## 1 Introduction
### Related Works and Motivation  
In the past, communication and decision fusion were taken as two separated parts in the application of the typical distributed detection, and the performance analysis for each part was carried out independently. For a distributed detection system with reliable communication, we assume that the local decisions are transmitted to the FC without errors. Based on such an assumption, numerical results on the distributed detection with decision fusion have been widely studied [8; 9; 10; 11; 12; 13; 14]. Under the assumption of conditional independence of the observations, the optimal decision rule at sensors and the optimal fusion rule at the FC are both likelihood-ratio tests (LRTs) [8]. Based on the LRT, several fusion statistics, including Chair-Varshney (CV) [9], generalized LRT [10], the Bayesian fusion statistic [11] and the counting fusion statistic [12; 13; 14], were proposed. These obtained results rely on the reliable communication between the FC and sensors. However, since the sensors are energy-limited, the transmission power of each sensor is not large enough to make the communication reliable. Usually, the communication channels are subject to channel fading, environmental noise and interferences. For the sake of simplicity, the parallel access channel (PAC) is assumed for information transmission. Each sensor transmits information to the FC across parallel and independent channels. This transmission can be realized through time division multiple access (TDMA), frequency division multiple access (FDMA) or code division multiple access (CDMA). Under the PAC assumption, several channel-aware fusion rules were developed to take the impairment from the channel fading and noise into account [15; 16; 17; 18; 19; 20]. In [15], an additive white Gaussian noise (AWGN) channel has been considered. A distributed binary detection under an _ad hoc_ network was investigated, and a mixed time scale recursive algorithm was proposed. The distributed detection model with fading and noisy channels is illustrated in Figure 1. Each sensor collects raw measurements, makes a local decision and then transmits it to an FC, where the global decision about the presence/absence of a target is given. The fading and noisy channel model considers the physical layer specifications. It can be regarded as a kind of modulation channel, where the waveform carrier modulation and the match-filtering are neglected. Based on this model, an LRT-based channel-aware fusion rule was derived in [17; 18]. Starting from the optimal LRT fusion rule, the authors also derived several suboptimal fusion rules: the maximum ratio combiner (MRC), the two-stage CV, and the equal gain combiner (EGC) fusion rules according to the availability of the channel state information (CSI) at the FC. Besides, a unified asymptotic decision rule based on the MRC fusion rule, namely generalized MRC, was proposed to detect weak signals in non-Gaussian noise [19]. These fusion rules significantly rely on the instantaneous CSI. However, acquiring such instantaneous information is cumbersome in resource-constrained applications and is not practical for some time-varying channel. An LRT fusion rule based on channel statistics (LRT-CS) was proposed in [20]. To avoid the requirement of the instantaneous CSI, the authors in [21, 22] investigated the impact of imperfect CSI on the fusion rule and detection performance by fixing the total transmission power of training and data symbols. Furthermore, coherent and noncoherent receptions were included in the analysis of the impact of channel uncertainty. Further, decision fusion with multi-hop transmissions [23] was developed to form a channel-aware fusion rule by considering large-scale sensor networks. These fusion rules based on the physical channel can completely characterize the channel error. However, in a decode-then-fuse system, each sensor observes the noisy measurements and obtains the local decisions according to given decision rules.