Distributed Compression and the Blind ZoneRecognition

Shaochuan Wu∗, Siqi Pan, Kangjian Ma, Yuming WeiSchool of Electronics and Information Engineering

Harbin Institute of TechnologyHarbin, China

Email: scwu@hit.edu.cn, elithebeth.pan@gmail.com, kangjianma@gmail.com, wym1674@163.com

Abstract—Wireless sensor network (WSN) technology providessupport infrastructure to solve blind zone coverage. Previousstudies mainly focus on deployment and management of sensornodes for enhancing signal coverage on a specific zone. But few ofthem are on coordinating several nodes to recognize blind zone.Moreover, it is a great challenge how to extract, transmit andprocess information energy efficiently in a decentralized way. Inthis paper, we propose a distributed algorithm for convenientand efficient blind zone recognition based on the early workon the unbiased broadcast gossip algorithm (UBGA) and thecompressed sensing. Using UBGA, a faster consensus algorithm,the system can simultaneously compute the compressed databased on compressed sensing technique and disseminate themthroughout the network. From these compressed data one canreconstruct the original data with approximate accuracy byquerying any sensor values using the gradient projection strategy.Simulation results illustrate that our solution, proposed in thispaper, can effectively recognize blind zone.

Keywords-Distributed Compression; Blind Zone Recogni-tion; Gossip Algorithm

I. INTRODUCTION

The developing trends of mobile communication system arethe wider bandwidth and the higher frequency. Resulting ofthe higher frequency and the faster signal attenuation, therewill be more blind zones where the received signal strength(RSS) does not meet the required QoS. Therefore, the blindzones become the key factor in restricting the coverage ofmobile communication system. Traditionally, the detection ofthe blind zones is mainly implemented by professional equip-ments and technical personnel, which leads to inflexibilityand increased cost. In this paper, we propose a distributedalgorithm based on the early work on unbiased broadcastgossip algorithm (UBGA) [1] and the compressed sensing [2]for convenient and efficient blind zone recognition. The nodes,equipped with sensing processors and transceivers with limitedcommunication ranges, act as supporting infrastructures, i.e. awireless sensor network, of the proposed algorithm [3], [4].Each node can obtain RSS in its local position and exchangeinformation wirelessly with its neighbors while a large numberof nodes in the network exchanging information directly willbring much communication overhead and energy consumption.Furthermore, when signal strength varies slowly in space, RSSat nearby sensors will be highly correlated [5]. Thereforecompressed sensing technology can be a wise choice to reduceoverhead and energy consumption under such circumstances.

Gossip algorithms for in-network signal processing aresolutions to consensus problems that a team of sensors requireachieving a consistent opinion throughout the network bylocal information exchanges with their neighbors [6]. Thegoal of the canonical gossip algorithm is reaching the averageconsensus where every node achieves the average of allinitial values after limited message exchanges [7]. Essentially,both gossip algorithms and the compressed sensing are lineartransformations. Gossip algorithms can be implemented indistributed compressed sensing [8]. Therefore, we can inferthat the proposed distributed algorithm can perform the recog-nition task successfully by only local sensing and interactionsamong nodes and its neighbors. There is no need to storeand maintain specific routes. In addition, the system with theproposed algorithm remains robust with failures of single nodeor unreliable transmission links.

The contributions of this paper are mainly in three aspects.First, we design a spatially distributed algorithm that allow ateam of sensors to gather data and reconstruct an image ofthe blind zones i.e. accomplishing blind zones recognition ina distributed way. Second, the proposed distributed algorithmis applicable in both directed and undirected networks, in con-trast with previous similar approaches [8], [9] only availablefor undirected networks. Third, the proposed algorithm canachieve convergence with fewer gossip iterations resulting ina lower overhead and energy consumption, i.e. the proposedalgorithm can accomplish the distributed compression andblind recognition more efficiently [10], [11].

This paper is organized as follows. In Section II we brieflyintroduce the problem and the framework of UBGA. SectionIII illustrates how UBGA can be used for the distributed com-pression and the blind zone recognition. Simulation results andperformance analysis are illustrated in Section IV comparedwith a pairwise gossip algorithm (PGA) based distributedcompression and blind zone recognition algorithm. Section Vconcludes the paper.

II. FRAMEWORK FOR UBGA

We take time selective and frequency selective fading intoaccount both of which obey non-zero mean Gaussian distri-bution. A single node sometimes fails to know whether thelow received signal strength is caused by deep fading or theblind zone. Even though the node knows that it is located in

978-1-5090-0304-4/16/$31.00 ©2016 IEEE 304

a blind zone, we are still not able to know the whole range ofblind zones and the distribution of according received signalstrength. In addition, the positions with the low received signalstrength possess time-varying characteristics as a result of thetime-varying channel.

According to ITU-R M.1225, the following model can beused for depicting the outdoor pedestrian test environment:

L = 40 log10 R+ 30 log10 f + 49 (1)

where L denotes path loss (dB) with R denoting the distancebetween base station and node (km) and f denoting the carrierfrequency (MHz). Then the noise following the Gaussian dis-tribution with non-zero mean added to path loss is illustratedin Fig. 1. Resulting of multipath noise, the distribution ofreceived signal strength is no longer a smooth function.

Fig. 1. Distribution of received signal power in WSN.

Received signal strength values are regarded as states ofnodes and exchanged according to UBGA which can guaranteean average consensus on the different initial values both inexpectation and in the second moment. We take the symmetricnetwork into consideration which can be abstracted as anundirected graph for the convenience of analysis. UBGA isintroduced as follows. Please refer to [1] for more details.

Compared with previous pair-wise gossip algorithm withonly one variable for each node, there are two variables for anynode i in UBGA where xi(t) denotes the state of node i after titerations, and yi(t) denotes the according companion variableinitialized as yi(0) = 0. At any random time slot determinedby the clock on every node, one node already initiated ischosen to broadcast its state and companion variables toits neighbors according to the underlying connection graph,followed by the update of state and companion variables ofdifferent nodes throughout the network. According to UBGA,the update of the states xj(t) and companion variables yj(t)of receiving nodes j ∈ N−

k is as follows.

xj(t+ 1) = 0.5xj(t) + 0.5xk(t) +ε

δ+j(2)

yj(t+ 1) = 0.5xj(t)− 0.5xk(t) + (1− ε

δ+j)yj(t)

+1

δ−kyk(t) (3)

where δ−k = |N−k | denotes the number of out-neighbors of

node k with δ+j = |N+j | the number of in-neighbors of node

j.By defining the best perturbation parameter ε = ε∗ (see

details in [1]) then the state and companion variables of thetransmitting node k update as

xk(t+ 1) = xk(t) (4)

yk(t+ 1) = 0 (5)

and the state and companion variables of idle nodes i /∈k ∪N−

k update as

xi(t+ 1) = xi(t) (6)

yi(t+ 1) = yi(t) (7)

UBGA is certified to achieve an average consensus oninitial states of different nodes after limited enough exchangesthroughout the network in expectation and in the secondmoment sense [1].

III. UBGA FOR DISTRIBUTED COMPRESSION AND BLINDZONE RECOGNITION

With UBGA we can simultaneously compute and transferrandom projections or compressed data. The noise followsnon-zero mean distribution the mean of which can be regardedas the path loss and the rest part of the noise can be seen aswhite Gaussian noise. So that we can still use the distributedcompressed sensing technology to reconstruct the originalsignal from the noisy one.

We exploit the eigenvectors of the graph Laplacian as atransform of the original signal [6]. In general networks,compressed data of any node i can be expressed as yi =∑n

j=1 Ai,jxj , a simple linear combination of RSS at all nodes,which can be accomplished in a distributed way. Every nodej = 1, . . . , n locally computes its i = 1, . . . , k parallelinitial values xi,j(0) = nAi,jxj , i = 1, . . . , k, j = 1, . . . , nby multiplying its measurement xj with the correspondingelement of the compressing matrix Ai,j which are the i.i.d.and zero-mean normal random variables with variance 1/n.The distribution of local terms nAi,jxj , i = 1, . . . , k, j =1, . . . , n and the compression of the data of any node i i.e.yi =

∑nj=1 Ai,jxj can be accomplished simultaneously across

the network by UBGA. At the end of UBGA iterations, thestates of every node reaches a consensus on the averageof weighted initial states of every nodes xj , j = 1, . . . , nrespectively as yi =

∑nj=1 Ai,jxj , i = 1, . . . , k. Stacking

305

yi, i = 1, . . . , k into a vector y, one can recover an estimateof the signal x by querying any sensor node i their statesi.e. {yi,j(t)}, j = 1, . . . , n in the network. The resultingobjective function can is min

θ‖y − ATT θ‖2 + τ

∑ni=1 |θi|,

where τ > 0 denotes a regularization parameter with T thecompressing transformation matrix. We exploit the gradientprojection strategy to accomplish the reconstruction of thecompressed sensing, i.e. the convex optimization problemaccording to the objective function as in yi.

IV. PERFORMANCE ANALYSIS

In this section, we focus on the theoretical analysis andcomputer simulations on the performance of UBGA baseddistributed compression and the blind zone recognition. Allsimulations are based on undirected random geometric graphabstracted from the underlying network topology with thestandard connectivity radius

√2 log n/n, where n denotes the

number of sensor nodes randomly distributed within a unitsquare zone.

A. Field Estimation

Firstly, we theoretically analyze the performance of thedistributed compression and blind zone recognition algorithm.We are trying to obtain accurate reconstruction xi of thetrue xi by solving optimizing problem min

θ‖y − ATT θ‖2 +

τ∑n

i=1 |θi|. Suppose that x is compressible in the orthogonalbasis {ϕi}ni=1. Let x(m) denote the best m-term approximationof x. The noise power is denoted as σ2. As noted above,x = [x1, . . . , xn]

T can be expressed as x =∑n

i=1 θiϕi

where θi = xTϕi is the ith coefficient of x in the basis{ϕi}ni=1, resulting in x = Tθ where θ = col{θ1, . . . θn}and T denotes the collection of transform matrices. DefineΘ = {x : ‖Tθ‖2 ≤ nB2}(θi is uniformly quantized tonp levels) where B denotes a known constant meaning theboundary of the average element energy. We regard x as α-compressible if the best m-term approximation error obeys thefollowing limitation for some α > 0.

‖x− x(m)‖2n

≤ const · m−2α (8)

Sampling the signal i.e. the elements of the signal vectorat k(k � n) random locations, we can assume that the bestm-term approximation error obeys the oracle inequality [11]as

E

[‖x− x(m)‖2n

]≤ C1 min

θ∈Θ

{‖θ − θ∗‖2n

+c(θ) log 2 + 4

kε

}(9)

where

C1 =(27− 4e)S2 + (50− 4

√2)S + 26

(23− 4e)S2 + (50− 4√2)S + 24

(S = B/σ) (10)

and

c(x) = c(θ) = (1+p) log(n)‖θ‖0, ε = 1/(50(B+σ)2) (11)

Let the vector θ(m) denote the collection of coefficientsin the best m-term approximation of x and θ

(m)q denote the

nearest element in Θ. Substitute θ = θ(m)q in the oracle bound

(9), we can get

‖θ(m)q − θ∗‖2 = ‖θ(m)

q − θm + θm − θ∗‖2≤ ‖θ(m)

q − θm‖2 + 2‖θ(m)q − θm‖2

· ‖θ(m) − θ∗‖+ ‖θ(m) − θ∗‖2

≤ CH

n2p−2+ 2m−α

√nCACH

n2p−2

+ nCAm−2α (12)

We can deduce the following inequality with the factc(θ(m)q

)= (1 + p)m log(n).

E

[‖x− x(m)‖2n

]≤ C1 min

m

{CH

n2p−1+ 2m−α

√nCACH

n2p−2

+ CAm−2α +

(1 + p)m logn log 2

kε

+4

kε

}(13)

Balancing the third and fourth terms gives

m =

((1 + p) log 2

εCA

) −12α+1

(k

log n

) 12α+1

(14)

then we can get

CAm−2α = CA

((1 + p) log 2

εCA

) 2α2α+1

(k

log n

) −2α2α+1

(15)

With k > 1 and n > e, the following inequality holds.

E

[‖x− x(m)‖2n

]≤ const ·

(k

logn

) −2α2α+1

(16)

It can be shown that the best m-term approximation oforiginal signal can be recovered from k random projectionswith an average square error upper bounded by a constant

multiplying(

kl log n

) −2α2α+1

. The result theoretically guaranteesthe performance of the consistent estimation of the originalsignal even when the received noisy signal is only approxi-mately sparse.

Next, we take computer simulations to demonstrate theperformance of the proposed approach for distributed com-pression and the blind zone recognition. Fig. 2(a) shows theactual received signal power across the network. Fig. 2(b)shows the sampled original signal power and the best m-termapproximation coefficients in the transform basis respectivelyfrom which it is clear that a small number of coefficientscan represent the majority of the energy or information of

306

the original signal, so that the original signal is compressiblein such transform basis. In addition, the original signal iscompressible when the collection of the eigenvectors of thegraph Laplacian are adopted as the transform basis [6].

(a)

0 50 100 150 200 250 3000

100

200

300

400

500

600

700

800

900

1000

Number of terms in approximation(m)

Mea

n−Sq

uare

d A

ppro

xim

atio

n Er

ror

Orignal SignalTransformed Signal

(b)Fig. 2. Example illustrating compression of a signal using the eigenvectors ofthe graph Laplacian as a transform. (a) The actual received signal power acrossthe network. (b) The original signal in the original basis and the transformedcoefficient (sorted) in the transformation basis.

Fig. 3 shows the continuous reconstructed image obtainedby the image interpolation with k = 100. We define the blindzones where the received signal strength (RSS) do not meet therequired QoS. If we determine a threshold, we can obtain thecorresponding blind zone in Fig. 3. We adopt the mean squared

error 1n

∥∥∥x(t)− 1n11Tx(0)

∥∥∥22

(MSE) as an error metric. Notethat the more random linear projections (larger k) are used thelower the reconstruction and approximation error are, whilethe number of measurements k used is much smaller than thenumber of sensor nodes across the network n, i.e. k � n. TheMSEs of the blind zone recognition are listed in Table I withk = 10, 50, 100, 150 respectively, which indicates the tradeoffbetween k and the accuracy of the blind zone recognition.

B. Comparison Setup

We compare the proposed solution in this paper with aprevious distributed compression and blind zone recognition

Fig. 3. k = 100 the continuous reconstructed image.

approach [8] using pair-wise gossip algorithm (PGA). Fig. 4(a)

shows the variance q(t) = 1n

∥∥∥x(t)− 1n11Tx(t)

∥∥∥22

whichindicates how quickly the algorithms converge to a consensus.It is clear that UBGA converges to the average consensusfaster than PGA. In addition, lower variance means lower datavariation of different nodes in the network. Then, reconstructedimages from different nodes have less difference. Namely,blind zone recognition results from different node will becloser, as expected. From fig. 4(b) shows that when the numberof gossip iterations is fewer, UBGA is better than PGA in bothconvergence rate and convergence accuracy. When the numberof gossip iterations is more, the follow-up convergence rateof UBGA is slower than PGA. However, this two algorithmsalmost have achieved convergence at this time. Hence, theperformance difference under more gossip iterations is oflimited significance.

TABLE IMSE OF BLIND ZONE RECOGNITION

k 10 50 100 150

mse 1.1222 0.00082465 0.00030909 0.00018726

V. CONCLUSION

In this paper, an efficient, robust and scalable spatiallydistributed algorithm on compression and the blind zonerecognition in wireless fading channel is proposed whose per-formance is verified by theory and simulations. One can obtainsufficient information to make an accurate approximation ofthe entire network data and recognize blind zone by queryingany node in the network by the proposed distributed algorithm.More parallel initial values k for reconstruction can be adoptedto improve the distributed system performance at the cost ofcomputation according to the required recognition accuracy.The distributed signal coverage algorithms, which enhance thereceived signal strength in the blind zones recognized by thedistributed algorithm proposed in this paper, will be developedin the near future.

307

(a)

(b)Fig. 4. (a) the variance of UBGA with pair-wise gossip algorithm. (b)the comparison of UBGA with pair-wise gossip algorithm with respect toestimation MSE when k = 100 after finite gossip iterations.

ACKNOWLEDGMENTS

This work was supported by the National Basic ResearchProgram of China (No. 2013CB329003) and the NationalScience Foundation of China under grant (No. 61201147).

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