On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables

Handle URI:
http://hdl.handle.net/10754/621649
Title:
On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables
Authors:
Al-Naffouri, Tareq Y.; Moinuddin, Muhammed; Ajeeb, Nizar; Hassibi, Babak; Moustakas, Aris L.
Abstract:
© 2015 IEEE. In this work, we propose a unified approach to evaluating the CDF and PDF of indefinite quadratic forms in Gaussian random variables. Such a quantity appears in many applications in communications, signal processing, information theory, and adaptive filtering. For example, this quantity appears in the mean-square-error (MSE) analysis of the normalized least-meansquare (NLMS) adaptive algorithm, and SINR associated with each beam in beam forming applications. The trick of the proposed approach is to replace inequalities that appear in the CDF calculation with unit step functions and to use complex integral representation of the the unit step function. Complex integration allows us then to evaluate the CDF in closed form for the zero mean case and as a single dimensional integral for the non-zero mean case. Utilizing the saddle point technique allows us to closely approximate such integrals in non zero mean case. We demonstrate how our approach can be extended to other scenarios such as the joint distribution of quadratic forms and ratios of such forms, and to characterize quadratic forms in isotropic distributed random variables.We also evaluate the outage probability in multiuser beamforming using our approach to provide an application of indefinite forms in communications.
KAUST Department:
Electrical Engineering Program
Citation:
Al-Naffouri TY, Moinuddin M, Ajeeb N, Hassibi B, Moustakas AL (2016) On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables. IEEE Transactions on Communications 64: 153–165. Available: http://dx.doi.org/10.1109/TCOMM.2015.2496592.
Publisher:
Institute of Electrical and Electronics Engineers (IEEE)
Journal:
IEEE Transactions on Communications
Issue Date:
30-Oct-2015
DOI:
10.1109/TCOMM.2015.2496592
Type:
Article
ISSN:
0090-6778
Sponsors:
This publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No. URF/1/2221-01.
Appears in Collections:
Articles; Electrical Engineering Program

Full metadata record

DC FieldValue Language
dc.contributor.authorAl-Naffouri, Tareq Y.en
dc.contributor.authorMoinuddin, Muhammeden
dc.contributor.authorAjeeb, Nizaren
dc.contributor.authorHassibi, Babaken
dc.contributor.authorMoustakas, Aris L.en
dc.date.accessioned2016-11-03T13:21:51Z-
dc.date.available2016-11-03T13:21:51Z-
dc.date.issued2015-10-30en
dc.identifier.citationAl-Naffouri TY, Moinuddin M, Ajeeb N, Hassibi B, Moustakas AL (2016) On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables. IEEE Transactions on Communications 64: 153–165. Available: http://dx.doi.org/10.1109/TCOMM.2015.2496592.en
dc.identifier.issn0090-6778en
dc.identifier.doi10.1109/TCOMM.2015.2496592en
dc.identifier.urihttp://hdl.handle.net/10754/621649-
dc.description.abstract© 2015 IEEE. In this work, we propose a unified approach to evaluating the CDF and PDF of indefinite quadratic forms in Gaussian random variables. Such a quantity appears in many applications in communications, signal processing, information theory, and adaptive filtering. For example, this quantity appears in the mean-square-error (MSE) analysis of the normalized least-meansquare (NLMS) adaptive algorithm, and SINR associated with each beam in beam forming applications. The trick of the proposed approach is to replace inequalities that appear in the CDF calculation with unit step functions and to use complex integral representation of the the unit step function. Complex integration allows us then to evaluate the CDF in closed form for the zero mean case and as a single dimensional integral for the non-zero mean case. Utilizing the saddle point technique allows us to closely approximate such integrals in non zero mean case. We demonstrate how our approach can be extended to other scenarios such as the joint distribution of quadratic forms and ratios of such forms, and to characterize quadratic forms in isotropic distributed random variables.We also evaluate the outage probability in multiuser beamforming using our approach to provide an application of indefinite forms in communications.en
dc.description.sponsorshipThis publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No. URF/1/2221-01.en
dc.publisherInstitute of Electrical and Electronics Engineers (IEEE)en
dc.subjectCorrelated Gaussian random vectorsen
dc.subjectMulti-user diversityen
dc.subjectWeighted norms of Gaussian variablesen
dc.subjectWireless communicationsen
dc.titleOn the Distribution of Indefinite Quadratic Forms in Gaussian Random Variablesen
dc.typeArticleen
dc.contributor.departmentElectrical Engineering Programen
dc.identifier.journalIEEE Transactions on Communicationsen
dc.contributor.institutionDepartment of Electrical Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabiaen
dc.contributor.institutionDepartment of Electrical and Computer Engineering, King Abdulaziz University, Jeddah, Saudi Arabiaen
dc.contributor.institutionCenter of Excellence in Intelligent Engineering Systems (CEIES), King Abdulaziz University, Jeddah, Saudi Arabiaen
dc.contributor.institutionDepartment of Electrical and Computer Engineering, American University of Beirut, Beirut, Lebanonen
dc.contributor.institutionDepartment of Electrical Engineering, California Institute of Technology, Pasadena, CA, United Statesen
dc.contributor.institutionDepartment of Physics, National Kapodistrian University of Athens, Panepistimiopolis, Athens, Greeceen
kaust.authorAl-Naffouri, Tareq Y.en
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