystematic information of length 4m-1. The last two bits are parity checks on the two (4m-1) length parity sequences. The final information bit complements one of the extended parity sequences of length 4m. Solomon and van Tilborg have developed algorithms to generate these for the Quadratic Residue (QR) Codes of lengths 48 and beyond. For these codes and reasonable constraint lengths, there are sequential decodings for both hard and soft decisions. There are also possible Viterbi-type decodings that may be simple, as in a convolutional encoding/decoding of the extended Golay Code. In addition, the previously found constraint length K = 9 for the QR (48, 24;12) Code is lowered here to K = 8.
Solomon and van Tilborg have developed convolutional encoding algorithms for quadratic residue (QR) codes of lengths 47 and beyond. For these codes and reasonable constraint lengths, there are sequential decodings that may be simple, as in a convolutional encoding/decoding of the extended Golay Code. In addition, the previously found constraint length K = 9 for the (48, 24; 12) QR code was lower to K = 8 by Solomon. In our search for the smallest possible constraint lengths K for (80, 40; 16) self-dual quadratic residue and nonquadratic residue codes, we have found the constraint lengths K = 14 and K = 13, respectively. We have discovered a K = 21 convolutional encoding for the (104, 52; 20) QR code; there may be a smaller K for a (104, 52; 20) self-dual code that is not a quadratic residue code. The smaller the K, the less complex the sequential or Viterbi decoder.