IEEE Signal Processing - May 2018 - 119

denote random vectors, whereas uppercase italic letters, e.g., X, denote scalar
random variables. Lowercase boldfaced
letters, e.g., x, denote real vectors (including samples of random vectors), and lowercase italic letters, e.g., x, denote scalar
real numbers (including samples of scalar
random variables). Boldfaced capital letters, e.g., F, are used to denote real matrices. Uppercase italic letters, e.g., N, may be
also used to denote the dimensions of vectors, matrices, and Euclidean spaces and to
denote the number of elements in finite sets;
in those cases, the distinction with scalar
random variables is either implied in
context or noted explicitly in the text.
Let {X n}, n $ 0 be a sequence of
hidden (i.e., unobserved) continuous random vectors taking values in 0 N . Likewise, let {Yn, r}, n $ 0 be a sequence of
observed continuous random vectors taking values in 0 L such that the sequence
{Yn, r} is available at the rth node of a network of R sensors. The sequences {X n}
and {Yn, r} are related, for n $ 0 and
r ! {1, 2, f, R}, by the discrete-time,
stochastic state-space model
X n +1 = fn (X n) + G n U n
Yn, r = h n, r (X n) + Vn, r,

(1)
(2)

where the continuous random vectors
X 0, {U n} and {Vn, r} are assumed mutually independent for all n $ 0 and all
r ! {1, 2, f, R}; U n and Vn, r have zero
mean and covariance matrices respectively Q n and R n, r; and fn: 0 N " 0 N
and h n, r: 0 N " 0 L are arbitrary (possibly nonlinear) functions that are assumed
known for all n $ 0 and all indexes
r ! {1, 2, f, R}. In (1) and (2), U n and
Vn, r take values, respectively, in 0 M and
0 L, and G n is a real matrix of dimension
N # M that is known for all n $ 0.
Figure 1 presents the corresponding
hidden Markov model for (1) and (2) at
node r.
Our goal is to recursively estimate
the value x n assumed by X n at instant
n given the observed vector y 0: n, 1: R that
collects the values assumed by {Yk, r}
for k = 0, 1, f, n and r = 1, 2, f, R.
Assuming continuous-valued hidden
states, the aforementioned estimation task
requires the recursive computation of the
global network posterior probability den-

sity function (pdf) p (x n | y 0: n, 1: R) at each
time instant n $ 0. The desired minimum
mean-square error (MMSE) state estimate
is then given by xt n | n = E {X n|y 0: n, 1: R},
where E {X | y} denotes the expected value
of the random vector X given the observed
real vector y.
A special case of the general statespace model in (1) and (2), which is of particular interest in this lecture note, is the
linear state-space model
X n +1 = Fn X n + G n U n

and
p (x n |y 0: n, 1: R) ? = % p (y n, r | x n)G
R

r =1

# p (x n | y 0: n -1, 1: R),

where the symbol ? here denotes proportional to with the implied normalization
constant such that the function on the lefthand side of the symbol integrates to one.
When observations y 0, 1:R are available,
the recursions in (5) and (6) are initialized
by making p (x 0 |y 0: -1, 1:R) = p (x 0).
Equation (6) can be implemented either
in a data fusion center that has access to all
network measurements at each time instant
or by computing the products on the righthand side of (6) over the network using
a distributed processing algorithm such
as iterative flooding [8]. Although more
efficient than average or min/max consensus algorithms, the exact fully distributed
implementation of the optimal network filter using flooding still has a very large internode communication cost. Furthermore, as
each node is a potential point of failure in a
flooding protocol, the method lacks robustness. To circumvent those limitations, we
turn our attention in the next sections to
alternative diffusion filters.

(3)

Yn, r = H n, r X n + Vn, r,

(4)

where the random vectors X 0, {U n} and
{Vn, r}, in addition to being mutually independent, are also assumed jointly Gaussian for n $ 0 and r ! {1, 2, 3, f, R},
and Fn and H n, r are, respectively, N # N
and L # N real matrices that are again
assumed known for all n $ 0 and all
r ! {1, 2, f, R} .

Solution
Optimal solution
Under the mutual independence as sumptions described in the "Problem
Statement" section, it follows, using
the total probability theorem and Bayes
law, that the desired posterior pdf
p (x n | y 0: n, 1: R) is obtained recursively
from p (x n -1 |y 0: n -1, 1: R) in two steps: the
prediction step shown in (5) and the
update step shown in (6); see also [7]:
p (x n | y 0: n -1, 1: R) =

#0

N

ATC diffusion filter
In the sequel, we review the Bayesian formulation of the ATC diffusion filter as
proposed in [4]. The ATC methodology
assumes that, at instant n - 1, each network node r has an available posterior pdf
p n -1 | n -1, r (x n) that actually depends on all
network measurements that have contributed to its computation from instant zero

6 p (x n | x n -1)

# p (x n -1 |y 0: n -1, 1: R)@ dx n -1

(5)

Un - 2

Un - 1

Gn - 2
fn - 2 (.)

Un

Gn - 1
fn - 1 (.)

Xn - 1

Vn - 1, r

(6)

Gn
Xn

fn (.)

Vn, r

Xn + 1

fn + 1 (.)

Vn + 1, r

hn - 1, r (.)

hn , r (.)

Yn - 1, r

Yn , r

hn + 1, r (.)
Yn + 1, r

Figure 1. The hidden Markov model.
IEEE Signal Processing Magazine

|

May 2018

|

119



Table of Contents for the Digital Edition of IEEE Signal Processing - May 2018

Contents
IEEE Signal Processing - May 2018 - Cover1
IEEE Signal Processing - May 2018 - Cover2
IEEE Signal Processing - May 2018 - Contents
IEEE Signal Processing - May 2018 - 2
IEEE Signal Processing - May 2018 - 3
IEEE Signal Processing - May 2018 - 4
IEEE Signal Processing - May 2018 - 5
IEEE Signal Processing - May 2018 - 6
IEEE Signal Processing - May 2018 - 7
IEEE Signal Processing - May 2018 - 8
IEEE Signal Processing - May 2018 - 9
IEEE Signal Processing - May 2018 - 10
IEEE Signal Processing - May 2018 - 11
IEEE Signal Processing - May 2018 - 12
IEEE Signal Processing - May 2018 - 13
IEEE Signal Processing - May 2018 - 14
IEEE Signal Processing - May 2018 - 15
IEEE Signal Processing - May 2018 - 16
IEEE Signal Processing - May 2018 - 17
IEEE Signal Processing - May 2018 - 18
IEEE Signal Processing - May 2018 - 19
IEEE Signal Processing - May 2018 - 20
IEEE Signal Processing - May 2018 - 21
IEEE Signal Processing - May 2018 - 22
IEEE Signal Processing - May 2018 - 23
IEEE Signal Processing - May 2018 - 24
IEEE Signal Processing - May 2018 - 25
IEEE Signal Processing - May 2018 - 26
IEEE Signal Processing - May 2018 - 27
IEEE Signal Processing - May 2018 - 28
IEEE Signal Processing - May 2018 - 29
IEEE Signal Processing - May 2018 - 30
IEEE Signal Processing - May 2018 - 31
IEEE Signal Processing - May 2018 - 32
IEEE Signal Processing - May 2018 - 33
IEEE Signal Processing - May 2018 - 34
IEEE Signal Processing - May 2018 - 35
IEEE Signal Processing - May 2018 - 36
IEEE Signal Processing - May 2018 - 37
IEEE Signal Processing - May 2018 - 38
IEEE Signal Processing - May 2018 - 39
IEEE Signal Processing - May 2018 - 40
IEEE Signal Processing - May 2018 - 41
IEEE Signal Processing - May 2018 - 42
IEEE Signal Processing - May 2018 - 43
IEEE Signal Processing - May 2018 - 44
IEEE Signal Processing - May 2018 - 45
IEEE Signal Processing - May 2018 - 46
IEEE Signal Processing - May 2018 - 47
IEEE Signal Processing - May 2018 - 48
IEEE Signal Processing - May 2018 - 49
IEEE Signal Processing - May 2018 - 50
IEEE Signal Processing - May 2018 - 51
IEEE Signal Processing - May 2018 - 52
IEEE Signal Processing - May 2018 - 53
IEEE Signal Processing - May 2018 - 54
IEEE Signal Processing - May 2018 - 55
IEEE Signal Processing - May 2018 - 56
IEEE Signal Processing - May 2018 - 57
IEEE Signal Processing - May 2018 - 58
IEEE Signal Processing - May 2018 - 59
IEEE Signal Processing - May 2018 - 60
IEEE Signal Processing - May 2018 - 61
IEEE Signal Processing - May 2018 - 62
IEEE Signal Processing - May 2018 - 63
IEEE Signal Processing - May 2018 - 64
IEEE Signal Processing - May 2018 - 65
IEEE Signal Processing - May 2018 - 66
IEEE Signal Processing - May 2018 - 67
IEEE Signal Processing - May 2018 - 68
IEEE Signal Processing - May 2018 - 69
IEEE Signal Processing - May 2018 - 70
IEEE Signal Processing - May 2018 - 71
IEEE Signal Processing - May 2018 - 72
IEEE Signal Processing - May 2018 - 73
IEEE Signal Processing - May 2018 - 74
IEEE Signal Processing - May 2018 - 75
IEEE Signal Processing - May 2018 - 76
IEEE Signal Processing - May 2018 - 77
IEEE Signal Processing - May 2018 - 78
IEEE Signal Processing - May 2018 - 79
IEEE Signal Processing - May 2018 - 80
IEEE Signal Processing - May 2018 - 81
IEEE Signal Processing - May 2018 - 82
IEEE Signal Processing - May 2018 - 83
IEEE Signal Processing - May 2018 - 84
IEEE Signal Processing - May 2018 - 85
IEEE Signal Processing - May 2018 - 86
IEEE Signal Processing - May 2018 - 87
IEEE Signal Processing - May 2018 - 88
IEEE Signal Processing - May 2018 - 89
IEEE Signal Processing - May 2018 - 90
IEEE Signal Processing - May 2018 - 91
IEEE Signal Processing - May 2018 - 92
IEEE Signal Processing - May 2018 - 93
IEEE Signal Processing - May 2018 - 94
IEEE Signal Processing - May 2018 - 95
IEEE Signal Processing - May 2018 - 96
IEEE Signal Processing - May 2018 - 97
IEEE Signal Processing - May 2018 - 98
IEEE Signal Processing - May 2018 - 99
IEEE Signal Processing - May 2018 - 100
IEEE Signal Processing - May 2018 - 101
IEEE Signal Processing - May 2018 - 102
IEEE Signal Processing - May 2018 - 103
IEEE Signal Processing - May 2018 - 104
IEEE Signal Processing - May 2018 - 105
IEEE Signal Processing - May 2018 - 106
IEEE Signal Processing - May 2018 - 107
IEEE Signal Processing - May 2018 - 108
IEEE Signal Processing - May 2018 - 109
IEEE Signal Processing - May 2018 - 110
IEEE Signal Processing - May 2018 - 111
IEEE Signal Processing - May 2018 - 112
IEEE Signal Processing - May 2018 - 113
IEEE Signal Processing - May 2018 - 114
IEEE Signal Processing - May 2018 - 115
IEEE Signal Processing - May 2018 - 116
IEEE Signal Processing - May 2018 - 117
IEEE Signal Processing - May 2018 - 118
IEEE Signal Processing - May 2018 - 119
IEEE Signal Processing - May 2018 - 120
IEEE Signal Processing - May 2018 - 121
IEEE Signal Processing - May 2018 - 122
IEEE Signal Processing - May 2018 - 123
IEEE Signal Processing - May 2018 - 124
IEEE Signal Processing - May 2018 - 125
IEEE Signal Processing - May 2018 - 126
IEEE Signal Processing - May 2018 - 127
IEEE Signal Processing - May 2018 - 128
IEEE Signal Processing - May 2018 - 129
IEEE Signal Processing - May 2018 - 130
IEEE Signal Processing - May 2018 - 131
IEEE Signal Processing - May 2018 - 132
IEEE Signal Processing - May 2018 - Cover3
IEEE Signal Processing - May 2018 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201809
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201807
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201805
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201803
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201801
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1117
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0917
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0717
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0517
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0317
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0117
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1116
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0916
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0716
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0516
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0316
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0116
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1115
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0915
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0715
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0515
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0315
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0115
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1114
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0914
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0714
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0514
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0314
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0114
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1113
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0913
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0713
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0513
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0313
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0113
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1112
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0912
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0712
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0512
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0312
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0112
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1111
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0911
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0711
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0511
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0311
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0111
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1110
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0910
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0710
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0510
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0310
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0110
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1109
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0909
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0709
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0509
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0309
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0109
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1108
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0908
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0708
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0508
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0308
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0108
https://www.nxtbookmedia.com