Change search

The Linear Model under Gaussian Mixture Inputs: Selected Problems in Communications
Norwegian University of Science and Technology, Faculty of Information Technology, Mathematics and Electrical Engineering, Department of Electronics and Telecommunications.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

Consider a linear model, y = Hx + n, where y is an observed vector, H is a known matrix, x is the unknown vector of interest, and n is noise. Such a linear model can describe, or approximate, a multitude of systems.

In this thesis, it is assumed that x and n are distributed as independent Gaussian mixtures (GM). Besides their ability to approximate other distributions, Gaussian mixtures can account for asymmetry, heavy tails and/or multi modality. They can in principle model any random variable, and therefore Gaussian mixtures provide great realism.

Assuming mixed Gaussian inputs, we study problems related to the minimum mean square error (MMSE) when estimating x from the observation y. Characterizing or manipulating the MMSE is non-trivial, mainly for the following reason. In the special case when both x and n are purely Gaussian inputs, then both the MMSE estimator and the MMSE have analytical, closed form expressions. In the more general case, however, when one or both of the inputs are multi-component Gaussian mixtures, then the MMSE estimator remains analytical, but the MMSE does not. The implication is that the optimal estimator can be implemented, but its performance cannot be exactly characterized.

One consequence, is that implementing MMSE reducing measures becomes a quite difficult task. For example, again in the purely Gaussian setting, an MMSE reducing linear precoder can be derived as the solution of a convex program. When inputs are Gaussian mixtures, however, this task is much more difficult. Then the problem not only turns non-convex, but the objective function (the MMSE) takes the form of a non-analytical integral.

Among the contributions of the thesis, two important ones can be summarized as follows: (i) We bound the MMSE, both from above and below, by analytical expressions. We also show that these bounds approach each other with increasing signal to noise ratio (SNR). Therefore, from moderate to high SNR, the MMSE can be bracketed rather accurately. (ii)We describe a procedure for designing the matrixH, so as to minimize MMSE. This design problem is motivated by two applications in signal processing. One concerns the design of error-reducing precoders; the other deals with selection of pilot signals for channel estimation.

NTNU, 2013.
##### Series
Doctoral theses at NTNU, ISSN 1503-8181 ; 2013:157
##### National Category
Telecommunication
##### Identifiers
ISBN: 978-82-471-4421-3 (printed ver.)ISBN: 978-82-471-4422-0 (electronic ver.)OAI: oai:DiVA.org:ntnu-21494DiVA: diva2:638099
##### Public defence
2013-06-13, 13:15
Available from: 2013-07-26 Created: 2013-07-25 Last updated: 2013-07-26Bibliographically approved
##### List of papers
1. GAUSSIAN MIXTURE MODELING FOR SOURCE LOCALIZATION
Open this publication in new window or tab >>GAUSSIAN MIXTURE MODELING FOR SOURCE LOCALIZATION
2011 (English)In: 2011 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, IEEE conference proceedings, 2011, 2604-2607 p.Conference paper (Refereed)
##### Abstract [en]

Exploiting prior knowledge, we use Bayesian estimation to localize a source heard by a fixed sensor network. The method has two main aspects: Firstly, the probability density function (PDF) of a function of the source location is approximated by a Gaussian mixture model (GMM). This approximation can theoretically be made arbitrarily accurate, and allows a closed form minimum mean square error (MMSE) estimator for that function. Secondly, the source location is retrieved by minimizing the Euclidean distance between the function and its MMSE estimate using a gradient method. Our method avoids the issues of a numerical MMSE estimator but shows comparable accuracy.

##### Place, publisher, year, edition, pages
IEEE conference proceedings, 2011
##### Series
, International Conference on Acoustics Speech and Signal Processing ICASSP, ISSN 1520-6149
##### Keyword
Localization, Sensor Networks, Closed form MMSE estimators, Gaussian mixture models
##### Identifiers
urn:nbn:no:ntnu:diva-21490 (URN)10.1109/ICASSP.2011.5947018 (DOI)000296062403005 ()978-1-4577-0539-7 (ISBN)
##### Conference
IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), MAY 22-27, 2011, Prague, CZECH REPUBLIC
##### Note

(c) 2013 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.

Available from: 2013-07-25 Created: 2013-07-25 Last updated: 2013-07-26Bibliographically approved
2. On MMSE Estimation: A Linear Model Under Gaussian Mixture Statistics
Open this publication in new window or tab >>On MMSE Estimation: A Linear Model Under Gaussian Mixture Statistics
2012 (English)In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 60, no 7, 3840-3845 p.Article in journal (Refereed) Published
##### Abstract [en]

In a Bayesian linear model, suppose observation y = Hx + n stems from independent inputs x and n which are Gaussian mixture (GM) distributed. With known matrix H, the minimum mean square error (MMSE) estimator for x, has analytical form. However, its performance measure, the MMSE itself, has no such closed form. Because existing Bayesian MMSE bounds prove to have limited practical value under these settings, we instead seek analytical bounds for the MMSE, both upper and lower. This paper provides such bounds, and relates them to the signal-to-noise-ratio (SNR).

##### Place, publisher, year, edition, pages
IEEE Signal Processing Society, 2012
##### Keyword
Gaussian mixture distribution, linear model, minimum mean square error estimation
##### Identifiers
urn:nbn:no:ntnu:diva-21491 (URN)10.1109/TSP.2012.2192112 (DOI)000305578800041 ()
##### Note

(c) 2013 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.

Available from: 2013-07-25 Created: 2013-07-25 Last updated: 2013-07-26Bibliographically approved
3. The Linear Model under Mixed Gaussian Inputs: Designing the Transfer Matrix
Open this publication in new window or tab >>The Linear Model under Mixed Gaussian Inputs: Designing the Transfer Matrix
##### Identifiers
urn:nbn:no:ntnu:diva-21492 (URN)
##### Note

This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible

Available from: 2013-07-25 Created: 2013-07-25 Last updated: 2013-07-26Bibliographically approved
4. Pilot Design for MIMO channel estimation: An Alternative to the Kronecker Structure Assumption
Open this publication in new window or tab >>Pilot Design for MIMO channel estimation: An Alternative to the Kronecker Structure Assumption
2013 (English)In: Accepted for IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) 2013, 2013Conference paper (Other academic)
##### Identifiers
urn:nbn:no:ntnu:diva-21493 (URN)
##### Conference
IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)2013
##### Note

(c) 2013 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.

Available from: 2013-07-25 Created: 2013-07-25 Last updated: 2013-07-26Bibliographically approved

#### Open Access in DiVA

##### File information
File name FULLTEXT01.pdfFile size 1404 kBChecksum SHA-512
Type fulltextMimetype application/pdf
##### By organisation
Department of Electronics and Telecommunications
##### On the subject
Telecommunication