# 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).