## 3 Multichannel Signal Acquisition
### Multichannel Combining  
Matched filtering is a useful approach for acquisition and packet detection in communication receivers. By appending a known signal \(s[i]\), such as an LFM signal, to the beginning of the packet and then monitoring the output of a receiving filter matched to the signal, acquisition can be accomplished. To analyze the signal acquisition performance of the multichannel ring vector sensor receiver, we consider a multichannel combining approach where each channel of the vector sensor is fed into a matched filter, matched to \(s[i]\), and then the outputs of the three matched filters are combined. If \(z_{k}\) denotes the output of the \(k\)th matched filter sampled at \(i=N-1\), then the signal acquisition decision statistic \(\Lambda\) of the ring vector sensor receiver can be written as  
\[\Lambda=\sum_{k=1}^{3}\Lambda_{k}\,,\ \Lambda_{k}=\left|z_{k}\right|^{2},\ z_{k}= \sum_{i=0}^{N-1}s^{*}[i]r_{k}[i], \tag{12}\]  
where \({}^{*}\) represents the conjugate. A flowchart of this method is presented in Figure 2a. Following the Neyman-Pearson theorem [20], the signal is successfully acquired with probability \(P_{A}\), if \(\Lambda>\gamma\), where \(\gamma\) is the decision threshold determined by the probability of false acquisition \(P_{FA}\). Since \(N\) is usually large, \(z_{k}\) in (12) can be considered to follow a complex normal distribution, based on the central limit theorem. Let \(H_{1}\) and \(H_{0}\) represent the signal-in-noise and noise-only scenarios, respectively. Additionally, \(\rho_{k}=E[|z_{k}|^{2}|H_{1}]=(d_{k}+1)\sigma_{k}^{2}\mathbb{E}\) and \(\tilde{\xi}_{k}=E[|z_{k}|^{2}|H_{0}]=\sigma_{k}^{2}\mathbb{E}\), where \(d_{k}=(\eta_{k}^{2}/\sigma_{k}^{2})\overline{\mathbb{E}}\) and \(\overline{\mathbb{E}}\) is energy of the signal \(s[i]\). The characteristic function (CF) of \(\Lambda_{k}\) in (12), \(\Theta_{\Lambda_{k}}(\omega)=E[\exp(j\omega\Lambda_{k})]\) with \(j^{2}=-1\), can be shown to be \((1-j\rho_{k}\omega)^{-1}\) and \((1-j\tilde{\xi}_{k}\omega)^{-1}\) under \(H_{1}\) and \(H_{0}\), respectively. Upon applying partial fraction expansion to the CF of \(\Lambda\), \(\Theta_{\Lambda}(\omega)=\prod_{k=1}^{3}\Theta_{\Lambda_{k}}(\omega)\), followed by inverse Fourier transform to obtain probability density function (PDF) from CF [20], PDF of \(\Lambda\) can be obtained and integrated, which result in the following performance probabilities  
\[P_{A}=\Pr(\Lambda>\gamma|H_{1})=\sum_{k=1}^{3}U_{k}\exp(-\gamma/\rho_{k}), \tag{13}\]  
\[P_{FA}=\Pr(\Lambda>\gamma|H_{0})=\sum_{k=1}^{3}V_{k}\exp(-\gamma/\tilde{\xi}_ {k}). \tag{14}\]  
In the above equations, we have \(U_{k}=\prod_{k^{\prime}=1,k^{\prime}\
eq k}^{3}(1-\rho_{k^{\prime}}\rho_{k}^ {-1})^{-1}\) and \(V_{k}=\prod_{k^{\prime}=1,k^{\prime}\
eq k}^{3}(1-\xi_{k^{\prime}}\tilde{\xi} _{k}^{-1})^{-1}\) as the partial fraction expansion coefficients of \(\Theta_{\Lambda}(\omega)\) under \(H_{1}\) and \(H_{0}\), respectively.  
To implement this multichannel signal acquisition method for the ring vector sensor receiver, three matched filters are needed. In the next subsection, we consider a simpler method that requires only one matched filter to implement, and then compare its performance with the implementation that needs three matched filters.