Iterative Fusion of Distributed Decisions over the Gaussian Multiple-Access Channel Using Concatenated BCH-LDGM Codes
© Javier Del Ser et al. 2011
Received: 30 November 2010
Accepted: 20 January 2011
Published: 27 January 2011
This paper focuses on the data fusion scenario where nodes sense and transmit the data generated by a source to a common destination, which estimates the original information from more accurately than in the case of a single sensor. This work joins the upsurge of research interest in this topic by addressing the setup where the sensed information is transmitted over a Gaussian Multiple-Access Channel (MAC). We use Low Density Generator Matrix (LDGM) codes in order to keep the correlation between the transmitted codewords, which leads to an improved received Signal-to-Noise Ratio (SNR) thanks to the constructive signal addition at the receiver front-end. At reception, we propose a joint decoder and estimator that exchanges soft information between the LDGM decoders and a data fusion stage. An error-correcting Bose, Ray-Chaudhuri, Hocquenghem (BCH) code is further applied suppress the error floor derived from the ambiguity of the MAC channel when dealing with correlated sources. Simulation results are presented for several values of and diverse LDGM and BCH codes, based on which we conclude that the proposed scheme outperforms significantly (by up to 6.3 dB) the suboptimum limit assuming separation between Slepian-Wolf source coding and capacity-achieving channel coding.
One of the first contributions in this area was done by Lauer et al. in , who extended classical results from decision theory to the case of distributed correlated signals. Subsequently, Ekchian and Tenney  formulated the distributed detection problem for several network topologies. Later, in  Chair and Varshney derived an optimum data fusion rule which combines individually performed decisions on the data sensed at every sensor. This data fusion rule was shown to minimize the end-to-end probability of error of the overall system. More recently, several contributions have tackled the data fusion problem in diverse uncoded communication scenarios, for example, multihop networks subject to fading [16–18] and delays , parallel channels subject to fading [20–22], and asynchronous multiple-access channels [23, 24], among others.
On the other hand, when dealing with coded scenarios over noisy channels, it is important to point out that the data fusion problem can be regarded as a particular case of the so-called distributed joint source-channel coding of correlated sources, since the nonzero probability of sensing error imposes a spatial correlation among the data registered by the sensors. In the last decade, intense research effort has been conducted towards the design of practical iteratively-decodable (i.e., Turbo-like) joint source-channel coding schemes for the transmission of spatially and temporally correlated sources over diverse communication channels, for example, see [25–31] and references therein. However, these contributions address the reliable transmission of the information generated by a set of correlated sensors, whereas the encoded data fusion paradigm focuses on the reliable communication of an information source read by a set of sensors subject to a nonzero probability of sensing error; based on this, a certain error tolerance can be permitted when detecting the data registered by a given sensor. In this encoded data fusion setup, different Turbo-like codes have been proposed for iterative decoding and data fusion of multiple-sensor scenarios for the simplistic case of parallel AWGN channels, for example, Low Density Generator Matrix (LDGM) , Irregular Repeat-Accumulate (IRA) , and concatenated Zigzag  codes. In such references, it was shown that an iterative joint decoding and data fusion strategy performs better than a sequential scheme where decoding and data fusion are separately executed.
Following this research trend, this paper considers the data fusion scenario where the data sensed by nodes is transmitted to a common receiver over a Gaussian Multiple-Access Channel (MAC). In this scenario, it is well known that the spatial correlation between the data registered by the sensors should be preserved between the transmitted signals so as to maximize the effective signal-to-noise ratio (SNR) at the receiver. On this purpose, correlation-preserving LDGM codes have been extensively studied for the problem of joint source-channel coding of correlated sensors over the MAC [35–38]. In these references, it was shown that concatenated LDGM schemes permit to drastically reduce the error floor inherent to LDGM codes. Inspired by this previous work, in this paper we take a step further by analyzing the performance of concatenated BCH-LDGM codes for encoded data fusion over the Gaussian MAC. Specifically, our contribution is twofold: on one hand, we design an iterative receiver that jointly performs LDGM decoding and data fusion based on factor graphs and the Sum Product Algorithm. On the other hand, we show that for the particular data fusion scenario under consideration, the error statistics in the decoded information from the sensors allow for the concatenation of BCH codes [39, 40] in order to decrease the aforementioned intrinsic error floor of single LDGM codes. Extensive Monte Carlo simulations will verify that the proposed concatenated BCH-LDGM codes not only outperform vastly the suboptimum limit assuming separation between distributed source and channel coding, but also reaches the theoretical residual error bound derived by assuming errorless detection and decoding of the sensor data.
The rest of the paper is organized as follows: Section 2 delves into the system model of the considered encoded data fusion scenario, whereas Section 3 elaborates on the design of the iterative decoding and data fusion procedure. Next, Section 4 discusses Monte Carlo simulation results and finally, Section 5 ends the paper by drawing some concluding remarks.
2. System Model
where stands for the BPSK modulation mapping, and represents the average energy per channel symbol and sensor. The Gaussian MAC considered in this work assumes and , whereas are i.i.d. circularly symmetric complex Gaussian random variables with zero mean and variance per dimension . Nevertheless, explanations hereafter will make no assumptions on the value of the MAC coefficients. The joint receiver must estimate the original information generated by as based on the received sequence . This will be done by applying the message-passing Sum-Product Algorithm (SPA, see  and references therein) over the whole factor graph describing the statistical dependence between and , as will be explained in next section.
3. Iterative Joint Decoding and Data Fusion
that is, by symbolwise majority voting over the estimated sensor sequences. Notice that this practical scheme performs sequentially channel detection, LDGM decoding, BCH decoding, and fusion of the decoded data.
for . As widely evidenced in the literature related to the transmission of correlated information sources (see references in Section 1), this correlation should be exploited at the receiver in order to enhance the reliability of the fused sequence . In other words, the considered scenario should take advantage of this correlation, not only by means of an enhanced effective SNR at the receiver thanks to the correlation-preserving properties of LDGM codes, but also through the exploitation of the statistical relation between sequences corresponding to different sensors . The latter dependence between and can be efficiently capitalized by (1) describing the joint probability distribution of all the variables involved in the system by means of factor graphs and (2) marginalizing for via the message-passing Sum-Product Algorithm (SPA). This methodology allows decreasing the computational complexity with respect to a direct marginalization based on exhaustive evaluation of the entire joint probability distribution. Particularly, the statistical relation between sensor sequences is exploited in one of the compounding factor subgraphs of the receiver, as will be later detailed.
In regard to Figure 3(b), observe that a set of switches controlled by binary variables and drive the connection/disconnection of systematic ( ) and parity ( ) variable nodes from the MAC subgraph. The reason being that, as later detailed in Section 4, the degradation of the iterative SPA due to short-length cycles in the underlying factor graph can be minimized by properly setting these switches.
: a posteriori soft information for the value of the node , which is computed, at iteration and , as the product of the a posteriori soft information rendered by the SPA when applied to MAC and LDGM subgraphs.
: similar to the previously defined , this notation refers to the a posteriori information for the value of node , which is calculated, at iteration and , as the product of the corresponding a posteriori information produced at both MAC and LDGM subgraphs.
: refined a posteriori soft information of node for the value , which is produced as a consequence of the processing stage in Figure 3(c).
which is performed for . It is interesting to observe that in this expression, all those indices in error detected by the BCH decoder will consequently drive a flip in the soft information fed to the SENSING subgraph.
4. Simulation Results
To verify the performance of the proposed system, extensive Monte Carlo simulations have been performed for sensors and a sensing error probability set, without loss of generality, to for all sensors. The experiments have been divided in two different sets so as to shed light on the aforementioned statistics of the number of errors per iterations. Accordingly, the first set does not consider any outer BCH coding, and only identical LDGM codes of rate (input symbols per coded symbol), variable and check degree distributions , and input blocklength are utilized at every sensor. The number of iterations for the proposed iterative receiver has been set equal to . The metric adopted for the performance evaluation is the End-to-End Bit Error Rate (BER) between and , which is averaged over 2000 different information sequences per simulated point and plotted versus the ratio per sensor (energy per bit to noise power spectral density ratio). Gaussian MAC is considered in all simulations by imposing .
It is also important to remark that the results plotted in Figure 4 have been obtained by setting the variables controlling the switches from Figure 3(b) to during the first iteration, while for the remaining iterations (i.e., the MAC subgraph is disconnected and does not participate in the message passing procedure). The rationale behind this setup lies on the length-4 loop connecting variable nodes , ( ), and for , which degrades significantly the performance of the message-passing SPA. Further simulations have been carried out to assess this degradation, which are omitted for the sake of clarity in the present discussion. Based on this result, all simulations henceforth will utilize the same switch schedule as the one used for this first set of simulations.
5. Concluding Remarks
In this paper, we have investigated the performance of concatenated BCH-LDGM codes for iterative data fusion of distributed decisions over the Gaussian MAC. The use of LDGM codes permits to efficiently exploit the intrinsic spatial correlation between the information registered by the sensors, whereas BCH codes are selected to lower the error floor due to the MAC ambiguity about the transmitted symbols. Specifically, we have designed an iterative receiver comprising channel detection, BCH-LDGM decoding, and data fusion, which have been thoroughly detailed by means of factor graphs and the Sum-Product Algorithm. Furthermore, a specially tailored soft information flipping technique based on the output of the BCH decoding stage has also been included in the proposed iterative receiver. Extensive computer simulations results obtained for varying number of sensors, LDGM, and BCH codes have revealed that (1) our scheme outperforms significantly the suboptimum limit assuming separation between distributed source and capacity-achieving channel coding and (2) the obtained end-to-end error rate performance attains the theoretical lower bound assuming perfect recovery of the sensor sequences.
This work was supported in part by the Spanish Ministry of Science and Innovation through the CONSOLIDER-INGENIO (CSD200800010) and the Torres-Quevedo (PTQ-09-01-00740) funding programs and by the Basque Government through the ETORTEK programme (Future Internet EI08-227 project).
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