Optimal Multidimensional Signal Processing

in Wireless Sensor Networks

Anatoli Torokhti

and Stanley Miklavcic

School of Mathematics and Statistics, University of South Australia, Mawson Lakes, SA, Australia

Phenomics and Bioinformatics Research Centre, School of Mathematics and Statistics,

University of South Australia, Mawson Lakes, SA, Australia

Keywords:

Multidimensional Signal Processing.

Abstract:

Wireless sensor networks involve a set of spatially distributed sensors and a fusion center. Three methods for

ﬁnding models of the sensors and the fusion center are proposed.

1 INTRODUCTION

Wireless sensor networks (WSNs) have recently

emerged as a promising technology for a wide range

of multimedia applications (Vaseghi, 2007). A related

scenario involves a set of spatially distributed sensors

making local observations y

correlated with a signal

of interest x. Due to some external and instrumental

factors, observations are noisy. Each sensor Q

trans-

mits information about its measurements to a fusion

center P whose primary goal is to recover the original

signal within a prescribed accuracy. Fig 1 illustrates

the case.

Figure 1: Block diagram of the WSN. Here, N designates a

noisy environment,

, ...,

are estimations of x

,... , x

It is widely recognized that efﬁcient transmission

strategies should reduce (compress) the amount of in-

formation transmitted by sensors. In this paper, the

above-mentioned efﬁcient transmission strategies are

studied. We propose a novel approach based on a

reduction of the multidimensional signal processing

problem in WSNs to the new optimization problem.

We adopt a transform-based approach to deter-

mine the optimal transmission strategies in WSNs.

More precisely, each sensor applies a suitable linear

transform Q

to its random observation vector y

with

components so as to reduce its dimensionality to r

components. The fusion center applies a linear trans-

form P to reconstruct the random source vector of

interest x with m components. Thus, Q

and P are

given by matrices Q

∈ R

×n

and P ∈ R

m×r

, respec-

tively, where r

≤ n

, r = r

+ . .. + r

and r ≤ m.

Let us write (Ω,Σ,µ) for a probability space. For

i = 1,.. ., p, let x

∈ L

(Ω,R

) be a random signal

with realizations x

= x

(ω) ∈ R

. We denote

x =













and kx(·)k

Ω

kx(ω)k

dµ(ω)

(1)

where x ∈ L

(Ω,R

), m= m

= ...+m

and kx(ω)k

is the Euclidean norm of x(ω) ∈ R

. We also denote

y =













where y

∈ L

(Ω,R

) and n = n

+. .. +

Let us deﬁne a sensor model Q

by the relation

)](ω) = Q

(ω)] (2)

where Q

: L

(Ω,R

) → L

(Ω,R

) and Q

is a ma-

trix, Q

∈ R

×n

. For

r = r

+ . .. + r

, (3)

a fusion center model, P : L

(Ω,R

) → L

(Ω,R

), is

deﬁned similarly to (2), by matrix P ∈ R

m×r

Problem 1. For j = 1, .. ., p, let x

and y

be refer-

ence signals and observed data, respectively. Find

126

Torokhti A. and Miklavcic S..

Optimal Multidimensional Signal Processing in Wireless Sensor Networks.

DOI: 10.5220/0004056101260129

In Proceedings of the International Conference on Signal Processing and Multimedia Applications and Wireless Information Networks and Systems

(SIGMAP-2012), pages 126-129

ISBN: 978-989-8565-25-9

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

models of the sensors, Q

,... ,Q

, and a model of the

fusion center, P , that provide

min

P ,Q

,...,Q



x− P







)









Ω

. (4)

2 MAIN RESULTS

2.1 First Method: WSN Equipped with

Orthogonal Data Convertors

Let us extend the original problem (4) to the problem

equipped with additional data converters, G

,...,G

such that they transform observations y

,... ,y

vectors with the special property givenby Deﬁnition 1

below. This property allows us to determine solution

in a quite simple way.

For x and y presented by

x = [x

(1)

,... ,x

(m)

]

and y = [y

(1)

,... ,y

(n)

]

with x

(ℓ)

∈ L

(Ω,R) and y

(q)

∈ L

(Ω,R) where ℓ =

1,. .. ,m and q = 1,...,n, respectively, we write

E[xy

] = E

(ℓ)

(q)

m,n

ℓ,q=1

∈ R

m×n

and

(ℓ)

(q)

Ω

(ℓ)

(ω)y

(q)

(ω) dµ(ω).

Deﬁnition 1. For i = 1,... , p, let

= G

)

where G

: L

(Ω,R

) → L

(Ω,R

). The data con-

verters G

,...,G

are called pairwise orthogonal if

= O when i 6= j, (5)

where O is the zero matrix.

The determination of the pairwise orthogonal data

converters G

,... ,G

is given in Lemma 1 below.

Let us now extend problem (4) by including data

converters G

,... ,G

Problem 2. For i = 1, ..., p, ﬁnd models of the sen-

sors, Q

,... ,Q

, and a model of the fusion center, P ,

that provide

min

P ,Q

,...,Q



x− P







)









Ω

. (6)

Let us denote by M

†

the Moor-Penrose pseudo-

inverse of a matrix M.

First, we give the models of orthogonal data con-

verters G

,...,G

that satisfy (5) as follows.

Lemma 1. Let u

= G

) for i = 1, .. ., p and let

,... ,G

be such that

) = y

and G

) = y

−

i−1

∑

k=1

) (7)

for i = 2,..., p with Z

: L

(Ω,R

) → L

(Ω,R

)

deﬁned by

= E

†

+ M

(I − E

†

) (8)

with M

∈ R

n×n

arbitrary. Then the G

,... ,G

are

pairwise orthogonal data converters.

Next, to ﬁnd a solution of Problem 2, we write

P = [P

... P

] where, for j = 1,..., p, P

is deﬁned

by matrix P

∈ R

m×r

. Then



x− [P

... P

]







)









Ω



x− [F

,... ,F

](u)



Ω

, (9)

where

= P

and u = [u

,... ,u

]

. (10)

Thus, problem (6) is reduced to the equivalent prob-

lem of ﬁnding F

,... ,F

that solve

min

,...,F



x− [F

,... ,F

](u)



Ω

(11)

subject to

rank F

≤ r

, ..., rank F

≤ r

. (12)

To ﬁnd a solution of problem (11)–(12) we write



x− [F

,... ,F

](u)



Ω

= kE

1/2

− kE

1/2

)

†

+ kE

1/2

)

†

− FE

1/2

Here, the only term that depends on F

,...,F

1/2

)

†

− [F

,...,F

1/2

= kA− [F

,... ,F

]Ck

(13)

where A = E

1/2

)

†

and C = E

1/2

. Due to the

property (5), matrix E

is block-diagonal,







O ... O

O E

... O

... ... ... ...

O O ... E







Therefore, matrix C is also is block-diagonal,

C =







O .. . O

O C

... O

... ... ... . ..

O O .. . C







OptimalMultidimensionalSignalProcessinginWirelessSensorNetworks

127

If we write A = [A

...A

] where, for j = 1,..., p,

∈ R

m×n

, then it follows from (13) that

kA− [F

,... ,F

]Ck

∑

j=1

− F

j j

Thus, problem (11)–(12) is reduced to p individ-

ual problems of ﬁnding F

, for j = 1,.. ., p, that solves

min

− F

j j

with rank F

≤ r

. (14)

The solution has been given in (Torokhti and

Friedland, 2009) as follows.

2.1.1 Best Rank-constrained Matrix

Approximation

Let C

m×n

be a set of m× n complex valued matrices,

and denote by R (m,n,r) ⊆ C

m×n

the variety of all

m× n matrices of rank r at most. Fix A = [a

]

m,n

i, j=1

∈

m×n

. Then A

∗

∈ C

n×m

is the conjugate transpose of

A. Let the singular value decomposition (SVD) of A

be given by

A = U

∗

, (15)

where U

∈ C

m×m

, V

∈ C

n×n

are unitary matrices,

:= diag(σ

(A),... ,σ

min(m,n)

(A)) ∈ C

m×n

is a gen-

eralized diagonal matrix, with the singular values

(A) ≥ σ

(A) ≥ .. . ≥ 0 on the main diagonal.

Let U

= [u

...u

] and V

= [v

...v

] be

the representations of U and V in terms of their m and

n columns, respectively. Let

A,L

rank A

∑

i=1

∗

∈ C

m×m

and P

A,R

rank A

∑

i=1

∗

∈ C

n×n

(16)

be the orthogonal projections on the range of A

and A

∗

, correspondingly. Deﬁne a truncated SVD,

{A}

, of matrix A by

{A}

∑

i=1

(A)u

∗

= U

∗

∈ C

m×n

(17)

for r = 1,. .. ,rank A, where

= [u

...u

], Σ

= diag(σ

(A),... ,σ

(A))

and V

= [v

...v

]. (18)

For r > rank A, we write {A}

:= A (or{A}

{A}

rank A

). For 1 ≤ r < rank A, the matrix {A}

uniquely deﬁned if and only if σ

(A) > σ

r+1

(A).

Recall that A

†

:= V

†

∗

∈ C

n×m

is the Moore-

Penrose generalized inverse of A, where Σ

†

diag



(A)

,... ,

rank A

(A)

,0, .. .,0



∈ C

n×m

Henceforth k ·k designates the Frobenius norm.

Theorem 1 below provides a solution to the prob-

lem of ﬁnding a matrix F that solves

min

F∈R (p,q,r)

||A− BFC||. (19)

Theorem 1. (Friedland and Torokhti, 2007) Let A ∈

m×n

, B ∈ C

m×p

andC ∈ C

q×n

be given matrices. Let

= (I

− P

B,R

)S and L

= T(I

− P

C,L

) (20)

where S ∈ C

p×p

and T ∈ C

q×q

are any matrices, and

is the p × p identity matrix. Then the matrix

F = (I

+ L

†

B,L

C,R

}

†

+ L

) (21)

is a minimizing matrix for the minimal problem (19).

Any minimizing F has the above form.

2.1.2 Determination of Models of Sensors and

Fusion Center that Satisfy (6)

It follows from (19), (21), that a solution of the prob-

lem in (14) is a particular case of Theorem 1.

Indeed if, in (19)–(21), we write A

, F

, C

j j

and r

instead of A, F, C and r, respectively, and set n = n

p = m, q= n

and B = I then (14) coincides with (19).

Its solution follows from (21) in the form

= {A

}

†

j j

+ L

), (22)

where similarly to L

in (20), L

= T

− P

)

with T

to be any n

×n

matrix. The solution of prob-

lem (11)–(12) is given by (22) as well.

Since (11)–(12) is equivalent to (6), it remains to

show that models of sensors, Q

,... ,Q

, and a model

of the fusion center, P , that satisfy (6), follow from

(22). To this end, we recall that by (10),

= P

where F

, P

and Q

are deﬁned by matrices F

∈

m×n

, P

∈ R

m×r

and Q

∈ R

×n

, respectively,

where F

= P

. The matrices P

and Q

are deter-

mined as follows. Let us write the SVD of F

in (22)

= U

(23)

where matrices

= [u

,... ,u

] ∈ R

m×m

= diag(σ

),... , σ

min(m,n

)

)) ∈ R

m×n

and V

= [v

,... ,v

] ∈ R

×n

are similar to matrices U

, Σ

and V

for the SVD of

matrix A in (15), respectively. In particular, σ

, .. . ,

jmin(m,n

)

are the associated singular values. Let

= [u

,... ,u

] ∈ R

m×r

, (24)

= diag(σ

),... ,σ

)) ∈ R

×r

(25)

and V

= [v

,... ,v

] ∈ R

n×r

. (26)

SIGMAP2012-InternationalConferenceonSignalProcessingandMultimediaApplications

128

Then F

in (22) can be written in form F

= P

where, for j = 1,.. ., p,

= U

, Q

= V

, (27)

= U

∈ R

m×r

, Q

= Σ

. (28)

Thus, we have proved the following.

Theorem 2. The models of the sensors and the fu-

sion center that satisfy (6) are given by matrices

,... ,Q

and P = [P

,... ,P

], respectively, deter-

mined by (27) or (28).

2.2 Second Method: Direct Solution of

WSN Problem (4)

Here, we consider a way to determine models of the

sensors, Q

,... ,Q

, and the fusion center, P , for the

case when the orthogonal data converters, G

,... ,G

(see (6), Deﬁnition 1 and Lemma 1), are not used, i.e.

when Q

,...,Q

and P should satisfy (4).

In this case, similar to (9) and (10), we have



x− [P

... P

]







)









Ω

= kE

1/2

− kE

1/2

)

†

+kE

1/2

)

†

− FE

1/2

where, as before, for j = 1,. .., p, F

= P

. Here,

the only term that depends on F

,... ,F

1/2

)

†

−[F

,... ,F

1/2

= kA−[F

,... ,F

]Ck

where A = E

1/2

)

†

and C = E

1/2

. Thus, problem

(4) is reduced to ﬁnding F

, for j = 1, .. ., p, that solve

min

,...,F

kA− [F

,... ,F

]Ck

(29)

subject to

rank F

≤ r

, ..., rank F

≤ r

. (30)

A difference from (13) is that in (29), matrix C is

not block-diagonal. In this general case, a solution

to problem (29)–(30), F

,... ,F

, follows from the ex-

tension of Theorem 1. This result will be provided at

the conference. Then, for j = 1,... , p, each matrix F

that satisﬁes (29)–(30) is presented in the form (27)

or (28).

Thus, in this case, the models of the sensors and

the fusion center that satisfy (4) are given by matrices

,... ,Q

and P = [P

,... ,P

], respectively, deter-

mined by( 27) or (28) provided that F

,... ,F

solve

(29)–(30).

2.3 Third Method: Approximate

Solution of WSN Problem (4)

Here, we consider a method which represents a com-

promise between the ﬁrst and second methods. In

(29), matrices A = [A

,... ,A

] and C can be repre-

sented in the form

A =

+ . .. +

and C = [C

,... ,C

]

, (31)

respectively, where

= [A

,O, .. .,O], .. .,

[O,. .. ,O, A

] and, for j = 1, ..., p, C

∈ R

×n

is a

block of C. Then

kA− [F

,... ,F

]Ck

≤

∑

j=1

−

∑

j=1

. (32)

The latter motivates ﬁnding models of the sensors,

,... ,Q

, and the fusion center, P = [P

,... ,P

], in

the form F

= P

,... ,F

= P

, where F

,... ,F

are determined from p individual problems of ﬁnding

, for j = 1, .. ., p, that solves

min

− F

with rank F

≤ r

. (33)

A direct comparison with (14) shows that the problem

in (33) is different from that in (14). This is because,

for j = 1,..., p, matrices A

, C

j j

and

, C

are dif-

ferent. Nevertheless, formally, the problems in (14)

and (33) are similar. Therefore, the solution of (33))

is given in the form (22) where the notation should be

changed in accordance with that in (31)–(33).

As a result, the following theorem is true.

Theorem 3. The models of the sensors and the fu-

sion center of the WSN that approximate the opti-

mal models are given by matrices Q

,... ,Q

and

P = [P

,... ,P

] determined by (27) or (28), where A

andC

j j

must be replaced with

andC

, respectively.

REFERENCES

Friedland, S. and Torokhti, A. (2007). Generalized rankcon-

strained matrix approximations. SIAM J. Matrix Anal.

Appl., 29:656659.

Torokhti, A. and Friedland, S. (2009). Towards theory of

generic principal component analysis. J. Multivariate

Analysis, 100:661669.

Vaseghi, S. V. (2007). Multimedia signal processing: the-

ory and applications in speech, music and communi-

cations. John Wiley and Sons.

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