## 4 Receive Diversity  
It can be generally supposed that the signal transmitted by the cluster head travels through several resolvable discrete multipaths and arrives at the receiver arrays, each multipath having its own independent DOA, time delay, and amplitude. For example, the sensors are deployed in an open field where they collect data and send to the cluster head. The collected data is sampled and modulatedusing BPSK modulation and converted into a serial bit stream. This data bit stream needs to be transmitted to the sink to be analyzed. Assuming that perfect channel state information (CSI) is available at the receiver, if at any time \(t\), \(s(t)\) is the transmitted signal across all links, then the transmitted signals are received over \(N_{r}\) independent and identically distributed GBSB channels corrupted by complex Gaussian noise, the received signal \(y(t)\) can be represented as:  
\[y(t)=\sum_{l}\alpha_{l}(t)s(t-\tau_{l}(t))+n(t) \tag{9}\]  
where \(s(t)\) is the input signal, \(n(t)\) is the additive white Gaussian noise, \(\alpha_{l}(t)\) is the attenuation factor for the signal received on the \(l^{th}\) path. As per antenna array theory, each multipath signal brings multiple signals at the receiving array. The effect of every individual multipath signal on every element of the antenna array can be equalized to multiply by \(a_{k}(\theta_{l})\), known as the steering vector of antenna array where \(k\) represents the index of antenna array. For an N-element linear antenna array the channel impulse response of the \(i^{th}\) user can be expressed as:  
\[h_{i}(t)=\sum_{l=1}^{L}\alpha_{l,i}(t)\,a\big{(}\theta_{l,i}\big{)}\delta\big{(} t-\tau_{l,i}\big{)} \tag{10}\]  
Thus, the output received at the sink is given by \(y(t)=\sum_{l}h_{i}(t)s(t-\tau_{l}(t))+n(t)\), and the N\(\times\)l array response vector or the steering vector \(a\big{(}\theta_{l,i}\big{)}\) is defined as:  
\[a\big{(}\theta_{l,i}\big{)}=\left[1\ e^{-j2\pi\frac{d}{\lambda}sin\big{(} \theta_{l,i}+\Delta_{l}\big{)}}\...\ e^{-j2\pi\frac{d}{\lambda}(N-1)sin\big{(}\theta_{l,i}+\Delta_{l}\big{)}} \right]^{T} \tag{11}\]  
where \(d\) is the element spacing and \(\Delta_{l}\)is the angle spread of the \(i^{th}\) user. The noise on each diversity branch is assumed to be uncorrelated. The collection of independently fading signal branches can be combined in a variety of ways to improve the received SNR. Since the chance of having two deep fades from two uncorrelated signals at any instant is rare, combining them can reduce the effect of the fades. Diversity is a powerful communication receiver technique that provides wireless link improvement at relatively low cost. It exploits the random nature of radio propagation by finding independent signal paths for communication. In virtually all applications, diversity decisions are made by the receiver, and are unknown to the transmitter. The diversity concept can be explained simply. If one radio path undergoes a deep fade, another independent path may have a strong signal. By having more than one path to select from, both the instantaneous and average SNRs at the receiver may be improved. There are a variety of ways in which the independently fading signal branches can be combined; hence, the three most prevalent space diversity-combining techniques used in this paper are the Maximal Ratio Combining (MRC) [15] Equal Gain Combining (EGC) [16,17], and Selection Combining (SC) [18,19]. For example, the received signals are combined at the receiver using MRC to maximize the SNR and give the following expression:  
\[\tilde{y}(t)=\sum_{j=1}^{n_{R}}h_{j}^{*}y_{j}(t)=s(t)\sum_{j}^{n_{R}}\big{|} h_{j}\big{|}^{2}+n^{`}(t) \tag{12}\]  
In terms of the weight vector \(w\), where \(w=h^{H}\), the output x at the receiver is given by:  
\[\text{x=}\sqrt{E_{s}}h^{H}\,hs+h^{H}n \tag{13}\]  
where \(h^{H}h=\sum_{j}^{n_{R}}\big{|}h_{j}\big{|}^{2}\) is the sum of the channel powers across all the receive antennas.