Source Extraction in a Bandwidth Constrained Wireless Sensor Network Hongbin Chen,1,2 Chi K. Tse1 and Jiuchao Feng2 Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong, China 2 College of Electronic and Information Engineering, South China University of Technology, Guangzhou 510641, China Email: {enhbchen, cktse}@eie.polyu.edu.hk &
[email protected] 1
Keywords: Sensor network, source extraction, cluster, quantization, correlation coefficient
parameter has been considered in a statistical sense (using the mean-squared error criterion). Each sensor collects one noisecorrupted data, quantizes the collected data to 1-bit, and sends a binary message to the fusion center. Then, the fusion center estimates the parameter based on the received messages from all sensors. However, messages originating from all sensors are assumed to be perfectly received by the fusion center. Energy-efficient decentralized estimation schemes in a wireless sensor network are proposed in [8,9]. The communications between sensors and the fusion center are over AWGN channels with channel path fading. Moreover, the fusion center is assumed to know a priori the noise levels of sensor observations and the channel path fading coefficients. This may not hold true in practical wireless communication environments. Suppose a sensor network is applied for monitoring hazardous fields or fields that are unreachable to people. There may exist several unknown source signals in the field. For example, when a sensor network is used for chemical pollution monitoring, possibly more than one kind of pollution sources are observed. Furthermore, a sensor network with multi-functional sensors may be used to monitor heterogeneous fields where multi-modal source signals exist. In this paper, we propose a blind source extraction scheme in a bandwidth constrained wireless sensor network. Suppose a remote sink wants to extract multiple source signals in a sensing field. Each sensor observes a linear mixture of the source signals and their observations are corrupted by AWGN. A sensor in the network acts as a cluster head, performs local extraction of the source signals based on its own observation and the received noisy quantized data from other sensors. Then, the cluster head quantizes the extracted signals and sends the quantized data to the sink in a multi-hop way through a fading channel. Finally, the sink performs global extraction of the source signals. In order to improve the accuracy of source extraction, a multi-cluster structure of the sensor network is also proposed. The network is partitioned into several clusters. In each cluster, local extraction is performed and the following process is the same as that in the singlecluster case. The proposed scheme can be used for sensor networks that are deployed to monitor a sensing field where multiple source signals coexist and the field is hazardous or unreachable to people. The main advantage of our scheme over other schemes [3-
Abstract In this paper, we propose a blind source extraction scheme in a bandwidth constrained wireless sensor network. We consider the extraction of multiple source signals in a sensing field. Each sensor observes a linear mixture of the source signals and their observations are corrupted by additive white Gaussian noise (AWGN). First a sensor in the network acts as a cluster head and performs local extraction of the source signals using a fast fixed-point algorithm based on its own observation and the received noisy quantized data from other sensors. Then, the extracted signals are quantized using a pulse code modulation (PCM) method and the quantized data are sent to the sink in a multi-hop way through a fading channel. Finally the sink performs global extraction of the source signals using a constant modulus algorithm (CMA) and PCM decoding. In order to improve the accuracy of source extraction, a multi-cluster structure is also proposed. Simulation results confirm effective extraction of the source signals by comparing with the case of source extraction without quantization of the sensed data.
1. INTRODUCTION In the past few decades, we have witnessed a rapid progress in the development of smart electronic devices and wireless communication technologies [1]. This brought about the advent of low-cost sensors and wireless sensor networks. Densely distributed sensors in a sensing field are connected to form a sensor network via robust wireless links. These sensors are small and equipped with four basic components: sensing, data processing, radio communication, and power supply. They are self-organized and co-operate to accomplish some complex tasks [2]. The ultimate aim of a sensor network is to monitor a sensing field and to retrieve useful information from it. In practice, the design of sensor networks is under several constraints including limited available energy, limited communication bandwidth of sensors, noise corruption of data collected by sensors and distortion of the wireless communication channels. Recently, Some decentralized estimation schemes have been proposed for bandwidth constrained sensor networks [3-7]. The problem of estimating one unknown
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signals and is represented by
9] is that we can extract multiple source signals in a sensing field. We assume that the communications between sensors and the cluster heads are over AWGN channels and the communications between the cluster heads and the sink are over fading channels [10]. In addition we need not know the noise levels and channel path fading coefficients. This paper is organized as follows. In Section 2, the problem of blind source extraction in a bandwidth constrained wireless sensor network is formulated. In Section 3, a new two-stage algorithm is derived. In Section 4, a multi-cluster structure of the sensor network is proposed. Numerical simulation results illustrating the performance of the scheme is presented in Section 5. Section 6 gives some concluding remarks and forecasts some future works.
yn (l) =
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where yn and s j are the nth observed signal and the jth source signal, respectively, wn is the nth AWGN which is uncorrelated with wm (m = n) and also uncorrelated with s j ; αn j is the unknown mixing coefficient; l denotes the discrete time (n = 1, · · · , N, l = 0, · · · , L − 1) [14]. In addition, we assume that the mixing is time-invariant in one sampling duration (l = 0, · · · , L − 1). Sensors are densely distributed in the sensing field, so that N > J. The available energy and communication bandwidth of all sensors are limited. However, the cluster head must have more resources than other sensors in the network. Moreover, the sink can have sufficient energy and communication bandwidth and be controlled by people. Suppose the sink wants to extract the source signals in the sensing field. The source extraction process is divided into two stages. In the first stage, sensor n in the network acts as the cluster head and performs local extraction of the source signals based on its own observation and the received noisy quantized data from other sensors. In the second stage, the extracted signals are quantized and the quantized data are sent to the sink in a multi-hop way through a fading channel. Finally, the sink performs global extraction of the source signals. Note that any sensor in the network can play the role of cluster head, provided that it has enough resources.
2. PROBLEM FORMULATION In a wireless sensor network, sensors are densely distributed in a sensing field, and there is a lot of redundant data collected by sensors. Also, the sink is remote from the sensing field. It is thus very energy consuming for sensors to directly send their collected data to the sink [11]. A clusterbased structure is more feasible for effective use of the limited resources. In a cluster-based structure, the cluster head aggregates the data from other sensors, and then sends the aggregated data to the sink [12]. In this paper, we will study a cluster-based sensor network.
2.1. Cluster-Based Sensor Networks We begin with the description of a single cluster wireless sensor network [13] which is shown in Fig.1. The field is ob-
2.2. The Benchmarking Case For the purpose of comparison and evaluation, we consider the condition where the data collected by sensors are not quantized and sent to the cluster head through AWGN channels. Then, the cluster head performs local extraction and sends the quantized data to a processing center through a fading channel. The processing center eventually performs global source extraction based on the received data. Without quantization of the sensed data, the performance is expected to be superior, and we will therefore use this case as a benchmarking case for evaluating our extraction scheme.
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We will derive the source extraction algorithm in this section. Assume that the source signals are statistically independent and J is known a priori. The aim is to estimate s j , j = 1, · · · , J at the sink in a statistical sense (using the correlation coefficient criterion to measure the similarity between the globally extracted signals and the source signals). The whole procedure is divided into the following two stages.
Figure 1. A single cluster wireless sensor network served simultaneously by a network of N distributed sensors. Sensor n is the cluster head and the other sensors are cluster members. The signal observed by the sensors is assumed to be an unknown instantaneous linear mixture of the source
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• Estimate the noise variance which is the average of the ˆ N − J smallest eigenvalues of R(0) and denote it by σˆ 2 . 2 −1/2 h1 , · · · , (λJ − σˆ 2 )−1/2 hJ ]T . Then P = [(λ1 − σˆ )
3.1. The First Stage In this stage, the cluster head performs local extraction of the source signals based on its own observation and the received noisy quantized data from other sensors. Suppose the signals observed by sensors are bounded over the interval [−W,W ] and the number of quantization levels for sensor i is Mi (1 ≤ i ≤ N, i = n). First, yi is normalized to the range [0, 1] by a linear transformation y˜i = (W + yi )/2W.
The prewhitened vector is X = PY . The fast fixed-point algorithm is constructed by the following steps: • Choose a random initial vector u(0) of norm 1. Set τ = 1. • Start iteration: u(τ) = E{X(u(τ − 1)T X)3 } − 3u(τ − 1), where E(·) is an expectation operator and numerically calculated by the average of data samples.
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Then, local independent quantizers Qi : y˜i → Fi (y˜i , Mi ) are designed and y˜i is quantized sample by sample, where Fi (y˜i , Mi ) is a discrete data of Mi bits and can be represented by y˜i − qi
=
• Divide u(τ) by its norm. • If |u(τ)T u(τ − 1)| is not close enough to 1, let τ = τ + 1, and go back to iterate. Otherwise, stop iteration and output the vector u(τ). u(τ)T X equals one of the source signals.
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To extract J source signals, the sink sends request messages to the cluster head. The cluster head runs this algorithm J times and each time sends one extracted signal to the sink.
The quantization noise qi are statistically independent because quantizations are done separately in each sensor [5]. The quantized data are then sent to the cluster head bit by bit. Suppose sensor n has received a total of N −1 noisy quantized data which are denoted by zi,k = Fi (y˜i , Mi )k + vi,k , k = 1, · · · , Mi
3.2. The Second Stage In this stage, the cluster head quantizes the extracted signals and sends the quantized data to the sink in a multi-hop way through a fading channel. We denote the locally extracted source signals by sˆ j , j = 1, · · · , J. The range of sˆj is normalized to [−1, 1] for the convenience of PCM encoding in the following. First the normalized sˆ j is quantized sample by sample using a pulse code modulation (PCM) method [19]. PCM is the simplest waveform coding technique for digitizing analog signals. As 8-bit is adequate to represent the source signal, we design an uniform PCM quantizer: sˆj → Fj (sˆj , 8), where Fj (sˆj , 8) is a discrete data of 8 bits. Divide [−1, 1] into 256 even intervals and these intervals are denoted by b8 · · · b1 from “00000000” to “11111111”. Map each sample point value of the normalized sˆ j to the intervals we get the PCM codes of each sample point. Replace “0” with “-1” of the codes (to facilitate the application of CMA in the following) and rearrange the bits in order of sample points into a sequence b. Then, sensor n sends the quantized data bit by bit to the sink in a multi-hop way through a fading channel. We assume the communications between the cluster head and the sink are over flat fading channels. The signal received by the sink can be written as
(5)
where vi,k is AWGN. The original mixed signals yi are reconstructed at sensor n and denoted by z˜i , where z˜i = 2W
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Then, sensor n performs local extraction of the source signals based on yn and the reconstructed mixed signals z˜i (distorted, z˜i ≈ yi ). Though the mixed signals are distorted, their waveforms retain many original features, thus making local extraction possible. There are many classical algorithms for source extraction. We choose the fast fixed-point algorithm [15-17] because it is very simple and fast to converge. In each run it can extract one independent non-Gaussian source signal from the mixture, no matter what their probability distributions are. Let Y = [yn , z˜1 , · · · , z˜N ]T . Before source extraction, to reduce the effect of noise, Y is centered (subtracting its mean value) and prewhitened by applying a whitening matrix P obtained by the method proposed in [18]. The prewhitening process is shown as follows:
rε = gε bε + ωε , ε = 1, · · · , 8L
ˆ • Estimate the sample covariance R(0) from L data samples of the centered Y .
where gε is the total fading channel gain with gε > 0 and ωε is the total AWGN [20]. Though the bits are transmitted in a multi-hop way, it is easy to see that the form of (7) does not change. The distorted sequence received by the sink is denoted by r. At the sink first the constant modulus algorithm
ˆ • Take an eigen-decomposition of R(0). Denote the J ˆ largest eigenvalues of R(0) by λ1 , · · · , λJ and the corresponding eigenvectors by h1 , · · · , hJ .
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(CMA) [21] is used for channel equalization and bit recovery. The CMA is a stochastic gradient algorithm and is given by c(t + 1) = c(t) − μr(t) d(t) (|d(t)|2 − 1),
from either cluster head, the source signals are said to be extracted successfully.
(8)
5. SIMULATION RESULTS In this section, we provide an example to illustrate the performance of the proposed scheme. The mixing factors are randomly generated and chosen from a standard normal distribution. We assume that all noise levels in the two stages and the quantization levels of sensor observations are equal. The correlation coefficient criterion is adopted to evaluate the performance of the scheme, which is commonly used in source separation or extraction schemes [22,23] and is defined by
where t denotes the iteration step, μ is a small positive parameter, and d = c r. This algorithm is easy to implement and has good convergence property provided that c(0) and μ are carefully chosen. Generally c(0) is a large positive value. The ˆ After iteration, deterrecovered bit sequence is denoted by b. ˆ = 1; else b(t) ˆ = −1. Then, PCM mine: if d(t) > 0, then b(t) decoding is done based on the recovered bits (to replace “1” with “0” in advance). Rearranging these bits into 8 × L sequences and mapping each group of 8 codes to the intervals, we get each sample point value as the average of the two borders of the corresponding interval. The globally extracted source signal is denoted by γ j .
| ∑L−1 γ j (l)s j (l)| ζ j = l=0 . L−1 2 L−1 2 γ j (l) ∑l=0 s j (l) ∑l=0
The correlation coefficient of source signals and locally extracted signals are denoted by ζe and the correlation coefficient of source signals and globally extracted signals are denoted by ζr . Computer simulations are run independently for β times. If ζe ≥ 0.9 and ζr ≥ 0.8, we consider that source signals are successfully extracted. This criterion is based on our observation that if ζe ≥ 0.9 and ζr ≥ 0.8, the globally extracted signal is sufficiently close to the source signal. The total number of successful extraction is evaluated. The success rate ρ is defined as the ratio of the total number of successful extraction versus β. The average value of successful ζe and ζr are also computed. In our study, we choose β = 50.
4. MULTI-CLUSTER STRUCTURE To improve the accuracy of source extraction, we consider creating K (K ≥ 2) clusters in the sensor network. The sensing field, sensor network and wireless communication environments are the same with that in the single cluster case. A two-cluster sensor network is shown in Fig.2. In a K-cluster
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Two chaotic signals are used as source signals in the simulation , one being generated by logistic map and the other by Chebyshev map, i.e., s1 (l) = 1 − 2 s21(l − 1), s2 (l) = cos(4 arccos(s2 (l − 1))). 1) With N = 30, Mi = 4 and SNR = 20 dB, the source signals are plotted in the first row of Fig.3, the locally extracted signals sˆ1 and sˆ2 in the second row, and the globally extracted signals γ1 and γ2 in the third row. Resemblance of the locally extracted signals and the globally extracted signals with the source signals are clearly evident from Fig.3. 2) With Mi = 4 and SNR = 20 dB, the success rate ρ and correlation coefficients (ζe and ζr ) versus N are shown in Fig.4. From Fig.4, we see that ρ, ζe and ζr generally increase with N. But as N gets larger (≥ 30), no noticeable improvement is observed. 3) With N = 30 and SNR = 20 dB, the success rate ρ and correlation coefficients (ζe and ζr ) versus Mi are shown in Fig.5. We observe from Fig.5 that ρ, ζe and ζr increase with Mi , but becomes saturated as Mi reaches 4. When Mi = 4, the results are comparable to that of the benchmarking case described in Section 2.2.
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Figure 2. A two-cluster sensor network sensor network, K sensors act as the cluster heads, others are the cluster members. Assuming that the K cluster heads have δ common overlapping cluster members. Then, each of them has N−K−δ non-overlapping cluster members. Each cluster K head performs local extraction of the source signals based on its own observation and the received noisy quantized data from cluster members in its cluster. Then, each cluster head quantizes the extracted signal and sends the quantized data to the sink separately, in a multi-hop way through a fading channel. The sink performs global extraction of the source signals based on the received signal originating from each cluster head. If the source signals are globally extracted successfully at the sink based on the received signal originating
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4) With N = 30 and Mi = 4, the success rate ρ and correlation coefficients (ζe and ζr ) versus SNR are shown in Fig.6. We can see from Fig.6 that ρ, ζe and ζr generally increase with SNR, but when SNR becomes larger, little further improvement can be gained.
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Figure 4. Success rate ρ and correlation coefficients (ζe and ζr ) versus the number of sensors N (single cluster). Solid line with ∗ corresponds to the proposed scheme and line with corresponds to the benchmarking case
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The source signals used in the simulation are the same as those used in the single-cluster case. 1) With N = 30, K = 2, Mi = 4 and SNR = 20 dB, the success rate ρ versus δ is shown in the first row of Fig.7.
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2) With N = 30, K = 3, Mi = 4 and SNR = 20 dB, the success rate ρ versus δ is shown in the second row of Fig.7. The results of the single-cluster case are also plotted in Fig.7 for comparison. We can see clearly from Fig.7 that the multicluster case outperforms the single-cluster case. The performance of the three-cluster case is better than that of the twocluster case. When δ = N − K, the performance of both the multi-cluster case and the single-cluster case are almost the same. For the multi-cluster case, ρ decreases fast with δ. 3) With N = 30, Mi = 4, K = 1, 2, 3, 5, 6, 10, SNR = 20 dB, and with all sensors are uniformly distributed in all clusters without overlapping, the success rate ρ versus K is shown in the third row of Fig.7. The results of the single-cluster case are also plotted in Fig.7 for comparison. We can see from Fig.7 that the success rate ρ increases with K, and approaches 1 when K reaches about 6.
and SNR reach certain thresholds, no further improvement can be gained. This agrees with our intuition. When N is sufficiently large, the redundant information provided by sensors is saturated, and further increase of N has little effect on the performance. Similarly, when Mi is large enough, z˜i is very similar to yi , and there is very little distortion. Further increase in Mi may induce over-representation of the original mixed signals. Also, when SNR is sufficiently high, the effect of noise is relatively small in comparison with that of mixing. The reduction in the noise level brings very little performance gain. To improve the accuracy of source extraction, a multi-cluster structure of the sensor network has been proposed. Simulation results show that the multi-cluster case outperforms the single-cluster case obviously. The success rate ρ in the multi-cluster case increases with increasing K but decreases with increasing δ. The reasons are that when K increases, the sink has more references of the source signal. When δ increases, the redundant information provided by sensors decreases. Our future work will involve evaluation of the scheme based on data collected in real environments. We may also consider some algorithms in the multi-cluster case that can possibly optimize a predefined cost function in terms of performance and energy consumption, under some choice of parameters.
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ACKNOWLEDGMENT
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This work is supported by Hong Kong Polytechnic University under Research Grant No. 1-BBZA.
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REFERENCES
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[1] Yadid-Pecht, O., Zaghloul, M., and Wilson, D., 2007, “Welcome to the special section on smart sensors,” IEEE Transactions on Circuits and Systems–I, 54, no.1, (January): 1-3. [2] Akyildiz, I.F., Su, W., Sankarasubramaniam, Y., and Cayirci, E., 2002, “A survey on sensor networks,” IEEE Communications Magazine, (August): 102-114. [3] Luo, Z., 2005, “Universal decentralized estimation in a bandwidth constrained sensor network,” IEEE Transactions on Information Theory, 51, no.6, (June): 2210-2219. [4] Luo, Z., 2005, “An isotropic universal decentralized estimation scheme for a bandwidth constrained ad hoc sensor network,” IEEE Journal on Selected Areas in Communications, 23, no.4, (April): 735-744. [5] Xiao, J., and Luo, Z., 2005, “Decentralized estimation in an inhomogeneous sensing environment,” IEEE Transactions on Information Theory, 51, no.10, (October): 35643575. [6] Ribeiro, A., and Giannakis, G.B., 2006, “Bandwidthconstrained distributed estimation for wireless sensor networks—Part I: Gaussian case,” IEEE Transactions on Signal Processing, 54, no.3, (March): 1131-1143.
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Figure 7. Success rate ρ versus the number of overlapping cluster members δ (first and second row) and the number of clusters K (third row) with N = 30, Mi = 4 and SNR = 20 dB. Solid line with ∗ corresponds to the single-cluster case, line with corresponds to K = 2 and line with ◦ corresponds to K=3
6. CONCLUSION In this paper, we have proposed a blind source extraction scheme in a bandwidth constrained wireless sensor network. Simulation results show that our proposed scheme can extract the source signals effectively. The performance of the proposed scheme is close to the benchmarking performance. In the single-cluster case, the success rate ρ and correlation coefficients (ζe and ζr ) generally increase with the number of sensors N, the quantization level Mi and SNR, but as N, Mi
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[7] Ribeiro, A., and Giannakis, G.B., 2006, “Bandwidthconstrained distributed estimation for wireless sensor networks—Part II: Unknown probability density function,” IEEE Transactions on Signal Processing, 54, no.7, (July): 2784-2796. [8] Xiao, J., Cui, S., Luo, Z., and Goldsmith, A.J., 2006, “Power scheduling of universal decentralized estimation in sensor networks,” IEEE Transactions on Signal Processing, 54, no.2, (February): 413-422. [9] Cui, S., Xiao, J., Goldsmith, A.J., Luo, Z., and H.V. Poor. 2005, “Energy-efficient joint estimation in sensor networks: analog vs. digital,” in Proceedings of 2005 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP’05, (Philadelphia, USA, March 18-23). IEEE SPS, New York, USA, 745-748. [10] Jayaweera, S.K., 2006, “Virtual MIMO-based cooperative communication for energy-constrained wireless sensor networks,” IEEE Transactions on Wireless Communications, 5, no.5, (May): 984-989. [11] Wang, A., and Chandrakasan, A., 2002, “Energyefficient DSPs for Wireless Sensor Networks,” IEEE Signal Processing Magazine, (July): 68-78. [12] Shih, T.-F., 2006, “Particle swarm optimization algorithm for energy-efficient cluster-based sensor networks,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E89–A, no.7, (July): 19501958. [13] Tubaishat, M., and Madria, S., 2003, “Sensor networks: An overview,” IEEE Potentials, (April/May): 20-23. [14] Hanna, M.T., 2003, “Multiple Signal Extraction by Multiple Interference Attenuation in the Presence of Random Noise in Seismic Array Data,” IEEE Transactions on Signal Processing, 51, no.7, (July): 1683-1694. [15] Hyv¨arinen, A., and Oja, E., 1997, “A fast fixed-point algorithm for independent component analysis,” Neural Computation, 9, 1483-1492. [16] Hyv¨arinen, A., 1999, “Fast and robust fixed-point algorithms for independent component analysis,” IEEE Transactions on Neural Networks, 10, no.5, (May): 626-634. [17] Hyv¨arinen, A., and Oja, E., 2000, “Independent component analysis: algorithms and applications,” Neural Networks, 13, (April/May): 411-430. [18] Belouchrani, A., Abed-Meraim, K., Cardoso. J.-F., and Moulines, E., 1997, “A blind source separation technique using second-order statistics,” IEEE Transactions on Signal
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Processing, 45, no.2, (February): 434-444. [19] Jayant, N.S., 1974, “Digital coding of speech waveforms: PCM, DPCM, and DM quantizers,” Proceedings of the IEEE, 62, no.5, (May): 611-633. [20] Chen, B., Jiang, R.X., Kasetkasem, T., and Varshney, P.K., 2004, “Channel aware decision fusion in wireless sensor networks,” IEEE Transactions on Signal Processing, 52, no.12, (December): 3454-3458. [21] Godard, D.N., 1980, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Transactions on Communications, 28, no.11, (November): 1867-1875. [22] Farina, D., F´evotte, C., Doncarli, C., and Merletti, R., 2004, “Blind separation of linear instantaneous mixtures of nonstationary surface myoelectric signals,” IEEE Transactions on Biomedical Engineering, 51, no.9, (September): 1555-1567. [23] Tesfayesus, W., and D.M.Durand. 2006. “Blind source separation of neural recordings and control signals,” in Proceedings of IEEE 2006 International Conference of the Engineering in Medicine and Biology Society, EMBC’06, (New York, USA, August 30-September 3). IEEE EMBS, New York, USA, 731-734.
Biographies Hongbin Chen is a Research Assistant with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong, and at the same time he is enrolling in a PhD program with the South China University of Technology, Guangzhou, China. His research interests include sensor networks, blind signal processing, and Kalman filter applications. Chi K. Tse is presently Chair Professor and Head of Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong. His research interests include sensor networks, complex networks and nonlinear systems. He is an IEEE Fellow and was an IEEE Distinguished Lecturer. Jiuchao Feng is a Professor at the School of Electronic and Information Engineering, South China University of Technology, Guangzhou, China. His interests include signal processing, Kalman filter applications, and blind signal extraction. He was awarded the title of Zhu Jiang Scholar Professor by the Provincial Ministry of Education in Guangdong Province, China.
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