ABSTRACT

Indefinite Quadratic Forms in Gaussian Random Variables: Distribution, Scaling, and Applications

Many applications in statistics, signal processing, and communications deal with quadratic forms in Gaussian random variables. In this talk, we study the distribution and scaling of indefinite quadratic forms in Gaussian random variables and apply that to study the scaling of broadcast channels.

In the first part of the talk, we show how to use complex integration to derive the distribution of an arbitrary indefinite quadratic form of Gaussian variables. For zero mean circularly symmetric Gaussian variables, the distribution is obtained in closed form. When the variables are real and/or nonzero mean, the distribution can be expressed as a 1-dimensional unconstrained integral. Our approach can be naturally extended to obtain the joint distribution of two or more indefinite quadratic forms. We also show how the approach can be extended to non-Gaussian variables as well.

In the second part of the talk, we use some of the results of the first part to study the effect of spatial correlation between transmit antennas on the sum-rate capacity of the MIMO broadcast channel (i.e., downlink of a cellular system). Specifically, for a system with a large number of users n, we analyze the scaling laws of the sum-rate for the dirty paper coding (DPC) and for different types of beamforming transmission schemes. When the channel is i.i.d., it has been shown that for large number of users n, the sum rate is equal to M*loglog(n) + M*log SNR where M is the number of transmit antennas. When the channel exhibits some spatial correlation with a covariance matrix R, we show that this results in an SNR hit that depends on 1) the multiuser broadcast technique and 2) on the eigenvalues of the correlation matrix R. We quantify this hit for DPC and various beamforming techniques.

As a second application of the developed quadratic forms theory, we briefly demonstrate how the developed theory allows us to perform closed form mean-square analysis of the normalized LMS adaptive algorithm.

Part of this work was done Babak Hassibi (California Institute of Technology).