@article{3035985,
    title = "On the distribution of indefinite quadratic forms in Gaussian random variables",
    author = "Al-Naffouri, T.Y. and Moinuddin, M. and Ajeeb, N. and Hassibi, B. and Moustakas, A.L.",
    journal = "IEEE Transactions on Communications",
    year = "2016",
    volume = "64",
    number = "1",
    pages = "153-165",
    publisher = "Institute of Electrical and Electronics Engineers, Inc. (IEEE)",
    issn = "0090-6778, 1558-0857",
    doi = "10.1109/TCOMM.2015.2496592",
    keywords = "Adaptive algorithms;  Adaptive filters;  Beamforming;  Function evaluation;  Gaussian distribution;  Information filtering;  Information theory;  Mean square error;  Number theory;  Probability;  Probability density function;  Probability distributions;  Random variables;  Signal interference;  Signal processing;  Spurious signal noise;  Wireless telecommunication systems, Applications in communications;  Complex integrations;  Gaussian random variable;  Gaussian random vectors;  Gaussian variables;  Multiuser beamforming;  Multiuser diversity;  Wireless communications, Gaussian beams",
    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-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. © 2015 IEEE."
}