Applied Mathematics
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Applied Mathematics at the University of Adelaide is one of the leading groups in teaching and research in Australia. Members conduct internationally recognized research into a wide range of Applied Mathematics. The research within Applied Mathematics can be loosely grouped into the following areas.
- Biomedical Engineering
- Fluid Dynamics
- Optimisation
- Stochastic Modelling
- Teletraffic Research
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Browsing Applied Mathematics by Author "Abbott, D."
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Item Metadata only A characterization of suprathreshold stochastic resonance in an array of comparators by correlation coefficient(World Scientific Publishing Co. Pty. Ltd., 2002) McDonnell, M.; Abbott, D.; Pearce, C.Suprathreshold Stochastic Resonance (SSR), as described recently by Stocks, is a new form of Stochastic Resonance (SR) which occurs in arrays of nonlinear elements subject to aperiodic input signals and noise. These array elements can be threshold devices or FitzHugh-Nagumo neuron models for example. The distinguishing feature of SSR is that the output measure of interest is not maximized simply for nonzero values of input noise, but is maximized for nonzero values of the input noise to signal intensity ratio, and the effect occurs for signals of arbitrary magnitude and not just subthreshold signals. The original papers described SSR in terms of information theory. Previous work on SR has used correlation based measures to quantify SR for aperiodic input signals. Here, we argue the validity of correlation based measures and derive exact expressions for the cross-correlation coefficient in the same system as the original work, and show that the SSR effect also occurs in this alternative measure. If the output signal is thought of as a digital estimate of the input signal, then the output noise can be considered simply as quantization noise. We therefore derive an expression for the output signal to quantization noise ratio, and show that SSR also occurs in this measure.Item Metadata only An analysis of noise enhanced information transmission in an array of comparators(Elsevier Sci Ltd, 2002) McDonnell, M.; Abbott, D.; Pearce, C.; School of Electrical and Electronic EngineeringAn array of N comparators subject to the same input signal and independent additive noise, with the outputs from each comparator summed, is a useful noise model for a range of systems including flash analog-to-digital converters, Digital Multibeam Steering sonar arrays and parallel neurons. It has previously been shown that for certain threshold configurations the transmitted information through such an array is maximised for non-zero noise. This behaviour has been termed Suprathreshold Stochastic Resonance (SSR) [1] and in this paper we show that SSR occurs for a number of different signal and noise distributions. Also presented is an analysis of the variance of the quantisation error incurred when all thresholds are set equal to the signal mean, for Gaussian and uniform distributions. It is shown that the minimum error variance is given by a non-zero value of noise.Item Metadata only An overview of wavelets for image processing for wireless applications(SPIE, 2002) Osborne, D.; Rogers, D.; Mazumdar, J.; Coutts, R.; Abbott, D.; International Symposium on Smart Materials, Nano- and Micro-Smart Systems (2002 : Melbourne, Australia); Harvey, E.C.; Abbott, D.; Varadan, V.K.The implementation of wavelets in differing areas of signal processing has been a popular research area over the last decade. However, utilising this technology in compressing two dimensional signals, such as digital images is relatively new. Wavelet compression has many distinct advantages over earlier compression methods, the most important of which is suitability to error protection as well as the ability to precisely truncate the compressed bitstream to achieve a desired bit rate for transmission. In this paper some of the recently emerging technologies pertaining to wavelet coding of images will be reviewed, particularly with the use of wireless channels. These developments include techniques to filter images that have been degraded through the addition of noise as well as reconstructing parts of images that have been lost as a result of the fading that characterises wireless mobile environments.Item Metadata only Cross-spectral measurement of neural signal transfer(SPIE, 2004) McDonnell, M.; Sethuraman, S.; Kish, L.; Abbott, D.; Fluctuations and Noise (2004 : Gran Canaria Island, Spain); Kish, L.B.The phenomenon of noise enhanced signal transfer, or stochasticresonance, has been observed in many nonlinear systems such asneurons and ion channels. Initial studies of stochastic resonancefocused on systems driven by a periodic signal, and hence used asignal to noise ratio based measure for comparison between theinput and output of the system. It has been pointed out that forthe more general case of aperiodic signals other measures arerequired, such as cross-correlation or information theoreticaltools. In this paper we present simulation results obtained in amodel neural system driven by a broadband aperiodic signal, andproducing a signal imitating neural spikes. The system is analyzedby using cross-spectral measures.Item Metadata only Maximising information transfer through nonlinear noisy devices(SPIE, 2002) McDonnell, M.; Stocks, N.; Pearce, C.; Abbott, D.; International Symposium on Smart Materials, Nano- and Micro-Smart Systems (2002 : Melbourne, Australia); Nicolau, D.V.; Lee, A.P.Consider an array of parallel comparators (threshold devices) receiving the same input signal, but subject to independent noise, where the output from each device is summed to give an overall output. Such an array is a good model of a number of nonlinear systems including flash analogue to digital converters, sonar arrays and parallel neurons. Recently, this system was analysed by Stocks in terms of information theory, who showed that under certain conditions the transmitted information through the array is maximised for non-zero noise. This phenomenon was termed Suprathreshold Stochastic Resonance (SSR). In this paper we give further results related to the maximisation of the transmitted information in this system.Item Metadata only Neural mechanisms for analog to digital conversion(SPIE, 2004) McDonnell, M.; Abbott, D.; Pearce, C.; BioMEMS and Nanotechnology (1st : 2003 : Perth, Australia); Faraone, L.; Varadan, V.K.Consider an array of threshold devices, such as neurons orcomparators, where each device receives the same input signal, butis subject to independent additive noise. When the output fromeach device is summed to give an overall output, the system actsas a noisy Analog to Digital Converter (ADC). Recently, such asystem was analyzed in terms of information theory, and it wasshown that under certain conditions the transmitted informationthrough the array is maximized for non-zero noise. Such aphenomenon where noise can be of benefit in a nonlinear system istermed Stochastic Resonance (SR). The effect in the array ofthreshold devices was termed Suprathreshold Stochastic Resonance(SSR) to distinguish it from conventional forms of SR, in whichusually a signal needs to be subthreshold for the effect to occur.In this paper we investigate the efficiency of the analog todigital conversion when the system acts like an array of simple neurons, by analyzing the average distortion incurred and comparing this distortion to that of a conventional flash ADC.Item Metadata only Optimal quantization and suprathreshold stochastic resonance(SPIE, 2005) McDonnell, M.; Stocks, N.; Pearce, C.; Abbott, D.; Fluctuations and noise in biological, biophysical, and biomedical systems (24 May 2005 - 26 May 2005 : Austin, Texas, USA); Stocks, N.G.; Abbott, D.; Morse, R.P.It is shown that Suprathreshold Stochastic Resonance (SSR) iseffectively a way of using noise to perform quantization or lossysignal compression with a population of identical threshold-baseddevices. Quantization of an analog signal is a fundamentalrequirement for its efficient storage or compression in a digitalsystem. This process will always result in a loss of quality,known as distortion, in a reproduction of the original signal. Thedistortion can be decreased by increasing the number of statesavailable for encoding the signal (measured by the rate, or mutualinformation). Hence, designing a quantizer requires a tradeoffbetween distortion and rate. Quantization theory has recently beenapplied to the analysis of neural coding and here we examine thepossibility that SSR is a possible mechanism used by populationsof sensory neurons to quantize signals. In particular, we analyzethe rate-distortion performance of SSR for a range of input SNR'sand show that both the optimal distortion and optimal rate occursfor an input SNR of about 0 dB, which is a biologically plausiblesituation. Furthermore, we relax the constraint that allthresholds are identical, and find the optimal threshold valuesfor a range of input SNRs. We find that for sufficiently smallinput SNRs, the optimal quantizer is one in which all thresholdsare identical, that is, the SSR situation is optimal in this case.Item Metadata only Optimal quantization for energy-efficient information transfer in a population of neuron-like devices(SPIE, 2004) McDonnell, M.; Stocks, N.; Pearce, C.; Abbott, D.; Fluctuations and Noise (2004 : Gran Canaria Island, Spain); Kish, L.Suprathreshold Stochastic Resonance (SSR) is a recently discoveredform of stochastic resonance that occurs in populations of neuron-like devices. A key feature of SSR is that all devices in the population possess identical threshold nonlinearities. It haspreviously been shown that information transmission through such asystem is optimized by nonzero internal noise. It is also clearthat it is desirable for the brain to transfer information in anenergy efficient manner. In this paper we discuss the energy efficient maximization of information transmission for the case ofvariable thresholds and constraints imposed on the energy available to the system, as well as minimization of energy for the case of a fixed information rate. We aim to demonstrate that under certain conditions, the SSR configuration of all devices having identical thresholds is optimal. The novel feature of this work is that optimization is performed by finding the optimal threshold settings for the population of devices, which is equivalent to solving a noisy optimal quantization problem.Item Metadata only Stochastic resonance and data processing inequality(IEE-Inst Elec Eng, 2003) McDonnell, M.; Stocks, N.; Pearce, C.; Abbott, D.The data processing inequality of information theory proves that no more information can be obtained out of a set of data than was there to begin with. However, many papers in the field of stochastic resonance report signal to noise ratio gains in some nonlinear systems due to the addition of noise. Such an observation appears on the surface to contradict the data processing inequality. It is demonstrated that the data processing inequality is upheld for the case of a periodic input signalsItem Metadata only The data processing inequality and stochastic resonance(SPIE, 2003) McDonnell, M.; Stocks, N.; Pearce, C.; Abbott, D.; Noise in Complex Systems and Stochastic Dynamics (2003 : Santa Fe, New Mexico, USA); Jaenisch, H.M.; Handley, J.W.The data processing inequality of information theory states that given random variables X, Y and Z which form a Markov chain in the order X-->Y-->Z, then the mutual information between X and Y is greater than or equal to the mutual information between X and Z. That is I(X) >= I(X;Z) . In practice, this means that no more information can be obtained out of a set of data then was there to begin with, or in other words, there is a bound on how much can be accomplished with signal processing. However, in the field of stochastic resonance, it has been reported that a signal to noise ratio gain can occur in some nonlinear systems due to the addition of noise. Such an observation appears to contradict the data processing inequality. In this paper, we investigate this question by using an example model system.