TY - JOUR

T1 - On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables

AU - Al-Naffouri, Tareq Y.

AU - Moinuddin, Muhammed

AU - Ajeeb, Nizar

AU - Hassibi, Babak

AU - Moustakas, Aris L.

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): URF/1/2221-01
Acknowledgements: 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.

PY - 2015/10/30

Y1 - 2015/10/30

N2 - © 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.

AB - © 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.

UR - http://hdl.handle.net/10754/621649

UR - http://ieeexplore.ieee.org/document/7312953/

UR - http://www.scopus.com/inward/record.url?scp=84958172625&partnerID=8YFLogxK

U2 - 10.1109/TCOMM.2015.2496592

DO - 10.1109/TCOMM.2015.2496592

M3 - Article

VL - 64

SP - 153

EP - 165

JO - IEEE Transactions on Communications

JF - IEEE Transactions on Communications

SN - 0090-6778

IS - 1

ER -