## 4 Decision Fusion Rule with Channel Errors Characterized by BER  
This section gives the optimal fusion rule based on the non-ideal channel characterized by the probability \(P(y_{i}|u_{i})\). It can be written as a function of BER. The BER is able to describe the data uncertainty at the FC due to channel errors, as well as modulations/demodulations. By taking different physical layer specifications, including modulation, reception mode and channel characteristics between sensors and the FC, the BER is derived. In this paper, we consider the flat and slow fading channel. This means that the channel fading is roughly equal across the entire signal bandwidth for a narrowband signal transmission [27]. For a decode-then-fuse distributed detection system, the FC decodes the received data and obtains the binary quantization results. Thus, the wireless and non-ideal channel can be modeled as the probability\(P(y_{i}|u_{i})\). The probability of a \(k\) bit error, when \(n\) bit data are transmitted, follows a binomial distribution, given as [28]:  
\[P(n,k)=\binom{n}{k}(P_{e})^{k}(1-P_{e})^{(n-k)} \tag{6}\]  
where \(P_{e}\) is the channel BER. We have \(P(y_{i}=0|u_{i}=1)=P(y_{i}=1|u_{i}=0)=P(1,1)=P_{e}\). \(P(1,1)\) denotes the probability of one-bit error for one-bit transmission. As we know, the BER depends on several factors of the communication channel, including the reception mode at the FC, the signal modulation/demodulation method at sensors and the FC and the statistic characteristics of the non-ideal channel. Thus, the overall impact from the non-ideal channel and the different modulation/demodulation schemes on the received data can be characterized by \(P_{e}\). As such, the fusion rule of the decode-then-fuse collaborative detection system and the physical layer specifications are related to each other by \(P_{e}\). For a flat and slow Rayleigh fading channel, the BER with coherent binary frequency-shift keying (BFSK or 2 FSK) modulation, coherent BPSK modulation and noncoherent differential phase-shift keying (DPSK) modulation can be addressed as the following relations (see [27] pp: 170\(\sim\)176):  
\[P_{e}= \begin{cases}\frac{1}{2}\left(1-\sqrt{\frac{\gamma_{b}}{2+\gamma_{b}}} \right)&\text{coherent 2 FSK modulation}\\ \frac{1}{2}\left(1-\sqrt{\frac{\gamma_{b}}{1+\gamma_{b}}}\right)&\text{coherent BPSK modulation}\\ \frac{1}{2(1+\overline{\gamma_{b}})}&\text{noncoherent DPSK modulation}\\ \end{cases}  \tag{7}\]  
where \(\bar{\gamma}_{b}\) is the average received SNR of the communication channel, defined as:  
\[\bar{\gamma}_{b}=\frac{P_{r}}{N_{0}B}=\frac{P_{r}}{\sigma^{2}} \tag{8}\]  
where \(N_{0}B=N_{0}/2\times 2B\) is the total noise power within the bandwidth \(2B\) and \(\sigma^{2}=N_{0}/2\times 2B\). \(P_{r}\) is the received signal power. Furthermore, the received SNR can be regarded as \(\bar{\gamma}_{b}=E_{b}/(N_{0}BT_{b})=E_{b}/N_{0}\) for data pulses with \(T_{b}=1/B\)[27]. For a flat and slow Ricean fading channel with different \(K\) factors, the BER with coherent 2 FSK modulation, coherent BPSK modulation and noncoherent DPSK modulation can be written as:  
\[P_{e}= \begin{cases}\frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}\frac{(1+K)\sin^{2} \phi}{(1+K)\sin^{2}\phi+0.5\overline{\gamma}_{b}}\exp\left(-\frac{0.5K\overline {\gamma}_{b}}{(1+K)\sin^{2}\phi+0.5\overline{\gamma}_{b}}\right)\mathrm{d} \phi\\ \text{coherent 2 FSK modulation}\\ \frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}\frac{(1+K)\sin^{2}\phi}{(1+K)\sin^{2} \phi+\overline{\gamma}_{b}}\exp\left(-\frac{K\overline{\gamma}_{b}}{(1+K)\sin ^{2}\phi+\overline{\gamma}_{b}}\right)\mathrm{d}\phi\\ \text{coherent BPSK modulation}\\ \frac{1+K}{2(1+K+\overline{\gamma}_{b})}\exp\left(-\frac{K\overline{\gamma}_{b }}{1+K+\overline{\gamma}_{b}}\right)\\ \text{noncoherent DPSK modulation}\\ \end{cases}  \tag{9}\]  
where \(K\) is the ratio of the powers of the line of sight component to the scattered component of the received signal.