HOW RANDOM-SET THEORY
CAN HELP
WIRELESS COMMUNICATIONS
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(or, solving estimation problems in wireless
communications where one of the things
you do not know is the number of things
you do not know)
EZIO
BIGLIERI
Ezio Biglieri
1
Universitat Pompeu Fabra, Barcelona, Spain
WPMC 2008, LAPLAND, FINLAND
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MOTIVATIONS
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PROBLEM I: MULTIUSER DETECTION
multiuser detection
A multiuser system is described by the set
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where k is the number of active interferers,
and
xi are the state vectors of the individual
interferers
(k=0 yields the empty set, and
corresponds to no interferer)
multiuser detection
Problem: Estimate Xt when the number of
interferers is not known a priori
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(Xt is a random set, that is, a set
whose randomness is not only in the
elements, but also in the number of
elements.)
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PROBLEM II: NEIGHBOR DISCOVERY
neighbor discovery
 In a wireless network, neighbor discovery
(ND) is the detection of all neighbors with
which a given reference node may
communicate directly.
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 ND may be the first algorithm run in
a network, and the basis of medium
access, clustering, and routing algorithms.
neighbor discovery
TD
#1
#2
#3
#4
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T
receive interval of reference user
transmit interval of neighboring users
 Structure of a discovery session
(Ephremides et al., 2008)
neighbor discovery
Signal collected from all potential neighbors
during receiving slot t :
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extended to a random
number of users
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signature of user k
amplitude of user k
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PROBLEM III:
MULTIPATH CHANNEL ESTIMATION
channel model (OFDM)
channel frequency response
additive noise
observed signal
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discrete time
delays
training sequence
(diagonal matrix)
path gains
random sets
The channel response at time t is modeled
by the random set
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where Lmax is the maximum number of
active interferers
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APPLY RANDOM –SET THEORY
random set theory
Random Set Theory
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 RST is a probability theory of finite sets
that exhibit randomness not only in each
element, but also in the number of
elements
random set theory
Random Set Theory
We define a random closed set
as a map between a sample
space and the family of closed
subsets of a space
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random set theory
Define the Belief Function
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where C is a closed subset of
random set theory
Decomposition of a belief function
into a sum of simpler belief functions:
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random set theory
A special case shows that the
belief function generalizes
standard probability measures.
For X = {x}, x a random vector:
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random set theory
However, belief functions
are not probability measures:
C1  C2 =  implies
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but
example of belief function
(“Singleton-or-empty” sets).
Let
We obtain
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set derivative
Define the set derivative of  at {}:
and
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set integral
Define the set integral of  at S:
where S is a closed set,  a measure,
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and
fundamental theorem
Set derivative and set integral
are the inverse of each other.
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RANDOM-SET ESTIMATION THEORY
random-set estimation theory
 The Belief density is the set
derivative of the belief function:
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 It can be used as a density in
ordinary detection/estimation
theory.
Bayesian estimators
A Bayesian set estimator is
generated by
 Choosing a cost function
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 Finding the estimator that
minimizes the cost function
Bayesian estimators
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Choosing a cost function may not be
an easy task: think for example of a
situation in which we must estimate
the number of interferers and their
power in a multiple-access
environment.
Bayesian estimators
Bayesian set estimator I
 First, estimate the cardinality of the set:
 Next, find
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Bayesian estimators
Bayesian set estimator II
 Find
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where c is a small constant, expressing how
close X and its estimate must be to have
cost = 0 (different values of c correspond
to different cost functions, and hence to
different estimators).
Bayesian estimators
Bayesian set estimator III
 Estimate first the cardinality of the set
and the identities of its elements, then
their continuous parameters
using a posteriori expectations.
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SYSTEM MODELING
modeling observations
Ingredients
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Description of measurement process
(the “channel”)
modeling the dynamics
Ingredients
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Evolution of random set with time
(Markovian assumption)
Bayesian filtering equations
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 Integrals are “set integrals”
(the inverses of set derivatives)
 Closed form in the finite-set case
 Otherwise, use “particle filtering”
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APPLICATIONS TO WIRELESS COMMUNICATIONS:
MULTIUSER DETECTION
multiuser detection
Description
multiuser
systems
A multiuserofsystem
with
unknown
number of interferers is described
by the random set
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where k is the number of active interferers,
and
xi are the state vectors of the individual
interferers
(k=0 corresponds to no interferer)
previous work
 Previous work (Mitra, Poor, Halford, Brandt-Pierce,…)
focused on activity detection, addition of a single user.
 It was recognized that certain detectors
suffer from catastrophic error
if a new user enter the system.
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 Wu, Chen (1998) advocate a two-step
detection algorithm:
 MUSIC algorithm estimates active users
 MUD is used on estimated number of users
environment:
static, deterministic
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Static, random channel, 3 users:
Classic ML vs. joint ML detection of data and # of interferers
environment: static, random
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Static, random channel, 3 users:
Joint ML detection of data and # of interferes vs. MAP
environment: dynamic, random
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multiuser dynamics
random set:
users at time t
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users at time t-1
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surviving users
users surviving
from time t-1
new users
new users
all potential users
surviving users
 = probability of persistence
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B
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C
new users
 = activity factor
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B
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C
surviving users + new users
Derive the belief density of
through the “generalized convolution”
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lesson learned

MUD receivers must know the number of interferers,
otherwise performance is impaired.
 Introducing a priori information about
the number of active users improves
MUD performance and robustness.
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 A priori information may include activity factor.
 A priori information may also include
a model of users’ motion.
detection and estimation
 In addition to detecting the number of
active users and their data, one may
want to estimate their parameters
(e.g., their power)
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 A Markov model of power evolution
is needed
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APPLICATIONS TO WIRELESS COMMUNICATIONS:
CHANNEL ESTIMATION
new approach
 Impulse responses have unknown
number of samples L
do not force L = Lmax
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 Impulse responses may change
little from one frame to the next
exploit dynamic model
of impulse response evolution
new approach
Cost function for Bayesian estimate:
Wrong estimate of set of active paths:
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cost is Q
Discrepancy between estimated and
true continuous parameters:
cost is function g of the difference
To simplify, estimate the set of
active paths, and choose a quadratic g
performance of GMAP-III
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Real part, SNR = 20 dB
performance of Kalman filter
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Real part, SNR = 20 dB
MIMO-OFDM
MIMO-OFDM with:
K = 64 subcarriers
N = 2, M = 3
Lmax = 4
Gauss-Markov path gains
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(path amplitudes are normal,
path amplitude evolution is
Markov and Gaussian)
MIMO-OFDM
LS assumes
all paths active;
neglects dynamic
model
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KF assumes
all paths active;
tracks their
variations
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BIBLIOGRAPHY
■ R. P. S. Mahler, Statistical Multisource-Multitarget Information Fusion.
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Artech House, 2007.
■ H. T. Nguyen, An Introduction to Random Sets. Chapman & Hall, 2006.
■ I. R. Goodman, R. P. S. Mahler, and H. T. Nguyen, Mathematics
of Data Fusion. Kluwer, 1997.
■ E. Biglieri and M. Lops, “Multiuser detection in a dynamic
environment. Part I: User identification and data detection,”
IEEE Trans. Inform. Theory, September 2007.
■ D. Angelosante, E. Biglieri, and M. Lops, “Multipath channel
tracking in OFDM systems,” PIMRC 2007, Athens, Greece,
September 2--6, 2007.
■ D. Angelosante, E. Biglieri, and M. Lops, “Multiuser detection
in a dynamic environment: Joint user identification and parameter
estimation,” IEEE Int. Symp. Inform. Theory (ISIT 2007),
Nice, France, 2007.
■ D. Angelosante, E. Biglieri, and M. Lops, “Sequential estimation
of time-varying multipath channel for MIMO-OFDM systems,”
IEEE IEEE Int. Symp. Inform. Theory (ISIT 2008),
Toronto, ON, July 6--11, 2008.