Performance Analysis and Code Design for
Minimum Hamming Distance Fusion in
Wireless Sensor Networks
Chien Yao, Po-Ning Chen, Senior Member, IEEE, Tsang-Yi Wang, Member, IEEE,
Yunghsiang S. Han, Member, IEEE, and Pramod K. Varshney, Fellow, IEEE
Abstract—Distributed classification fusion using
error-cor-recting codes (DCFECC) has recently been proposed for wireless sensor networks operating in a harsh environment. It has been shown to have a considerably better capability against unexpected sensor faults than the optimal likelihood fusion. In this paper, we analyze the performance of a DCFECC code with minimum Hamming distance fusion. No assumption on identical distribution for local observations, as well as common marginal distribution for the additive noises of the wireless links, is made. In addition, sensors are allowed to employ their own local classification rules. Upper bounds on the probability of error that are valid for any finite number of sensors are derived based on large deviations technique. A necessary and sufficient condition under which the minimum Hamming distance fusion error vanishes as the number of sensors tends to infinity is also established. With the necessary and sufficient condition and the upper error bounds, the relation between the fault-tolerance capability of a DCFECC code and its pair-wise Hamming distances is characterized, and can be used together with any code search criterion in finding the code with the desired fault-tolerance capability. Based on the above results, we further propose a code search criterion of much less complexity than the minimum Hamming distance fusion error criterion adopted earlier by the authors. This makes the code construction with acceptable fault-tolerance capability for a net-work with over a hundred of sensors practical. Simulation results show that the code determined based on the new criterion of much less complexity performs almost identically to the best code that minimizes the minimum Hamming distance fusion error. Also simulated and discussed are the performance trends of the codes
Manuscript received November 5, 2005; revised January 17, 2007. This work was supported in part by the Research Center of Wireless Network and Multi-media Communication under “Aim for Top University Plan” project of NSYSU and the Ministry of Education, Taiwan, and the NSC of Taiwan, R.O.C., under Grants NSC 95-2221-E-009-247 and NSC 95-2221-E-009-563. The material in this paper was presented in part at the IEEE International Symposium on Infor-mation Theory, Seattle, WA, July 2006.
C. Yao is with Department of Electronics Engineering, National Chiao-Tung University, Hsinchu, Taiwan 300, R.O.C. (e-mail: [email protected]).
P.-N. Chen is with the Department of Communication Engineering, National Chiao-Tung University, Hsinchu, Taiwan 300, R.O.C. (e-mail: qponing@mail. nctu.edu.tw).
T.-Y. Wang is with the Institute of Communications Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan 804, R.O.C. (e-mail: tcwang@mail. nsysu.edu.tw).
Y. S. Han is with the Graduate Institute of Communication Engineering, Na-tional Taipei University, Taipei, Taiwan 237, R.O.C. (e-mail:[email protected]. edu.tw).
P. K. Varshney is with the Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244 USA (e-mail: [email protected]).
Communicated by A. Høst-Madsen, Associate Editor for Detection and Estimation.
Digital Object Identifier 10.1109/TIT.2007.894670
searched based on the new simpler criterion with respect to the network size and the number of hypotheses.
Index Terms—Coding, detection, classification, information
fu-sion, fault tolerance, wireless sensor networks.
I. INTRODUCTION
R
ECENT advances in processor, radio, and memory tech-nology have generated a great interest in the notion of de-ploying a large number of networked sensors for applications such as environment monitoring. The classification of target ob-jects, as well as their tracking, are the fundamental requirements in these applications [1], [2], [8], [10], [11], [17]. In this paper, we consider a wireless sensor network (WSN) that consists of geographically dispersed sensors, wireless (and hence noisy) one-way communication links, and a fusion center. Limitation on the communication bandwidth in wireless links due to the consideration of economical energy consumption at local sen-sors prevents the system from conveying raw observation data to the fusion center. A local compression on the raw observa-tion data thus has to be employed at each sensor. Usually, the information content of the compressed outputs from local sen-sors is of fewer bits than in a WSN, where is the number of object classes to be distinguished. In this work, we are specifically concerned with the case where the sensor nodes only send out binary decisions to the fusion center at which they are fused to produce the final -ary decision.Another issue that may be encountered in a WSN is that sen-sors are prone to be blocked or even damaged when they are de-ployed in a harsh environment [1]. In addition, a low-cost sensor that is manufactured by a simple technology may suffer from hardware, as well as software, malfunctions after deployment. As a result, the fault-tolerance capability to protect against un-expected sensor failures is also of equal importance to the per-formance and complexity of a WSN.
To fulfill the above mentioned requirements, a distributed classification fusion approach using error correcting codes (DCFECC) has been proposed to provide good fault-tolerance capability under feasible system complexity [18]. In the pro-posed approach, an error-correcting code matrix is first designed by either simulated annealing or cyclic column replacement, where each row of bits forms a codeword that corresponds to one of hypotheses. Each local sensor then outputs the respective code bit of the codeword corresponding to the declared hypothesis that is locally determined based
Fig. 1. System model for DCFECC.
on its own observation. Upon receipt of the binary outputs from sensors, the fusion center makes a multiclass decision based on minimum Hamming distance decoding. Unlike the conventional approach that employs the optimal maximum
a posteriori probability (MAP) fusion rule, it was shown in
[18] that with adequately high probability, the decision made by the minimum Hamming distance fusion rule can fall into the correct acceptance region even if several sensor faults are present.
In [7], we have provided the performance analysis of the minimum Hamming distance fusion approach when the number of sensors is sufficiently large. This asymptotic performance analysis for the distributed binary detection/classification problem considered here is different from those investigated by [4], [6], [16], where the MAP fusion rule rather than the minimum Hamming distance fusion rule was used.
In this work, we extend our analysis in [7] by relaxing the as-sumptions of common distribution for all local observations and identical local classification rule for all sensors. Also, the addi-tive noises over the wireless links is allowed to be independent but nonidentical in statistics. Contrary to the requirement of suf-ficiently large number of sensors in [7], the probability bounds obtained in this work are now valid for any finite number of sensors. In particular, a necessary and sufficient condition under which the minimum Hamming distance fusion error vanishes as the number of sensors tends to infinity is established. With the necessary and sufficient condition and the upper bounds on the error probability, the relation between the fault-tolerance capa-bility of a DCFECC code and its pair-wise Hamming distances can be analytically characterized. It can thereby be used to-gether with any code search criterion for finding the code matrix with the desired fault-tolerance capability. Most importantly, a code search criterion of much less complexity than the min-imum Hamming distance fusion error criterion adopted in [18] is proposed. Based on this, code construction with acceptable fault-tolerance capability for a network with over a hundred of sensors becomes possible. Simulation results show that the code determined based on the new criterion with much less com-plexity performs almost identically to the best code that min-imizes the minimum Hamming distance fusion error. Also sim-ulated and discussed are the performance trends of the codes searched based on the new simpler criterion with respect to the number of sensors and the number of hypotheses. Detailed dis-cussions are provided in subsequent sections.
This paper is organized in the following fashion. The system model is described in detail in Section II. The error bounds are derived using large deviations technique in Section III, followed by the establishment of the necessary and sufficient condition under which the fusion error vanishes as the number of sensors becomes large. In Section IV, we characterize the fault-tol-erance capability of a DCFECC code with minimum Hamming distance fusion. With the availability of the upper bounds on the error probability and the characterization of the fault-tolerance capability, the new code search criterion is presented in Sec-tion V. SecSec-tion VI summarizes and discusses the simulaSec-tion and numerical results obtained in this work. Section VII concludes the paper.
For better readability, the proofs of the supportive lemmas are deferred till the Appendix.
II. SYSTEMMODEL
As depicted in Fig. 1, we consider the distributed -ary clas-sification problem in a parallel fusion network, which is perhaps the topology that has received the most attention in the area of WSNs [3], [4], [9], [10], [11], [12], [14], [19]. In this problem, all the local sensors observe the same phenomenon that statis-tically belongs to one of the possible classes. Independent interferences are assumed present at the local sensors, which, in mathematics, makes the local observations condi-tionally independent across sensors given each hypothesis. Also assume that each local sensor classifies its own observation, in-dependent of all others, to one of the hypotheses using its own decision rule. In other words, the local sensor nodes need not employ identical decision rules. We then denote by the probability of classifying given that is the true hypoth-esis for sensor .
After the observation is locally classified at sensor , a local output bit is transmitted through a noisy channel to the fu-sion center at which place the received bits are combined to yield the fusion decision. Due to channel transmission errors,
the word received at the fusion center
may not equal the transmitted word . In
this paper, we assume that the event of link error, i.e., , is independent for all the communication links between sensors and the fusion center, and is also independent of the observa-tions as well as the true hypothesis , and its
Ideally, the objective of the distributed classification in par-allel fusion networks is to determine the local classification rules and the fusion rule such that the minimum fusion error can be achieved. For this reason, the optimal MAP fusion rule is often employed at the fusion center. However, such an optimal design was shown to degrade drastically in performance when unex-pected sensor failures are present [18]. This results in the mo-tivation behind the DCFECC codes in [18], i.e., to borrow the technique of error-correcting (namely, fault-correcting) codes to design a wireless sensor network system that is much less sen-sitive, and hence, more robust, to sensor faults that are unseen by the fusion center.
By following this motivation, a code matrix of which the design methodology will be covered in Section V is specified in advance in the DCFECC system as shown on the right side of Fig. 1. This code matrix is an matrix with element
, where and .
Each hypothesis is associated with a row in the code ma-trix. Each column of stands for the local binary outputs cor-responding to the locally classified hypotheses at the respective sensor. Thus, sensor transmits , if is declared to be true locally. Clearly, at least sensor nodes are required for the identification of hypotheses, and it requires more sensor nodes to provide the coding redundancy for fault
toler-ance. For notational convenience, is
used to denote the row of corresponding to the hypothesis . The redundancy that constitutes the desired fault tolerance comes from the adopted minimum distance fusion rule, or
specifically, , where is
the Hamming distance [18]. The tie-break rule is to randomly pick a codeword from those with the same smallest Hamming distance to the received vector .
III. PERFORMANCEANALYSIS
In this section, we first derive a large deviation probability bound for finite sample size. Based on the probability bound, we analyze the performance of the distributed -ary classifica-tion system using minimum Hamming distance fusion. A nec-essary and sufficient condition under which the error rate of the DCFECC codes vanishes as the number of sensors tends to in-finity is then established.
A. Large Deviation Probability Bound for Finite Sample Size
From the minimum Hamming distance fusion, i.e.,
where “ ” denotes the exclusive–OR operation and
, it can be anticipated that the analysis of the system performance relies completely on the probabilities of
events and . Note that given the true
hypothesis is , the more negative the quantity , the smaller the fusion error is. This induces the necessity of finding a good probability bound for the sum of independent random variables given is the true hypothesis in
the following. For notational simplicity, we drop the redundant subscript in in the derivation that follows.
Lemma 1: Let be independent antipodal random
variables with
and Define
and
Then, if
Remark: Since remains the same if we
rede-fine as . Hence, with the assumption
of , the result of Lemma 1 can be re-expressed as
(1)
The probability bound in (1) does not exhibit any apparent relation with , namely, the average of the means of . This can be amended by the next lemma.
Lemma 2: If , then
B. Performance Analysis for Distributed -Ary Classification
Fusion System With Minimum Hamming Distance Fusion
Based on the probability bounds obtained in the previous sub-section, we can upper-bound the error probability of the dis-tributed -ary classification system using minimum Hamming distance fusion rule by the following theorem.
Theorem 1: Let be the average probability of minimum
Hamming distance fusion error given as fusion decision
If for every
where and then (3) Proof: fusion decision Observe that (4) and are independent across sensors given is true. Therefore, by Lemma 1
which results in
Theorem 1 provides an upper bound on the probability of error by means of Lemma 1. Based on Lemma 2, the next corol-lary shows that the upper bound in (3) can be further upper-bounded by quantities that are only functions of the negative quantity defined in (2). As a result, the intuition that a DCFECC code with larger pair-wise Hamming distances is expected to perform better can be justified.
Corollary 1: Under condition (2), the average probability of
minimum Hamming distance fusion error can also be bounded above by
(5)
(6)
where , and
(7) Notably, by condition (2), and
Proof: The proof of inequality (5) follows a similar
pro-cedure as in Theorem 1 except that Lemma 2 is used instead of Lemma 1. Inequality (6) is a direct consequence of in-equality (5).
With the above corollary, we figure that if for some for all sufficiently large , the DCFECC decoding error vanishes exponentially as approaches infinity. Since under a fixed number of hypotheses, usually grows linearly with the number of sensors for typical DCFECC codes, we conclude that the average error probability for the distributed -ary classification system using minimum Hamming distance fusion can be made zero asymptotically as goes to infinity, and the error exponent is bounded below by
as long as . Next, we will show that the
assumption that leads to a nonvanishing
, and hence, establish the necessary and sufficient condition under which vanishes.
Theorem 2: If is bounded away
from zero infinitely often in number of sensors.
Proof: The assumption that
im-plies the existence of such that for infinitely many . Hence, for any satisfying , there exist
and such that
(8) By defining the same as in Theorem 1, we obtain
As a result
fusion decision
approaches infinity
where the last step follows from the central limit theorem for sum of independent bounded variables. Thus, the claim of the theorem holds for the case that tends to infinity.
In situations when is bounded as approaches in-finity (in which case a bad DCFECC code design results), the theorem is trivially valid.
The final lemma in this section shows that the upper bounds in Corollary 1, as well as the expression of , can be greatly simplified for identical sensor systems.
Lemma 3: Suppose that for , where
, and is the same for all sensors. Then,
if , we have
(9) (10) where is simplified to
Lemma 3 indicates that under an identical sensor system as-sumption, the two upper bounds in Corollary 1 are reduced to functions of pair-wise Hamming distances. This greatly simpli-fies their evaluation.
By denoting
we observe that is given by the product of
and , where the former only depends on the noises corresponding to communication links between sensors and the fusion center, while the latter is only a function of the local classification accuracy and the adopted DCFECC code. The upper bound in (10) can then be rewritten as . Hence, the effects of link noise and local classification accuracy can be separately considered through the help of bound (10).
For antipodal transmission over additive white Gaussian
noise (AWGN) channels, we have , where
is the complementary error function, and is the signal-to-noise ratio of the communication link. Fig. 2 then shows that the error bound in (10) reaches its ultimate floor
value when is larger than 7 dB, which
corresponds to , and this threshold is indepen-dent of the local classification accuracy and the DCFECC codes adopted.
We can similarly characterize the effect of local classification accuracy through the help of the simple probability bound in (10). A usual assumption on the statistics of local observation is that is Gaussian distributed with mean and variance given that hypothesis is true. Define the local classification rule as is declared true if
Then
if if if
where , and is the standard normal
cumulative distribution function. As a result, approaches as for , and , otherwise, and
Hence, the ultimate floor values of bound (10) for the codes in Table I (see Section VI-C) tend to
when grows beyond a constant threshold 17 dB as depicted in Fig. 3.
IV. FAULT-TOLERANCECAPABILITY
The wireless sensor network considered is likely to contain faulty sensor nodes due to harsh environmental conditions. Faults may include all misbehaviors, ranging from simple
random sensor faults or stuck-at faults1to sensors that behave
1By “random sensor fault,” we mean that the sensor sends out1 or 0
ran-domly regardless of the local observation. Also, a sensor with stuck-at-one (re-spectively, stuck-at-zero) fault will always transmit one (re(re-spectively, zero) to the fusion center, and neglect the local observation it sensed.
Fig. 2. Bound (10) for the codes listed in Table I (see Section VI-C) at = 6 dB, where we assume that the local observation y is Gaussian distributed with mean` and variance 1= given hypothesis H is true.
arbitrarily or maliciously. In this section, we remark on the fault-tolerance capability for the system employing minimum Hamming distance fusion rule according to Corollary 1.
From the upper bound in (6) and the definition of in (7), we observe that the smaller the , the more negative is, which in turn gives a smaller upper probability bound. When sensor faults (SF) occur, is no longer given by (4), but be-comes a function of the new statistics of owing to sensor faults. For example, when stuck-at-one fault occurs at sensor
for . Hence
and has nothing to do with the local classification accuracy. Sim-ilarly, for stuck-at-zero fault
In case a random fault occurs, in which
. Accordingly, may range from
to . As no prior information on the sensor faults type as well as the indices of faulty sensors is assumed known at the fusion center, it is reasonable to consider the fault-tolerance
capability of the system by the worst case scenario in which . Thus
and sensor fault under the worst case surely degrades the system performance bound in (6). Then, the next corollary, which is a straightforward extension of Corollary 1 based on the above discussion, can be used to characterize the fault-tolerance capa-bility of a DCFECC coding system.
Corollary 2: Let be the set of indices of faulty sensors.
Then, if , we have the expression at the bottom of the following page, where the superscript “ ” denotes the set complement operation and
Fig. 3. Bound (10) for the codes in Table I (see Section VI-C) at = 0 dB.
we can verify based on the above corollary that
(11)
where is the sorted counterpart of , satisfying
In order to guarantee that vanishes, it suffices to have (12)
For an identical sensor system where for , this condition reduces to
(13) As
for an identical sensor system, we have
(14) It is worth mentioning that the condition that was used as a code search requirement in [18] resembles the inter-pretation for conventional coding techniques, which states that a
code with minimum pair-wise Hamming distance can tol-erate around errors. However, inequality (14) hints that
a larger than instead of may be
nec-essary for an (identical) fault-tolerant sensor system due to the local classification inaccuracy. Thus, in the worst case, where the fusion center has no information on both the sensor fault types and indices of faulty sensors, the number of faulty sensors allowable for the codes in Table I (see Section VI-C) is only
two-thirds of as on an
average in this table. Inequality (14) also interestingly indicates that for an identical sensor system, the worst case fault-tolerance requirement has nothing to do with the link noise. Inequality (14) will reduce to the heuristic constraint of when all the misclassification probabilities become zero (in which
case for , and hence
regardless of the codes adopted).
Summarizing the above discussion, we may define the
fault-tolerance capability of a DCFECC code as the maximal number
of faulty sensors allowable subject to the validity of (12) as par-allel to the usual definition of error-correcting capability (i.e., ) of block codes ([13, p. 65]). As a consequence of the definition, a sequence of DCFECC codes guarantee to have van-ishing fusion error as goes to infinity if the number of faulty sensors is always restricted within the fault-tolerance capabili-ties of these codes. The guarantee to have asymptotic zero-fu-sion-error is analogous to that in digital communication, when the number of errors is less than the error-correcting capabili-ties of concerned block codes, error-free transmission of these codes is guaranteed. Based on this definition, the fault-tolerance capability of a DCFECC code is equal to
for identical sensor systems.
In the next section, we will set the target fault-tolerance
capa-bility as an auxiliary constraint, in addition to the minimization
of (5), during the search of a fault-tolerant DCFECC code. V. COMPUTERSEARCH OFDCFECC CODES
Computer search for a DCFECC code based on the minimum probability-of-fusion-error criterion for a large sensor network system is infeasible due to the prohibitive algorithmic com-plexity even by the simulated annealing or the cyclic column replacement algorithms recommended in [18]. However, a large sensor network that consists of either a large number of sensors or a large number of hypotheses under classification may still be encountered in practice. This raises the research issue on how to find a DCFECC code that performs well for a large sensor network.
At a first glance, one may think that the minimization of defined in Corollary 1 is a good alternative criterion for code search for a large sensor network. Yet, a simple example with
and
indicates that the code matrix that minimizes the resultant
should satisfy and , and can be
shown straightforwardly not to be the most fault-tolerant design
of for . It can be
conjectured from the error bound in (6) that a good criterion to be minimized should be a function of both and . The above example shows the validity of the conjecture, and the code
matrix with does minimize the
error bound .
A better criterion than , especially for a non-identical sensor network in which some sensors or some hy-potheses have much larger than the others, is the upper prob-ability bound in (5). A simple calculation shows that the number of exclusive-OR, multiplication and addition operations required to determine (5) for a selected code matrix is of the order .2However, the required operations for the determina-tion of that are necessary for the determination of the minimum Hamming distance fusion error, which can be closely approximated as
(15) where
and
is the decision partition for hypothesis , are of the order . This order is much greater than that required by the criterion of (5). Simulations in the next section show that the DCFECC code obtained by minimizing (5) performs almost identically to the optimal DCFECC code that directly mini-mizes the fusion error. This justifies the feasibility of the use of criterion (5), in terms of both complexity and performance, for a large sensor network.
As far as the fault tolerance capability is concerned, another condition given in (12) should also be incorporated to constrain the minimum required in the code search process. Without this minimum constraint, the code that minimizes (5) may end up with limited fault-tolerance capability. Specifically, for the setting in Section VI-A, the code matrix obtained by minimizing (5) without the minimum constraint turns out to have a small (cf. Table I (see Section VI-C)), even though it has good pair-wise Hamming distances between
adjacent hypotheses as . Since the
2The calculation of eachq given in (4) needsM exclusive-ORoperations,
M + 1 multiplications, and M additions, and there are NM of q ’s to deter-mine. Hence, this step requiresO(NM ) operations. It can be shown that with the availability offq g , the computation of (5) requires the same order of operations.
Fig. 4. Simulated performances of three8 2 20 codes that minimize (9) without the minimum d constraint. The code search for these three codes are, respectively, performed at( ; ) = (1; 6 dB); (0 dB, 1) and (0 dB, 6 dB). The error probabilities in this figure are simulated at = 0 dB.
local classification error for distant hypotheses, such as and , is very small when no sensors are faulty, it is unnecessary to provide a large pair-wise Hamming distance between distant hy-potheses. However, as sensor faults can greatly enlarge the mis-classification error for distant hypotheses, such code becomes less robust to sensor misbehaviors.
Hence, in what follows, we provide the search algorithm for the DCFECC code that minimizes (5) subject to the minimum constraint in (12) for given system statistics, as well as sensor network size and number of hypotheses .
1. Initialization. For and ,
assign3
where
and is the probability density function of local ob-servation at sensor given hypothesis is true.
3Here, the adopted local classification rules are not the system-wide globally
optimal ones as those used in [18], but the locally optimal ones. Note that as the smallest size of WSNs considered in this paper, i.e., 20 sensors, is still much larger than the 10-sensor system considered in [18], global optimization of both local classification rules and the DCFECC code under minimum Hamming dis-tance fusion becomes computational intractable. Nonetheless, with the tractable and easy-to-obtain locally optimal classification rules, the desired system per-formance, as well as the target fault-tolerance capability, can be achieved by de-ploying sufficient number of sensors, using the approach proposed in this work.
2. For the previously assigned , find
by simulated annealing algorithm the DCFECC code that minimizes (5) subject to the constraint of (12) for a target fault-tolerance capability (or equivalently, a target sensor fault ratio defined as ).
A final note in this section is that according to Lemma 3, the computational complexity of criterion (5) can be greatly re-duced if the code search is applied to the simpler identical sensor system, where the quantities involved in computations are only the pair-wise Hamming distances.
VI. NUMERICAL ANDSIMULATIONRESULTS
In this section, we examine the performance and fault-tol-erance capability of the DCFECC codes that are constructed through the code search algorithm of the previous section.
A. Identical Sensor System
Assume that each communication link employs an-tipodal transmission over an AWGN channel; hence,
for , where is
the complementary error function, and is the signal-to-noise ratio of the communication link. In addition, each local ob-servation is assumed to be Gaussian distributed with mean and variance given that hypothesis is true, where the signal-to-noise ratio for sensor observations, i.e., , is the square of the minimum difference in these Gaussian means,
Fig. 5. Simulated performances of the two8 2 20 codes that respectively minimizes (9) (code 1) and (15) (code 2) at = 0 dB. The code search is performed for = 0 dB and = 6 dB. Performance bounds for code 1 are also illustrated.
divided by the common observation variance. In this case, the code search criterion of (5) is reduced to (9).
Take and . We first search for the DCFECC code that minimizes the minimum Hamming distance fusion error defined in (15), and compare it with the best code that minimizes (9) without the minimum constraint. It needs to be pointed out that the best codes that respectively minimize (9) and (15) may be different for different target signal-to-noise ratios (see Fig. 4).4Here, we choose 0 dB and 6 dB as the target signal-to-noise ratios during the code search, which
corresponds to and
..
. ... ... . .. ...
4One can observe from Fig. 4 that the three8 2 20 codes obtained for
different target signal-to-noise ratios have different performances. Their ( ; d ) are (00:640353; 7); (00:523694; 10), and (00:57259;7), respectively, for target signal-to-noise ratios ( ; ) = (1; 6 dB); (0 dB; 1); and (0 dB , 6 dB). As anticipated, the code obtained at target ( ; ) = (0 dB; 1dB) performs the best at high since its target signal-to-noise ratios are closest to the operational ones at this range, i.e., = 0 dB and 8 dB. Additionally, the code for target ( ; ) = (0 dB,6 dB) performs only slightly better than the code for target( ; ) = (1 dB,6 dB), and both of them outperform the third code at low . From this figure, we can infer that the deviation of the target from the operational ones affects the resultant performance more than that of .
..
. ... ... . .. ...
..
. ... ... . .. ...
where , and is the standard normal
cumulative distribution function.
As shown in Fig. 5, code 1 and code 2 that are, respectively, obtained by minimizing (9) and (15), have almost indistinguish-able performances even when the number of sensors is small. This is somewhat surprising because the performance bounds derived based on large deviations technique usually have a vis-ible difference from the true performance for small sample size. This implies that the code search criterion (9) is not only much less complex than criterion (15), but can indeed yield a code that performs very close to the optimal minimum-fusion-error code. Furthermore, it can be observed that bounds (3) and (9) almost coincide with each other for all ranges of . In fact, it will be shown by subsequent simulations that the agreement be-tween bounds (3) and (9) is not only true for small network size, but remains so as network size further increases. A final
obser-Fig. 6. Simulated performances and performance bounds of the8 2 600 code that minimizes (9) subject to 10% sensor fault ratio at = 0 dB. The code search is performed for = 0 dB and = 6 dB. The type of the simulated sensor fault is stuck-at-one.
vation from Fig. 5 is that the fusion error of the obtained code achieves a floor value of at large . Such an error-floor phenomenon is due to the existence of the communication link noise, and the floor value can be reduced when a larger is employed.
Next, we demonstrate that criterion (9) can be used for code search for a large sensor network. By taking and , we search for the DCFECC code that minimizes the upper probability bound in (9) subject to 10% sensor fault ratio according to (12). Again, we choose 6 dB and 0 dB during the code search. The and of the obtained code are, respectively, and . These two values, as expected, satisfy the minimum constraint with 10% sensor fault ratio, i.e.,
%
The simulation results and performance bounds for this code are summarized in Figs. 6 and 7. Note that it is infeasible to perform the code search for minimum fusion error criterion defined in (15) for such a large sensor network.
From Fig. 6, we observe again that bound (3) is almost iden-tical to bound (9), and is about 1.5 dB better than bound (10) at . This indicates that it is not beneficial in terms of performance to replace the code search criterion (9) by the more complex (3), and adopting the simpler (10) as a new code search
criterion may result in a code with degraded performance. Sec-ondly, the simulated performance for the searched code with 10% stuck-at-one sensor fault ratio almost follows the curve of its search criterion (9) as we have anticipated, and is only 0.5 dB inferior to its performance without sensor faults at . Also illustrated in this figure is that another 0.75-dB perfor-mance degradation will occur if the stuck-at-one sensor fault ratio increases up to 20%.
Observations similar to Fig. 6 can be made from Fig. 7 except that the performance degradation from zero fault to 10% sensor fault ratio is doubled when it is measured in rather in (specifically, 1 dB in Fig. 7, but only 0.5 dB in Fig. 6). The 3-dB performance difference between bound (9) and bound (10), when it is measured in , also becomes twice of the 1.5-dB difference for measure at .
Finally, Fig. 8 compares the performances of four codes, re-spectively, obtained by minimizing bound (9) subject to zero sensor fault ratio, bound (9) subject to 10% sensor fault ratio, bound (10) subject to zero sensor fault ratio, and bound (10) sub-ject to 10% sensor fault ratio. The target signal-to-noise ratios for these code searches are again 0 dB and 6 dB. We then gradually increase the number of faulty sensors to examine the robustness of these four codes. We observe that the code ob-tained by minimizing (9) subject to 10% sensor fault ratio per-forms the best when nearly 10% of sensors are faulty, and still remains the most robust when the number of faulty sensors is further increased. This result suggests again that bound (9) is a more suitable criterion to be minimized as far as a fault-toler-ance WSN is concerned.
Fig. 7. Simulated performances and performance bounds of the8 2 600 code that minimizes (9) subject to 10% sensor fault ratio at = 6 dB. The code search is carried out at = 0 dB and = 6 dB. The type of the simulated sensor fault is stuck-at-one.
B. Nonidentical Sensor System
In this simulation, the sensors are divided into three equal groups. Each sensor group forms an identical sensor subsystem as in the previous subsection. The signal-to-noise ratios for the
three groups are, respectively, and
. In other words, some sensors have more accurate local classification but poor communication links, while some other sensors may be short in local classification accuracy but have less noisy communication links. A potential scenario for the above system configuration is that a group of sensors is closer to the target but farther from the fusion center, and another group of sensors is located distant from the target but is near the fusion center. Again, the code search is performed under 6 dB and 0 dB, and the target sensor fault ratio allowable is 10%. The and of the obtained code are, respectively, and . The simulation results and performance bounds for this code are summarized in Figs. 9 and 10, and sim-ilar conclusions can be drawn as those for the identical sensor system. As a summary, in both figures, bound (5) coincides with bound (3), and the performance of the code found with the target sensor fault ratio follows the curve of its searched criterion (5). In addition, the performance degradation due to a decrement in
is larger than that due to the same decrement in .
It should be pointed out that although bounds (3) and (5) (equivalently, bound (9) for the identical sensor system) coin-cide in the ranges in Figs. 7–10, they actually deviate from each
other when and become much larger.5For example, when is large, bound (3) achieves a floor value of
in Fig. 7, while the ultimate floor value for bound (5) (namely, (9)) equals in the same figure. A more clear differ-ence in ultimate floor values can be obtained in Figs. 9 and 10, where bounds (3) and (5), respectively, achieve
and at large in Fig. 9, and, respectively,
ap-proach and at large in Fig. 10. In
addition, the ultimate floor values of bound (6) are
and for Figs. 6, 7, 9, and 10, respectively. Since these floor values indeed imply ex-tremely small error performances that are of very minor interest in practice, we simply neglect the illustration of them in these figures.
We end this subsection by pointing out that similar observa-tions can be made when the difference between the best and the worst sensors is further increased. As an example, by setting the signal-to-noise ratios, respectively, for the three groups as
and , searching the code
under 8 dB and 0 dB subject to 10% sensor fault
5In Fig. 6, bounds (3) and (5) remain close even when is large. Since q
converges to as " 1, it can be derived that bounds (3) and (5) (namely, (9)) actually achieve the same floor value of
1
M (1 0 (1 0 2) ) = 1:79 2 10 when = for all 1 j N.
Fig. 8. Simulated performances of four8 2 100 codes that minimizes bound (9), (10), (9) subject to 10% sensor fault ratio and bound (10) subject to 10% sensor fault ratio. The code search is performed at = 0 dB and = 6 dB. The type of the simulated sensor fault is stuck-at-one.
Fig. 9. Simulated performances and performance bounds of the8 2 600 code that minimizes (5) subject to 10% sensor fault ratio at = 0 dB. The code search is performed at = 0 dB and = 6 dB. The type of the simulated sensor fault is stuck-at-one.
Fig. 10. Simulated performances and performance bounds of the8 2 600 code that minimizes (5) subject to 10% sensor fault ratio at = 6 dB. The code search is performed at = 0 dB and = 6 dB. The type of the simulated sensor fault is stuck-at-one.
Fig. 11. Simulated performances and performance bounds of the8 2 600 code that minimizes (5) subject to 10% sensor fault ratio at = 0 dB. The code search is performed at = 0 dB and = 8 dB. The type of the simulated sensor fault is stuck-at-one.
Fig. 12. Simulated performances and performance bounds of the 82 600 code that minimizes (5) subject to 10% sensor fault ratio at = 8 dB. The code search is performed at = 0 dB and = 8 dB. The type of the simulated sensor fault is stuck-at-one.
TABLE I
LIST OFEXPONENTS, ANDd FOR THEBESTCODESTHATMINIMIZE(9)INFIG. 13
ratio yields Figs. 11 and 12. The and of the obtained code are, respectively, and . It can then be ob-served the two figures that the results due to a larger difference in signal-to-noise ratios between the best and the worst sensors are actually very similar to those limited to 6-dB difference.
C. Fusion Error Versus Network Size and Number of Hypotheses
In this subsection, the best codes for the identical sensor system specified in Section VI-A are determined for different
network sizes under 6 dB, and 0 dB.
Since what we are mainly concerned with is the performance trend with respect to the network size, no minimum con-straint is set in this code search. Also simulated is the relation between resulting fusion error and number of hypothesis under fixed . The results are summarized in Figs. 13 and 14.
We observe from Fig. 13 that the performance bounds for the best codes decrease exponentially with respect to the network size , which hints that the true performances of the found
Fig. 13. Simulated performances and performance bounds for the family of best codes that minimize (9). The code search is performed atM = 8; = 0 dB, and = 6 dB.
TABLE II
LIST OF ANDd FOR THEBESTCODESTHATMINIMIZE(9)INFIG. 14
DCFECC codes should at least follow the same trend of expo-nential decay. As shown in Table I, the exponents for the simu-lated performances, bound (9) and bound (10) are, respectively, and on an average. Also listed in Table I are the and for these searched codes. The list indi-cates that for the best code is almost independent of the
network size, and remains around for most . In addi-tion, increases linearly with respect to .
Fig. 14 summarizes the simulated relationship between fusion error and the number of hypotheses. Recall that the code rate of an code matrix is conventionally
de-fined as . Thus, the code rates of the
DCFECC codes simulated previously are prohibitively small when they are compared with the code rates of the traditional error-correcting codes. Since the traditional error-correcting codes with larger code rates (or equivalently, larger ) can still yield a good (e.g., there exists rate-Bose–Chaudhuri–Hocquenghem (BCH) code with
[15, p. 438]), it is not surprising to obtain that at , the fusion error only mildly increases with respect to when . Even with a scale of exponentially increasing up to , the growth rate of the fusion errors only sightly increases. It can be observed from Table II that the of the best codes that minimize (9) does not tend to decrease in for
(as it should do for very large ). Again, remains almost constant for all , and has the average around . This result indicates that the number of hypotheses for the DCFECC codes can actually be of an exponential order in the number of sensors (specifically, ).
VII. CONCLUSION
The envisioned use of sensor networks for fault-tolerant clas-sification applications in a harsh environment calls for the
de-Fig. 14. Simulated performances and performance bounds for the family of best codes that minimize (9). The code search is performed atN = 100; = 0 dB, and = 6 dB.
Fig. 15. Equivalent serial-connected binary channel model specifically for wireless sensor networks. The channel noisefn g for memoryless BSC is inde-pendent and identically distributed withPrfn = 1g = . The transition probability Prfu jc g is given by (16), and is a function of the codebook CCC.
sign of collaborative classification using coding theory, e.g., the DCFECC approach with minimum Hamming distance fusion. In this paper, we have analytically characterized its performance in both fault-free and faulty situations. Also, characterized is the fault-tolerance capability of a DCFECC code, which is subse-quently used together with the newly proposed code search cri-terion to find the code with the desired fault-tolerant robustness. Our results indicate, as anticipated, the strong relation between the fault-tolerance capability and the pair-wise Hamming dis-tances of a DCFECC code. Although we mostly emphasize that our code construction approach is specifically suitable for net-works with a large number of sensors owing to its simplicity, the simulation result shown in Fig. 5 implies its suitability in terms of performance for networks with only tens of sensors.
The coding problem considered in this paper can actually be transformed into one for the memoryless binary-symmetric channel (BSC) with unreliable bit-by-bit postulate encoders as shown in Fig. 15, if the link noises have common marginal dis-tribution. We can further consider the memoryless BSC channel with unreliable bitwise postulate encoders as a serial connec-tion of two binary channels, in which the first channel suffers
code-dependent noises that give
and the second channel is the memoryless BSC channel. The case of sensor faults under the equivalent channel model be-comes that of turning independent of (and hence, code independent) without notifying the fusion center. Our results then indicate that the constraint that the number of code-inde-pendent bits in (i.e., the number of faulty sensors) is less than is sufficient to guarantee a vanishing de-coding error for such a serially connected binary channel. This bound is derived based on the pessimistic view when both in-dices of faulty sensors and sensor fault types are unknown to the fusion center, or equivalently, the decoder is aware of neither the index of every faulty bit nor its resultant code-independent distribution. In the extreme case that and are completely de-pendent, which should occur when for every
, the constraint reduces to the conventional for coding techniques since as we have an-ticipated. This observation hints that in a channel suffering from code-dependent noises, a code that makes (channel output) and (channel input) more “dependent” (and thus, the channel output has more information about the input) is expected to be a better and more robust code, which is exactly the under-lying concept behind the Shannon baptized “channel capacity.” It would be interesting to research along this line, and determine the capacity of the postulate code-dependent channels.
APPENDIX
A. Proof of Lemma 1
1) Property of : is convex since it is the
pointwise supremum of a collection of affine functions. Also,
because for . By Jensen’s
inequality
which implies . Hence, gives its
min-imum value at .
2) Support Line of : Let be the
sup-port line of the convex , which passes through the point
. The convexity of and imply that
, and
3) Probability Bound:
B. Proof of Lemma 2
Let , and note that .
Therefore, the assumption of the lemma is equivalent to .
The validity of the lemma for can be proved by Jensen’s inequality in terms of the upper bound in (1) as follows:
where the last equality takes the optimizer
for
In case , we have and
C. Proof of Lemma 3
Under the assumption of an identical sensor system
Accordingly
and the lemma is verified.
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