On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables

Software Description: 

This software evaluate 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 this software is to replace inequalities that appear in the CDF calculation with unit step functions and to use complex integral representation of 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.

The software be used for the joint distribution of quadratic forms and ratios of such forms, and to characterize quadratic forms in isotropic distributed random variables.

The code also evaluates the outage probability in multiuser beamforming using our  approach to provide an application of indefinite forms in communications.

Please send commnets to Muhammed Moinuddin

Reference: 

T. Y. Al-Naffouri and M. Moinuddin and N. Ajeeb and B. Hassibi and A. L. Moustakas, "On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables", IEEE Transactions on Communications.   vol. 64 , pp. 153-165, Jan 2016

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