## 2 Statement of the Problem  
Consider an WSN-based target detection system consisting of \(N\) spatially-distributed sensors and an FC. There are two hypotheses \(H_{0}\) and \(H_{1}\) under test. \(H_{0}\) corresponds to target-absence and \(H_{1}\) corresponds to target-presence. Let \(X_{i}\) be the observed data at the \(i\)-th sensor (\(i=1,\ldots,N\)). Note that \(X_{i}\) can be either a random variable or a random vector. The observed data across the whole network are \(\mathbf{x}=[X_{1},X_{2},\ldots,X_{N}]^{T}\), where the superscript \(T\) denotes the transpose. A local decision rule \(\gamma_{i}(\cdot)\) is applied to the observed data, and a local decision \(u_{i}\) is available, _i.e._, \(u_{i}=\gamma_{i}(X_{i})\), \(\forall:1{\leq}i{\leq}N\). The local decisions can be written in a vector as \(\mathbf{u}=[u_{1},u_{2},\ldots,u_{N}]^{T}\). The local decision \(u_{i}\) is coded, modulated and then is transmitted to the FC via a wireless channel with channel errors. The FC decodes, demodulates the received signal and then derives binary corrupted results from sensors. The received data can be distorted by the non-ideal communication channels due to noise and the multipath effect. Furthermore, errors can be brought in by using different modulation/demodulation schemes and reception modes. Assuming that \(y_{i}\) is the demodulated data of \(u_{i}\) at the FC, the complete impact of the transmission channel, the modulation/demodulation scheme and reception mode can be characterized by\(P(y_{i}|u_{i})\). Then, all of these results \(\mathbf{y}=[y_{1},y_{2},\cdots,y_{N}]^{T}\) from local sensors are combined according to the proposed fusion rule \(\gamma_{0}(\cdot)\) to give a final detection result \(u_{0}\) about the presence/absence of a target, _i.e._, \(u_{0}=\gamma_{0}(\mathbf{y})\). Here, \(u_{0}\in\{0,1\}\) is the final decision of the detection system, where '0' denotes target absence and '1' denotes target presence. Figure 2 gives an illustration of such a decision fusion system. It can be shown that the fusion model forms a Markov chain, _i.e._, \(H_{0}/H_{1}\rightarrow\mathbf{x}\rightarrow\mathbf{u}\rightarrow\mathbf{y}\to u_{0}\) is a Markovian process. Since the system detection performance is related to the specifications, including modulation/demodulation schemes, reception modes and the characteristics of the non-ideal communication channel, the more practical BER-based decision fusion strategy for the decode-then-fuse target detection system will be developed and studied in this paper. To meet the bandwidth limitation of the wireless channel and power-limited budgets of sensors, we consider one-bit quantization according to the local decision rule \(u_{i}=\gamma_{i}(X_{i})\), _i.e._, \(u_{i}\in\{0,1\}\). We denote \(P_{fa_{i}}\) and \(P_{d_{i}}\), respectively, as false alarm and detection probabilities corresponding to the local decision \(u_{i}\) for the \(i\)-th sensor. In this paper, we focus on the derivation of the optimal fusion rule at the FC and assume that the local decisions are known, and each sensor detection performance index, the probability of detection and the probability of false alarm, is assumed to be predefined. The local detection performance indices can be determined according to a predefined detection range with a given local decision rule or can be estimated by a recursive algorithm, e.g., joint probability estimation [25]. The proposed fusion strategy is implemented as follows. Each sensor observes noisy measurements and derives a local binary decision. Then, the local decision result is coded and modulated before signal transmission. The fusion strategy in the FC is performed after demodulation and decoding the corrupted decisions from sensors. Thus, the fusion rule can be regarded as a decode-then-fuse one. The main goal of this manuscript is to provide a fusion strategy at the FC based on the demodulated or decoded data. Hence, the intermediate process of the practical modulation/demodulation and coding/decoding is not mentioned.