# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### Multi-Frequency Direction of Arrival Estimation by Low-Rank Approximation

(2016) In Master's Theses in Mathematical Sciences FMA820 20161
Mathematics (Faculty of Engineering)
Abstract
We consider the problem of estimating the location of a number of wave-emitting sources, known as the direction of arrival problem. We consider both the two- and three-dimensional case. The method we use works for equally as well as unequally spaced sensors, that are located in (possibly) non-square regions. In the case of wideband signals we show how measurements from multiple frequencies can be used to improve the estimation. To make the direction of arrival estimation we formulate a low-rank optimization problem over so called Hankel matrices. The low-rank optimization problem is solved by a recent fixed-point algorithm, which is based on convex optimization. We also show how a function that is a sum of plane waves can be reconstructed... (More)
We consider the problem of estimating the location of a number of wave-emitting sources, known as the direction of arrival problem. We consider both the two- and three-dimensional case. The method we use works for equally as well as unequally spaced sensors, that are located in (possibly) non-square regions. In the case of wideband signals we show how measurements from multiple frequencies can be used to improve the estimation. To make the direction of arrival estimation we formulate a low-rank optimization problem over so called Hankel matrices. The low-rank optimization problem is solved by a recent fixed-point algorithm, which is based on convex optimization. We also show how a function that is a sum of plane waves can be reconstructed from sparse measurements, once the directions of the waves have been estimated. We test the methods by numerical simulations. The improved performance when using multiple frequencies is clearly demonstrated in one of the examples. (Less)
Popular Abstract
In this thesis we consider the direction of arrival problem, where the goal is to estimate the directions in which a number of wave-emitting sources are located. This problem is of interest in many areas such as seismology, telecommunication, radar and astrophysics. In two dimensions, the waves are measured by sensors that are placed on a line. The directions of the waves control how much the measurements at the different sensors will differ. This makes it possible to use the measurements to estimate the directions of the waves. In three dimensions the sensors have to placed in a plane, since the problem then has another degree of freedom.

A common way to make direction of arrival estimation is to study waves with a fixed frequency. In... (More)
In this thesis we consider the direction of arrival problem, where the goal is to estimate the directions in which a number of wave-emitting sources are located. This problem is of interest in many areas such as seismology, telecommunication, radar and astrophysics. In two dimensions, the waves are measured by sensors that are placed on a line. The directions of the waves control how much the measurements at the different sensors will differ. This makes it possible to use the measurements to estimate the directions of the waves. In three dimensions the sensors have to placed in a plane, since the problem then has another degree of freedom.

A common way to make direction of arrival estimation is to study waves with a fixed frequency. In this thesis we want to improve the estimation by using waves with multiple frequencies, so called wideband signals. We also want to handle the case when the sensors are unevenly spaced.

There are a number of methods to make direction of arrival estimation.
The method in this thesis uses a fairly new approach which is based on mathematical optimization. The optimization method relies on some special properties of matrices. A matrix is a rectangular table of numbers and there is much theory about their properties. The function that we want to minimize uses a property of matrices that does not behave like ordinary functions. For example it has the same value for almost all points, except the points in an infinitely much smaller set. The solution that we are looking for is in this smaller set. To find the solution, the problem is reformulated using theory about convex functions. Convex functions are functions that have many desirable properties when doing optimization, such as often having a unique minimum.

The method was tested on synthetically generated test data. The method proved to be effective in using wideband signals in order to increase the accuracy of the estimations. The method with multiple frequencies was able to get high precision estimations, in cases where the method with a single frequency would fail, or only estimate one of many directions correctly. The method also proved to be flexible in handling unevenly spaced sensors. (Less)
author
supervisor
organization
course
FMA820 20161
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Direction of arrival estimation, Hankel matrices, low rank, optimization
publication/series
Master's Theses in Mathematical Sciences
report number
LUTFMA-3294-2016
ISSN
1404-6342
other publication id
2016:E21
language
English
id
8882670
2016-08-25 15:12:30
date last changed
2016-08-25 15:12:30
```@misc{8882670,
abstract     = {We consider the problem of estimating the location of a number of wave-emitting sources, known as the direction of arrival problem. We consider both the two- and three-dimensional case. The method we use works for equally as well as unequally spaced sensors, that are located in (possibly) non-square regions. In the case of wideband signals we show how measurements from multiple frequencies can be used to improve the estimation. To make the direction of arrival estimation we formulate a low-rank optimization problem over so called Hankel matrices. The low-rank optimization problem is solved by a recent fixed-point algorithm, which is based on convex optimization. We also show how a function that is a sum of plane waves can be reconstructed from sparse measurements, once the directions of the waves have been estimated. We test the methods by numerical simulations. The improved performance when using multiple frequencies is clearly demonstrated in one of the examples.},
author       = {Jangland, Linus},
issn         = {1404-6342},
keyword      = {Direction of arrival estimation,Hankel matrices,low rank,optimization},
language     = {eng},
note         = {Student Paper},
series       = {Master's Theses in Mathematical Sciences},
title        = {Multi-Frequency Direction of Arrival Estimation by Low-Rank Approximation},
year         = {2016},
}

```