research article a sparse signal reconstruction algorithm...

Research ArticleA Sparse Signal Reconstruction Algorithm inWireless Sensor Networks

Zhi Zhao and Jiuchao Feng

School of Electronic and Information Engineering South China University of Technology Guangzhou 510641 China

Correspondence should be addressed to Zhi Zhao zhaozhiperfect163com

Received 29 December 2015 Revised 15 April 2016 Accepted 20 April 2016

Academic Editor Xiumei Li

Copyright copy 2016 Z Zhao and J FengThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study reconstruction of time-varying sparse signals in a wireless sensor network where the bandwidth and energy constraintsare considered severely A novel particle filter algorithm is proposed to deal with the coarsely quantized innovation To recover thesparse pattern of estimate by particle filter we impose the sparsity constraint on the filter estimate by means of two methodsSimulation results demonstrate that the proposed algorithms provide performance which is comparable to that of the fullinformation (ie unquantized) filtering schemes even in the case where only 1 bit is transmitted to the fusion center

1 Introduction

In recent years wireless sensor networks (WSNs) have beenwidely applied in many areas A WSN system is composedof a large number of battery-powered sensors via wirelesscommunication Reconstruction of time-varying signals isa key technology for WSNs and plays an important role inmany applications ofWSNs (see eg [1ndash3] and the referencestherein) As we know the lifetime of a WSN depends on thelifespan of its sensors which are battery-powered To prolongthe lifespan of sensors sensors are often allowed to transmitonly partial (eg quantizedencoded) information to a fusioncenter Therefore quantization of sensor measurements hasbeen widely taken into account in the practical applications[4ndash6] Moreover it is maybe infeasible to quantize andtransmit sensor measurements directly This is because forunstable systems while the states will become unboundeda large number of quantization bits may be needed andso higher bandwidth and rate of quantizer are used bysensors However as demonstrated in [1ndash3] the filteringschemes relying on the quantized innovations can providethe performance which is comparable to that of the full (egunquantizeduncoded) information filtering schemes

On the other hand due to sparseness of signals exhibitedinmany applications recently developed compressed sensingtechniques have been extensively applied in WSNs [7ndash9]This enables reconstruction of sparse signal from far fewer

measurements Therefore the demands for communicationbetween sensors and the fusion center will be lessened byexploiting sparseness of signals in WSNs so as to saveboth bandwidth and energy [9 10] Reconstruction of time-varying sparse signals in WSNs has been recently studiedin [11] by using the group lasso and fused lasso techniquesThis is a batch algorithm which relies on quadratic pro-gramming to recover the unknown signal A computationallyefficient recursive lasso algorithm (R-lasso) was introducedin [12] for estimating recursively the sparse signal at eachpoint in time In [13] the SPARLS algorithm relies on theexpectation-maximization technique to find estimates of thetap-weight vector output stream from its noisy observationsRecently many researchers have attempted to solve theproblem in the classic framework of signal estimation suchas Kalman filtering (KF) and its variants [14ndash16] The KF-based approaches can be divided into two classes the hybridand self-reliant For the former the peripheral optimizationschemes were employed to estimate the support set of asparse signal and then a reduced-order Kalman filter wasused to reconstruct the signal [14] Meanwhile for the latterthe sparsity constraint is enforced via the so-called pseudo-measurement (PM) [15] In [15] two stages of Kalmanfiltering are employed one for tracking the temporal changesand the other for enforcing the sparsity constraint at eachstage In [16] an unscented Kalman filter for the pseudo-measurement update stage was proposed To the best of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 9410873 11 pageshttpdxdoiorg10115520169410873

2 Mathematical Problems in Engineering

the authorsrsquo knowledge there is limited work on recursivecompressed sensing techniques considering quantization asa mean of further reduction of the required bandwidthand power resources However an increased attention hasbeen paid to develop algorithms for reconstructing sparsesignals using quantized observations recently [17ndash21] In [17]a convex relaxation approach for reconstructing sparse signalfrom quantized observations was proposed by using an ℓ

1-

norm regularization term and two convex cost functions In[18] Qiu and Dogandzic proposed an unrelaxed probabilisticmodel with ℓ

0-norm constrained signal space and derived

an expectation-maximization algorithm In addition [19ndash21]have investigated the reconstruction of sparse source from 1-bit quantized measurement in the extreme case

In this paper we study reconstruction of time-varyingsparse signals in WSNs by using quantized measurementsOur contributions are as follows

(1) We propose an improved particle filter algorithmwhich extends the fundamental results [2] to amultiple-observation case by employing informationfilter form The algorithm in [2] is derived under theassumption that the fusion center has access to onlyonemeasurement source at each time stepThe exten-sion to a multiple-measurement scenario is straight-forward but in generalmay lead to a computationallyinvolved scheme In contrast the proposed algorithmcan be implemented in a computationally efficientsequential processing form and avoids any matrixinversion Meanwhile the proposed algorithm hasan advantage over numerical stability for inaccurateinitialization

(2) We propose a new method to impose sparsity con-straint on estimator by the particle filter algorithmCompared to the iterative method in [15] the result-ing method is noniterative and easy to implement

In particular the system has an underlying state-spacestructure where the state vector is sparse In each timeinterval the fusion center transmits the predicted signalestimate and its corresponding error covariance to a selectedsubset of sensors The selected sensors compute quantizedinnovations and transmit them to the fusion center Thefusion center reconstructs the sparse state by employing theproposed particle filter algorithm and sparse cubature pointfilter method

This paper is organized as follows Section 2 gives abrief overview of basic problems in compressed sensing andintroduces the sparse signal recovery method using Kalmanfiltering with embedded pseudo-measurement In Section 3we describe the system model Section 4 develops a particlefilter with quantized innovation To recover the sparsitypattern of the state estimate by particle filter a sparse cubaturepoint filter method is developed with lower complexitycompared to reiterative PM update method in Section 5The intact version of adaptively recursive reconstructionalgorithm for sparse signals with quantized innovations andthe analysis of their complexity are presented in Section 6

Section 7 contains simulation results and the conclusions areconcluded in Section 8

NotationN119889(120583 Σ) denotes 119889-dimensional Gaussian rv with

mean 120583 and variance Σ the 119889-dimensional Gaussian prob-ability density with mean 120583 and variance Σ is denoted by120601119889(sdot 120583 Σ) Φ(119909 120583 Σ) denotes Gaussian probability distribu-

tion with mean 120583 and variance Σ and Φ(S 120583 Σ) denotesthe truncated probability where S belongs to Borel 120590-fieldu119894119895

denotes the collection of random 119906119894 119906

119895 Boldfaced

uppercase and lowercase symbols represent matrices andvectors respectively For a vector x x(119894) denotes its 119894thcomponent and Cov[x] denotes the error covariance 119864[(x minus119864x)(x minus 119864x)119879] For a matrix R R(119897 119897) denotes the (119897 119897) entryof R

2 Sparse Signal Reconstruction UsingKalman Filter

21 Sparse Signal Recovery Compressive sensing is a frame-work for signal sensing and compression that enables repre-sentation of a sensed signal with fewer samples than thoseones required by classical sampling theory Consider a sparserandom discrete-time process x

119896119896ge0

in 119877119873 where x1198960≪

119873 and x119896is called 119870-sparse if x

1198960= 119870 Assume x

119896evolves

according to the following dynamical equations

x119896+1

= F119896x119896+ w119896 (1)

y119896= H119896x119896+ k119896 (2)

where F119896isin 119877119873times119873 is the state transition matrix and H

119896isin

119877119872times119873 is the measurement matrix Moreover w

119896119896ge0

andk119896119896ge0

denote the zero meanrsquos white Gaussian sequence withcovariances W

119896⪰ 0 and R

119896⪰ 0 respectively y

119896is 119872-

dimensional linear measurement of x119896 When 119872 lt 119873 and

rank(H119896) lt 119873 it is noted that the reconstruction x

119896from

underdetermined system is an ill-posed problemHowever [22 23] have shown that x

119896can be accurately

reconstructed by solving the following optimization problem

minx119896isin119877119873

1003817100381710038171003817x11989610038171003817100381710038170

st 1003817100381710038171003817y119896 minusH119896x119896

1003817100381710038171003817

2

2le 120598

(3)

Unfortunately the above optimization problem is NP-hard and cannot be solved effectively Fortunately as shownin [23] if the measurement matrix H

119896obeys the so-called

restricted isometry property (RIP) the solution of (3) canbe obtained with overwhelming probability by solving thefollowing convex optimization

minx119896isin119877119873

1003817100381710038171003817x11989610038171003817100381710038171

st 1003817100381710038171003817y119896 minusH119896x119896

1003817100381710038171003817

2

2le 120598

(4)

This is a fundamental result in compressed sensing (CS)Moreover for reconstructing a119870-sparse signal x

119896isin 119877119873119872 ge

119862 sdot 119870 log(119873119870) linear measurements are needed where 119862 isa fixed constant

Mathematical Problems in Engineering 3

22 Pseudo-Measurement Embedded Kalman Filtering Forthe system given in (1) and (2) the estimation of x

119896provided

by Kalman filtering is equivalent to the solution of thefollowing unconstrained ℓ

2minimization problem

minx119896isin119877119873

119864 [1003817100381710038171003817x119896 minus x

119896

1003817100381710038171003817

2

2| Y119896] (5)

where 119864[sdot | Y119896] is the conditional expectation of the given

measurementsY119896≜ y1 y

119896

As shown in [15] the stochastic case of (4) is as follows

minx119896isin119877119873

1003817100381710038171003817x11989610038171003817100381710038171

st 119864 [1003817100381710038171003817x119896 minus x

119896

1003817100381710038171003817

2

2| Y119896] le 120598119896

(6)

and its dual problem is

minx119896isin119877119873

119864 [1003817100381710038171003817x119896 minus x

119896

1003817100381710038171003817

2

2| Y119896]

1003817100381710038171003817x11989610038171003817100381710038171 le 120598119896

(7)

In [15] the authors incorporate the inequality constraintx1198961le 120598119896into the filtering process using a fictitious pseudo-

measurement equation

0 = H119896x119896minus 1205981015840

119896 (8)

where H119896

= sign(x119879119896) and 120598

1015840

119896sim N(0 119877

1015840

120598) serves as

the fictitious measurement noise constrained optimizationproblem (7) can be solved in the framework of Kalmanfiltering and the specificmethod has been summarized as CS-embedded KF (CSKF) algorithm with ℓ

1-norm constraint in

[15] It is apparent from (8) that themeasurementmatrix H119896is

state-dependent and can be approximated by H119896= sign(x119879

119896)

where sign(sdot) is the sign function The pseudo-measurementequation was interpreted in Bayesian filtering frameworkand a semi-Gaussian prior distribution was discussed in [15]Furthermore 1198771015840

120598is a tuning parameter which regulates the

tightness of ℓ1-norm constraint on the state estimate x

119896

3 System Model and Problem Statement

Consider a WSN configured in the star topology (see Fig-ure 1 for an example topology) In the star topology thecommunication is established between sensors and a singlecentral controller called the fusion center (FC) The FC ismains powered while the sensors are battery-powered andbattery replacement or recharging in relatively short intervalsis impractical The data is exchanged only between the FCand a sensor In our application 119872 sensors observe linearcombinations of sparse time-varying signals and send theobservations to a fusion center for signal reconstructionHere our attention is focused on Gaussian state-space mod-els that is for sensor 119897 the signal and the observation satisfythe following discrete-time linear system

x119896+1

= F119896x119896+ w119896

119910119897119896= h119897119896x119896+ V119897119896

(9)

Fusion centerSensorCommunication flow

Figure 1 Network topology

where h119897119896

isin 1198771times119873 is the local observation matrix and

x119896isin 119877119873 denotes time-varying state vector which is sparse

in some transform domain that is x119896= Ψs

119896 where the

majority of components of s119896are zero andΨ is an appropriate

basis Without loss of generality we assume that x119896itself

is sparse and has at most 119870 nonzero components whoselocations are unknown (119870 ≪ 119873) The fusion center gathersobservations at all 119872 sensors in the 119872-dimensional globalreal-valued vector y

119896and preserves the global observation

matrixH119896= [ h1198791119896

h1198792119896

sdot sdot sdot h119879119872119896

]119879isin 119877119872times119873 which satisfies

the so-called restricted isometry property (RIP) imposed inthe compressed sensing Then the global observation modelcan be described in (2) All the sensors are unconcerned aboutthe sparsity Moreover w

119896and k119896are uncorrelated Gaussian

white noise with zero mean and covariances W119896and R

119896

respectivelyThe goal of theWSN is to forman estimate of sparse signal

x119896at the fusion center Due to the energy and bandwidth

constraint inWSNs the observed analogmeasurements needto be quantizedcoded before sending them to the fusioncenter Moreover the quantized innovation scheme also canbe used At time 119896 the 119897th sensor observes a measurement119910119897119896

and computes the innovation 119890119897119896= 119910119897119896minus h119897x119896|119896minus1

whereh119897x119896|119896minus1

together with the variance of innovation Cov[119890119897119896]

is received from the fusion center Then the innovation 119890119897119896

is quantized to 119902119897119896

and sent to the fusion center As thefusion center has enough energy and enough transmissionbandwidth the data transmitted by the fusion center donot need to be quantized The decision of which sensoris active at time 119896 and consequently which observationinnovation 119890

119897119896gets transmitted depends on the underlying

scheduling algorithmThe quantized transmission of 119890119897119896also

implies that 119902119897119896can be viewed as a nonlinear function of the

sensorrsquos analog observation The aforementioned procedureis illustrated in Figure 2

4 A Particle Filter Algorithm with CoarselyQuantized Observations

Most of the earlier works for estimation using quantizedmea-surements concentrated upon using numerical integration


Processxk

y1k y2k yMk

S1 S2 SM

q1k q2k qMk

Feedback

Feedback

Fusioncenter Feedback

middot middot middot

Figure 2 System model

methods to approximate the optimal state estimate and makean assumption that the conditional density is approximatelyGaussian However this assumption does not hold forcoarsely quantized measurements as demonstrated in thefollowing

Firstly suppose x119896 and 119910

119897119896 are jointly Gaussian then

it is well known that the probability density of x119899conditioned

on y1198970119896

is a Gaussian with the following parameters

x119896| y1198970119896

sim 120578119896+ Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896

where 120578119896simN119889(0Σx119896 minus Σx119896y1198970119896Σ

minus1

y1198970119896Σy1198970119896x119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

≜ΣΔ

x119896y1198970119896

)

(10)

where y1198970119896

≜ 1199101198970 119910

119897119896 When x

119896 and 119910

119897119896 follow the

linear dynamical equations defined in (9) it is well knownthat the covariance ΣΔx119896y1198970119896 ≜ P

119896|119896= Cov[x

119896minus119864[x119896| y119897119896]] can

be propagated by Riccati recursion equation of KF Let 119902119897119896

denote the quantized measurements obtained by quantizing119910119897119896 that is 119902

119897119896 is a measurable function of 119910

1198970119896 It will

be shown that the probability density of x119896conditioned on the

quantizedmeasurements q1198970119896

≜ 1199021198970 119902

119897119896 has the similar

characterization as (10) The result is stated in the followinglemma

Lemma 1 (akin to Lemma 31 in [2]) The state x119896conditioned

on the quantized measurements q1198970119896

can be given by a sum oftwo independent random variables as follows

x119896| q1198970119896

sim 120578119896+ Σx119896y1198970119896Σ

minus1

y1198970119896 [y1198970119896 | q1198970119896]

where 120578119896simN119889(0ΣΔ

x119896y1198970119896) (11)

Proof See Appendix

It should be noted that the difference between (10) and (11)is the replacement of y

1198970119896by random variable y

1198970119896| q1198970119896

Apparently y

1198970119896| q1198970119896

is a multivariate Gaussian random

variable truncated to lie in the region defined by q1198970119896

So thecovariance of x

119896| q1198970119896

can be expressed as

Cov [x119896| q1198970119896]

= ΣΔ

x119896y1198970119896

+ Σx119896y1198970119896Σminus1

y1198970119896Cov [y1198970119896 | q1198970119896]Σminus1

y1198970119896Σy1198970119896x119896

(12)

Under an environment of high rate quantization it isapparent that y

1198970119896| q1198970119896

converges to y1198970119896

and x119896| q1198970119896

approximates Gaussian From Lemma 1 we note that x119896|

q1198970119896

is not Gaussian For nonlinear and non-Gaussian signalreconstruction problems a promising approach is particlefiltering [24] The particle filtering is based on sequentialMonte Carlo methods and the optimal recursive Bayesianfiltering It uses a set of particles with associated weights toapproximate the posterior distribution As a bootstrap thegeneral shape of standard particle filtering is outlined below

Algorithm 0 (standard particle filtering (SPF))

(1) Initialization Initialize the119873119901particles x119894

0|minus1sim 119901(x

0)

and x0|minus1

= 0(2) At time 119896 using measurement 119902

119897119896= 119876(119910

119897119896) the

importance weights are calculated as follows 120596119894119896=

119901(119902119897119896| x119896= x119894119896|119896minus1

q1198970119896minus1

)(3) Measurement update is given by

x119901119891119896|119896=

119873119901

sum

119894=1

120596119894

119896x119894119896|119896minus1

(13)

where 120596119894119896are the normalized weights that is

120596119894

119896=

120596119895

119896

sum119873119901

119894=1120596119894

119896

(14)

(4) Resample 119873119901particles with replacement as follows

Generate iid random variables 119869120580119873119901

120580=1such that

119875(119869120580= 119894) = 120596

119894

119896

x120580119896|119896= x119869120580119896 (15)

(5) For 119894 = 1 119873119901 predict new particles according to

x119895119896+1|119896

sim 119901 (x119896+1

| x119896= x119894119896|119896 q1198970119896)

ie x119895119896+1|119896

= F119896x119894119896|119896

(16)

(6) Consider x119901119891119896+1|119896

= F119896x119901119891119896|119896 Also set 119896 = 119896 + 1 and

iterate from Step (2)

Assume that the channel between the sensor and fusioncenter is rate-limited severely and the sign of innovationscheme is employed (ie 119902

119897119896= sign(119910

119897119896minus 119897119896|119896minus1

))


Obviously the importance weights are given by 120596119894

119896=

Φ(119902119897119896h119897(x119894119896|119896minus1

minus x119896|119896minus1

) 0R119890119896(119897 119897)) We note that 119864[x

119896|

q1198970119896] = Σx119896y1198970119896Σ

minus1

y1198970119896119864[y1198970119896 | q1198970119896] Therefore it will be

sufficient to propagate particles that are distributed as 120585119896|

q1198970119896

where

120585119896= Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896 (17)

In addition note that the quantizer output 119902119897119896at time 119896

is calculated by quantizing a scalar valued function of 119910119897119896

q1198970119896minus1

So on receipt of 119902119897119896

and by using the previouslyreceived quantized values q

1198970119896minus1 some Borel measurable set

containing 119910119897119896 that is 119910

119897119896isin S119896q1198970119896 can be inferred at the

fusion centerIn order to develop a particle filter to propagate 120585

119896| q1198970119896

we need to give the measurement update of the probabilitydensity 119901(120585

119896minus1| q1198970119896minus1

) rarr 119901(120585119896minus1

| q1198970119896) and time update of

the probability density 119901(120585119896minus1

| q1198970119896) rarr 119901(120585

119896| q1198970119896) which

are described by Lemmas 2 and 3 respectively

Lemma 2 The likelihood ratio between the conditional laws of120585119896minus1

| q1198970119896

and 120585119896minus1

| q1198970119896minus1

is given by

119901 (120585119896minus1

| q1198970119896)

119901 (120585119896minus1

| q1198970119896minus1

)prop Φ(S

119896q1198970119896 h119897119896F119896120585119896minus1R119890119896 (119897 119897)) (18)

So if 120585119894119896minus1|119896minus1

119894is a set of particles distributed by the law

119901(120585119896minus1

| q1198970119896minus1

) then from Lemma 2 a new set of particles120585120580

119896minus1|119896120580can be generated For each particle 120585119894

119896minus1|119896minus1 associate

a weight 120596119894 = Φ(S119896q1198970119896 h119897119896F119896120585119896minus1R119890119896(119897 119897)) generate iid

random variables 119869120580 such that 119875(119869

120580= 119894) prop 120596

119894 and set120585120580

119896minus1|119896= 120585119869120580

119896minus1|119896minus1 This is the standard resampling technique

from Steps (3) and (4) of Algorithm 0 [25] It should be notedthat this is equivalent to ameasurement update since we updatethe conditional law 119901(120585

119896minus1| q1198970119896minus1

) by receiving the newmeasurement 119902

119897119896

Lemma 3 The random variable 119910119897119896| 120585119896minus1 q1198970119896

is a truncatedGaussian and its probability density function can be expressedas 120601(S

119896q1198970119896 h119897119896F119896120585119896minus1R119890119896(119897 119897))This result should be rather obvious Here one can observe

that 120585119896is the MMSE estimate of the state x

119896given y

1198970119896 Since

x119896 and 119910

119897119896 have the state-space structure Kalman filter can

be employed to propagate 120585119896recursively as follows

120585119896|119896= F119896120585119896|119896minus1

+ K119896(119910119897119896minus h119897119896F119896120585119896minus1|119896minus1

)

K119896=

P119896|119896minus1

h119879119897119896

h119897119896P119896|119896minus1

h119879119897119896+ R119896(119897 119897)

(19)

However the information filter (IF) which utilizes the infor-mation states and the inverse of covariance rather than thestates and covariance is the algebraically equivalent form ofKalman filter Compared with the KF the information filteris computationally simpler and can be easily initialized withinaccurate a priori knowledge [26] Moreover another greatadvantage of the information filter is its ability to deal with

multisensor data fusion [27] The information from differentsensors can be easily fused by simply adding the informationcontributions to the information matrix and information state

Hence we substitute the information form for (19) asfollows

Y119896|119896= Y119896|119896minus1

+I119896

z119896|119896= z119896|119896minus1

+ i119896

(20)

where Y119896|119896

= Pminus1119896|119896

and z119896|119896

= Y119896|119896120585119896|119896

are the informa-tion matrix and information state respectively In additionthe covariance matrix and state can be recovered by usingMATLABrsquos leftdivide operator that is P

119896|119896= Y119896|119896 I119873and

120585119896|119896

= Y119896|119896

z119896|119896 where I

119873denotes an 119873 times 119873 identity

matrixThe information state contribution i119896and its associated

information matrixI119896are

I119896= h119879119897119896Rminus1119896(119897 119897) h

119897119896

i119896= h119879119897119896Rminus1119896(119897 119897) 119910

119897119896

(21)

Together with (20) Lemma 3 completely describes thetransition from 119901(120585

119896minus1| q1198970119896) to 119901(120585

119896| q1198970119896) Following

suit with Step (5) of Algorithm 0 suppose 120585120580119896minus1|119896

120580is a set

of particles distributed as 119901(120585119896minus1

| q1198970119896) then a new set of

particles 120585119894119896|119896119894 which are distributed as 119901(120585

119896| q1198970119896) can be

obtained as follows For each 120585120580119896minus1|119896

generate 119910119894119897119896|119896 by the law

described as follows

119901 (119910119897119896| 120585120580

119896minus1|119896 q1198970119896)

= 120601 (S119896q1198970119896 h119897F119896120585

120580

119896minus1|119896R119890119896(119897 119897))

(22)

Also set z119894119896|119896= z120580119896|119896minus1

+ h119879119897119896Rminus1119896(119897 119897)119910119894

119897119896|119896

From the above the particle filter using coarsely quan-tized innovation (QPF) has been derived for individual sen-sor The extension of multisensor scenario will be describedin Section 6

5 Sparse Signal Recovery

To ensure that the proposed quantized particle filteringscheme recovers sparsity pattern of signals the sparsityconstraints should be imposed on the fused estimate thatis (x119896|119896) Here we can make the sparsity constraint enforced

either by reiterating pseudo-measurement update [15] or viathe proposed sparse cubature point filter method

51 Iterative Pseudo-Measurement Update Method As statedin Section 2 the sparsity constraint can be imposed at eachtime point by bounding the ℓ

1-norm of the estimate of the

state vector x119896|1198961le 120598119896 This constraint is readily expressed

as a fictitious measurement 0 = x119896|1198961minus 1205981015840

119896 where 120598

119896

can be interpreted as a measurement noise [15 28] Now


we construct an auxiliary state-space model of the form asfollows

120574120591+1

= 120574120591

0 = H120591120574120591minus 120598120591

(23)

where 1205741|1

= x119896|119896

and P1199011198981|1

= P119896|119896

and H120591

=

[sign(1120591|120591

) sdot sdot sdot sign(119873120591|120591

)] 120591 = 1 2 119871 119895120591|120591

denotesthe 119895th component of the least-mean-square estimate of 120574

120591

(obtained via Kalman filter) Finally we reassign x119896|119896

= 119871|119871

and P119896|119896

= P119901119898119871|119871

where the time-horizon of auxiliary state-space model (23) 119871 is chosen such that

119871|119871minus 119871minus1|119871minus1

2 is

below some predetermined threshold This iterative proce-dure is formalized below

120591+1|120591+1

= 120591|120591minus

P119901119898120591|120591

sign (120591|120591)10038171003817100381710038171003817120591|120591

100381710038171003817100381710038171

sign (120591|120591)119879

P119901119898120591|120591

sign (120591|120591) + 1198771015840120598

(24)

P119901119898120591+1|120591+1

= P119901119898120591|120591minus

P119901119898120591|120591

sign (120591|120591) sign (

120591|120591)119879

P119901119898120591|120591

sign (120591|120591)119879

P119901119898120591|120591

sign (120591|120591) + 1198771015840120598

(25)

52 Sparse Cubature Point Filter Method Suppose a novelrefinement method based on cubature Kalman filter (CKF)Compared to the iterative method described above theresulting method is noniterative and easy to implement

It is well known that unscented Kalman filter (UKF) isbroadly used to handle generalized nonlinear process andmeasurement models It relies on the so-called unscentedtransformation to compute posterior statistics of ℏ isin 119877119898 thatare related to x by a nonlinear transformation ℏ = f(x) Itapproximates themean and the covariance of ℏ by a weightedsum of projected sigma points in the 119877119898 space However thesuggested tuning parameter for UKF is 120581 = 3 minus 119873 For ahigher order system the number of states is far more thanthree so the tuning parameter 120581 becomes negative and mayhalt the operation Recently a more accurate filter namedcubature Kalman filter has been proposed for nonlinear stateestimation which is based on the third-degree spherical-radial cubature rule [29] According to the cubature rule the2119873 sample points are chosen as follows

120594119904= x + radic119873(Sx)119904 120596

119904=

1

2119873

120594119873+119904

= x minus radic119873(Sx)119904 120596119873+119904

=1

2119873

(26)

where 119904 = 1 2 119873 and Sx isin 119877119873times119873 denotes the square-rootfactor of P that is P = SxS119879x Now the mean and covarianceof ℏ = f(x) can be computed by

119864 [x] = 1

2119873

2119873

sum

119904=1

120594lowast

119904

Cov [y] = 1

2119873

2119873

sum

119904=1

120594lowast

119904120594lowast119879

119904minus xx119879

(27)

where 120594lowast119904= f(120594119904) 119904 = 0 1 2119873

ek

z1k

zN119901

k

Resa

mpl

e

Pred

ictio

n

Method 1

ormethod 2

IF

IF

IF

Figure 3 Illustration of the proposed reconstruction algorithm

By the iterative PM method it should be noted that(24) can be seen as the nonlinear evolution process duringwhich the state gradually becomes sparse Motivated by thiswe employ CKF to implement the nonlinear refinementprocess Let P

119896|119896and 120594

119904119896|119896be the updated covariance and

the 119894th cubature point at time 119896 respectively (ie after themeasurement update) A set of sparse cubature points at time119896 is thus given by

120594lowast

119904119896|119896= 120594119904119896|119896

minus

P119896|119896

sign (120594119904119896|119896

)10038171003817100381710038171003817120594119904119896|119896

100381710038171003817100381710038171

sign (120594119904119896|119896

)119879

P119896|119896

sign (120594119904119896|119896

) + 120598

(28)

where 120598= O (

10038171003817100381710038171003817120594119904119896|119896

10038171003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598(29)

for 119904 = 1 2119873 where g isin 119877119873 is a tunable constant vectorOnce the set 120594lowast

119904119896|1198962119873

119904=1is obtained its sample mean and

covariance can be computed by (27) directly For readabilitywe defer the proof of (29) to Appendix

6 The Algorithm

We now summarize the intact algorithm as follows (illus-trated in Figure 3)

(1) Initialization at 119896 = 0 generate 119894

0|minus1 0|minus1

P0|minus1

then compute z119894

0|minus1 z0|minus1

Y0|minus1

(2) The fusion center transmits R

119890119896(119897 119897) which denote the

(119897 119897) entry of the innovation error covariance matrix(see (30)) and predicted observation

119897119896(see (31)) to

the 119897th sensor

R119890119896= H119896P119896|119896minus1

H119879119896+ R119896 (30)

119897119896|119896minus1

= h119897119896119896|119896minus1

(31)


(3) The 119897th sensor computes the quantized innovation(see (32)) and transmits it to the fusion center

119902119897119896= 119876(

119910119897119896minus 119897119896|119896minus1

R119890119896(119897 119897)

)R119890119896(119897 119897) (32)

(4) On receipt of quantized innovation (see (32)) theBorel 120590-field S

119896q1198970119896 can be inferred Then a set ofobservation particles (see (33)) and correspondingweights (see (34)) are generated by fusion center

119910119894

119897119896|119896sim 120601 (S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (33)

120596119894

119897119896sim Φ(S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (34)

(5) Run measurement updates in the information form(see (35)) using an observation particle y119894

119896|119896=

[119910119894

1119896|119896sdot sdot sdot 119910119872

119897119896|119896]119879 generated in Step (4)

Y119896|119896= Y119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) h

119897119896

z119894119896|119896= z119894119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) 119910

119894

119897119896|119896

(35)

(6) Resample the particles by using the normalizedweights

(7) Compute the fused filtered estimate z119896|119896

z119896|119896=

119873119901

sum

119894=1

120596119894

119896z119894119896|119896 (36)

where 120596119894119896= prod119872

119897=1120596119894

119897119896

(8) Impose the sparsity constraint on fused estimate 119896|119896

by either (a) or (b)

(a) iterative PM update method(b) sparse cubature point filter method

(9) Determine time updates z119894119896+1|119896

z119896+1|119896

Y119896+1|1

119897119896+1|119896

and P

119896+1|119896for the next time interval

z119894119896+1|119896

= F119896z119894119896|119896

z119896+1|119896

= F119896z119896|119896

Y119896+1|119896

= [F119896P119896|119896F119879119896+W119896+1]minus1

119896+1|119896

= Y119896+1|119896

z119896+1|119896

P119896+1|119896

= Y119896+1|119896

I119873

(37)

Remarks Here the use of symbol 120585 is just for algorithmdescription and also can be interchanged with x In additionit should be noted that the proposed algorithm amounts to119873119901Kalman filters running in parallel that are driven by the

observations 119910119894119897119896119873119901

119894=1

61 Computational Complexity The complexity of samplingStep (4) in the general algorithm is 119874(119873

119901) Measurement

update Step (5) is of the order119874(1198732119872)+119874(119873119873119901) resampling

Step (7) has a complexity 119874(119873119901) Step (9) has complexity

either 119874(1198732119871) or 119874(119873) The complexity of time update Step(10) is 119874(1198732119873

119901) + 119874(119873

2119872)

7 Simulation Results

In this section the performance of the proposed algo-rithms is demonstrated by using numerical experiment inwhich sparse signals are reconstructed from a series ofcoarsely quantized observations Without loss of generalitywe attempt to reconstruct a 10-sparse signal sequence x

119896 in

119877256 and assume that the support set of sequence is constant

The sparse signal consists of 10 elements that behave as arandom walk process The driving white noise covariance ofthe elements in the support of x

119896is set asW

119896(119894 119894) = 01

2 Thisprocess can be described as follows

x119896+1

(119894) =

x119896(119894) + w

119896(119894) if 119894 isin supp (x

119896)

0 otherwise(38)

where 119894 sim 119880int[1 256] and x0(119894) sim N(0 5

2) Both the

index 119894 sim supp(x119896) and the value of x

119896(119894) are unknown The

measurement matrix H isin 11987772 times 256 consists of entries that

are sampled according toN(0 172) This type of matrix hasbeen shown to satisfy the RIP with overwhelming probabilityfor sufficiently sparse signals The observation noise covari-ance is set as R

119896= 001

2I72 The other parameters are set as

x0|minus1

= 0 119873119901= 150 and 119871 = 100 There are two scenarios

considered in the numerical experiment The first one isconstant support and the second one is changing supportIn the first scenario we assume severely limited bandwidthresources and transmit 1-bit quantized innovations (iesign of innovation) We compare the performance of theproposed algorithms with the scheme considered in [15]which investigates the scenario where the fusion center hasfull innovation (unquantizeduncoded) For convenience werefer to the scheme in [15] as CSKF the proposed QPF withiterative PM update method and sparse cubature point filtermethod are referred to as Algorithms 1 and 2 respectively

Figure 4 shows how various algorithms track the nonzerocomponents of the signal The CSKF algorithm performsthe best since it uses full innovations Algorithm 1 per-forms almost as well as the CSKF algorithm The QPFclearly performs poorly while Algorithm 2 performs close toAlgorithm 1 gradually

Figure 5 gives a comparison of the instantaneous values ofthe estimates at time index 119896 = 100 All of three algorithmscan correctly identify the nonzero components

Finally the error performance of the algorithms is shownin Figure 6 The normalized RMSE (ie x

119896minus x1198962x1198962)

is employed to evaluate the performance As can be seenAlgorithm 1 performs better thanAlgorithm 2 and very closeto the CSKF before 119896 = 40 However the reconstructionaccuracies of all algorithms almost coincide with each other


0 20 40 60 80 100minus1

0

1

2

3

4

Time index Time index


0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

0 20 40 60 80 100minus15

minus1

minus05

0

05

0 20 40 60 80 100minus05

0

05

1

15

2

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

x 7x 7

7

x 27

x 141

Figure 4 Nonzero component tracking performance

50 100 150 200 250minus8

minus6

minus4

minus2

0

2

4

6

8

Support index

Am

plitu

de

TrueCSKF

Algorithm 1Algorithm 2

Figure 5 Instantaneous values at 119896 = 100

0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

QPFCSKF Algorithm 1

Algorithm 2

1

09

08

07

06

05

04

03

02

01

0

Figure 6 Normalized RMSE


0 20 40 60 80 1000

1

2

3

4

0 20 40 60 80 100minus1

0

1

2

3

0 20 40 60 80 100 0 20 40 60 80 100

x 4x 9

1

x 42

x 98

minus05

0

05

1

15

2

minus15

minus1

minus05

0

05

Time index

Time index

Time index

Time index

TrueAlgorithm 1Algorithm 2




Figure 7 Support change tracking

after roughly 119896 = 45 It is noted that the performance isachieved with far fewer measurements than the unknowns(lt30) In the example the complexity of Algorithm 2 isdominated by119874(1198732119872) = 4915 times 10

6 which is of the sameorder as that of QPF while the complexity of Algorithm 1 isdominated by119874(1198732119871) = 6553 times 10

6 It is maybe preferableto employ Algorithm 2

In the second scenario we verify the effectiveness ofthe proposed algorithm for sparse signal with slow changesupport set The simulation parameters are set as 119873 = 160119872 = 40 W

119896(119894 119894) = 001

2 and R119896= 025

2I40

and theothers are the same as the first scenario We assume thatthere are only 119870 = 4 possible nonzero components and thatthe actual number of the nonzero elements may change overtime In particular the component x4 is nonzero for theentire measurement interval x42 becomes zero at 119896 = 61x91 is nonzero from 119896 = 41 onwards and x98 is nonzerobetween 119896 = 41 and 119896 = 61 All the other components remainzero throughout the considered time interval Figure 7 showsthe estimator performance compared with the actual time

variations of the 4 nonzero components As can be seen thealgorithms have good performance for tracking the supportchanged slowly

In addition we study the relationship between quanti-zation bits and accuracy of reconstruction Figure 8 showsnormalized RMSE versus number of quantization bits when119896 = 100 The performance gets better but gains little as thequantization bits increase However more quantization bitswill bring greater overheads of communication computa-tion and storage resulting in more energy consumption ofsensors For this reason 1-bit quantization scheme has beenemployed in our algorithms and is enough to guarantee theaccuracy of reconstruction

Moreover it is noted that the information filter isemployed to propagate the particles in our algorithms Com-pared with KF apart from the ability to deal with multisensorfusion the IF also has an advantage over numerical stabilityIn Figure 9 we take Algorithm 1 for example and givethe comparison of two cases whether Algorithm 1 is with


1 2 3 40

002

004

006

008

01

Number of quantization bits

Nor

mal

ized

RM

SE


Figure 8 Normalized RMSE versus number of quantization bits

0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

2

18

16

14

12

1

08

06

04

02

0

Algorithm 1-IFAlgorithm 1-KF


KF or with IF As can be seen Algorithm 1-IF gives goodperformance but Algorithm 1-KF diverges

8 Conclusions

The algorithms for reconstructing time-varying sparse sig-nals under communication constraints have been proposedin this paper For severely bandwidth constrained (1-bit) sce-narios a particle filter algorithm based on coarsely quantizedinnovation is proposed To recover the sparsity pattern thealgorithm enforces the sparsity constraint on fused estimateby either iterative PM update method or sparse cubaturepoint filter method Compared with iterative PM updatemethod the sparse cubature point filter method is preferabledue to the comparable performance and lower complexityA numerical example demonstrated that the proposed algo-rithm is effective with a far smaller number of measurements

than the size of the state vector This is very promising inthe WSNs with energy constraint and the lifetime of WSNscan be prolonged Nevertheless the algorithm presented inthis paper is only suitable for the time-varying sparse signalwith an invariant or slowly changing support set and themore generalmethods should be combinedwith a support setestimator which will be discussed in our future work further

Appendix

Proof In order to prove Lemma 1 we will show that themoment generating function (MGF) of x

119896| q1198970119896

can beregarded as the product of two MGFs corresponding to thetwo random variables in (11) Recall that the MGF of a 119889-dimensional rv a is expressed as119872a(119904) = 119864[119890

119904119879a] forall119904 isin 119877

119889Note that

119901 (x119896| q1198970119896) = int119901 (x

119896 y1198970119896

| q1198970119896) 119889y1198970119896

(A1)

and119901(x119896| y1198970119896 q1198970119896) = 119901(x

119896| y1198970119896) so theMGFof x

119896| q1198970119896

can be given by

119864 [119890119904119879x119896 | q

1198970119896]

= int 119890119904119879x119896119901 (x

119896| y1198970119896) 119901 (y1198970119896

| q1198970119896) 119889x119896119889y1198970119896

= 119890(12)119904

119879ΣΔ

x119896y1198970119896119904int 119890119904119879Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896

119901 (y1198970119896

| q1198970119896) 119889y1198970119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

MGF of Σx119896y1198970119896Σminus1

y1198970119896y1198970119896|q1198970119896

997904rArr

119872x119896|q1198970119896 (119904) = 119872120578119896 (119904)119872y1198970119896|q1198970119896 (Σminus1

y1198970119896Σy1198970119896x119896119904)

(A2)

where 120578119896sim N119889(0ΣΔ

x119896y1198970119896) Here we used the fact that x119896|

y1198970119896

sim N119889(Σx119896y1198970119896Σ

minus1

y1198970119896 ΣΔ

x119896y1198970119896) and119872b(c119905) = 119872cb(119905) Thenthe result should be rather obvious from (A2)

Proof Consider the pseudo-measurement equation

0 =1003817100381710038171003817x11989610038171003817100381710038171 minus 120598

1015840

119896 (A3)

As x119896is unknown the relation x

119896= x119896+ x119896can be used to get

0 = [sign (x119896)]119879 x119896minus 1205981015840

119896= sign (x

119896)119879

+ g119879x119896minus 1205981015840

119896 (A4)

where g le 119888 almost surely Equation (A4) is the approx-imate pseudo-measurement with an observation noise

119896=

g119879x119896minus1205981015840

119896 Note that119864[1205981015840

119896] = 0 and119864[12059810158402

119896] = 1198771015840

120598 Since themean

of 119896cannot be obtained easily we approximate the second

moment

119864 [ 1205982

119896] = 119864 [g119879 (x

119896+ x119896) (x119896+ x119896)119879 g | Y

119896] + 1198771015840

120598 (A5)

Here we used the fact that x119896and 120598

1015840

119896are statistically

independent Substituting P119896|119896

obtained by KF for the errorcovariance in (A5) we can get

120598= 119864 [ 120598

2

119896]

= g119879 (x119896) (x119896)119879 g + g119879119864 [x

119896(x119896)119879

| Y119896] g + 1198771015840

120598

asymp O (1003817100381710038171003817x1198961003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598

(A6)


Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork was supported by the fund for theNational NaturalScience Foundation of China (Grants nos 60872123 and61101014) and Higher-Level Talent in Guangdong ProvinceChina (Grant no N9101070)

References

[1] A Ribeiro G B Giannakis and S I Roumeliotis ldquoSOI-KFdistributed Kalman filtering with low-cost communicationsusing the sign of innovationsrdquo IEEE Transactions on SignalProcessing vol 54 no 12 pp 4782ndash4795 2006

[2] R T Sukhavasi and B Hassibi ldquoThe Kalman-like particlefilter optimal estimation with quantized innovationsmeasure-mentsrdquo IEEE Transactions on Signal Processing vol 61 no 1 pp131ndash136 2013

[3] G Battistelli A Benavoli and L Chisci ldquoData-driven commu-nication for state estimationwith sensor networksrdquoAutomaticavol 48 no 5 pp 926ndash935 2012

[4] MMansouri L Khoukhi HNounou andMNounou ldquoSecureand robust clustering for quantized target tracking in wirelesssensor networksrdquo Journal of Communications andNetworks vol15 no 2 pp 164ndash172 2013

[5] M Mansouri O Ilham H Snoussi and C Richard ldquoAdaptivequantized target tracking in wireless sensor networksrdquoWirelessNetworks vol 17 no 7 pp 1625ndash1639 2011

[6] Y Zhou D Wang T Pei and Y Lan ldquoEnergy-efficient targettracking in wireless sensor networks a quantized measurementfusion frameworkrdquo International Journal of Distributed SensorNetworks vol 2014 Article ID 682032 10 pages 2014

[7] S Qaisar R M Bilal W Iqbal M Naureen and S LeeldquoCompressive sensing from theory to applications a surveyrdquoJournal of Communications andNetworks vol 15 no 5 pp 443ndash456 2013

[8] Y Liu H Yu and J Wang ldquoDistributed Kalman-consensusfiltering for sparse signal estimationrdquoMathematical Problems inEngineering vol 2014 Article ID 138146 7 pages 2014

[9] C Karakus A C Gurbuz and B Tavli ldquoAnalysis of energyefficiency of compressive sensing in wireless sensor networksrdquoIEEE Sensors Journal vol 13 no 5 pp 1999ndash2008 2013

[10] J Haupt W U Bajwa M Rabbat and R Nowak ldquoCompressedsensing for networked datardquo IEEE Signal Processing Magazinevol 25 no 2 pp 92ndash101 2008

[11] D Angelosante G B Giannakis and E Grossi ldquoCompressedsensing of time-varying signalsrdquo in Proceedings of the 16thInternational Conference on Digital Signal Processing (DSP rsquo09)pp 1ndash8 Santorini Greece July 2009

[12] D Angelosante and G B Giannakis ldquoRLS-weighted Lasso foradaptive estimation of sparse signalsrdquo in Proceedings of theIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo09) pp 3245ndash3248 Taipei Taiwan April2009

[13] B Babadi N Kalouptsidis and V Tarokh ldquoSPARLS the sparseRLS algorithmrdquo IEEE Transactions on Signal Processing vol 58no 8 pp 4013ndash4025 2010

[14] N Vaswani and W Lu ldquoModified-CS modifying compressivesensing for problems with partially known supportrdquo IEEETransactions on Signal Processing vol 58 no 9 pp 4595ndash46072010

[15] A Carmi P Gurfil and D Kanevsky ldquoMethods for sparsesignal recovery using Kalman filtering with embedded pseudo-measurement norms and quasi-normsrdquo IEEE Transactions onSignal Processing vol 58 no 4 pp 2405ndash2409 2010

[16] A Y Carmi L Mihaylova and D Kanevsky ldquoUnscentedcompressed sensingrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo12) pp 5249ndash5252 Kyoto Japan March 2012

[17] A Zymnis S Boyd and E Candes ldquoCompressed sensing withquantizedmeasurementsrdquo IEEE Signal Processing Letters vol 17no 2 pp 149ndash152 2010

[18] K Qiu and A Dogandzic ldquoSparse signal reconstruction fromquantized noisy measurements via GEM hard thresholdingrdquoIEEE Transactions on Signal Processing vol 60 no 5 pp 2628ndash2634 2012

[19] C-H Chen and J-Y Wu ldquoAmplitude-aided 1-bit compressivesensing over noisy wireless sensor networksrdquo IEEE WirelessCommunications Letters vol 4 no 5 pp 473ndash476 2015

[20] X Dong and Y Zhang ldquoA MAP approach for 1-Bit compressivesensing in synthetic aperture radar imagingrdquo IEEE Geoscienceand Remote Sensing Letters vol 12 no 6 pp 1237ndash1241 2015

[21] K Knudson R Saab and R Ward ldquoOne-bit compressive sens-ing with norm estimationrdquo IEEE Transactions on InformationTheory vol 62 no 5 pp 2748ndash2758 2016

[22] D L Donoho ldquoCompressed sensingrdquo IEEE Transactions onInformation Theory vol 52 no 4 pp 1289ndash1306 2006

[23] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationThe-ory vol 52 no 2 pp 489ndash509 2006

[24] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processing vol50 no 2 pp 174ndash188 2002

[25] T Li M Bolic and P M Djuric ldquoResampling Methods for Par-ticle Filtering classification implementation and strategiesrdquoIEEE Signal Processing Magazine vol 32 no 3 pp 70ndash86 2015

[26] S Wang J Feng and C K Tse ldquoA class of stable square-rootnonlinear information filtersrdquo IEEE Transactions on AutomaticControl vol 59 no 7 pp 1893ndash1898 2014

[27] K P B Chandra D-W Gu and I Postlethwaite ldquoSquare rootcubature information filterrdquo IEEE Sensors Journal vol 13 no 2pp 750ndash758 2013

[28] D Simon and T L I Chia ldquoKalman filtering with state equalityconstraintsrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 38 no 1 pp 128ndash136 2002

[29] I Arasaratnam and S Haykin ldquoCubature Kalman filtersrdquo IEEETransactions on Automatic Control vol 54 no 6 pp 1254ndash12692009

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Algebra

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Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the authorsrsquo knowledge there is limited work on recursivecompressed sensing techniques considering quantization asa mean of further reduction of the required bandwidthand power resources However an increased attention hasbeen paid to develop algorithms for reconstructing sparsesignals using quantized observations recently [17ndash21] In [17]a convex relaxation approach for reconstructing sparse signalfrom quantized observations was proposed by using an ℓ

1-

norm regularization term and two convex cost functions In[18] Qiu and Dogandzic proposed an unrelaxed probabilisticmodel with ℓ

0-norm constrained signal space and derived

an expectation-maximization algorithm In addition [19ndash21]have investigated the reconstruction of sparse source from 1-bit quantized measurement in the extreme case

In this paper we study reconstruction of time-varyingsparse signals in WSNs by using quantized measurementsOur contributions are as follows

(1) We propose an improved particle filter algorithmwhich extends the fundamental results [2] to amultiple-observation case by employing informationfilter form The algorithm in [2] is derived under theassumption that the fusion center has access to onlyonemeasurement source at each time stepThe exten-sion to a multiple-measurement scenario is straight-forward but in generalmay lead to a computationallyinvolved scheme In contrast the proposed algorithmcan be implemented in a computationally efficientsequential processing form and avoids any matrixinversion Meanwhile the proposed algorithm hasan advantage over numerical stability for inaccurateinitialization

(2) We propose a new method to impose sparsity con-straint on estimator by the particle filter algorithmCompared to the iterative method in [15] the result-ing method is noniterative and easy to implement

In particular the system has an underlying state-spacestructure where the state vector is sparse In each timeinterval the fusion center transmits the predicted signalestimate and its corresponding error covariance to a selectedsubset of sensors The selected sensors compute quantizedinnovations and transmit them to the fusion center Thefusion center reconstructs the sparse state by employing theproposed particle filter algorithm and sparse cubature pointfilter method

This paper is organized as follows Section 2 gives abrief overview of basic problems in compressed sensing andintroduces the sparse signal recovery method using Kalmanfiltering with embedded pseudo-measurement In Section 3we describe the system model Section 4 develops a particlefilter with quantized innovation To recover the sparsitypattern of the state estimate by particle filter a sparse cubaturepoint filter method is developed with lower complexitycompared to reiterative PM update method in Section 5The intact version of adaptively recursive reconstructionalgorithm for sparse signals with quantized innovations andthe analysis of their complexity are presented in Section 6

Section 7 contains simulation results and the conclusions areconcluded in Section 8

NotationN119889(120583 Σ) denotes 119889-dimensional Gaussian rv with

mean 120583 and variance Σ the 119889-dimensional Gaussian prob-ability density with mean 120583 and variance Σ is denoted by120601119889(sdot 120583 Σ) Φ(119909 120583 Σ) denotes Gaussian probability distribu-

tion with mean 120583 and variance Σ and Φ(S 120583 Σ) denotesthe truncated probability where S belongs to Borel 120590-fieldu119894119895

denotes the collection of random 119906119894 119906

119895 Boldfaced

uppercase and lowercase symbols represent matrices andvectors respectively For a vector x x(119894) denotes its 119894thcomponent and Cov[x] denotes the error covariance 119864[(x minus119864x)(x minus 119864x)119879] For a matrix R R(119897 119897) denotes the (119897 119897) entryof R

2 Sparse Signal Reconstruction UsingKalman Filter

21 Sparse Signal Recovery Compressive sensing is a frame-work for signal sensing and compression that enables repre-sentation of a sensed signal with fewer samples than thoseones required by classical sampling theory Consider a sparserandom discrete-time process x

119896119896ge0

in 119877119873 where x1198960≪

119873 and x119896is called 119870-sparse if x

1198960= 119870 Assume x

119896evolves

according to the following dynamical equations

x119896+1

= F119896x119896+ w119896 (1)

y119896= H119896x119896+ k119896 (2)

where F119896isin 119877119873times119873 is the state transition matrix and H

119896isin

119877119872times119873 is the measurement matrix Moreover w

119896119896ge0

andk119896119896ge0

denote the zero meanrsquos white Gaussian sequence withcovariances W

119896⪰ 0 and R

119896⪰ 0 respectively y

119896is 119872-

dimensional linear measurement of x119896 When 119872 lt 119873 and

rank(H119896) lt 119873 it is noted that the reconstruction x

119896from

underdetermined system is an ill-posed problemHowever [22 23] have shown that x

119896can be accurately

reconstructed by solving the following optimization problem

minx119896isin119877119873

1003817100381710038171003817x11989610038171003817100381710038170

st 1003817100381710038171003817y119896 minusH119896x119896

1003817100381710038171003817

2

2le 120598

(3)

Unfortunately the above optimization problem is NP-hard and cannot be solved effectively Fortunately as shownin [23] if the measurement matrix H

119896obeys the so-called

restricted isometry property (RIP) the solution of (3) canbe obtained with overwhelming probability by solving thefollowing convex optimization

minx119896isin119877119873

1003817100381710038171003817x11989610038171003817100381710038171

st 1003817100381710038171003817y119896 minusH119896x119896

1003817100381710038171003817

2

2le 120598

(4)

This is a fundamental result in compressed sensing (CS)Moreover for reconstructing a119870-sparse signal x

119896isin 119877119873119872 ge

119862 sdot 119870 log(119873119870) linear measurements are needed where 119862 isa fixed constant



119896provided



minx119896isin119877119873

119864 [1003817100381710038171003817x119896 minus x

119896

1003817100381710038171003817

2

2| Y119896] (5)



119896


minx119896isin119877119873

1003817100381710038171003817x11989610038171003817100381710038171

st 119864 [1003817100381710038171003817x119896 minus x

119896

1003817100381710038171003817

2

2| Y119896] le 120598119896

(6)


minx119896isin119877119873

119864 [1003817100381710038171003817x119896 minus x

119896

1003817100381710038171003817

2

2| Y119896]

1003817100381710038171003817x11989610038171003817100381710038171 le 120598119896

(7)



0 = H119896x119896minus 1205981015840

119896 (8)

where H119896

= sign(x119879119896) and 120598

1015840

119896sim N(0 119877

1015840

120598) serves as





119896)




119896



x119896+1

= F119896x119896+ w119896

119910119897119896= h119897119896x119896+ V119897119896

(9)



where h119897119896




119896 where the





matrixH119896= [ h1198791119896

h1198792119896

sdot sdot sdot h119879119872119896





119896





whereh119897x119896|119896minus1



is quantized to 119902119897119896









Processxk

y1k y2k yMk

S1 S2 SM

q1k q2k qMk

Feedback

Feedback








on y1198970119896


x119896| y1198970119896

sim 120578119896+ Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896


minus1

y1198970119896Σy1198970119896x119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

≜ΣΔ

x119896y1198970119896

)

(10)

where y1198970119896

≜ 1199101198970 119910

119897119896 When x

119896 and 119910

119897119896 follow the


119896|119896= Cov[x

119896minus119864[x119896| y119897119896]] can




1198970119896 It will



≜ 1199021198970 119902






x119896| q1198970119896

sim 120578119896+ Σx119896y1198970119896Σ

minus1

y1198970119896 [y1198970119896 | q1198970119896]

where 120578119896simN119889(0ΣΔ

x119896y1198970119896) (11)

Proof See Appendix



1198970119896| q1198970119896

Apparently y

1198970119896| q1198970119896




119896| q1198970119896

can be expressed as

Cov [x119896| q1198970119896]

= ΣΔ

x119896y1198970119896

+ Σx119896y1198970119896Σminus1

y1198970119896Cov [y1198970119896 | q1198970119896]Σminus1

y1198970119896Σy1198970119896x119896

(12)


1198970119896| q1198970119896


and x119896| q1198970119896


q1198970119896





0)

and x0|minus1


119897119896= 119876(119910

119897119896) the


119901(119902119897119896| x119896= x119894119896|119896minus1

q1198970119896minus1


x119901119891119896|119896=

119873119901

sum

119894=1

120596119894

119896x119894119896|119896minus1

(13)


120596119894

119896=

120596119895

119896

sum119873119901

119894=1120596119894

119896

(14)



120580=1such that

119875(119869120580= 119894) = 120596

119894

119896

x120580119896|119896= x119869120580119896 (15)


x119895119896+1|119896

sim 119901 (x119896+1

| x119896= x119894119896|119896 q1198970119896)

ie x119895119896+1|119896

= F119896x119894119896|119896

(16)

(6) Consider x119901119891119896+1|119896

= F119896x119901119891119896|119896 Also set 119896 = 119896 + 1 and



119897119896= sign(119910

119897119896minus 119897119896|119896minus1

))



119896=

Φ(119902119897119896h119897(x119894119896|119896minus1


) 0R119890119896(119897 119897)) We note that 119864[x

119896|

q1198970119896] = Σx119896y1198970119896Σ

minus1



q1198970119896

where

120585119896= Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896 (17)



q1198970119896minus1

So on receipt of 119902119897119896






119896| q1198970119896


119896minus1| q1198970119896minus1

) rarr 119901(120585119896minus1



| q1198970119896) rarr 119901(120585

119896| q1198970119896) which



| q1198970119896

and 120585119896minus1

| q1198970119896minus1

is given by

119901 (120585119896minus1

| q1198970119896)

119901 (120585119896minus1

| q1198970119896minus1

)prop Φ(S

119896q1198970119896 h119897119896F119896120585119896minus1R119890119896 (119897 119897)) (18)

So if 120585119894119896minus1|119896minus1


119901(120585119896minus1

| q1198970119896minus1






120580= 119894) prop 120596

119894 and set120585120580

119896minus1|119896= 120585119869120580



119896minus1| q1198970119896minus1


119897119896





119896given y

1198970119896 Since

x119896 and 119910



120585119896|119896= F119896120585119896|119896minus1


)

K119896=

P119896|119896minus1

h119879119897119896

h119897119896P119896|119896minus1

h119879119897119896+ R119896(119897 119897)

(19)




Y119896|119896= Y119896|119896minus1

+I119896

z119896|119896= z119896|119896minus1

+ i119896

(20)

where Y119896|119896

= Pminus1119896|119896

and z119896|119896

= Y119896|119896120585119896|119896


119896|119896= Y119896|119896 I119873and

120585119896|119896

= Y119896|119896

z119896|119896 where I




I119896= h119879119897119896Rminus1119896(119897 119897) h

119897119896

i119896= h119879119897119896Rminus1119896(119897 119897) 119910

119897119896

(21)


119896minus1| q1198970119896) to 119901(120585

119896| q1198970119896) Following


120580is a set




119896| q1198970119896) can be




119901 (119910119897119896| 120585120580

119896minus1|119896 q1198970119896)

= 120601 (S119896q1198970119896 h119897F119896120585

120580

119896minus1|119896R119890119896(119897 119897))

(22)

Also set z119894119896|119896= z120580119896|119896minus1

+ h119879119897119896Rminus1119896(119897 119897)119910119894

119897119896|119896









119896 where 120598

119896




120574120591+1

= 120574120591

0 = H120591120574120591minus 120598120591

(23)

where 1205741|1

= x119896|119896

and P1199011198981|1

= P119896|119896

and H120591

=

[sign(1120591|120591


)] 120591 = 1 2 119871 119895120591|120591


120591


= 119871|119871

and P119896|119896

= P119901119898119871|119871



2 is


120591+1|120591+1

= 120591|120591minus

P119901119898120591|120591

sign (120591|120591)10038171003817100381710038171003817120591|120591

100381710038171003817100381710038171

sign (120591|120591)119879

P119901119898120591|120591

sign (120591|120591) + 1198771015840120598

(24)

P119901119898120591+1|120591+1

= P119901119898120591|120591minus

P119901119898120591|120591

sign (120591|120591) sign (

120591|120591)119879

P119901119898120591|120591

sign (120591|120591)119879

P119901119898120591|120591

sign (120591|120591) + 1198771015840120598

(25)



120594119904= x + radic119873(Sx)119904 120596

119904=

1

2119873

120594119873+119904

= x minus radic119873(Sx)119904 120596119873+119904

=1

2119873

(26)


119864 [x] = 1

2119873

2119873

sum

119904=1

120594lowast

119904

Cov [y] = 1

2119873

2119873

sum

119904=1

120594lowast

119904120594lowast119879

119904minus xx119879

(27)

where 120594lowast119904= f(120594119904) 119904 = 0 1 2119873

ek

z1k

zN119901

k

Resa

mpl

e

Pred

ictio

n

Method 1

ormethod 2

IF

IF

IF



119896|119896and 120594



120594lowast

119904119896|119896= 120594119904119896|119896

minus

P119896|119896

sign (120594119904119896|119896

)10038171003817100381710038171003817120594119904119896|119896

100381710038171003817100381710038171

sign (120594119904119896|119896

)119879

P119896|119896

sign (120594119904119896|119896

) + 120598

(28)

where 120598= O (

10038171003817100381710038171003817120594119904119896|119896

10038171003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598(29)


119904119896|1198962119873



6 The Algorithm



0|minus1 0|minus1

P0|minus1


0|minus1 z0|minus1

Y0|minus1


119890119896(119897 119897) which denote the


119897119896(see (31)) to

the 119897th sensor

R119890119896= H119896P119896|119896minus1

H119879119896+ R119896 (30)

119897119896|119896minus1

= h119897119896119896|119896minus1

(31)



119902119897119896= 119876(

119910119897119896minus 119897119896|119896minus1

R119890119896(119897 119897)

)R119890119896(119897 119897) (32)



119910119894

119897119896|119896sim 120601 (S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (33)

120596119894

119897119896sim Φ(S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (34)


119896|119896=

[119910119894

1119896|119896sdot sdot sdot 119910119872


Y119896|119896= Y119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) h

119897119896

z119894119896|119896= z119894119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) 119910

119894

119897119896|119896

(35)



z119896|119896=

119873119901

sum

119894=1

120596119894

119896z119894119896|119896 (36)

where 120596119894119896= prod119872

119897=1120596119894

119897119896





z119896+1|119896

Y119896+1|1

119897119896+1|119896

and P


z119894119896+1|119896

= F119896z119894119896|119896

z119896+1|119896

= F119896z119896|119896

Y119896+1|119896

= [F119896P119896|119896F119879119896+W119896+1]minus1

119896+1|119896

= Y119896+1|119896

z119896+1|119896

P119896+1|119896

= Y119896+1|119896

I119873

(37)


observations 119910119894119897119896119873119901

119894=1


119901) Measurement




119901) + 119874(119873

2119872)



119896 in



119896is set asW

119896(119894 119894) = 01


x119896+1

(119894) =

x119896(119894) + w

119896(119894) if 119894 isin supp (x

119896)

0 otherwise(38)


2) Both the





119896= 001


x0|minus1






119896minus x1198962x1198962)



0 20 40 60 80 100minus1

0

1

2

3

4



0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

0 20 40 60 80 100minus15

minus1

minus05

0

05

0 20 40 60 80 100minus05

0

05

1

15

2

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

x 7x 7

7

x 27

x 141


50 100 150 200 250minus8

minus6

minus4

minus2

0

2

4

6

8

Support index

Am

plitu

de

TrueCSKF



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

QPFCSKF Algorithm 1

Algorithm 2

1

09

08

07

06

05

04

03

02

01

0



0 20 40 60 80 1000

1

2

3

4

0 20 40 60 80 100minus1

0

1

2

3

0 20 40 60 80 100 0 20 40 60 80 100

x 4x 9

1

x 42

x 98

minus05

0

05

1

15

2

minus15

minus1

minus05

0

05

Time index

Time index

Time index

Time index










119896(119894 119894) = 001

2 and R119896= 025

2I40






1 2 3 40

002

004

006

008

01


Nor

mal

ized

RM

SE



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

2

18

16

14

12

1

08

06

04

02

0




8 Conclusions



Appendix


119896| q1198970119896


119904119879a] forall119904 isin 119877

119889Note that

119901 (x119896| q1198970119896) = int119901 (x

119896 y1198970119896

| q1198970119896) 119889y1198970119896

(A1)

and119901(x119896| y1198970119896 q1198970119896) = 119901(x

119896| y1198970119896) so theMGFof x

119896| q1198970119896

can be given by

119864 [119890119904119879x119896 | q

1198970119896]

= int 119890119904119879x119896119901 (x

119896| y1198970119896) 119901 (y1198970119896

| q1198970119896) 119889x119896119889y1198970119896

= 119890(12)119904

119879ΣΔ

x119896y1198970119896119904int 119890119904119879Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896

119901 (y1198970119896

| q1198970119896) 119889y1198970119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

MGF of Σx119896y1198970119896Σminus1

y1198970119896y1198970119896|q1198970119896

997904rArr

119872x119896|q1198970119896 (119904) = 119872120578119896 (119904)119872y1198970119896|q1198970119896 (Σminus1

y1198970119896Σy1198970119896x119896119904)

(A2)

where 120578119896sim N119889(0ΣΔ


y1198970119896

sim N119889(Σx119896y1198970119896Σ

minus1

y1198970119896 ΣΔ



0 =1003817100381710038171003817x11989610038171003817100381710038171 minus 120598

1015840

119896 (A3)



0 = [sign (x119896)]119879 x119896minus 1205981015840

119896= sign (x

119896)119879

+ g119879x119896minus 1205981015840

119896 (A4)


119896=

g119879x119896minus1205981015840

119896 Note that119864[1205981015840

119896] = 0 and119864[12059810158402

119896] = 1198771015840



moment

119864 [ 1205982

119896] = 119864 [g119879 (x

119896+ x119896) (x119896+ x119896)119879 g | Y

119896] + 1198771015840

120598 (A5)


1015840




120598= 119864 [ 120598

2

119896]

= g119879 (x119896) (x119896)119879 g + g119879119864 [x

119896(x119896)119879

| Y119896] g + 1198771015840

120598

asymp O (1003817100381710038171003817x1198961003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598

(A6)


Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896provided



minx119896isin119877119873

119864 [1003817100381710038171003817x119896 minus x

119896

1003817100381710038171003817

2

2| Y119896] (5)



119896


minx119896isin119877119873

1003817100381710038171003817x11989610038171003817100381710038171

st 119864 [1003817100381710038171003817x119896 minus x

119896

1003817100381710038171003817

2

2| Y119896] le 120598119896

(6)


minx119896isin119877119873

119864 [1003817100381710038171003817x119896 minus x

119896

1003817100381710038171003817

2

2| Y119896]

1003817100381710038171003817x11989610038171003817100381710038171 le 120598119896

(7)



0 = H119896x119896minus 1205981015840

119896 (8)

where H119896

= sign(x119879119896) and 120598

1015840

119896sim N(0 119877

1015840

120598) serves as





119896)




119896



x119896+1

= F119896x119896+ w119896

119910119897119896= h119897119896x119896+ V119897119896

(9)



where h119897119896




119896 where the





matrixH119896= [ h1198791119896

h1198792119896

sdot sdot sdot h119879119872119896





119896





whereh119897x119896|119896minus1



is quantized to 119902119897119896









Processxk

y1k y2k yMk

S1 S2 SM

q1k q2k qMk

Feedback

Feedback








on y1198970119896


x119896| y1198970119896

sim 120578119896+ Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896


minus1

y1198970119896Σy1198970119896x119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

≜ΣΔ

x119896y1198970119896

)

(10)

where y1198970119896

≜ 1199101198970 119910

119897119896 When x

119896 and 119910

119897119896 follow the


119896|119896= Cov[x

119896minus119864[x119896| y119897119896]] can




1198970119896 It will



≜ 1199021198970 119902






x119896| q1198970119896

sim 120578119896+ Σx119896y1198970119896Σ

minus1

y1198970119896 [y1198970119896 | q1198970119896]

where 120578119896simN119889(0ΣΔ

x119896y1198970119896) (11)

Proof See Appendix



1198970119896| q1198970119896

Apparently y

1198970119896| q1198970119896




119896| q1198970119896

can be expressed as

Cov [x119896| q1198970119896]

= ΣΔ

x119896y1198970119896

+ Σx119896y1198970119896Σminus1

y1198970119896Cov [y1198970119896 | q1198970119896]Σminus1

y1198970119896Σy1198970119896x119896

(12)


1198970119896| q1198970119896


and x119896| q1198970119896


q1198970119896





0)

and x0|minus1


119897119896= 119876(119910

119897119896) the


119901(119902119897119896| x119896= x119894119896|119896minus1

q1198970119896minus1


x119901119891119896|119896=

119873119901

sum

119894=1

120596119894

119896x119894119896|119896minus1

(13)


120596119894

119896=

120596119895

119896

sum119873119901

119894=1120596119894

119896

(14)



120580=1such that

119875(119869120580= 119894) = 120596

119894

119896

x120580119896|119896= x119869120580119896 (15)


x119895119896+1|119896

sim 119901 (x119896+1

| x119896= x119894119896|119896 q1198970119896)

ie x119895119896+1|119896

= F119896x119894119896|119896

(16)

(6) Consider x119901119891119896+1|119896

= F119896x119901119891119896|119896 Also set 119896 = 119896 + 1 and



119897119896= sign(119910

119897119896minus 119897119896|119896minus1

))



119896=

Φ(119902119897119896h119897(x119894119896|119896minus1


) 0R119890119896(119897 119897)) We note that 119864[x

119896|

q1198970119896] = Σx119896y1198970119896Σ

minus1



q1198970119896

where

120585119896= Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896 (17)



q1198970119896minus1

So on receipt of 119902119897119896






119896| q1198970119896


119896minus1| q1198970119896minus1

) rarr 119901(120585119896minus1



| q1198970119896) rarr 119901(120585

119896| q1198970119896) which



| q1198970119896

and 120585119896minus1

| q1198970119896minus1

is given by

119901 (120585119896minus1

| q1198970119896)

119901 (120585119896minus1

| q1198970119896minus1

)prop Φ(S

119896q1198970119896 h119897119896F119896120585119896minus1R119890119896 (119897 119897)) (18)

So if 120585119894119896minus1|119896minus1


119901(120585119896minus1

| q1198970119896minus1






120580= 119894) prop 120596

119894 and set120585120580

119896minus1|119896= 120585119869120580



119896minus1| q1198970119896minus1


119897119896





119896given y

1198970119896 Since

x119896 and 119910



120585119896|119896= F119896120585119896|119896minus1


)

K119896=

P119896|119896minus1

h119879119897119896

h119897119896P119896|119896minus1

h119879119897119896+ R119896(119897 119897)

(19)




Y119896|119896= Y119896|119896minus1

+I119896

z119896|119896= z119896|119896minus1

+ i119896

(20)

where Y119896|119896

= Pminus1119896|119896

and z119896|119896

= Y119896|119896120585119896|119896


119896|119896= Y119896|119896 I119873and

120585119896|119896

= Y119896|119896

z119896|119896 where I




I119896= h119879119897119896Rminus1119896(119897 119897) h

119897119896

i119896= h119879119897119896Rminus1119896(119897 119897) 119910

119897119896

(21)


119896minus1| q1198970119896) to 119901(120585

119896| q1198970119896) Following


120580is a set




119896| q1198970119896) can be




119901 (119910119897119896| 120585120580

119896minus1|119896 q1198970119896)

= 120601 (S119896q1198970119896 h119897F119896120585

120580

119896minus1|119896R119890119896(119897 119897))

(22)

Also set z119894119896|119896= z120580119896|119896minus1

+ h119879119897119896Rminus1119896(119897 119897)119910119894

119897119896|119896









119896 where 120598

119896




120574120591+1

= 120574120591

0 = H120591120574120591minus 120598120591

(23)

where 1205741|1

= x119896|119896

and P1199011198981|1

= P119896|119896

and H120591

=

[sign(1120591|120591


)] 120591 = 1 2 119871 119895120591|120591


120591


= 119871|119871

and P119896|119896

= P119901119898119871|119871



2 is


120591+1|120591+1

= 120591|120591minus

P119901119898120591|120591

sign (120591|120591)10038171003817100381710038171003817120591|120591

100381710038171003817100381710038171

sign (120591|120591)119879

P119901119898120591|120591

sign (120591|120591) + 1198771015840120598

(24)

P119901119898120591+1|120591+1

= P119901119898120591|120591minus

P119901119898120591|120591

sign (120591|120591) sign (

120591|120591)119879

P119901119898120591|120591

sign (120591|120591)119879

P119901119898120591|120591

sign (120591|120591) + 1198771015840120598

(25)



120594119904= x + radic119873(Sx)119904 120596

119904=

1

2119873

120594119873+119904

= x minus radic119873(Sx)119904 120596119873+119904

=1

2119873

(26)


119864 [x] = 1

2119873

2119873

sum

119904=1

120594lowast

119904

Cov [y] = 1

2119873

2119873

sum

119904=1

120594lowast

119904120594lowast119879

119904minus xx119879

(27)

where 120594lowast119904= f(120594119904) 119904 = 0 1 2119873

ek

z1k

zN119901

k

Resa

mpl

e

Pred

ictio

n

Method 1

ormethod 2

IF

IF

IF



119896|119896and 120594



120594lowast

119904119896|119896= 120594119904119896|119896

minus

P119896|119896

sign (120594119904119896|119896

)10038171003817100381710038171003817120594119904119896|119896

100381710038171003817100381710038171

sign (120594119904119896|119896

)119879

P119896|119896

sign (120594119904119896|119896

) + 120598

(28)

where 120598= O (

10038171003817100381710038171003817120594119904119896|119896

10038171003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598(29)


119904119896|1198962119873



6 The Algorithm



0|minus1 0|minus1

P0|minus1


0|minus1 z0|minus1

Y0|minus1


119890119896(119897 119897) which denote the


119897119896(see (31)) to

the 119897th sensor

R119890119896= H119896P119896|119896minus1

H119879119896+ R119896 (30)

119897119896|119896minus1

= h119897119896119896|119896minus1

(31)



119902119897119896= 119876(

119910119897119896minus 119897119896|119896minus1

R119890119896(119897 119897)

)R119890119896(119897 119897) (32)



119910119894

119897119896|119896sim 120601 (S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (33)

120596119894

119897119896sim Φ(S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (34)


119896|119896=

[119910119894

1119896|119896sdot sdot sdot 119910119872


Y119896|119896= Y119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) h

119897119896

z119894119896|119896= z119894119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) 119910

119894

119897119896|119896

(35)



z119896|119896=

119873119901

sum

119894=1

120596119894

119896z119894119896|119896 (36)

where 120596119894119896= prod119872

119897=1120596119894

119897119896





z119896+1|119896

Y119896+1|1

119897119896+1|119896

and P


z119894119896+1|119896

= F119896z119894119896|119896

z119896+1|119896

= F119896z119896|119896

Y119896+1|119896

= [F119896P119896|119896F119879119896+W119896+1]minus1

119896+1|119896

= Y119896+1|119896

z119896+1|119896

P119896+1|119896

= Y119896+1|119896

I119873

(37)


observations 119910119894119897119896119873119901

119894=1


119901) Measurement




119901) + 119874(119873

2119872)



119896 in



119896is set asW

119896(119894 119894) = 01


x119896+1

(119894) =

x119896(119894) + w

119896(119894) if 119894 isin supp (x

119896)

0 otherwise(38)


2) Both the





119896= 001


x0|minus1






119896minus x1198962x1198962)



0 20 40 60 80 100minus1

0

1

2

3

4



0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

0 20 40 60 80 100minus15

minus1

minus05

0

05

0 20 40 60 80 100minus05

0

05

1

15

2

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

x 7x 7

7

x 27

x 141


50 100 150 200 250minus8

minus6

minus4

minus2

0

2

4

6

8

Support index

Am

plitu

de

TrueCSKF



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

QPFCSKF Algorithm 1

Algorithm 2

1

09

08

07

06

05

04

03

02

01

0



0 20 40 60 80 1000

1

2

3

4

0 20 40 60 80 100minus1

0

1

2

3

0 20 40 60 80 100 0 20 40 60 80 100

x 4x 9

1

x 42

x 98

minus05

0

05

1

15

2

minus15

minus1

minus05

0

05

Time index

Time index

Time index

Time index










119896(119894 119894) = 001

2 and R119896= 025

2I40






1 2 3 40

002

004

006

008

01


Nor

mal

ized

RM

SE



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

2

18

16

14

12

1

08

06

04

02

0




8 Conclusions



Appendix


119896| q1198970119896


119904119879a] forall119904 isin 119877

119889Note that

119901 (x119896| q1198970119896) = int119901 (x

119896 y1198970119896

| q1198970119896) 119889y1198970119896

(A1)

and119901(x119896| y1198970119896 q1198970119896) = 119901(x

119896| y1198970119896) so theMGFof x

119896| q1198970119896

can be given by

119864 [119890119904119879x119896 | q

1198970119896]

= int 119890119904119879x119896119901 (x

119896| y1198970119896) 119901 (y1198970119896

| q1198970119896) 119889x119896119889y1198970119896

= 119890(12)119904

119879ΣΔ

x119896y1198970119896119904int 119890119904119879Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896

119901 (y1198970119896

| q1198970119896) 119889y1198970119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

MGF of Σx119896y1198970119896Σminus1

y1198970119896y1198970119896|q1198970119896

997904rArr

119872x119896|q1198970119896 (119904) = 119872120578119896 (119904)119872y1198970119896|q1198970119896 (Σminus1

y1198970119896Σy1198970119896x119896119904)

(A2)

where 120578119896sim N119889(0ΣΔ


y1198970119896

sim N119889(Σx119896y1198970119896Σ

minus1

y1198970119896 ΣΔ



0 =1003817100381710038171003817x11989610038171003817100381710038171 minus 120598

1015840

119896 (A3)



0 = [sign (x119896)]119879 x119896minus 1205981015840

119896= sign (x

119896)119879

+ g119879x119896minus 1205981015840

119896 (A4)


119896=

g119879x119896minus1205981015840

119896 Note that119864[1205981015840

119896] = 0 and119864[12059810158402

119896] = 1198771015840



moment

119864 [ 1205982

119896] = 119864 [g119879 (x

119896+ x119896) (x119896+ x119896)119879 g | Y

119896] + 1198771015840

120598 (A5)


1015840




120598= 119864 [ 120598

2

119896]

= g119879 (x119896) (x119896)119879 g + g119879119864 [x

119896(x119896)119879

| Y119896] g + 1198771015840

120598

asymp O (1003817100381710038171003817x1198961003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598

(A6)


Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Processxk

y1k y2k yMk

S1 S2 SM

q1k q2k qMk

Feedback

Feedback








on y1198970119896


x119896| y1198970119896

sim 120578119896+ Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896


minus1

y1198970119896Σy1198970119896x119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

≜ΣΔ

x119896y1198970119896

)

(10)

where y1198970119896

≜ 1199101198970 119910

119897119896 When x

119896 and 119910

119897119896 follow the


119896|119896= Cov[x

119896minus119864[x119896| y119897119896]] can




1198970119896 It will



≜ 1199021198970 119902






x119896| q1198970119896

sim 120578119896+ Σx119896y1198970119896Σ

minus1

y1198970119896 [y1198970119896 | q1198970119896]

where 120578119896simN119889(0ΣΔ

x119896y1198970119896) (11)

Proof See Appendix



1198970119896| q1198970119896

Apparently y

1198970119896| q1198970119896




119896| q1198970119896

can be expressed as

Cov [x119896| q1198970119896]

= ΣΔ

x119896y1198970119896

+ Σx119896y1198970119896Σminus1

y1198970119896Cov [y1198970119896 | q1198970119896]Σminus1

y1198970119896Σy1198970119896x119896

(12)


1198970119896| q1198970119896


and x119896| q1198970119896


q1198970119896





0)

and x0|minus1


119897119896= 119876(119910

119897119896) the


119901(119902119897119896| x119896= x119894119896|119896minus1

q1198970119896minus1


x119901119891119896|119896=

119873119901

sum

119894=1

120596119894

119896x119894119896|119896minus1

(13)


120596119894

119896=

120596119895

119896

sum119873119901

119894=1120596119894

119896

(14)



120580=1such that

119875(119869120580= 119894) = 120596

119894

119896

x120580119896|119896= x119869120580119896 (15)


x119895119896+1|119896

sim 119901 (x119896+1

| x119896= x119894119896|119896 q1198970119896)

ie x119895119896+1|119896

= F119896x119894119896|119896

(16)

(6) Consider x119901119891119896+1|119896

= F119896x119901119891119896|119896 Also set 119896 = 119896 + 1 and



119897119896= sign(119910

119897119896minus 119897119896|119896minus1

))



119896=

Φ(119902119897119896h119897(x119894119896|119896minus1


) 0R119890119896(119897 119897)) We note that 119864[x

119896|

q1198970119896] = Σx119896y1198970119896Σ

minus1



q1198970119896

where

120585119896= Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896 (17)



q1198970119896minus1

So on receipt of 119902119897119896






119896| q1198970119896


119896minus1| q1198970119896minus1

) rarr 119901(120585119896minus1



| q1198970119896) rarr 119901(120585

119896| q1198970119896) which



| q1198970119896

and 120585119896minus1

| q1198970119896minus1

is given by

119901 (120585119896minus1

| q1198970119896)

119901 (120585119896minus1

| q1198970119896minus1

)prop Φ(S

119896q1198970119896 h119897119896F119896120585119896minus1R119890119896 (119897 119897)) (18)

So if 120585119894119896minus1|119896minus1


119901(120585119896minus1

| q1198970119896minus1






120580= 119894) prop 120596

119894 and set120585120580

119896minus1|119896= 120585119869120580



119896minus1| q1198970119896minus1


119897119896





119896given y

1198970119896 Since

x119896 and 119910



120585119896|119896= F119896120585119896|119896minus1


)

K119896=

P119896|119896minus1

h119879119897119896

h119897119896P119896|119896minus1

h119879119897119896+ R119896(119897 119897)

(19)




Y119896|119896= Y119896|119896minus1

+I119896

z119896|119896= z119896|119896minus1

+ i119896

(20)

where Y119896|119896

= Pminus1119896|119896

and z119896|119896

= Y119896|119896120585119896|119896


119896|119896= Y119896|119896 I119873and

120585119896|119896

= Y119896|119896

z119896|119896 where I




I119896= h119879119897119896Rminus1119896(119897 119897) h

119897119896

i119896= h119879119897119896Rminus1119896(119897 119897) 119910

119897119896

(21)


119896minus1| q1198970119896) to 119901(120585

119896| q1198970119896) Following


120580is a set




119896| q1198970119896) can be




119901 (119910119897119896| 120585120580

119896minus1|119896 q1198970119896)

= 120601 (S119896q1198970119896 h119897F119896120585

120580

119896minus1|119896R119890119896(119897 119897))

(22)

Also set z119894119896|119896= z120580119896|119896minus1

+ h119879119897119896Rminus1119896(119897 119897)119910119894

119897119896|119896









119896 where 120598

119896




120574120591+1

= 120574120591

0 = H120591120574120591minus 120598120591

(23)

where 1205741|1

= x119896|119896

and P1199011198981|1

= P119896|119896

and H120591

=

[sign(1120591|120591


)] 120591 = 1 2 119871 119895120591|120591


120591


= 119871|119871

and P119896|119896

= P119901119898119871|119871



2 is


120591+1|120591+1

= 120591|120591minus

P119901119898120591|120591

sign (120591|120591)10038171003817100381710038171003817120591|120591

100381710038171003817100381710038171

sign (120591|120591)119879

P119901119898120591|120591

sign (120591|120591) + 1198771015840120598

(24)

P119901119898120591+1|120591+1

= P119901119898120591|120591minus

P119901119898120591|120591

sign (120591|120591) sign (

120591|120591)119879

P119901119898120591|120591

sign (120591|120591)119879

P119901119898120591|120591

sign (120591|120591) + 1198771015840120598

(25)



120594119904= x + radic119873(Sx)119904 120596

119904=

1

2119873

120594119873+119904

= x minus radic119873(Sx)119904 120596119873+119904

=1

2119873

(26)


119864 [x] = 1

2119873

2119873

sum

119904=1

120594lowast

119904

Cov [y] = 1

2119873

2119873

sum

119904=1

120594lowast

119904120594lowast119879

119904minus xx119879

(27)

where 120594lowast119904= f(120594119904) 119904 = 0 1 2119873

ek

z1k

zN119901

k

Resa

mpl

e

Pred

ictio

n

Method 1

ormethod 2

IF

IF

IF



119896|119896and 120594



120594lowast

119904119896|119896= 120594119904119896|119896

minus

P119896|119896

sign (120594119904119896|119896

)10038171003817100381710038171003817120594119904119896|119896

100381710038171003817100381710038171

sign (120594119904119896|119896

)119879

P119896|119896

sign (120594119904119896|119896

) + 120598

(28)

where 120598= O (

10038171003817100381710038171003817120594119904119896|119896

10038171003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598(29)


119904119896|1198962119873



6 The Algorithm



0|minus1 0|minus1

P0|minus1


0|minus1 z0|minus1

Y0|minus1


119890119896(119897 119897) which denote the


119897119896(see (31)) to

the 119897th sensor

R119890119896= H119896P119896|119896minus1

H119879119896+ R119896 (30)

119897119896|119896minus1

= h119897119896119896|119896minus1

(31)



119902119897119896= 119876(

119910119897119896minus 119897119896|119896minus1

R119890119896(119897 119897)

)R119890119896(119897 119897) (32)



119910119894

119897119896|119896sim 120601 (S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (33)

120596119894

119897119896sim Φ(S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (34)


119896|119896=

[119910119894

1119896|119896sdot sdot sdot 119910119872


Y119896|119896= Y119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) h

119897119896

z119894119896|119896= z119894119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) 119910

119894

119897119896|119896

(35)



z119896|119896=

119873119901

sum

119894=1

120596119894

119896z119894119896|119896 (36)

where 120596119894119896= prod119872

119897=1120596119894

119897119896





z119896+1|119896

Y119896+1|1

119897119896+1|119896

and P


z119894119896+1|119896

= F119896z119894119896|119896

z119896+1|119896

= F119896z119896|119896

Y119896+1|119896

= [F119896P119896|119896F119879119896+W119896+1]minus1

119896+1|119896

= Y119896+1|119896

z119896+1|119896

P119896+1|119896

= Y119896+1|119896

I119873

(37)


observations 119910119894119897119896119873119901

119894=1


119901) Measurement




119901) + 119874(119873

2119872)



119896 in



119896is set asW

119896(119894 119894) = 01


x119896+1

(119894) =

x119896(119894) + w

119896(119894) if 119894 isin supp (x

119896)

0 otherwise(38)


2) Both the





119896= 001


x0|minus1






119896minus x1198962x1198962)



0 20 40 60 80 100minus1

0

1

2

3

4



0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

0 20 40 60 80 100minus15

minus1

minus05

0

05

0 20 40 60 80 100minus05

0

05

1

15

2

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

x 7x 7

7

x 27

x 141


50 100 150 200 250minus8

minus6

minus4

minus2

0

2

4

6

8

Support index

Am

plitu

de

TrueCSKF



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

QPFCSKF Algorithm 1

Algorithm 2

1

09

08

07

06

05

04

03

02

01

0



0 20 40 60 80 1000

1

2

3

4

0 20 40 60 80 100minus1

0

1

2

3

0 20 40 60 80 100 0 20 40 60 80 100

x 4x 9

1

x 42

x 98

minus05

0

05

1

15

2

minus15

minus1

minus05

0

05

Time index

Time index

Time index

Time index










119896(119894 119894) = 001

2 and R119896= 025

2I40






1 2 3 40

002

004

006

008

01


Nor

mal

ized

RM

SE



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

2

18

16

14

12

1

08

06

04

02

0




8 Conclusions



Appendix


119896| q1198970119896


119904119879a] forall119904 isin 119877

119889Note that

119901 (x119896| q1198970119896) = int119901 (x

119896 y1198970119896

| q1198970119896) 119889y1198970119896

(A1)

and119901(x119896| y1198970119896 q1198970119896) = 119901(x

119896| y1198970119896) so theMGFof x

119896| q1198970119896

can be given by

119864 [119890119904119879x119896 | q

1198970119896]

= int 119890119904119879x119896119901 (x

119896| y1198970119896) 119901 (y1198970119896

| q1198970119896) 119889x119896119889y1198970119896

= 119890(12)119904

119879ΣΔ

x119896y1198970119896119904int 119890119904119879Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896

119901 (y1198970119896

| q1198970119896) 119889y1198970119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

MGF of Σx119896y1198970119896Σminus1

y1198970119896y1198970119896|q1198970119896

997904rArr

119872x119896|q1198970119896 (119904) = 119872120578119896 (119904)119872y1198970119896|q1198970119896 (Σminus1

y1198970119896Σy1198970119896x119896119904)

(A2)

where 120578119896sim N119889(0ΣΔ


y1198970119896

sim N119889(Σx119896y1198970119896Σ

minus1

y1198970119896 ΣΔ



0 =1003817100381710038171003817x11989610038171003817100381710038171 minus 120598

1015840

119896 (A3)



0 = [sign (x119896)]119879 x119896minus 1205981015840

119896= sign (x

119896)119879

+ g119879x119896minus 1205981015840

119896 (A4)


119896=

g119879x119896minus1205981015840

119896 Note that119864[1205981015840

119896] = 0 and119864[12059810158402

119896] = 1198771015840



moment

119864 [ 1205982

119896] = 119864 [g119879 (x

119896+ x119896) (x119896+ x119896)119879 g | Y

119896] + 1198771015840

120598 (A5)


1015840




120598= 119864 [ 120598

2

119896]

= g119879 (x119896) (x119896)119879 g + g119879119864 [x

119896(x119896)119879

| Y119896] g + 1198771015840

120598

asymp O (1003817100381710038171003817x1198961003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598

(A6)


Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896=

Φ(119902119897119896h119897(x119894119896|119896minus1


) 0R119890119896(119897 119897)) We note that 119864[x

119896|

q1198970119896] = Σx119896y1198970119896Σ

minus1



q1198970119896

where

120585119896= Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896 (17)



q1198970119896minus1

So on receipt of 119902119897119896






119896| q1198970119896


119896minus1| q1198970119896minus1

) rarr 119901(120585119896minus1



| q1198970119896) rarr 119901(120585

119896| q1198970119896) which



| q1198970119896

and 120585119896minus1

| q1198970119896minus1

is given by

119901 (120585119896minus1

| q1198970119896)

119901 (120585119896minus1

| q1198970119896minus1

)prop Φ(S

119896q1198970119896 h119897119896F119896120585119896minus1R119890119896 (119897 119897)) (18)

So if 120585119894119896minus1|119896minus1


119901(120585119896minus1

| q1198970119896minus1






120580= 119894) prop 120596

119894 and set120585120580

119896minus1|119896= 120585119869120580



119896minus1| q1198970119896minus1


119897119896





119896given y

1198970119896 Since

x119896 and 119910



120585119896|119896= F119896120585119896|119896minus1


)

K119896=

P119896|119896minus1

h119879119897119896

h119897119896P119896|119896minus1

h119879119897119896+ R119896(119897 119897)

(19)




Y119896|119896= Y119896|119896minus1

+I119896

z119896|119896= z119896|119896minus1

+ i119896

(20)

where Y119896|119896

= Pminus1119896|119896

and z119896|119896

= Y119896|119896120585119896|119896


119896|119896= Y119896|119896 I119873and

120585119896|119896

= Y119896|119896

z119896|119896 where I




I119896= h119879119897119896Rminus1119896(119897 119897) h

119897119896

i119896= h119879119897119896Rminus1119896(119897 119897) 119910

119897119896

(21)


119896minus1| q1198970119896) to 119901(120585

119896| q1198970119896) Following


120580is a set




119896| q1198970119896) can be




119901 (119910119897119896| 120585120580

119896minus1|119896 q1198970119896)

= 120601 (S119896q1198970119896 h119897F119896120585

120580

119896minus1|119896R119890119896(119897 119897))

(22)

Also set z119894119896|119896= z120580119896|119896minus1

+ h119879119897119896Rminus1119896(119897 119897)119910119894

119897119896|119896









119896 where 120598

119896




120574120591+1

= 120574120591

0 = H120591120574120591minus 120598120591

(23)

where 1205741|1

= x119896|119896

and P1199011198981|1

= P119896|119896

and H120591

=

[sign(1120591|120591


)] 120591 = 1 2 119871 119895120591|120591


120591


= 119871|119871

and P119896|119896

= P119901119898119871|119871



2 is


120591+1|120591+1

= 120591|120591minus

P119901119898120591|120591

sign (120591|120591)10038171003817100381710038171003817120591|120591

100381710038171003817100381710038171

sign (120591|120591)119879

P119901119898120591|120591

sign (120591|120591) + 1198771015840120598

(24)

P119901119898120591+1|120591+1

= P119901119898120591|120591minus

P119901119898120591|120591

sign (120591|120591) sign (

120591|120591)119879

P119901119898120591|120591

sign (120591|120591)119879

P119901119898120591|120591

sign (120591|120591) + 1198771015840120598

(25)



120594119904= x + radic119873(Sx)119904 120596

119904=

1

2119873

120594119873+119904

= x minus radic119873(Sx)119904 120596119873+119904

=1

2119873

(26)


119864 [x] = 1

2119873

2119873

sum

119904=1

120594lowast

119904

Cov [y] = 1

2119873

2119873

sum

119904=1

120594lowast

119904120594lowast119879

119904minus xx119879

(27)

where 120594lowast119904= f(120594119904) 119904 = 0 1 2119873

ek

z1k

zN119901

k

Resa

mpl

e

Pred

ictio

n

Method 1

ormethod 2

IF

IF

IF



119896|119896and 120594



120594lowast

119904119896|119896= 120594119904119896|119896

minus

P119896|119896

sign (120594119904119896|119896

)10038171003817100381710038171003817120594119904119896|119896

100381710038171003817100381710038171

sign (120594119904119896|119896

)119879

P119896|119896

sign (120594119904119896|119896

) + 120598

(28)

where 120598= O (

10038171003817100381710038171003817120594119904119896|119896

10038171003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598(29)


119904119896|1198962119873



6 The Algorithm



0|minus1 0|minus1

P0|minus1


0|minus1 z0|minus1

Y0|minus1


119890119896(119897 119897) which denote the


119897119896(see (31)) to

the 119897th sensor

R119890119896= H119896P119896|119896minus1

H119879119896+ R119896 (30)

119897119896|119896minus1

= h119897119896119896|119896minus1

(31)



119902119897119896= 119876(

119910119897119896minus 119897119896|119896minus1

R119890119896(119897 119897)

)R119890119896(119897 119897) (32)



119910119894

119897119896|119896sim 120601 (S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (33)

120596119894

119897119896sim Φ(S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (34)


119896|119896=

[119910119894

1119896|119896sdot sdot sdot 119910119872


Y119896|119896= Y119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) h

119897119896

z119894119896|119896= z119894119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) 119910

119894

119897119896|119896

(35)



z119896|119896=

119873119901

sum

119894=1

120596119894

119896z119894119896|119896 (36)

where 120596119894119896= prod119872

119897=1120596119894

119897119896





z119896+1|119896

Y119896+1|1

119897119896+1|119896

and P


z119894119896+1|119896

= F119896z119894119896|119896

z119896+1|119896

= F119896z119896|119896

Y119896+1|119896

= [F119896P119896|119896F119879119896+W119896+1]minus1

119896+1|119896

= Y119896+1|119896

z119896+1|119896

P119896+1|119896

= Y119896+1|119896

I119873

(37)


observations 119910119894119897119896119873119901

119894=1


119901) Measurement




119901) + 119874(119873

2119872)



119896 in



119896is set asW

119896(119894 119894) = 01


x119896+1

(119894) =

x119896(119894) + w

119896(119894) if 119894 isin supp (x

119896)

0 otherwise(38)


2) Both the





119896= 001


x0|minus1






119896minus x1198962x1198962)



0 20 40 60 80 100minus1

0

1

2

3

4



0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

0 20 40 60 80 100minus15

minus1

minus05

0

05

0 20 40 60 80 100minus05

0

05

1

15

2

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

x 7x 7

7

x 27

x 141


50 100 150 200 250minus8

minus6

minus4

minus2

0

2

4

6

8

Support index

Am

plitu

de

TrueCSKF



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

QPFCSKF Algorithm 1

Algorithm 2

1

09

08

07

06

05

04

03

02

01

0



0 20 40 60 80 1000

1

2

3

4

0 20 40 60 80 100minus1

0

1

2

3

0 20 40 60 80 100 0 20 40 60 80 100

x 4x 9

1

x 42

x 98

minus05

0

05

1

15

2

minus15

minus1

minus05

0

05

Time index

Time index

Time index

Time index










119896(119894 119894) = 001

2 and R119896= 025

2I40






1 2 3 40

002

004

006

008

01


Nor

mal

ized

RM

SE



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

2

18

16

14

12

1

08

06

04

02

0




8 Conclusions



Appendix


119896| q1198970119896


119904119879a] forall119904 isin 119877

119889Note that

119901 (x119896| q1198970119896) = int119901 (x

119896 y1198970119896

| q1198970119896) 119889y1198970119896

(A1)

and119901(x119896| y1198970119896 q1198970119896) = 119901(x

119896| y1198970119896) so theMGFof x

119896| q1198970119896

can be given by

119864 [119890119904119879x119896 | q

1198970119896]

= int 119890119904119879x119896119901 (x

119896| y1198970119896) 119901 (y1198970119896

| q1198970119896) 119889x119896119889y1198970119896

= 119890(12)119904

119879ΣΔ

x119896y1198970119896119904int 119890119904119879Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896

119901 (y1198970119896

| q1198970119896) 119889y1198970119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

MGF of Σx119896y1198970119896Σminus1

y1198970119896y1198970119896|q1198970119896

997904rArr

119872x119896|q1198970119896 (119904) = 119872120578119896 (119904)119872y1198970119896|q1198970119896 (Σminus1

y1198970119896Σy1198970119896x119896119904)

(A2)

where 120578119896sim N119889(0ΣΔ


y1198970119896

sim N119889(Σx119896y1198970119896Σ

minus1

y1198970119896 ΣΔ



0 =1003817100381710038171003817x11989610038171003817100381710038171 minus 120598

1015840

119896 (A3)



0 = [sign (x119896)]119879 x119896minus 1205981015840

119896= sign (x

119896)119879

+ g119879x119896minus 1205981015840

119896 (A4)


119896=

g119879x119896minus1205981015840

119896 Note that119864[1205981015840

119896] = 0 and119864[12059810158402

119896] = 1198771015840



moment

119864 [ 1205982

119896] = 119864 [g119879 (x

119896+ x119896) (x119896+ x119896)119879 g | Y

119896] + 1198771015840

120598 (A5)


1015840




120598= 119864 [ 120598

2

119896]

= g119879 (x119896) (x119896)119879 g + g119879119864 [x

119896(x119896)119879

| Y119896] g + 1198771015840

120598

asymp O (1003817100381710038171003817x1198961003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598

(A6)


Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120574120591+1

= 120574120591

0 = H120591120574120591minus 120598120591

(23)

where 1205741|1

= x119896|119896

and P1199011198981|1

= P119896|119896

and H120591

=

[sign(1120591|120591


)] 120591 = 1 2 119871 119895120591|120591


120591


= 119871|119871

and P119896|119896

= P119901119898119871|119871



2 is


120591+1|120591+1

= 120591|120591minus

P119901119898120591|120591

sign (120591|120591)10038171003817100381710038171003817120591|120591

100381710038171003817100381710038171

sign (120591|120591)119879

P119901119898120591|120591

sign (120591|120591) + 1198771015840120598

(24)

P119901119898120591+1|120591+1

= P119901119898120591|120591minus

P119901119898120591|120591

sign (120591|120591) sign (

120591|120591)119879

P119901119898120591|120591

sign (120591|120591)119879

P119901119898120591|120591

sign (120591|120591) + 1198771015840120598

(25)



120594119904= x + radic119873(Sx)119904 120596

119904=

1

2119873

120594119873+119904

= x minus radic119873(Sx)119904 120596119873+119904

=1

2119873

(26)


119864 [x] = 1

2119873

2119873

sum

119904=1

120594lowast

119904

Cov [y] = 1

2119873

2119873

sum

119904=1

120594lowast

119904120594lowast119879

119904minus xx119879

(27)

where 120594lowast119904= f(120594119904) 119904 = 0 1 2119873

ek

z1k

zN119901

k

Resa

mpl

e

Pred

ictio

n

Method 1

ormethod 2

IF

IF

IF



119896|119896and 120594



120594lowast

119904119896|119896= 120594119904119896|119896

minus

P119896|119896

sign (120594119904119896|119896

)10038171003817100381710038171003817120594119904119896|119896

100381710038171003817100381710038171

sign (120594119904119896|119896

)119879

P119896|119896

sign (120594119904119896|119896

) + 120598

(28)

where 120598= O (

10038171003817100381710038171003817120594119904119896|119896

10038171003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598(29)


119904119896|1198962119873



6 The Algorithm



0|minus1 0|minus1

P0|minus1


0|minus1 z0|minus1

Y0|minus1


119890119896(119897 119897) which denote the


119897119896(see (31)) to

the 119897th sensor

R119890119896= H119896P119896|119896minus1

H119879119896+ R119896 (30)

119897119896|119896minus1

= h119897119896119896|119896minus1

(31)



119902119897119896= 119876(

119910119897119896minus 119897119896|119896minus1

R119890119896(119897 119897)

)R119890119896(119897 119897) (32)



119910119894

119897119896|119896sim 120601 (S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (33)

120596119894

119897119896sim Φ(S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (34)


119896|119896=

[119910119894

1119896|119896sdot sdot sdot 119910119872


Y119896|119896= Y119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) h

119897119896

z119894119896|119896= z119894119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) 119910

119894

119897119896|119896

(35)



z119896|119896=

119873119901

sum

119894=1

120596119894

119896z119894119896|119896 (36)

where 120596119894119896= prod119872

119897=1120596119894

119897119896





z119896+1|119896

Y119896+1|1

119897119896+1|119896

and P


z119894119896+1|119896

= F119896z119894119896|119896

z119896+1|119896

= F119896z119896|119896

Y119896+1|119896

= [F119896P119896|119896F119879119896+W119896+1]minus1

119896+1|119896

= Y119896+1|119896

z119896+1|119896

P119896+1|119896

= Y119896+1|119896

I119873

(37)


observations 119910119894119897119896119873119901

119894=1


119901) Measurement




119901) + 119874(119873

2119872)



119896 in



119896is set asW

119896(119894 119894) = 01


x119896+1

(119894) =

x119896(119894) + w

119896(119894) if 119894 isin supp (x

119896)

0 otherwise(38)


2) Both the





119896= 001


x0|minus1






119896minus x1198962x1198962)



0 20 40 60 80 100minus1

0

1

2

3

4



0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

0 20 40 60 80 100minus15

minus1

minus05

0

05

0 20 40 60 80 100minus05

0

05

1

15

2

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

x 7x 7

7

x 27

x 141


50 100 150 200 250minus8

minus6

minus4

minus2

0

2

4

6

8

Support index

Am

plitu

de

TrueCSKF



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

QPFCSKF Algorithm 1

Algorithm 2

1

09

08

07

06

05

04

03

02

01

0



0 20 40 60 80 1000

1

2

3

4

0 20 40 60 80 100minus1

0

1

2

3

0 20 40 60 80 100 0 20 40 60 80 100

x 4x 9

1

x 42

x 98

minus05

0

05

1

15

2

minus15

minus1

minus05

0

05

Time index

Time index

Time index

Time index










119896(119894 119894) = 001

2 and R119896= 025

2I40






1 2 3 40

002

004

006

008

01


Nor

mal

ized

RM

SE



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

2

18

16

14

12

1

08

06

04

02

0




8 Conclusions



Appendix


119896| q1198970119896


119904119879a] forall119904 isin 119877

119889Note that

119901 (x119896| q1198970119896) = int119901 (x

119896 y1198970119896

| q1198970119896) 119889y1198970119896

(A1)

and119901(x119896| y1198970119896 q1198970119896) = 119901(x

119896| y1198970119896) so theMGFof x

119896| q1198970119896

can be given by

119864 [119890119904119879x119896 | q

1198970119896]

= int 119890119904119879x119896119901 (x

119896| y1198970119896) 119901 (y1198970119896

| q1198970119896) 119889x119896119889y1198970119896

= 119890(12)119904

119879ΣΔ

x119896y1198970119896119904int 119890119904119879Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896

119901 (y1198970119896

| q1198970119896) 119889y1198970119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

MGF of Σx119896y1198970119896Σminus1

y1198970119896y1198970119896|q1198970119896

997904rArr

119872x119896|q1198970119896 (119904) = 119872120578119896 (119904)119872y1198970119896|q1198970119896 (Σminus1

y1198970119896Σy1198970119896x119896119904)

(A2)

where 120578119896sim N119889(0ΣΔ


y1198970119896

sim N119889(Σx119896y1198970119896Σ

minus1

y1198970119896 ΣΔ



0 =1003817100381710038171003817x11989610038171003817100381710038171 minus 120598

1015840

119896 (A3)



0 = [sign (x119896)]119879 x119896minus 1205981015840

119896= sign (x

119896)119879

+ g119879x119896minus 1205981015840

119896 (A4)


119896=

g119879x119896minus1205981015840

119896 Note that119864[1205981015840

119896] = 0 and119864[12059810158402

119896] = 1198771015840



moment

119864 [ 1205982

119896] = 119864 [g119879 (x

119896+ x119896) (x119896+ x119896)119879 g | Y

119896] + 1198771015840

120598 (A5)


1015840




120598= 119864 [ 120598

2

119896]

= g119879 (x119896) (x119896)119879 g + g119879119864 [x

119896(x119896)119879

| Y119896] g + 1198771015840

120598

asymp O (1003817100381710038171003817x1198961003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598

(A6)


Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119902119897119896= 119876(

119910119897119896minus 119897119896|119896minus1

R119890119896(119897 119897)

)R119890119896(119897 119897) (32)



119910119894

119897119896|119896sim 120601 (S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (33)

120596119894

119897119896sim Φ(S

119896q1198970119896 h119897119896119894

119896|119896minus1R119890119896(119897 119897)) (34)


119896|119896=

[119910119894

1119896|119896sdot sdot sdot 119910119872


Y119896|119896= Y119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) h

119897119896

z119894119896|119896= z119894119896|119896minus1

+

119872

sum

119897=1

h119879119897119896Rminus1119896(119897 119897) 119910

119894

119897119896|119896

(35)



z119896|119896=

119873119901

sum

119894=1

120596119894

119896z119894119896|119896 (36)

where 120596119894119896= prod119872

119897=1120596119894

119897119896





z119896+1|119896

Y119896+1|1

119897119896+1|119896

and P


z119894119896+1|119896

= F119896z119894119896|119896

z119896+1|119896

= F119896z119896|119896

Y119896+1|119896

= [F119896P119896|119896F119879119896+W119896+1]minus1

119896+1|119896

= Y119896+1|119896

z119896+1|119896

P119896+1|119896

= Y119896+1|119896

I119873

(37)


observations 119910119894119897119896119873119901

119894=1


119901) Measurement




119901) + 119874(119873

2119872)



119896 in



119896is set asW

119896(119894 119894) = 01


x119896+1

(119894) =

x119896(119894) + w

119896(119894) if 119894 isin supp (x

119896)

0 otherwise(38)


2) Both the





119896= 001


x0|minus1






119896minus x1198962x1198962)



0 20 40 60 80 100minus1

0

1

2

3

4



0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

0 20 40 60 80 100minus15

minus1

minus05

0

05

0 20 40 60 80 100minus05

0

05

1

15

2

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

x 7x 7

7

x 27

x 141


50 100 150 200 250minus8

minus6

minus4

minus2

0

2

4

6

8

Support index

Am

plitu

de

TrueCSKF



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

QPFCSKF Algorithm 1

Algorithm 2

1

09

08

07

06

05

04

03

02

01

0



0 20 40 60 80 1000

1

2

3

4

0 20 40 60 80 100minus1

0

1

2

3

0 20 40 60 80 100 0 20 40 60 80 100

x 4x 9

1

x 42

x 98

minus05

0

05

1

15

2

minus15

minus1

minus05

0

05

Time index

Time index

Time index

Time index










119896(119894 119894) = 001

2 and R119896= 025

2I40






1 2 3 40

002

004

006

008

01


Nor

mal

ized

RM

SE



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

2

18

16

14

12

1

08

06

04

02

0




8 Conclusions



Appendix


119896| q1198970119896


119904119879a] forall119904 isin 119877

119889Note that

119901 (x119896| q1198970119896) = int119901 (x

119896 y1198970119896

| q1198970119896) 119889y1198970119896

(A1)

and119901(x119896| y1198970119896 q1198970119896) = 119901(x

119896| y1198970119896) so theMGFof x

119896| q1198970119896

can be given by

119864 [119890119904119879x119896 | q

1198970119896]

= int 119890119904119879x119896119901 (x

119896| y1198970119896) 119901 (y1198970119896

| q1198970119896) 119889x119896119889y1198970119896

= 119890(12)119904

119879ΣΔ

x119896y1198970119896119904int 119890119904119879Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896

119901 (y1198970119896

| q1198970119896) 119889y1198970119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

MGF of Σx119896y1198970119896Σminus1

y1198970119896y1198970119896|q1198970119896

997904rArr

119872x119896|q1198970119896 (119904) = 119872120578119896 (119904)119872y1198970119896|q1198970119896 (Σminus1

y1198970119896Σy1198970119896x119896119904)

(A2)

where 120578119896sim N119889(0ΣΔ


y1198970119896

sim N119889(Σx119896y1198970119896Σ

minus1

y1198970119896 ΣΔ



0 =1003817100381710038171003817x11989610038171003817100381710038171 minus 120598

1015840

119896 (A3)



0 = [sign (x119896)]119879 x119896minus 1205981015840

119896= sign (x

119896)119879

+ g119879x119896minus 1205981015840

119896 (A4)


119896=

g119879x119896minus1205981015840

119896 Note that119864[1205981015840

119896] = 0 and119864[12059810158402

119896] = 1198771015840



moment

119864 [ 1205982

119896] = 119864 [g119879 (x

119896+ x119896) (x119896+ x119896)119879 g | Y

119896] + 1198771015840

120598 (A5)


1015840




120598= 119864 [ 120598

2

119896]

= g119879 (x119896) (x119896)119879 g + g119879119864 [x

119896(x119896)119879

| Y119896] g + 1198771015840

120598

asymp O (1003817100381710038171003817x1198961003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598

(A6)


Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100minus1

0

1

2

3

4



0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

0 20 40 60 80 100minus15

minus1

minus05

0

05

0 20 40 60 80 100minus05

0

05

1

15

2

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

TrueCSKFAlgorithm 1

Algorithm 2QPF

x 7x 7

7

x 27

x 141


50 100 150 200 250minus8

minus6

minus4

minus2

0

2

4

6

8

Support index

Am

plitu

de

TrueCSKF



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

QPFCSKF Algorithm 1

Algorithm 2

1

09

08

07

06

05

04

03

02

01

0



0 20 40 60 80 1000

1

2

3

4

0 20 40 60 80 100minus1

0

1

2

3

0 20 40 60 80 100 0 20 40 60 80 100

x 4x 9

1

x 42

x 98

minus05

0

05

1

15

2

minus15

minus1

minus05

0

05

Time index

Time index

Time index

Time index










119896(119894 119894) = 001

2 and R119896= 025

2I40






1 2 3 40

002

004

006

008

01


Nor

mal

ized

RM

SE



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

2

18

16

14

12

1

08

06

04

02

0




8 Conclusions



Appendix


119896| q1198970119896


119904119879a] forall119904 isin 119877

119889Note that

119901 (x119896| q1198970119896) = int119901 (x

119896 y1198970119896

| q1198970119896) 119889y1198970119896

(A1)

and119901(x119896| y1198970119896 q1198970119896) = 119901(x

119896| y1198970119896) so theMGFof x

119896| q1198970119896

can be given by

119864 [119890119904119879x119896 | q

1198970119896]

= int 119890119904119879x119896119901 (x

119896| y1198970119896) 119901 (y1198970119896

| q1198970119896) 119889x119896119889y1198970119896

= 119890(12)119904

119879ΣΔ

x119896y1198970119896119904int 119890119904119879Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896

119901 (y1198970119896

| q1198970119896) 119889y1198970119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

MGF of Σx119896y1198970119896Σminus1

y1198970119896y1198970119896|q1198970119896

997904rArr

119872x119896|q1198970119896 (119904) = 119872120578119896 (119904)119872y1198970119896|q1198970119896 (Σminus1

y1198970119896Σy1198970119896x119896119904)

(A2)

where 120578119896sim N119889(0ΣΔ


y1198970119896

sim N119889(Σx119896y1198970119896Σ

minus1

y1198970119896 ΣΔ



0 =1003817100381710038171003817x11989610038171003817100381710038171 minus 120598

1015840

119896 (A3)



0 = [sign (x119896)]119879 x119896minus 1205981015840

119896= sign (x

119896)119879

+ g119879x119896minus 1205981015840

119896 (A4)


119896=

g119879x119896minus1205981015840

119896 Note that119864[1205981015840

119896] = 0 and119864[12059810158402

119896] = 1198771015840



moment

119864 [ 1205982

119896] = 119864 [g119879 (x

119896+ x119896) (x119896+ x119896)119879 g | Y

119896] + 1198771015840

120598 (A5)


1015840




120598= 119864 [ 120598

2

119896]

= g119879 (x119896) (x119896)119879 g + g119879119864 [x

119896(x119896)119879

| Y119896] g + 1198771015840

120598

asymp O (1003817100381710038171003817x1198961003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598

(A6)


Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 1000

1

2

3

4

0 20 40 60 80 100minus1

0

1

2

3

0 20 40 60 80 100 0 20 40 60 80 100

x 4x 9

1

x 42

x 98

minus05

0

05

1

15

2

minus15

minus1

minus05

0

05

Time index

Time index

Time index

Time index










119896(119894 119894) = 001

2 and R119896= 025

2I40






1 2 3 40

002

004

006

008

01


Nor

mal

ized

RM

SE



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

2

18

16

14

12

1

08

06

04

02

0




8 Conclusions



Appendix


119896| q1198970119896


119904119879a] forall119904 isin 119877

119889Note that

119901 (x119896| q1198970119896) = int119901 (x

119896 y1198970119896

| q1198970119896) 119889y1198970119896

(A1)

and119901(x119896| y1198970119896 q1198970119896) = 119901(x

119896| y1198970119896) so theMGFof x

119896| q1198970119896

can be given by

119864 [119890119904119879x119896 | q

1198970119896]

= int 119890119904119879x119896119901 (x

119896| y1198970119896) 119901 (y1198970119896

| q1198970119896) 119889x119896119889y1198970119896

= 119890(12)119904

119879ΣΔ

x119896y1198970119896119904int 119890119904119879Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896

119901 (y1198970119896

| q1198970119896) 119889y1198970119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

MGF of Σx119896y1198970119896Σminus1

y1198970119896y1198970119896|q1198970119896

997904rArr

119872x119896|q1198970119896 (119904) = 119872120578119896 (119904)119872y1198970119896|q1198970119896 (Σminus1

y1198970119896Σy1198970119896x119896119904)

(A2)

where 120578119896sim N119889(0ΣΔ


y1198970119896

sim N119889(Σx119896y1198970119896Σ

minus1

y1198970119896 ΣΔ



0 =1003817100381710038171003817x11989610038171003817100381710038171 minus 120598

1015840

119896 (A3)



0 = [sign (x119896)]119879 x119896minus 1205981015840

119896= sign (x

119896)119879

+ g119879x119896minus 1205981015840

119896 (A4)


119896=

g119879x119896minus1205981015840

119896 Note that119864[1205981015840

119896] = 0 and119864[12059810158402

119896] = 1198771015840



moment

119864 [ 1205982

119896] = 119864 [g119879 (x

119896+ x119896) (x119896+ x119896)119879 g | Y

119896] + 1198771015840

120598 (A5)


1015840




120598= 119864 [ 120598

2

119896]

= g119879 (x119896) (x119896)119879 g + g119879119864 [x

119896(x119896)119879

| Y119896] g + 1198771015840

120598

asymp O (1003817100381710038171003817x1198961003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598

(A6)


Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 40

002

004

006

008

01


Nor

mal

ized

RM

SE



0 10 20 30 40 50 60 70 80 90 100Time index

Nor

mal

ized

RM

SE

2

18

16

14

12

1

08

06

04

02

0




8 Conclusions



Appendix


119896| q1198970119896


119904119879a] forall119904 isin 119877

119889Note that

119901 (x119896| q1198970119896) = int119901 (x

119896 y1198970119896

| q1198970119896) 119889y1198970119896

(A1)

and119901(x119896| y1198970119896 q1198970119896) = 119901(x

119896| y1198970119896) so theMGFof x

119896| q1198970119896

can be given by

119864 [119890119904119879x119896 | q

1198970119896]

= int 119890119904119879x119896119901 (x

119896| y1198970119896) 119901 (y1198970119896

| q1198970119896) 119889x119896119889y1198970119896

= 119890(12)119904

119879ΣΔ

x119896y1198970119896119904int 119890119904119879Σx119896y1198970119896Σ

minus1

y1198970119896y1198970119896

119901 (y1198970119896

| q1198970119896) 119889y1198970119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

MGF of Σx119896y1198970119896Σminus1

y1198970119896y1198970119896|q1198970119896

997904rArr

119872x119896|q1198970119896 (119904) = 119872120578119896 (119904)119872y1198970119896|q1198970119896 (Σminus1

y1198970119896Σy1198970119896x119896119904)

(A2)

where 120578119896sim N119889(0ΣΔ


y1198970119896

sim N119889(Σx119896y1198970119896Σ

minus1

y1198970119896 ΣΔ



0 =1003817100381710038171003817x11989610038171003817100381710038171 minus 120598

1015840

119896 (A3)



0 = [sign (x119896)]119879 x119896minus 1205981015840

119896= sign (x

119896)119879

+ g119879x119896minus 1205981015840

119896 (A4)


119896=

g119879x119896minus1205981015840

119896 Note that119864[1205981015840

119896] = 0 and119864[12059810158402

119896] = 1198771015840



moment

119864 [ 1205982

119896] = 119864 [g119879 (x

119896+ x119896) (x119896+ x119896)119879 g | Y

119896] + 1198771015840

120598 (A5)


1015840




120598= 119864 [ 120598

2

119896]

= g119879 (x119896) (x119896)119879 g + g119879119864 [x

119896(x119896)119879

| Y119896] g + 1198771015840

120598

asymp O (1003817100381710038171003817x1198961003817100381710038171003817

2

2+ g119879P

119896|119896g) + 1198771015840

120598

(A6)


Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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