Statistical mechanics of low-density parity check error-correcting codes over Galois fields

A variation of low-density parity check (LDPC) error-correcting codes defined over Galois fields (GF(q)) is investigated using statistical physics. A code of this type is characterised by a sparse random parity check matrix composed of C non-zero elements per column. We examine the dependence of the code performance on the value of q, for finite and infinite C values, both in terms of the thermodynamical transition point and the practical decoding phase characterised by the existence of a unique (ferromagnetic) solution. We find different q-dependence in the cases of C = 2 and C ≥ 3; the analytical solutions are in agreement with simulation results, providing a quantitative measure to the improvement in performance obtained using non-binary alphabets.

Error-correcting codes on a Bethe-like lattice

We analyse Gallager codes by employing a simple mean-field approximation that distorts the model geometry and preserves important interactions between sites. The method naturally recovers the probability propagation decoding algorithm as a minimization of a proper free-energy. We find a thermodynamical phase transition that coincides with information theoretical upper-bounds and explain the practical code performance in terms of the free-energy landscape.

Finite-connectivity spin-glass phase diagrams and low-density parity check codes

We obtain phase diagrams of regular and irregular finite-connectivity spin glasses. Contact is first established between properties of the phase diagram and the performance of low-density parity check (LDPC) codes within the replica symmetric (RS) ansatz. We then study the location of the dynamical and critical transition points of these systems within the one step replica symmetry breaking theory (RSB), extending similar calculations that have been performed in the past for the Bethe spin-glass problem. We observe that the location of the dynamical transition line does change within the RSB theory, in comparison with the results obtained in the RS case. For LDPC decoding of messages transmitted over the binary erasure channel we find, at zero temperature and rate R=14, an RS critical transition point at pc 0.67 while the critical RSB transition point is located at pc 0.7450±0.0050, to be compared with the corresponding Shannon bound 1-R. For the binary symmetric channel we show that the low temperature reentrant behavior of the dynamical transition line, observed within the RS ansatz, changes its location when the RSB ansatz is employed; the dynamical transition point occurs at higher values of the channel noise. Possible practical implications to improve the performance of the state-of-the-art error correcting codes are discussed. © 2006 The American Physical Society.

Continuous phase synchronised drives (for a rod-making machine)

Traditional high speed machinery actuators are powered and coordinated by mechanical linkages driven from a central drive, but these linkages may be replaced by independently synchronised electric drives. Problems associated with utilising such electric drives for this form of machinery were investigated. The research concentrated on a high speed rod-making machine, which required control of high inertias (0.01-0.5kgm2), at continuous high speed (2500 r/min), with low relative phase errors between two drives (0.0025 radians). Traditional minimum energy drive selection techniques for incremental motions were not applicable to continuous applications which require negligible energy dissipation. New selection techniques were developed. A brushless configuration constant enabled the comparison between seven different servo systems; the rate earth brushless drives had the best power rates which is a performance measure. Simulation was used to review control strategies, such that a microprocessor controller with a proportional velocity loop within a proportional position loop with velocity feedforward was designed. Local control schemes were investigated as means of reducing relative errors between drives: the slave of a master/slave scheme compensates for the master's errors: the matched scheme has drives with similar absolute errors so the relative error is minimised, and the feedforward scheme minimises error by adding compensation from previous knowledge. Simulation gave an approximate velocity loop bandwidth and position loop gain required to meet the specification. Theoretical limits for these parameters were defined in terms of digital sampling delays, quantisation, and system phase shifts. Performance degradation due to mechanical backlash was evaluated. Thus any drive could be checked to ensure that the performance specification could be realised. A two drive demonstrator was commissioned with 0.01kgm2 loads. By use of simulation the performance of one drive was improved by increasing the velocity loop bandwidth fourfold. With the master/slave scheme relative errors were within 0.0024 radians at a constant 2500 r/min for two 0.01 kgm^2 loads.

Computing with noise: phase transitions in Boolean formulas

Computing circuits composed of noisy logical gates and their ability to represent arbitrary Boolean functions with a given level of error are investigated within a statistical mechanics setting. Existing bounds on their performance are straightforwardly retrieved, generalized, and identified as the corresponding typical-case phase transitions. Results on error rates, function depth, and sensitivity, and their dependence on the gate-type and noise model used are also obtained.

Statistical physics of error-correcting codes

In this thesis we use statistical physics techniques to study the typical performance of four families of error-correcting codes based on very sparse linear transformations: Sourlas codes, Gallager codes, MacKay-Neal codes and Kanter-Saad codes. We map the decoding problem onto an Ising spin system with many-spins interactions. We then employ the replica method to calculate averages over the quenched disorder represented by the code constructions, the arbitrary messages and the random noise vectors. We find, as the noise level increases, a phase transition between successful decoding and failure phases. This phase transition coincides with upper bounds derived in the information theory literature in most of the cases. We connect the practical decoding algorithm known as probability propagation with the task of finding local minima of the related Bethe free-energy. We show that the practical decoding thresholds correspond to noise levels where suboptimal minima of the free-energy emerge. Simulations of practical decoding scenarios using probability propagation agree with theoretical predictions of the replica symmetric theory. The typical performance predicted by the thermodynamic phase transitions is shown to be attainable in computation times that grow exponentially with the system size. We use the insights obtained to design a method to calculate the performance and optimise parameters of the high performance codes proposed by Kanter and Saad.

Phase shift keyed systems based on a gain switched laser transmitter

Return-to-Zero (RZ) and Non-Return-to-Zero (NRZ) Differential Phase Shift Keyed (DPSK) systems require cheap and optimal transmitters for widespread implementation. The authors report on a gain switched Discrete Mode (DM) laser that can be employed as a cost efficient transmitter in a 10.7 Gb/s RZ DPSK system and compare its performance to that of a gain switched Distributed Feed-Back (DFB) laser. Experimental results show that the gain switched DM laser readily provides error free performance and a receiver sensitivity of -33.1 dBm in the 10.7 Gbit/s RZ DPSK system. The standard DFB laser on the other hand displays an error floor at 10(-1) in the same RZ DPSK system. The difference in performance, between the two types of gain switched transmitters, is analysed by investigating their linewidths. We also demonstrate, for the first time, the generation of a highly coherent gain switched pulse train which displays a spectral comb of approximately 13 sidebands spaced by the 10.7 GHz modulation frequency. The filtered side-bands are then employed as narrow linewidth Continuous Wave (CW) sources in a 10.7 Gb/s NRZ DPSK system.

Phase synchronization scheme for a practical phase sensitive amplifier of ASK-NRZ signals

We present a phase locking scheme that enables the demonstration of a practical dual pump degenerate phase sensitive amplifier for 10 Gbit/s non-return to zero amplitude shift keying signals. The scheme makes use of cascaded Mach Zehnder modulators for creating the pump frequencies as well as of injection locking for extracting the signal carrier and synchronizing the local lasers. An in depth optimization study has been performed, based on measured error rate performance, and the main degradation factors have been identified.

Time resolved bit error rate analysis of a fast switching tunable laser for use in optically switched networks

We investigate the use of different direct detection modulation formats in a wavelength switched optical network. We find the minimum time it takes a tunable sampled grating distributed Bragg reflector laser to recover after switching from one wavelength channel to another for different modulation formats. The recovery time is investigated utilizing a field programmable gate array which operates as a time resolved bit error rate detector. The detector offers 93 ps resolution operating at 10.7 Gb/s and allows for all the data received to contribute to the measurement, allowing low bit error rates to be measured at high speed. The recovery times for 10.7 Gb/s non-return-to-zero on–off keyed modulation, 10.7 Gb/s differentially phase shift keyed signal and 21.4 Gb/s differentially quadrature phase shift keyed formats can be as low as 4 ns, 7 ns and 40 ns, respectively. The time resolved phase noise associated with laser settling is simultaneously measured for 21.4 Gb/s differentially quadrature phase shift keyed data and it shows that the phase noise coupled with frequency error is the primary limitation on transmitting immediately after a laser switching event.

Characterization of time-resolved laser differential phase using 3D complementary cumulative distribution functions

An experimental method for characterizing the time-resolved phase noise of a fast switching tunable laser is discussed. The method experimentally determines a complementary cumulative distribution function of the laser's differential phase as a function of time after a switching event. A time resolved bit error rate of differential quadrature phase shift keying formatted data, calculated using the phase noise measurements, was fitted to an experimental time-resolved bit error rate measurement using a field programmable gate array, finding a good agreement between the time-resolved bit error rates.

DPSK signal regeneration with a dual-pump nondegenerate phase-sensitive amplifier

We demonstrate, for the first time to our knowledge, regeneration of a 42.66-Gb/s differential phase-shift keyed signal using a dual-pump nondegenerate four-wave-mixing-based fiber-optic parametric amplifier. The regenerative performance of the subsystem is characterized in terms of bit-error rate against narrowband and wideband introduced noise. While a strong receiver sensitivity improvement, up to 20 dB, is noticed against narrowband noise, against quasi-random (wideband) noise we observe a regeneration of 2.7 dB.

Bit-error rate performance of 20 Gbit/s WDM RZ-DPSK non-slope matched submarine transmission systems

Applying direct error counting, we assess the performance of 20 Gbit/s wavelength-division multiplexing return-to-zero differential phase-shift keying (RZ-DPSK) transmission at 0.4 bit/(s Hz) spectral efficiency for application on installed non-zero dispersion-shifted fibre based transoceanic submarine systems. The impact of the pulse duty cycle on the system performance is investigated and the reliability of the existing theoretical approaches to the BER estimation for the RZ-DPSK format is discussed.

Patterns of refractive change in myopic subjects during the incipient phase of presbyopia:a preliminary study

Purpose: Changes in refractive error are well documented over the typical human lifespan. However, a relatively neglected period of investigation appears to be during the late fourth decade; this is at the incipient phase of presbyopia (IP), where the amplitude of accommodation is much reduced and approaches the level where a first reading addition is anticipated. Significantly, informal clinical observation has suggested a low incidence of an unexpected abrupt increase in myopia during IP. Methods: We investigated this alleged myopic shift retrospectively by mapping the longitudinal refraction histories of normally-sighted 35-44years old British White patients previously examined in routine optometric practice. The refractive trends in the right eyes of healthy myopic subjects (spherical equivalent refraction, SER =-0.50D: N=39) were analysed relative to that point at which a first near dioptric addition was considered to be clinically useful. Results: A refractive change was evident in some subjects during IP; viz, an abrupt increase in myopic SER of between -0.50 and -0.75D. These individuals (N=8) represented 20% of the study population of myopic incipient presbyopes. Beyond the pivotal point of the first near addition the longitudinal refraction stabilized in these subjects. In contrast, and as the extent of the available longitudinal data would permit, the remaining myopic eyes maintained an approximately stable refractive trend throughout IP and beyond. Conclusions: The anatomical or physiological basis of this specific late (non-developmental) abrupt myopic refractive change is an intriguing issue. Axial (vitreous chamber elongation), corneal (contour) and lenticular (profile and index) power bases, alone or in concert, might be considered candidates for this hitherto unexplored refractive phenomenon. Although necessarily obtained under conventional conditions of central (0deg) fixation, our data might also be a reflection of the recent recognition of the possible influence of the peripheral refraction upon the axial error. Consideration of this material provides an impetus for further research, including ocular biometry, a reappraisal of ciliary zonular functional anatomy, renewed investigation of the AC/A ratio, and the extent of a centripetal refractive influence on myopia development. © 2011 The College of Optometrists.

Multiple component patient safety intervention in English hospitals:controlled evaluation of second phase

Objective: To independently evaluate the impact of the second phase of the Health Foundation's Safer Patients Initiative (SPI2) on a range of patient safety measures. Design: A controlled before and after design. Five substudies: survey of staff attitudes; review of case notes from high risk (respiratory) patients in medical wards; review of case notes from surgical patients; indirect evaluation of hand hygiene by measuring hospital use of handwashing materials; measurement of outcomes (adverse events, mortality among high risk patients admitted to medical wards, patients' satisfaction, mortality in intensive care, rates of hospital acquired infection). Setting: NHS hospitals in England. Participants: Nine hospitals participating in SPI2 and nine matched control hospitals. Intervention The SPI2 intervention was similar to the SPI1, with somewhat modified goals, a slightly longer intervention period, and a smaller budget per hospital. Results: One of the scores (organisational climate) showed a significant (P=0.009) difference in rate of change over time, which favoured the control hospitals, though the difference was only 0.07 points on a five point scale. Results of the explicit case note reviews of high risk medical patients showed that certain practices improved over time in both control and SPI2 hospitals (and none deteriorated), but there were no significant differences between control and SPI2 hospitals. Monitoring of vital signs improved across control and SPI2 sites. This temporal effect was significant for monitoring the respiratory rate at both the six hour (adjusted odds ratio 2.1, 99% confidence interval 1.0 to 4.3; P=0.010) and 12 hour (2.4, 1.1 to 5.0; P=0.002) periods after admission. There was no significant effect of SPI for any of the measures of vital signs. Use of a recommended system for scoring the severity of pneumonia improved from 1.9% (1/52) to 21.4% (12/56) of control and from 2.0% (1/50) to 41.7% (25/60) of SPI2 patients. This temporal change was significant (7.3, 1.4 to 37.7; P=0.002), but the difference in difference was not significant (2.1, 0.4 to 11.1; P=0.236). There were no notable or significant changes in the pattern of prescribing errors, either over time or between control and SPI2 hospitals. Two items of medical history taking (exercise tolerance and occupation) showed significant improvement over time, across both control and SPI2 hospitals, but no additional SPI2 effect. The holistic review showed no significant changes in error rates either over time or between control and SPI2 hospitals. The explicit case note review of perioperative care showed that adherence rates for two of the four perioperative standards targeted by SPI2 were already good at baseline, exceeding 94% for antibiotic prophylaxis and 98% for deep vein thrombosis prophylaxis. Intraoperative monitoring of temperature improved over time in both groups, but this was not significant (1.8, 0.4 to 7.6; P=0.279), and there were no additional effects of SPI2. A dramatic rise in consumption of soap and alcohol hand rub was similar in control and SPI2 hospitals (P=0.760 and P=0.889, respectively), as was the corresponding decrease in rates of Clostridium difficile and meticillin resistant Staphylococcus aureus infection (P=0.652 and P=0.693, respectively). Mortality rates of medical patients included in the case note reviews in control hospitals increased from 17.3% (42/243) to 21.4% (24/112), while in SPI2 hospitals they fell from 10.3% (24/233) to 6.1% (7/114) (P=0.043). Fewer than 8% of deaths were classed as avoidable; changes in proportions could not explain the divergence of overall death rates between control and SPI2 hospitals. There was no significant difference in the rate of change in mortality in intensive care. Patients' satisfaction improved in both control and SPI2 hospitals on all dimensions, but again there were no significant changes between the two groups of hospitals. Conclusions: Many aspects of care are already good or improving across the NHS in England, suggesting considerable improvements in quality across the board. These improvements are probably due to contemporaneous policy activities relating to patient safety, including those with features similar to the SPI, and the emergence of professional consensus on some clinical processes. This phenomenon might have attenuated the incremental effect of the SPI, making it difficult to detect. Alternatively, the full impact of the SPI might be observable only in the longer term. The conclusion of this study could have been different if concurrent controls had not been used.

All-optical 42.6 Gbit/s NRZ to RZ format conversion by cross-phase modulation in single SOA

A 42.6 Gbit/s all-optical non-retum-to-zero (NRZ) to return-to-zero (RZ) format converter using a single SOA followed by an asymmetrical Mach-Zehnder interferometer is presented. The format converter generates a correctly-coded RZ signal with a controllable duty-cycle. It has the advantages of flexible input N RZ wavelength, preserved input polarity, negative bit error rate power penalty and low switching pulse energy (15fJ).