In order to estimate the aforementioned original information sequence
, the optimum joint receiver would symbolwise apply the Maximum A Posteriori (MAP) decision criterium, that is,
where
denotes conditional probability. To efficiently perform the above decision criterion, a suboptimum practical scheme would first compute the conditional probabilities of the encoded symbol
given the received sequence, which is given, for
and
, as
where the proportionality stands for
, and
denotes that all binary variables are included in the sum except
, that is, the sum is evaluated for all the
possible combinations of the set
. Once the
conditional probabilities for the
th sensor codeword
are computed, an estimation
of the original sensor sequence
would be obtained by performing (1) iterative LDGM decoding based on
in an independent fashion with respect to the LDGM decoding procedures of the other
sensors and (2) an outer BCH decoding based on the hard-decoded sequence at the output of the LDGM decoder. Finally, the
recovered sensor sequences
(
) would be fused to render the estimation
as
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.
However, the performance of the above separate approach can be easily outperformed if one notices that, since we assume
(see Section 2), the sensor sequences
are symbolwise spatially correlated, that is
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.
This factor graph is exemplified in Figure 3(a), where the graph structure of the joint detector, decoder, and data fusion scheme is depicted for
sensors. As shown in this plot, this graph is built by interconnecting different subgraphs: the graph modeling the statistical dependence between
and
for all
(labeled as SENSING), the factor graph that relates sensor sequence
to codeword
through the LDGM parity check matrix
and the BCH code (to be later detailed), and the relationship between the received sequence
and the
codewords
, with
(labeled as MAC). Observe that the interconnection between subgraphs is done via variable nodes corresponding to
and
. In this context, since the concatenation of the LDGM and BCH code is systematic, variable nodes
and
collapse into a single node
, which has not been shown in the plots for the sake of clarity. Before delving into each subgraph, it is also important to note that this interconnected set of subgraphs embodies an overall cyclic factor graph over which the SPA algorithm iterates—for a fixed number of iterations
—in the order MAC
LDGM
BCH
LDGM
LDGM
BCH
SENSING.
Let us start by analyzing the MAC subgraph, which is represented in Figure 3(b). Variable nodes
are linked to the received symbol
through the auxiliary variable node
, which stands for the noiseless version of the MAC output
as defined in expression (1). If we denote as
the set of
possible values of
determined by the
possible combinations of
and the MAC coefficients
, then the message
corresponding to
will be given by the conditional probability distribution of the AWGN channel, that is
where the value of the constant
is selected so as to satisfy
. On the other hand, the function associated to the check node connecting
to
is an indicator function defined as
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.
The analysis follows by considering Figure 3(c), where the block integrating the BCH decoder is depicted in detail. At this point it is worth mentioning that the rationale behind concatenating the BCH code with the LDGM code lies on the statistics of the errors per simulated block, as the simulation results in Section 4 will clearly show. Based on these statistics, it is concluded that such an error floor is due to most of the simulated blocks having a low number of symbols in error, rather than few blocks with errors in most of their constituent symbols. Consequently, a BCH code capable of correcting up to
errors can be applied to detect and correct such few errors per block at a small loss in performance. Having said this, the integration of the BCH decoder in the proposed iterative receiver requires some preliminary definitions.
-
(i)
: 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.
-
(ii)
: 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.
-
(iii)
: extrinsic soft information for
built upon the information provided by the rest of sensors at iteration
and time tick
.
-
(iv)
: refined a posteriori soft information of node
for the value
, which is produced as a consequence of the processing stage in Figure 3(c).
Under the above definitions, the processing scheme depicted in Figure 3(c) aims at refining the input soft information coming from the MAC and LDGM subgraphs by first performing a hard decision (HD) on the BCH encoded sequence based on
,
, and the information output from the SENSING subgraph in the previous iteration, that is,
. This is done
within the current iteration
. Once the binary estimated sequence
corresponding to the BCH encoded block at the
th sensor is obtained and decoded, the binary output
is utilized for adaptively refining the a posteriori soft information
as
under the flipping rule
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.
Finally we consider Figure 3(c) corresponding to the SENSING subgraph, where the refined soft information from all sensors is fused to provide an estimation of
as
. Let
denote the soft information on
(for the value
and computed for
) contributed by sensor
at iteration
. The SPA applied to this subgraph renders (see [41, equations (5) and (6)])
where
denotes the sensing error probability which in turn establishes the amount of correlation between sensors. Factors
account for the normalization of each pair of messages, that is,
for all
. The estimation
of
at iteration
is then given by
that is, by the product of all messages arriving to variable node
at iteration
. The iteration ends by computing the soft information fed back from the SENSING subgraph directly to the corresponding LDGM decoder, namely,
where as before,
represents a normalization factor for each message pair.