# 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 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 integral. Our approach can
be naturally extended to obtain the
joint distribution of two or more
indefinite quadratic forms.

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. We briefly
discuss precoding techniques for
reducing the hit on RBF in the presence
of correlation.

Part of
this work was done jointly with Masoud
Sharif (Boston University) and Babak
Hassibi (California Institute of
Technology).