Identification of nonlinear systems by correlation using pseudorandom signals

This thesis is concerned with the measurement of the characteristics of nonlinear systems by crosscorrelation, using pseudorandom input signals based on m sequences. The systems are characterised by Volterra series, and analytical expressions relating the rth order Volterra kernel to r-dimensional crosscorrelation measurements are derived. It is shown that the two-dimensional crosscorrelation measurements are related to the corresponding second order kernel values by a set of equations which may be structured into a number of independent subsets. The m sequence properties determine how the maximum order of the subsets for off-diagonal values is related to the upper bound of the arguments for nonzero kernel values. The upper bound of the arguments is used as a performance index, and the performance of antisymmetric pseudorandom binary, ternary and quinary signals is investigated. The performance indices obtained above are small in relation to the periods of the corresponding signals. To achieve higher performance with ternary signals, a method is proposed for combining the estimates of the second order kernel values so that the effects of some of the undesirable nonzero values in the fourth order autocorrelation function of the input signal are removed. The identification of the dynamics of two-input, single-output systems with multiplicative nonlinearity is investigated. It is shown that the characteristics of such a system may be determined by crosscorrelation experiments using phase-shifted versions of a common signal as inputs. The effects of nonlinearities on the estimates of system weighting functions obtained by crosscorrelation are also investigated. Results obtained by correlation testing of an industrial process are presented, and the differences between theoretical and experimental results discussed for this case;

Analysis and optimisation of the performance of nonlinear optical communication systems

We investigate the feasibility of simultaneous suppressing of the amplification noise and nonlinearity, representing the most fundamental limiting factors in modern optical communication. To accomplish this task we developed a general design optimisation technique, based on concepts of noise and nonlinearity management. We demonstrate the immense efficiency of the novel approach by applying it to a design optimisation of transmission lines with periodic dispersion compensation using Raman and hybrid Raman-EDFA amplification. Moreover, we showed, using nonlinearity management considerations, that the optimal performance in high bit-rate dispersion managed fibre systems with hybrid amplification is achieved for a certain amplifier spacing – which is different from commonly known optimal noise performance corresponding to fully distributed amplification. Required for an accurate estimation of the bit error rate, the complete knowledge of signal statistics is crucial for modern transmission links with strong inherent nonlinearity. Therefore, we implemented the advanced multicanonical Monte Carlo (MMC) method, acknowledged for its efficiency in estimating distribution tails. We have accurately computed acknowledged for its efficiency in estimating distribution tails. We have accurately computed marginal probability density functions for soliton parameters, by numerical modelling of Fokker-Plank equation applying the MMC simulation technique. Moreover, applying a powerful MMC method we have studied the BER penalty caused by deviations from the optimal decision level in systems employing in-line 2R optical regeneration. We have demonstrated that in such systems the analytical linear approximation that makes a better fit in the central part of the regenerator nonlinear transfer function produces more accurate approximation of the BER and BER penalty. We present a statistical analysis of RZ-DPSK optical signal at direct detection receiver with Mach-Zehnder interferometer demodulation

A common contrast pooling rule for suppression within and between the eyes

Recent work has revealed multiple pathways for cross-orientation suppression in cat and human vision. In particular, ipsiocular and interocular pathways appear to assert their influence before binocular summation in human but have different (1) spatial tuning, (2) temporal dependencies, and (3) adaptation after-effects. Here we use mask components that fall outside the excitatory passband of the detecting mechanism to investigate the rules for pooling multiple mask components within these pathways. We measured psychophysical contrast masking functions for vertical 1 cycle/deg sine-wave gratings in the presence of left or right oblique (645 deg) 3 cycles/deg mask gratings with contrast C%, or a plaid made from their sum, where each component (i) had contrast 0.5Ci%. Masks and targets were presented to two eyes (binocular), one eye (monoptic), or different eyes (dichoptic). Binocular-masking functions superimposed when plotted against C, but in the monoptic and dichoptic conditions, the grating produced slightly more suppression than the plaid when Ci $ 16%. We tested contrast gain control models involving two types of contrast combination on the denominator: (1) spatial pooling of the mask after a local nonlinearity (to calculate either root mean square contrast or energy) and (2) "linear suppression" (Holmes & Meese, 2004, Journal of Vision 4, 1080–1089), involving the linear sum of the mask component contrasts. Monoptic and dichoptic masking were typically better fit by the spatial pooling models, but binocular masking was not: it demanded strict linear summation of the Michelson contrast across mask orientation. Another scheme, in which suppressive pooling followed compressive contrast responses to the mask components (e.g., oriented cortical cells), was ruled out by all of our data. We conclude that the different processes that underlie monoptic and dichoptic masking use the same type of contrast pooling within their respective suppressive fields, but the effects do not sum to predict the binocular case.

Adaptation and gain pool summation:Alternative models and masking data

Foley [J. Opt. Soc. Am. A 11 (1994) 1710] has proposed an influential psychophysical model of masking in which mask components in a contrast gain pool are raised to an exponent before summation and divisive inhibition. We tested this summation rule in experiments in which contrast detection thresholds were measured for a vertical 1 c/deg (or 2 c/deg) sine-wave component in the presence of a 3 c/deg (or 6 c/deg) mask that had either a single component oriented at -45° or a pair of components oriented at ±45°. Contrary to the predictions of Foley's model 3, we found that for masks of moderate contrast and above, threshold elevation was predicted by linear summation of the mask components in the inhibitory stage of the contrast gain pool. We built this feature into two new models, referred to as the early adaptation model and the hybrid model. In the early adaptation model, contrast adaptation controls a threshold-like nonlinearity on the output of otherwise linear pathways that provide the excitatory and inhibitory inputs to a gain control stage. The hybrid model involves nonlinear and nonadaptable routes to excitatory and inhibitory stages as well as an adaptable linear route. With only six free parameters, both models provide excellent fits to the masking and adaptation data of Foley and Chen [Vision Res. 37 (1997) 2779] but unlike Foley and Chen's model, are able to do so with only one adaptation parameter. However, only the hybrid model is able to capture the features of Foley's (1994) pedestal plus orthogonal fixed mask data. We conclude that (1) linear summation of inhibitory components is a feature of contrast masking, and (2) that the main aftereffect of spatial adaptation on contrast increment thresholds can be assigned to a single site. © 2002 Elsevier Science Ltd. All rights reserved.

Sub-critical regime of femtosecond inscription

We apply well known nonlinear diffraction theory governing focusing of a powerful light beam of arbitrary shape in medium with Kerr nonlinearity to the analysis of femtosecond (fs) laser processing of dielectric in sub-critical (input power less than the critical power of selffocusing) regime. Simple analytical expressions are derived for the input beam power and spatial focusing parameter (numerical aperture) that are required for achieving an inscription threshold. Application of non-Gaussian laser beams for better controlled fs inscription at higher powers is also discussed. © 2007 Optical Society of America.

Nonlinear diffraction in sub-critical femtosecond inscription

Material processing using high-intensity femtosecond (fs) laser pulses is a fast developing technology holding potential for direct writing of multi-dimensional optical structures in transparent media. In this work we re-examine nonlinear diffraction theory in context of fs laser processing of silica in sub-critical (input power less than the critical power of self-focusing) regime. We have applied well known theory, developed by Vlasov, Petrishev and Talanov, that gives analytical description of the evolution of a root-mean-square beam (not necessarily Gaussian) width RRMS(z) in medium with the Kerr nonlinearity.

Third order intermodulation products generated on transmission through a nonlinear radio-on-fibre link

Two-tone intermodulation tests were simulated for an amplitude modulated radio-on-fibre link including fibre dispersion, nonlinearity and loss. The third-order intercept results are presented for varying fibre lengths and optical transmission powers.

Stationary scattering from a nonlinear network

Transmission through a complex network of nonlinear one-dimensional leads is discussed by extending the stationary scattering theory on quantum graphs to the nonlinear regime. We show that the existence of cycles inside the graph leads to a large number of sharp resonances that dominate scattering. The latter resonances are then shown to be extremely sensitive to the nonlinearity and display multistability and hysteresis. This work provides a framework for the study of light propagation in complex optical networks.

Unrepeatered transmission through ultra-long fiber laser cavities

We present a study on the potential use of ultra-longlasercavities for unrepeateredfiber communication, based on the theory of nonlinearity management. A comparison is offered between the performance of ultra-longlasers and standard bi-directional distributed amplification schemes in nonrepeated transmission. Links based on both traditional (SMF/DCF) and modern Ultrawave transmissionfibers are considered.

Non-Linear oil firm force coefficients for a journal bearing operating under aligned and misaligned conditions

It is well established that hydrodynamic journal bearings are responsible for self-excited vibrations and have the effect of lowering the critical speeds of rotor systems. The forces within the oil film wedge, generated by the vibrating journal, may be represented by displacement and velocity coefficient~ thus allowing the dynamical behaviour of the rotor to be analysed both for stability purposes and for anticipating the response to unbalance. However, information describing these coefficients is sparse, misleading, and very often not applicable to industrial type bearings. Results of a combined analytical and experimental investigation into the hydrodynamic oil film coefficients operating in the laminar region are therefore presented, the analysis being applied to a 120 degree partial journal bearing having a 5.0 in diameter journal and a LID ratio of 1.0. The theoretical analysis shows that for this type of popular bearing, the eight linearized coefficients do not accurately describe the behaviour of the vibrating journal based on the theory of small perturbations, due to them being masked by the presence of nonlinearity. A method is developed using the second order terms of Taylor expansion whereby design charts are provided which predict the twentyeight force coefficients for both aligned, and for varying amounts of journal misalignment. The resulting non-linear equations of motion are solved using a modified Newton-Raphson method whereby the whirl trajectories are obtained, thus providing a physical appreciation of the bearing characteristics under dynamically loaded conditions.

An advanced real-time predictive maintenance framework for large scale machine systems

This thesis introduces and develops a novel real-time predictive maintenance system to estimate the machine system parameters using the motion current signature. Recently, motion current signature analysis has been addressed as an alternative to the use of sensors for monitoring internal faults of a motor. A maintenance system based upon the analysis of motion current signature avoids the need for the implementation and maintenance of expensive motion sensing technology. By developing nonlinear dynamical analysis for motion current signature, the research described in this thesis implements a novel real-time predictive maintenance system for current and future manufacturing machine systems. A crucial concept underpinning this project is that the motion current signature contains infor­mation relating to the machine system parameters and that this information can be extracted using nonlinear mapping techniques, such as neural networks. Towards this end, a proof of con­cept procedure is performed, which substantiates this concept. A simulation model, TuneLearn, is developed to simulate the large amount of training data required by the neural network ap­proach. Statistical validation and verification of the model is performed to ascertain confidence in the simulated motion current signature. Validation experiment concludes that, although, the simulation model generates a good macro-dynamical mapping of the motion current signature, it fails to accurately map the micro-dynamical structure due to the lack of knowledge regarding performance of higher order and nonlinear factors, such as backlash and compliance. Failure of the simulation model to determine the micro-dynamical structure suggests the pres­ence of nonlinearity in the motion current signature. This motivated us to perform surrogate data testing for nonlinearity in the motion current signature. Results confirm the presence of nonlinearity in the motion current signature, thereby, motivating the use of nonlinear tech­niques for further analysis. Outcomes of the experiment show that nonlinear noise reduction combined with the linear reverse algorithm offers precise machine system parameter estimation using the motion current signature for the implementation of the real-time predictive maintenance system. Finally, a linear reverse algorithm, BJEST, is developed and applied to the motion current signature to estimate the machine system parameters.

Investigation of photorefractive subharmonics in the absence of wavemixing

Using a new optical configuration free from the influence of photorefractive optical nonlinearity, we investigate the main characteristics of the spatial subharmonic K/2 excited in a Bi12SiO20 crystal by a light-intensity pattern with wave vector K and frequency O. It is shown that in a large region of intensity and applied electric field the optimum value O of the frequency corresponds to the conditions of parametric excitation of the weakly damped eigenmodes of the medium: the space-charge waves. The threshold and above-threshold characteristics of the subharmonic regime are in good agreement with the theory.

Generation of spatial subharmonic gratings in the absence of photorefractive beam coupling

Using a new experimental geometry, we have proved for the first time that the generation of spatial subharmonic gratings in photorefractive crystals is not dependent on optical nonlinearity. We present results which confirm that the subharmonic gratings result from a parametric excitation of ultra low-frequency eigenmodes of a crystal by a time modulated fundamental grating.

Analysis of magnetoencephaoographic data as a nonlinear dynamical system

This thesis presents the results from an investigation into the merits of analysing Magnetoencephalographic (MEG) data in the context of dynamical systems theory. MEG is the study of both the methods for the measurement of minute magnetic flux variations at the scalp, resulting from neuro-electric activity in the neocortex, as well as the techniques required to process and extract useful information from these measurements. As a result of its unique mode of action - by directly measuring neuronal activity via the resulting magnetic field fluctuations - MEG possesses a number of useful qualities which could potentially make it a powerful addition to any brain researcher's arsenal. Unfortunately, MEG research has so far failed to fulfil its early promise, being hindered in its progress by a variety of factors. Conventionally, the analysis of MEG has been dominated by the search for activity in certain spectral bands - the so-called alpha, delta, beta, etc that are commonly referred to in both academic and lay publications. Other efforts have centred upon generating optimal fits of "equivalent current dipoles" that best explain the observed field distribution. Many of these approaches carry the implicit assumption that the dynamics which result in the observed time series are linear. This is despite a variety of reasons which suggest that nonlinearity might be present in MEG recordings. By using methods that allow for nonlinear dynamics, the research described in this thesis avoids these restrictive linearity assumptions. A crucial concept underpinning this project is the belief that MEG recordings are mere observations of the evolution of the true underlying state, which is unobservable and is assumed to reflect some abstract brain cognitive state. Further, we maintain that it is unreasonable to expect these processes to be adequately described in the traditional way: as a linear sum of a large number of frequency generators. One of the main objectives of this thesis will be to prove that much more effective and powerful analysis of MEG can be achieved if one were to assume the presence of both linear and nonlinear characteristics from the outset. Our position is that the combined action of a relatively small number of these generators, coupled with external and dynamic noise sources, is more than sufficient to account for the complexity observed in the MEG recordings. Another problem that has plagued MEG researchers is the extremely low signal to noise ratios that are obtained. As the magnetic flux variations resulting from actual cortical processes can be extremely minute, the measuring devices used in MEG are, necessarily, extremely sensitive. The unfortunate side-effect of this is that even commonplace phenomena such as the earth's geomagnetic field can easily swamp signals of interest. This problem is commonly addressed by averaging over a large number of recordings. However, this has a number of notable drawbacks. In particular, it is difficult to synchronise high frequency activity which might be of interest, and often these signals will be cancelled out by the averaging process. Other problems that have been encountered are high costs and low portability of state-of-the- art multichannel machines. The result of this is that the use of MEG has, hitherto, been restricted to large institutions which are able to afford the high costs associated with the procurement and maintenance of these machines. In this project, we seek to address these issues by working almost exclusively with single channel, unaveraged MEG data. We demonstrate the applicability of a variety of methods originating from the fields of signal processing, dynamical systems, information theory and neural networks, to the analysis of MEG data. It is noteworthy that while modern signal processing tools such as independent component analysis, topographic maps and latent variable modelling have enjoyed extensive success in a variety of research areas from financial time series modelling to the analysis of sun spot activity, their use in MEG analysis has thus far been extremely limited. It is hoped that this work will help to remedy this oversight.

Numerical and statistical analysis of long-haul undersea optical communication systems

Firstly, we numerically model a practical 20 Gb/s undersea configuration employing the Return-to-Zero Differential Phase Shift Keying data format. The modelling is completed using the Split-Step Fourier Method to solve the Generalised Nonlinear Schrdinger Equation. We optimise the dispersion map and per-channel launch power of these channels and investigate how the choice of pre/post compensation can influence the performance. After obtaining these optimal configurations, we investigate the Bit Error Rate estimation of these systems and we see that estimation based on Gaussian electrical current systems is appropriate for systems of this type, indicating quasi-linear behaviour. The introduction of narrower pulses due to the deployment of quasi-linear transmission decreases the tolerance to chromatic dispersion and intra-channel nonlinearity. We used tools from Mathematical Statistics to study the behaviour of these channels in order to develop new methods to estimate Bit Error Rate. In the final section, we consider the estimation of Eye Closure Penalty, a popular measure of signal distortion. Using a numerical example and assuming the symmetry of eye closure, we see that we can simply estimate Eye Closure Penalty using Gaussian statistics. We also see that the statistics of the logical ones dominates the statistics of the logical ones dominates the statistics of signal distortion in the case of Return-to-Zero On-Off Keying configurations.