Towards Deep Learning Accelerated Sparse Bayesian Frequency Estimation

Abstract

The Discrete Fourier Transform is the simplest way to obtain the spectrum of a discrete complex signal. This thesis concerns the case when the signal is known to contain a small (unknown) number of frequencies, not limited to the discrete Fourier frequencies, embedded in complex Gaussian noise. A typical signal is generated from a digital radar and the frequency components stem from point scatters, typically targets. The task is to estimate the frequencies and their respective amplitudes. This is done in a hierarchical Bayesian framework known from literature. It allows frequencies to be off the Fourier grid. The thesis contains the derivation of involved distributions, complementing the literature, being one of the results.

The resulting algorithm is a so-called hybrid-Gibbs sampler utilising a mix of conjugate priors and Markov Chain Monte Carlo. It constitutes a triple loop and is computationally heavy. The innermost loop is a Metropolis-Hastings sampler that samples a type of generalised (univariate and conditional) von Mises distribution. The main task of this thesis is to investigate the possibility of replacing it with a deep generative model. This would yield a significant acceleration. The model used in the investigation is a Continuous Conditional Generative Adversarial Network (CCGAN). Such networks can be used to sample synthetic images from highly multidimensional and complex distributions. Encouraged by such results it is easy to think that training a CCGAN to sample a univariate conditional distribution is easy. Counter-intuitively the opposite seems to be true.

The first numerical result in the thesis is the successful reproduction of a result from literature, the sampling from a two-dimensional Gaussian distribution (constant covariance) conditioned on the mean being on the unit circle. This gives confidence in the implementation. In a second step the sampling of a univariate Gaussian distribution, conditioned on both mean and variance, is investigated. Performance is not satisfactory despite the simple nature of the problem. Learning von Mises type distributions, which are more complicated and also conditioned on high-dimensional data, yield, not surprisingly, even less good results with CCGAN than the univariate Gaussian case. Suggestions for further development of the final CCGAN are given with the hope of making the CCGAN useful in practical Bayesian inference.

Popular Abstract

A digital radar generates huge amount of data every second. The data contains information about electromagnetic reflections from a varying number of targets that could correspond to aircrafts or boats, but could also correspond to reflections from ground, sea and rain. It is a challenging problem to process all the data in real time, to transform terabytes of raw data to high quality tracks indicating how the targets move. In a broad sense, this thesis examines the problem of how the process of signal data analysis can be accelerated. In practice, this would enable radars to faster detect, localise and keep track of fast moving threats in the form of hostile fighter jets or missiles, which is crucial for taking defensive actions. More generally, the methods examined in this thesis do not necessarily solely apply to radar applications, but also to sonar and other applications concerning sensing of electromagnetic signals.

Anything that can be measured over space and time can be seen as a signal. Some of the most common examples of signals are in the form of audio, image, or radar signals.
In radar applications, the radar sends out a beam of light, although of a wavelength not directly visible by humans, and receives its echoes from the environment. This is analogous to how the human eye perceives photons reflected off objects. Stronger echoes are received from targets such as aircraft or rain clouds. The composition of all these echoes which the radar receives make up the so-called target scene.

Using signal processing, information that is not directly observable from a signal can be extracted and analysed. One form of such processing is through the computation of a signal’s spectrum, namely the frequencies that constitute the signal. This is analogous to how visible white light is composed out of all the colours of the rainbow, which can be experimentally shown by the famous prism-experiment where a beam of white light passes through one side of a prism and exists as a rainbow (its spectrum, where one colour corresponds to one frequency). Using frequency estimation of a radar signal, the estimated frequency components can then be used to determine where targets are located.

The problem at hand is the following: given an observed signal, the target scene is to be obtained. In other words, given a signal, the number of targets and their locations are to be determined. As mentioned, this can be done by first extracting the frequency components of a signal. In this thesis, an observed signal is assumed to consist of a small number of frequencies that take values among a so-called frequency grid. The frequency grid is simply a set of frequencies from which a signal's frequency components can take their values. More specifically, the number of targets in the target scene is assumed to be small, where each target gives rise to one frequency component, making the signal so-called sparse. Each frequency gives information about the location of a target in the target scene. A probabilistic model that assigns how probable each of the frequencies on the frequency grid are is used. Frequencies with high probabilities can then be said to be frequency components of the observed signal. However, what if the true frequency components of the signal do not align exactly with the assumed frequency grid? This problem is alleviated by adjusting the signal model in such a way that the estimated frequencies are allowed to lie between and outside the frequencies of the frequency grid.

The algorithm for computing what frequencies a signal consist of is computationally heavy. The core of this thesis lies in the investigation of how this algorithm can be accelerated using novel deep learning methods. To become acquainted with the deep learning model used for this task, more specifically a continuous conditional generative adversarial network (CCGAN), a number of subproblems were more or less solved before attacking the main problem at hand, which was that of computing what frequencies a signal consists of. Even though the resulting CCGAN model showed poor results, knowledge in the area of CCGANs has been gained. Moreover, interesting problems to be further dealt with have been illuminated and suggestions for future work have been made. Further results of this thesis are more theoretical, consisting of mathematical derivations concerning the methods used in the thesis, essentially complementing the literature on the subject.

• author: Persson, Mika
• supervisor: Adam Andersson; Andreas Jakobsson
• organization: Mathematical Statistics
• course: FMSM01 20221
• year: 2022
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• keywords: Bayesian Statistics, Deep Learning, Frequency Estimation, Generative Adversarial Networks, Artificial Neural Networks, Statistical Modelling
• publication/series: Master's Theses in Mathematical Sciences
• report number: LUTFMS-3458-2022
• ISSN: 1404-6342
• other publication id: 2022:E68
• language: English
• id: 9102407
• date added to LUP: 2022-11-02 14:29:58
• date last changed: 2023-02-06 15:11:32