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On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables

Al-Naffouri, Tareq Y. and Moinuddin, Muhammed and Ajeeb, Nizar and Hassibi, Babak and Moustakas, Aris L. (2016) On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables. IEEE Transactions on Communications, 64 (1). pp. 153-165. ISSN 0090-6778. https://resolver.caltech.edu/CaltechAUTHORS:20160211-110501483

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Abstract

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-mean-square (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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1109/TCOMM.2015.2496592DOIArticle
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7312953PublisherArticle
ORCID:
AuthorORCID
Al-Naffouri, Tareq Y.0000-0001-6955-4720
Additional Information:© 2015 IEEE. Manuscript received October 12, 2014; revised March 21, 2015, August 3, 2015, and October 6, 2015; accepted October 9, 2015. Date of publication October 30, 2015; date of current version January 14, 2016. The associate editor coordinating the review of this paper and approving it for publication was T. Tsiftsis. 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.
Funders:
Funding AgencyGrant Number
King Abdullah University of Science and Technology (KAUST)UNSPECIFIED
Office of Naval Research (ONR)URF/1/2221-01
Subject Keywords:Wireless communications, correlated Gaussian random vectors, multi-user diversity, weighted norms of Gaussian variables
Issue or Number:1
Record Number:CaltechAUTHORS:20160211-110501483
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160211-110501483
Official Citation:Al-Naffouri, T.Y.; Moinuddin, M.; Ajeeb, N.; Hassibi, B.; Moustakas, A.L., "On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables," in Communications, IEEE Transactions on , vol.64, no.1, pp.153-165, Jan. 2016 doi: 10.1109/TCOMM.2015.2496592
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:64420
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:19 Feb 2016 00:12
Last Modified:09 Mar 2020 13:18

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