A Review of the Frequency Estimation and Tracking Problems

This report presents a concise review of some frequency estimation and frequency tracking problems. In particular, the report focusses on aspects of these problems which have been addressed by members of the Frequency Tracking and Estimation project of the Centre for Robust and Adaptive Systems. The report is divided into four parts: problem specification and discussion, associated problems, frequency estimation algorithms and frequency tracking algorithms. Part I begins with a definition of the various frequency estimation and tracking problems. Practical examples of where each problem may arise are given. A comparison is made between the frequency estimation and tracking problems. In Part II, block frequency estimation algorithms, fast block frequency estimation algorithms and notch filtering techniques for frequency estimation are dealt with. Frequency tracking algorithms are examined in Part III. Part IV of this report examines various problems associated with frequency estimation. Associated problems include Cramer-Rao lower bounds, theoretical algorithm performance, frequency resolution, use of the analytic signal and model order selection

Time-Frequency Analysis as Probabilistic Inference

This is the final published version. It was originally published by IEEE at http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6918491.This paper proposes a new view of time-frequency analysis framed in terms of probabilistic inference. Natural signals are assumed to be formed by the superposition of distinct time-frequency components, with the analytic goal being to infer these components by application of Bayes' rule. The framework serves to unify various existing models for natural time-series; it relates to both the Wiener and Kalman filters, and with suitable assumptions yields inferential interpretations of the short-time Fourier transform, spectrogram, filter bank, and wavelet representations. Value is gained by placing time-frequency analysis on the same probabilistic basis as is often employed in applications such as denoising, source separation, or recognition. Uncertainty in the time-frequency representation can be propagated correctly to application-specific stages, improving the handing of noise and missing data. Probabilistic learning allows modules to be co-adapted; thus, the time-frequency representation can be adapted to both the demands of the application and the time-varying statistics of the signal at hand. Similarly, the application module can be adapted to fine properties of the signal propagated by the initial time-frequency processing. We demonstrate these benefits by combining probabilistic time-frequency representations with non-negative matrix factorization, finding benefits in audio denoising and inpainting tasks, albeit with higher computational cost than incurred by the standard approach.Funding was provided by EPSRC (grant numbers EP/G050821/1 and EP/L000776/1) and Google (R.E.T.) and by the Gatsby Charitable Foundation (M.S.)

Measuring Instantaneous Frequency of Local Field Potential Oscillations using the Kalman Smoother

Rhythmic local field potentials (LFPs) arise from coordinated neural activity. Inference of neural function based on the properties of brain rhythms remains a challenging data analysis problem. Algorithms that characterize non-stationary rhythms with high temporal and spectral resolution may be useful for interpreting LFP activity on the timescales in which they are generated. We propose a Kalman smoother based dynamic autoregressive model for tracking the instantaneous frequency (iFreq) and frequency modulation (FM) of noisy and non-stationary sinusoids such as those found in LFP data. We verify the performance of our algorithm using simulated data with broad spectral content, and demonstrate its application using real data recorded from behavioral learning experiments. In analyses of ripple oscillations (100–250 Hz) recorded from the rodent hippocampus, our algorithm identified novel repetitive, short timescale frequency dynamics. Our results suggest that iFreq and FM may be useful measures for the quantification of small timescale LFP dynamics.National Institutes of Health (U.S.) (NIH/NIMH R01 MH59733)National Institutes of Health (U.S.) (NIH/NIHLB R01 HL084502)Massachusetts Institute of Technology (Henry E. Singleton Presidential Graduate Fellowship Award

AM-FM methods for image and video processing

This dissertation is focused on the development of robust and efficient Amplitude-Modulation Frequency-Modulation (AM-FM) demodulation methods for image and video processing (there is currently a patent pending that covers the AM-FM methods and applications described in this dissertation). The motivation for this research lies in the wide number of image and video processing applications that can significantly benefit from this research. A number of potential applications are developed in the dissertation. First, a new, robust and efficient formulation for the instantaneous frequency (IF) estimation: a variable spacing, local quadratic phase method (VS-LQP) is presented. VS-LQP produces much more accurate results than current AM-FM methods. At significant noise levels (SNR \u3c 30dB), for single component images, the VS-LQP method produces better IF estimation results than methods using a multi-scale filterbank. At low noise levels (SNR \u3e 50dB), VS-LQP performs better when used in combination with a multi-scale filterbank. In all cases, VS-LQP outperforms the Quasi-Eigen Approximation algorithm by significant amounts (up to 20dB). New least squares reconstructions using AM-FM components from the input signal (image or video) are also presented. Three different reconstruction approaches are developed: (i) using AM-FM harmonics, (ii) using AM-FM components extracted from different scales and (iii) using AM-FM harmonics with the output of a low-pass filter. The image reconstruction methods provide perceptually lossless results with image quality index values bigger than 0.7 on average. The video reconstructions produced image quality index values, frame by frame, up to more than 0.7 using AM-FM components extracted from different scales. An application of the AM-FM method to retinal image analysis is also shown. This approach uses the instantaneous frequency magnitude and the instantaneous amplitude (IA) information to provide image features. The new AM-FM approach produced ROC area of 0.984 in classifying Risk 0 versus Risk 1, 0.95 in classifying Risk 0 versus Risk 2, 0.973 in classifying Risk 0 versus Risk 3 and 0.95 in classifying Risk 0 versus all images with any sign of Diabetic Retinopathy. An extension of the 2D AM-FM demodulation methods to three dimensions is also presented. New AM-FM methods for motion estimation are developed. The new motion estimation method provides three motion estimation equations per channel filter (AM, IF motion equations and a continuity equation). Applications of the method in motion tracking, trajectory estimation and for continuous-scale video searching are demonstrated. For each application, we discuss the advantages of the AM-FM methods over current approaches

Statistical models for natural sounds

It is important to understand the rich structure of natural sounds in order to solve important tasks, like automatic speech recognition, and to understand auditory processing in the brain. This thesis takes a step in this direction by characterising the statistics of simple natural sounds. We focus on the statistics because perception often appears to depend on them, rather than on the raw waveform. For example the perception of auditory textures, like running water, wind, fire and rain, depends on summary-statistics, like the rate of falling rain droplets, rather than on the exact details of the physical source. In order to analyse the statistics of sounds accurately it is necessary to improve a number of traditional signal processing methods, including those for amplitude demodulation, time-frequency analysis, and sub-band demodulation. These estimation tasks are ill-posed and therefore it is natural to treat them as Bayesian inference problems. The new probabilistic versions of these methods have several advantages. For example, they perform more accurately on natural signals and are more robust to noise, they can also fill-in missing sections of data, and provide error-bars. Furthermore, free-parameters can be learned from the signal. Using these new algorithms we demonstrate that the energy, sparsity, modulation depth and modulation time-scale in each sub-band of a signal are critical statistics, together with the dependencies between the sub-band modulators. In order to validate this claim, a model containing co-modulated coloured noise carriers is shown to be capable of generating a range of realistic sounding auditory textures. Finally, we explored the connection between the statistics of natural sounds and perception. We demonstrate that inference in the model for auditory textures qualitatively replicates the primitive grouping rules that listeners use to understand simple acoustic scenes. This suggests that the auditory system is optimised for the statistics of natural sounds

A Kalman filter model for signal estimation in the auditory system

Using a Kalman filter that contains a forward-predictive model of a relevant system, to predict the states of that system by means of an analysis-by-synthesis implementation in order to evade significant time delays incurred by feedback mechanisms was previously applied to the coordinated movement of limbs by means of the cerebellum. In this dissertation, the same concept was applied to the auditory system in order to investigate if such a concept is a universal neurophysiological method for correctly estimating a state in a quick and reliable way. To test this assumption an auditory system model and Kalman estimator were designed, where the Kalman filter contained a stochastically equivalent forward-predictive model of the complete auditory system model. The Kalman filter was used to estimate the power found in a particular band of the frequency spectrum and its performance in the mean-squared error sense was compared to that of a simple postsynaptic current decoding filter under various types of neural channel noise. It was shown that the Kalman filter, containing a biologically plausible internal model could estimate the power better than a postsynaptic current decoding filter, proposed in the literature. When the just-noticeable difference in intensity discrimination, as reported in the literature, was compared to model-predictions, it was shown that a smaller mean-squared error results in the case of the designed auditory system model and Kalman estimator. This suggests that the application of the Kalman filter concept is important as it provides a bridge between measured data and the auditory system model. It was concluded that a Kalman filter model containing a biologically plausible internal model can explain some characteristics of the signal processing of the auditory system. The research suggests that the principle of an estimator that contains an internal model could be a universal neurophysiological method for the correct estimation of a desired state.Dissertation (MEng (Electronic Engineering))--University of Pretoria, 2006.Electrical, Electronic and Computer Engineeringunrestricte

Laser Doppler Velocimetry for Joint Measurements of Acoustic and Mean Flow Velocities : LMS-based Algorithm and CRB Calculation

This paper presents a least mean square (LMS) algorithm for the joint estimation of acoustic and mean flow velocities from laser doppler velocimetry (LDV) measurements. The usual algorithms used for measuring with LDV purely acoustic velocity or mean flow velocity may not be used when the acoustic field is disturbed by a mean flow component. The LMS-based algorithm allows accurate estimations of both acoustic and mean flow velocities. The Cram\'er-Rao bound (CRB) of the associated problem is determined. The variance of the estimators of both acoustic and mean flow velocities is also given. Simulation results of this algorithm are compared with the CRB and the comparison leads to validate this estimator