Bandwidth-Constrained MAP Estimation for
Wireless Sensor Networks
S. Faisal Ali Shah, Alejandro Ribeiro and Georgios B. Giannakis
Dept. of ECE, University of Minnesota
200 Union Street S.E., Minneapolis, MN 55455
Abstract— We deal with distributed parameter estimation
algorithms for use in wireless sensor networks (WSNs) with a
fusion center when only quantized observations are available due
to power/bandwidth constraints. The main goal of the paper
is to design efficient estimators when the parameter can be
modelled as random with a priori information. In particular,
we develop maximum a posteriori (MAP) estimators for dis-
tributed parameter estimation and formulate the problem under
different scenarios. We show that the pertinent objective function
is concave and hence, the corresponding MAP estimator can
be obtained efficiently through simple numerical maximization
algorithms.
I. I NTRODUCTION
Characterized by small low-cost, low-power devices
equipped with limited sensing, computational and commu-
nication capabilities, wireless sensor networks (WSNs) are
well motivated for environmental monitoring, industrial instru-
mentation and surveillance applications [7]. The distributed
topology of these networks along with their limited power
budget and communication resources gave rise to the area of
collaborative signal and information processing [3].
Bandwidth-constrained distributed estimation arises when
deploying a WSN for monitoring, in which case the estimation
of certain parameters of interest necessitates collection of
different sensor estimates. As sensor observations have to be
quantized, WSN-based parameter estimators must rely on (per-
haps severely) quantized observations [4], [5]. Interestingly,
it has been shown in [6] that when the noise variance is
comparable to the parameter’s dynamic range, even quanti-
zation to a single bit per observation leads to a small penalty
in estimation variance when comparing maximum likelihood
(ML) estimators based on quantized (binary) versus original
(analog-amplitude) observations. A characteristic common to
all these works is the assumption that some prior information
is available, at least to bound the range of possible parameter
values. This suggests naturally, a connection with Bayesian
estimation.
Building on this observation, the present paper addresses
the problem of maximum a posteriori (MAP) estimation
based on binary observations, and shows that the graceful
Work in this paper was prepared through collaborative participation in
the Communications and Networks Consortium sponsored by the U. S.
Army Research Laboratory under the Collaborative Technology Alliance
Program, Cooperative Agreement DAAD19-01-2-0011. The U. S. Government
is authorized to reproduce and distribute reprints for Government purposes
notwithstanding any copyright notation thereon.
Email at (sfaisal, aribeiro, georgios)@ece.umn.edu
Physical
Phenomenon
/Process
Estimator
(Fusion
Center)
Sensor nodes
Transducer
Quantizer/
Encoder
.. .
.. .
.. .
BW/Power
Constrained
Links
Fig. 1. Block diagram of wireless sensor network
performance degradation of ML estimators extends to their
MAP counterparts. Moreover, we establish that while MAP
estimators cannot be expressed in closed form, they are found
as the maximum of a concave function; thus ensuring the
convergence of fast descent algorithms e.g., Newton’s method.
II. PROBLEM FORMULATION
Consider a physical phenomenon characterized by a set of P
parameters that we lump in vector form as θ := [ θ
1
,...,θ
P
]
T
.
From available a priori knowledge, θ is modelled as a random
vector parameter with prior probability density function (pdf)
p
θ
( θ) and mean E( θ)= μ
θ
.
For measuring θ, we deploy a WSN composed of N sensors
{S
n
}
N-1
n=0
, with each sensor observing θ through a linear
transformation
x( n)= Hθ + w( n) , (1)
where x( n) := [ x
1
( n) ,...,x
K
( n)]
T
∈ R
K
is the measure-
ment vector at sensor S
n
, w( n) ∈ R
K
is zero-mean additive
noise with pdf p
w
( w) and the matrix H ∈ R
K×P
. We
denote the vector formed by concatenating all the observa-
tions {x( n) }
N-1
n=0
of N sensors as x
0:N-1
. For simplicity of
exposition, we assume that H and p
w
( w) are constant across
sensors, and we also assume that w( n
1
) is independent of
w( n
2
) for n
1
= n
2
.
A clairvoyant (CV) benchmark for estimators based on bi-
nary observations corresponds to having all analog-amplitude
observations x
0:N-1
be available at the fusion center. In
this case, a possible approach to estimate θ is the MAP
estimator [2]
ˆ
θ
CV
= arg max
θ
{p[ θ|x
0:N-1
] } , (2)
where p[ θ|x
0:N-1
] is the conditional pdf of θ given x
0:N-1
.
As discussed earlier, power and bandwidth constraints,
dictate the need of a quantizer mapping the analog-amplitude
observations x( n) to a finite set:
b( n) := q( x( n)) , with q : R
K
→ {−1 , 1 }
K
, (3)
215 1424401321/05/$20.00 ©2005 IEEE