Current Research Topics/Projects
The
line of
research
pursued
by my
research
group
focuses
on
developing
various
alternative
formulations
(preferably
convex)
for
some of
the
fundamental
system
design
and
real
time
processing
problems
in the
area of
communication
and
signal
processing,
and
designing
efficient
optimization
algorithms
to
solve
or
approximate
them.
The
research
outlined
here is
expected
to
advance
significantly
the
state
of the
art of
digital
signal
processing
and
data
communication,
both of
which
are
experiencing
tremendous
growth
worldwide.
The
major
problems
under
investigation
by my
research
group
are:
1.
The
application
of
Interior
Point
Least
Square
(IPLS)
algorithm
to
adaptive
filtering
A
novel
approach
to
adaptive
filtering
based
on
interior
point
optimization
techniques
has
been
proposed
recently
by my
research
group.
The
algorithm's
asymptotic
convergence
was
shown
and its
fast
transient
performance
was
demonstrated
(both
experimentally
and
theoretically)
in the
context
of a
system
identification
application.
It
turns
out
that
for
both
RLS
(Recursive
Least
Square)
and
IPLS,
the
transient
behavior
is
dominated
by the
mis-adjustment
due to
initialization
of the
regularization
term.
While
RLS
eliminates
the mis-adjustment
at rate
O(1/n),
with n
being
the
number
of
samples,
IPLS
does so
at an
exponential
rate.
To
demonstrate
the
full
potential
offered
by IPLS,
it will
be
necessary
to
further
reduce
the
per-sample
computational
complexity
so that
it is
comparable
to RLS
(currently
IPLS
has a
slightly
higher
theoretical
per-sample
complexity
than
RLS).
Also,
applications
of IPLS
to
equalization
(particularly
when
short
training
sequences
are
used),
to
adaptive
space-time
processing
for
multi-antenna
systems
and to
multi-target
tracking
are
promising
avenues
of
further
research.
2.
The
application
of
Semidefinite
Programming
(SDP)
to
robust
filtering
and
blind
equalization
My
research
group
proposes
to
apply
robust
optimization
techniques
to the
blind
beamforming
and
equalization
problems
in
digital
communication.
In
practical
communication
environment,
the
exact
knowledge
of
channel
state
information
and
signal
model
is
difficult,
if not
impossible,
to
obtain.
The
success
of
these
robust
methods
can
help
reduce
the
dependence
of the
standard
beamforming
and
equalization
algorithms
on the
knowledge
of the
channel
and
signal
model.
For
example,
in
designing
a CDMA
receiver,
one
often
only
knows
the
undistorted
signature
waveform
of the
desired
user.
In the
situation
where
unknown
multi-access
interference
is
present
and the
channel
experiences
multi-path
fading,
simply
using
the
desired
user’s
signature
waveform
to
linearly
de-correlate
the
received
signal
offers
a poor
performance.
In
fact,
even
using
the
minimum
output
energy
criterion
of
Verdu
to
design
the
linear
receiver
is
inadequate
because
of the
channel
distortion.
Fortunately
robust
optimization
techniques
can be
used to
design
a
robust
CDMA
receiver
that
minimizes
the
worst-case
performance
degradation.
Other
problems
currently
under
investigation
include
the use
of
Semidefinite
Programming
(SDP)
to
design
recursive
filters
that
are
robust
against
modeling
errors,
and
convex
reformulation
of the
CMA
(Constant
Modulus)
blind
equalizer.
The
latter
is a
powerful
and
popular
blind
equalization
approach
but
suffers
from
local
minimums
and
non-convergence.
A
convex
reformulation
of CMA
condition
can
mitigate
this
major
deficiency.
3.
The
application
of SDP
to
transceiver
design
for a
multi-user
communication
system
Multicarrier
modulation
includes
transmission
schemes
whose
main
feature
is the
decomposition
of any
wide-band
channel
into a
set of
independent
narrowband
parallel
channels.
Within
this
family,
the
most
commonly
used
schemes
are
Discrete
MultiTone
(DMT)
and
Orthogonal
Frequency
Division
Multiplexing
(OFDM).
These
schemes
employ
the
Fast
Discrete
Fourier
Transform
(DFT)
at the
transmitter
and the
receiver,
resulting
in a
low
cost,
high-performance
implementation.
Many of
the
xDSL
over
wired
media
use DMT
and
Digital
Audio
Broadcast
(DAB)
or
Digital
Video
Broadcast
(DVB)
for
wireless
communications
use
OFDM.
To
achieve
the
capacity
of a
spectrally
shaped
Gaussian
channel
in a
single
user
case,
it is
well
known
one
must
use
appropriate
bit and
power
allocation
among
the
subcarriers
for the
single
user in
a way
that
corresponds
to the
classic
water-filling
distribution.
However,
in the
multi-user
case,
achieving
multiuser
capacity
requires
more
sophisticated
resource
allocation.
Simply
using
TDMA or
FDMA
non-overlapping
resource
allocation
schemes
in an
arbitrary
way
will
result
in
multiuser
rate
far
below
capacity.
The
goal of
this
research
is to
devise
optimal
resource
allocation
schemes
in the
MMSE
(Minimum
Mean
Squared
Error)
sense
for
multi-user
communication
systems.
4.
Efficient
optimal/sub-optimal
solution
methods
for
space-time
processing
in
wireless
communication
Recently
the use
of
multiple
antennas
at both
the
transmitter
and the
receiver
has
been
shown
to
provide
several-fold
increase
in
capacity.
In a
rich
scattering
environment
such as
indoor
wireless
scenario,
the
channel
gains
between
any
pair of
transmitter
antenna
and
receiver
antenna
can be
assumed
independent.
In this
case,
the
capacity
increase
is
shown
to be
proportional
to the
number
of
transmit
antennas.
To reap
the
benefits
brought
by the
MIMO
(Multi-Input-Multi-Output)
channel,
one
must
develop
efficient
decoding
algorithm
used in
such a
system,
especially
for
cases
where
channel
matrix
is ill
conditioned.
My
research
group
will
explore
robust
optimal
designs
for the
space-time
linear
precoders
and
decoders
for a
MIMO
communication
system.
The
existing
V-Blast
type of
receiver
algorithm
developed
in Bell
Labs
suffers
from
significant
performance
loss
when
the
channel
matrix
is ill
conditioned.
My
group
proposes
to
investigate
new
transceiver/receiver
structures
and
algorithms
that
are
less
sensitive
to the
ill
conditioning
and the
estimation
errors
of the
channel
matrix
and yet
still
computationally
efficient.
The
state
of the
art
robust
optimization
techniques
(such
robust
linear
programming)
will be
used.
My
research
group
also
intends
to
explore
various
robust
space-time
transmitter/receiver
schemes
using
the
minimum
symbol
mean
square
error
and the
(approximate)
maximization
of the
minimum
distance
between
symbol
hypotheses.
Furthermore,
the
scalability
issue
with
respect
to the
number
of
antennas,
size of
the
coding
block
and
transmit
average/peak
power
will be
investigated.
The
goal is
to turn
these
robust
design
formulations
into
convex
optimization
problems
and
devise
highly
efficient
interior
point
algorithms
to
solve
them.
Some of
the
recent
successes
along
this
line
include
the
work
from
Cioffi's
group
at
Stanford
University.
The
design
criteria
to be
used
include
the
(weighted)
Minimum
Mean
Square
Error (MMSE)
as well
as Bit
Error
Rate (BER).
This
work
has
received
initial
funding
from
CITO
(Communications
Information
Technology
Ontario,
the
provincial
research
center
of
excellence).
5.
Data
fusion
via
convex
optimization
The
problem
of
data/information
fusion
arises
in many
applications
including,
for
example,
the
multi-sensor
target
tracking
whereby
sensors
generate
reports
on the
target
identities
that
are
subsequently
combined
in a
fusion
center.
Since
the
sensor
reports
are
typically
fuzzy,
``incomplete''
and
inconsistent,
the
fusion
of such
sensor
reports
becomes
a major
challenge.
My
group
is
currently
developing
new
identity
information
fusion
methods
based
on
several
criteria:
the
minimization
of
inconsistencies
between
the
sensor
reports
by
using
convex
Quadratic
Programming
(QP)
and
linear
programming
(LP),
the
minimization
of the
inconsistency
of
reports
augmented
with a
logarithmic
barrier
for the
nonnegativity
constraint,
and the
minimization
of a
regularized
Kullback-Leibler
distance.
In
contrast
to the
Dempster-Shafer's
evidential
reasoning
approach
that
suffers
from
exponentially
growing
complexity,
these
approaches
are
highly
efficient
(polynomial
time
solvable).
Moreover,
these
new
approaches
are
capable
of
fusing
``Ratio
type''
sensor
reports.
Thus
they
are
more
general
than
the
evidential
reasoning
theory.
When
the
sensor
reports
are
consistent,
the
solution
generated
by the
new
fusion
methods
can be
shown
to
converge
to the
true
probability
distribution.
Various
theoretical
properties
of
these
new
fusion
approaches
as well
as the
classical
Baysian
and
Dempster-Shafer
approaches
are
being
analyzed
and
compared.
This
work
has
been
funded
by CITO
and by
DREV
(Defense
Research
Establishment
at
Valcartier,
a
research
arm of
the
Department
of
Defense
of
Canada).
6.
Efficient
optimization
methods
for the
intelligent
provisioning
and
management
of
optical
networks
In
an
optical
network,
one
needs
to
accurately
assess
resource
availability
before
accepting
a new
service
order.
By
including
wavelengths
in the
resource
pool,
the
provider
can
make
transport
an
essential
part of
each
service.
An
optical
network
provisioning
system
can
check,
for
example,
the
availability
of
unused
wavelengths
and IP
address
blocks
before
turning
on a
broadband
Internet
service.
With
this
capability,
innovative
network
operators
can use
advanced
asset
management
to
create
a
variety
of new,
revenue-enhancing
services
and to
identify
and
resolve
problems
before
they
impede
service
delivery.
My
group
plans
to
develop
a
mathematical
model
for the
resource
allocation
problem
and
propose
efficient
algorithms
to
perform
real-time
asset
management
of
optical
networks.
A main
goal of
the
proposed
automated
real-time
asset
management
system
is to
help
service
providers
get the
most
out of
their
transport
networks.
Even
though
DWDM
(Dense
Wave
Division
Multiplexing)
adds
tremendous
capacity
to the
wide-area
infrastructure,
networks
will
still
be
congested.
By
maintaining
up-to-the-millisecond
records
of
bandwidth
utilization
and
dynamically
reconfiguring
the
networks
when
necessary,
real-time
asset
management
will
ensure
that
each
wavelength
is used
as
efficiently
as
possible.
My
group
will
examine
various
issues
relating
to
traffic
management.
For
example,
it
plans
to
investigate
efficient
ways of
dynamically
reconfiguring
a
network
in
order
to
reduce
network
congestion.
As a
start,
some
simple
network
structures
(such
as a
ring
network)
will be
considered
first.
Various
combinatorial
optimization
and
integer
programming
techniques
(such
as
linear
programming
relaxation,
Lagrangian
relaxation)
will be
explored
in
solving
some of
the
NP-hard
formulations
of the
optical
network
management
problems.
Software
prototypes
will
also be
developed
in
collaboration
with
Nortel
Networks.
There
are a
number
of
major
theoretical
issues
arising
from
the
investigation
of
above
problems,
three
of
which
are
listed
below.
·
Stochastic
convergence
and
rate of
convergence
of IPLS
(Interior
Point
Least
Square)
or its
variants
in a
time-varying
environment
The
objective
is to
provide
a
rigorous
and
comprehensive
study
of
advanced
interior
point
methods
in the
context
of real
time
signal
processing
and
communication
applications
where
data
are
collected
dynamically
and are
corrupted
by
noise.
Previously
interior
point
methods
have
only
been
studied
in the
noise-free
deterministic
setting.
To
demonstrate
the
effectiveness
of IPLS
algorithm
in a
time-varying
environment,
it is
proposed
that
the
tracking
behavior
of IPLS
be
theoretically
analyzed
for
some
specific
noise
and
system
model.
For
example,
one can
assume
the
noise
is
Gaussian
and the
system
undergoes
periodic
changes
with a
given
frequency.
For
such
environment,
it
should
be
possible
to
derive
the
tracking
error
estimates
for RLS
(Recursive
Least
Square)
and
IPLS in
terms
of the
sample
size
and the
rate at
which
the
system
varies.
It is
expected
that
IPLS
will
generate
a
smaller
tracking
error
than
RLS in
such
time-varying
environment.
·
Convex
reformulation
of
transceiver
design
problem
Substantial
progress
has
been
made by
my
research
group
in
solving
the
MMSE
transceiver
design
problem
for a
multi-access
network.
Indeed,
even
though
the
straightforward
formulation
of the
MMSE
(Minimum
Mean
Square
Error)
joint
transmitter-receiver
design
problem
for a
multi-access
communication
system
is
nonconvex,
various
alternative
(and
equivalent)
formulations
can be
constructed
using
techniques
of
Linear
Matrix
Inequalities
(LMIs),
Second
Order
Cone
programming
and
geometric
programming.
These
formulations
can be
solved
using
the
highly
efficient
interior
point
methods.
However,
for the
downlink
portion
of a
multi-user
communication
network,
the
current
MMSE
transceiver
design
results
no
longer
hold
due to
a
different
channel
model.
Also,
the
issue
of how
to deal
with
dynamic
changes
in the
network
and how
to
model
the QoS
(Quality
of
Service)
are all
important
questions
that
are
beginning
to be
addressed
by my
research
group.
·
Linear
Matrix
Inequality
(LMI)
formulation
of
certain
convex
cones
An
important
technique
in
reformulating
a
nonconvex
optimization
problem
as an
SDP is
the use
of the
classical
S-procedure
by
Yakubovich
in
system’s
theory.
In this
procedure,
one can
convert
the
problem
of
minimizing
a
nonconvex
quadratic
function
over a
region
described
by
another
quadratic
polynomial
inequality
into an
SDP.
The
S-procedure
can
also be
seen as
a
consequence
of
Nesterov’s
recent
result
that
gives a
sufficient
condition
on when
a
convex
cone
can be
described
by
LMIs.
In many
communications
applications,
one
needs
to
minimize
a
nonconvex
quadratic
polynomial
over
the
intersection
of two
ellipsoids.
It is
not
clear
when
and if
this
problem
can
also be
converted
to an
SDP and
thus
solved
in
polynomial
time.
The
answer
to this
question
is
related
to how
to
describe
the
convex
cone of
nonnegative
quadratic
polynomials
that
are
nonnegative
over
the
intersection
of two
ellipsoids
by
LMIs.
The
positive
resolution
to this
question
will
have
major
consequences
in
several
areas
including
control,
communication
system
design,
and the
study
of
trust
region
methods
in
optimization.