IEEE SIGNAL PROCESSING LETTERS, VOL. 21, NO. 2, FEBRUARY 2014 135

Optimal Prediction Likelihood Tree Based Source-Channel ML Decoder for Wireless Sensor Networks

C. Manoj and Aditya K. Jagannatham

Abstract—In this work, we develop a framework for optimaljoint Source-Channel Maximum Likelihood (SCML) decodingin Wireless Sensor Networks (WSNs). The proposed scheme em-ploys a novel Generalized Likelihood Ratio Test based PredictionLikelihood Tree (PLT) approach to exploit the spatio-temporalnarrowband properties of the sensor data for sequence detec-tion in wireless sensor networks. Further, analytical bounds arederived to characterize the performance of the low complexitydecision feedback and optimal Viterbi based Maximum Likeli-hood Sequence Detection (MLSD) for joint decoding over fadingwireless channels, where only ad hoc schemes exist in currentliterature. The PLT based SCML scheme, which has a low com-plexity, is ideally suited for implementation in practical wirelesssensor networks with limited computational power and achieves aperformance close to the optimal MLSD bound. Simulation resultsare presented to validate the performance of the SCML algorithmand the proposed analytical bounds for sensor data reception inWSNs.

Index Terms—Prediction likelihood tree, source-channel jointdecoding, wireless sensor networks.

I. INTRODUCTION

A WIRELESS sensor network (WSN) consists of a largenumber of sensors, which transmit the sensed data to a

central fusion center for subsequent processing of the collectedsensor data. Several algorithms have been proposed in existingliterature to address the problem of error correction in WSNs.However, a majority of such schemes are based on modifiedforms of error correcting codes [1], Automatic RetransmissionreQuest (ARQ) [2] or a hybrid of both these methods i.e. HARQ[3], which add redundancy to the transmitted data leading to sig-nificant overheads. In this context, a novel Prediction HistoryTree (PHT) algorithm has been proposed for low complexityWSN error correction based on the temporal narrowband prop-erties of the sensor data in [4]. However, the above PHT schemeis ad hoc in nature and does not develop a rigorous framework tocharacterize the performance of the proposedWSN error correc-tion methodology. In contrast, we propose an analytically rig-orous Source-Channel Maximum Likelihood (SCML) frame-work for error correction in multihop wireless sensor networks.The optimality of the proposed SCML scheme is demonstratedthrough a systematic characterization of the likelihood, which

Manuscript received October 02, 2013; revised December 02, 2013; acceptedDecember 05, 2013. Date of current version December 18, 2013. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Prof. Chao Tian.The authors are at the Department of Electrical Engineering, IIT Kanpur,

208016, India (e-mail: adityaj@iitk.ac.in).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/LSP.2013.2294794

can be naturally motivated as a Generalized Likelihood RatioTest (GLRT) based Prediction Likelihood Tree (PLT), whichchooses the optimal decoded vector at each decoding instant.Further, unlike the informal nature of the results in [4], we de-rive analytical bounds to characterize the MSE performance ofthe proposed decoding scheme. We validate the performance ofthe proposed algorithms and the analytical bounds through sim-ulation results.

II. WSN SYSTEM MODEL

Consider a wireless sensor network with sensors.Let , denote the data sensed bysensor at the time instant. The measurement vector

of the wireless sensor network at time can be de-fined as . The datasample is mapped to the transmit symbol vector

employing the encoding function as, where

each modulated symbol , , be-longs to the source constellation containing the symbols ,

. Let denote the fading channel coefficientbetween the th sensor and the corresponding next hop sensornode. The received symbol is given as,

where is the zero-mean circularly symmetric complexGaussian white noise of variance i.e. .The suboptimal wideband maximum likelihood (WML) schemefor decoding the transmitted symbol , which is agnosticto the narrowband source statistics is given as,

(1)

where is the decoded symbol corresponding tothe transmitted symbol . The net decoded symbol vector

is given as , fromwhich the sensor sample is reconstructed as

. However, the above decoder leads to a poor detec-tion performance since it does not consider the spatio-temporalnarrowband source properties of the wireless sensor network.In this context, the optimal joint source-channel statistics basedmaximum likelihood decoder can be developed as follows. Ob-serve that the decoupled decoding procedure given in (1) for theindividual symbols can be formulated as the jointlikelihood maximization,

where the concatenated receive symbol vector, ,

1070-9908 © 2013 IEEE

136 IEEE SIGNAL PROCESSING LETTERS, VOL. 21, NO. 2, FEBRUARY 2014

where denotes the identity matrix of size , anddenotes the constellation of dimensional

vector symbols. For a multihop WSN protocol such as PE-GASIS [5], the cost function for joint maximum likelihooddecoding at the sensor corresponding to the datareceived from the previous multihop sensors is given as,

(2)

where the spatially concatenated received symbol vector

and the diagonal matrix is de-fined as the block diagonal matrix, with the ma-trices along the diagonal and

. The decoded sensor data vectoris given as,

Similar to the temporal narrowband prediction model in [4], letthe spatio-temporal narrowband model for the predicted value

be given by the autoregressive model,

(3)

where represents the past data vectors, is the bias vector and denotes

the prediction function. The optimal coefficient matriceswhich minimize the prediction

error variance can be derived based on the MinimumMean Squared Error (MMSE) criterion using the proceduregiven in [6]. For ease of notation, we drop the index in theanalysis below and note that it is valid for decoding at eachmultihop sensor node with vectors of appro-priate size. The joint source-channel log-likelihood function

for the constellationvector corresponding to the obser-vation is given as,

In the likelihood cost function above, the termdenotes the channel likelihood

corresponding to the observed symbol vector similar to(2) and the term represents the sourcelikelihood component which captures the spatio-temporalnarrowband nature of the data being sensed across the wirelesssensor network. Hence, the optimal decoded vectorwhich maximizes the joint source-channel likelihood is givenas,

(4)

The Decision Feedback (DF) based joint source-channel de-coder can now be realized by replacing

with , where represents thepreviously decoded symbol vectors. Since each transmit vectorbelongs to a set of size , the complexity of DF based de-coding at sensor node is . The optimal maximum like-lihood sequence decoder for the joint detection of the vectors

can be expressed as,

(5)

The above source-channel likelihood maximization for de-tection of the transmit sequencecan be efficiently computed employing the trellis basedViterbi algorithm [7] for maximum likelihood sequencedecoding. The corresponding path metric is given as

. The statesof the trellis at each time instant can be denotedby where each

. Hence, the total number of statesat each time instant is . Transitions in the trellis arepossible between states of the formand at time instants andrespectively, thus leading to a total of transitions perstate. It can now be seen that even for modest values offor QPSK, , model order and decoding at themultihop sensor 3, corresponding to , the total number ofstates is which is intractable for optimal sequencedecoding. Hence, we present a PLT based scheme for lowcomplexity sequence detection which can be seen to achievea performance close to the MSE bound for optimal sequencedetection derived in Section V.

III. PREDICTION LIKELIHOOD TREE BASED JOINTSOURCE-CHANNEL DECODING

In this section, we describe a novel Prediction LikelihoodTree (PLT), similar to the Prediction History Tree (PHT) archi-tecture in [4], to decode the optimal vector correspondingto the joint source-channel maximum likelihood cost functiongiven in (4). However, while the scheme in [4] is ad hoc andbased simply on the mean squared prediction error metric, weconsider the optimal joint source channel likelihood metric foreach path. Consider a PLT with depth . The root of the PLTrepresents the last decoded data . The nodes, at each level , for each at level, belong to the set of constellation vectors . Due to lack

of space, a schematic figure of the PLT is given in [8]. We nowdescribe the Generalized Log Likelihood Ratio Test (GLRT)based scheme for optimal joint source-channel decoding of thevector at level 1. For eachat level 1 corresponding to the time instant , thegeneralized parameter estimate corresponding to the predictions

denoted byis given as (6), shown at the bottom of the page.Thus, the above GLRT cost function

MANOJ AND JAGANNATHAM: OPTIMAL PREDICTION LIKELIHOOD TREE BASED SOURCE- CHANNEL ML DECODER 137

represents the ability of the vector to accuratelypredict the observations at successive levels in theprediction likelihood tree similar to the approach employed inthe PHT. Hence, the decoded symbol vectorat time instant can be computed by minimizingthe GLRT based ML cost function given as (7), shown at thebottom of the page.Now, similar to the PHT approach, this is made the root node

at time instant and the procedure is repeated for timeinstant by adding the nodes corresponding to timeinstant . Also, it can be seen that the DF decoder is equiva-lent to the PLT based SCML decoder for . Moreover, thePLT decoder for corresponds to the optimal maximumlikelihood Viterbi decoder. The worst-case complexity of PLTbased decoding is . Further, this complexity canbe significantly reduced using the sphere decoder. A detailedreport of the number of operations required for the worst-caseand sphere decoding is given in [8].

IV. MSE PERFORMANCE ANALYSIS OF DF BASED JOINTSOURCE-CHANNEL DECODING

For ease of notation, we drop the time index in the anal-ysis below. Let the dimensional data vector be encodedas . The corresponding joint source model for thevector is given as,

Similarly, consider the sensor sample encoded asand defined correspondingly. Hence the pairwise de-

coding error probability is given as [9],

where the effective noise covariance is the block diagonalmatrix with along the block diagonaland denotes the Gaussian error function. Employing the ex-

pression for pairwise error probability for source-channel de-coding above, an upper bound on the MSE is given as

where the last inequality above follows from the Chernoff

Bound . Averaging over the distribution of thefading wireless channel, the union bound for the MSE of DFdecoding is derived as,

(9)

Due to lack of space, a detailed derivation is given in [8].

V. MSE PERFORMANCE ANALYSIS OF VITERBIDECODING BASED MLSD

We now derive the lower bound for the mean squared errorof the decoded sensor data using the optimal Viterbi algorithmbased maximum likelihood sequence detection. We computethe lower bound using the error probability of the shortestpossible error event of length . Consider a trellis within which the actual sensor data sample path corresponds to

. Let the corresponding trans-mitted and received symbol vectors beand respectively. Let the sensor datavectors be decoded correctly at all time instants, except atinstant . At time , let the true and erroneously decoded datavectors be given as and .This occurs when the minimum cost path deviates from thetrue path for states from to before rejoining

(6)

(7)

(8)

138 IEEE SIGNAL PROCESSING LETTERS, VOL. 21, NO. 2, FEBRUARY 2014

the true path. Let the decoded data vector be denoted bywhere . Employing

the spatio-temporal narrowband prediction model in (3), theprobability of the error event is given as(8), shown at the bottom of the page, where is theidentity matrix. Using steps similar to the previous derivation,the MSE bound for MLSD can be derived as

(10)

Since there are states with possible transitionseach, the complexity of Viterbi decoding is . Due tolack of space, a detailed derivation and the complexity in termsof the number of operations is given in [8].

VI. SIMULATION RESULTS

We consider a PEGASIS [5] protocol based multihop WSNscenario with a multihop radius of sensor nodes,and corresponding to QPSK. The noise power at thereceiver is varied from - to 5 dB. The wireless fading channelcoefficient is Rayleigh fading in nature with average power

. The parameters of the prediction model areset as , , with diagonal elements of equalto 0.6 and off diagonal elements set as 0.075. Similarly, the di-agonal and off diagonal elements of are set as 0.1, 0.0025respectively. We consider a three stage generation model with

in the first and third stages for ,and for

. The model noise variance is set as dB. Forthis simulation, we consider a PLT of depth levels. To re-duce the decoding complexity, the maximally likely vectors arechosen by employing the sphere decoding algorithm in [10]. InFig. 1, we plot the MSE for decoded data of sensor at thenext hop sensor against the noise power for variousdecoding techniques. The WML decoding algorithm, clearlyhas the highest MSE. The DF based source-channel detectionscheme outperforms the WML. The analytical MSE bound forDF derived in (9) closely follows the simulation result. TheMSE performance of the proposed PLT based SCML decodingand the corresponding bound for the optimal Viterbi algorithmbased joint source-channel decoding (10) are also given therein.It can be seen that the performance of the PLT for SCML isclose to that of the error bound for the optimal maximum like-lihood sequence decoder. This is also justified from the obser-vation in Section III that while the DF decoder corresponds to aPLT depth of , the optimal Viterbi decoder correspondsto . Hence, the PLT performance for various values of, , lies in between these two performance bounds.

The performance of the PHT based peer algorithm proposed in

Fig. 1. MSE of decoded sensor data of sensor at the second hop sensor.

Fig. 2. MSE of decoded sensor data received at the base station for sensors.

[4] is also plotted and it is clear that the PHT algorithm is subop-timal and performs poorly compared to the proposed techniques.Fig. 2 shows the MSE for sensor data decoding versus the noisepower for each of the sensors in the PEGASISmultihop WSN chain. The results show that the SCML basederror correction algorithm outperforms the conventional WMLdecoding scheme.

VII. CONCLUSION

In this letter, we presented a scheme for optimalML decodingin spatio-temporal narrowband WSNs. A novel prediction like-lihood tree based framework was developed for low complexitysequential decoding of the received sensor samples. The PLTscheme achieves joint source-channel likelihood maximizationin contrast to the ad hoc PHT based scheme in existing litera-ture. Further, analytical bounds were derived to characterize themean squared error performance of the DF and optimal MLSDschemes for joint source-channel decoding. Simulation resultshave been presented which demonstrate the superior perfor-mance of the proposed scheme.

MANOJ AND JAGANNATHAM: OPTIMAL PREDICTION LIKELIHOOD TREE BASED SOURCE- CHANNEL ML DECODER 139

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