## 2 Materials and Methods
### Proposed MP/DD Method  
Double stages are included in the proposed multi-peak double-dwell (MP/DD) acquisition method. In the first stage, multiple peaks of test variables are selected as candidates of the signal test variable; in the second stage, the only signal test variable is singled out from these candidates whose SNR is enhanced by the longer time integration. The acquisition method is described in detail as follows. **Step 1:** The object searching space \(S(\tau\_{f}\)) is divided to discrete space \(S_{ij}(\tau\_{i}\_{f}\_{j})\) including \(N\) searching cells, and test variables \(\xi_{ij}(\tau\_{i}\_{f}\_{j})\) are generated by the specified parameters \(\tau_{i}\) and \(f_{j}\) in each searching cell. These test variables are integrated over a period \(T_{int}\) which costs \(NT_{int}C_{cell}\) computation overhead. **Step 2:** The first dwell search is carried on \(\xi_{ij}(\tau\_{i}\_{f}\_{j})\), and the largest M test variables \(\xi^{(1)}_{ij}(\tau\_{i}\_{f}\_{j})\) are picked up, whose searching cell \(\text{S}^{(1)}_{ij}(\tau\_{i}\_{f}\_{j})\) will be used in the next dwell. The computation overhead of selecting the largest M \(\xi^{(1)}_{ij}\)(\(\tau_{i}\)_f\(j\)) among all test variables is \(C_{s}\), and it is small enough to be ignored compared to \(C_{cell}\). M is a number whose value can be configured;  
**Step 3**: The second dwell is carried only on the sets \(\xi^{(1)}_{ij}\)(\(\tau_{i}\)_f\(j\)), and double integration time \(2T_{int}\) is used to generate test variables \(\xi^{(2)}_{ij}\)(\(\tau_{i}\)_f\(j\)) with \(2T_{int}MC_{cell}\) computation overhead. **Step 4:** The largest \(\xi^{(2)}_{ij}\)(\(\tau_{i}\)_f\(j\)) is singled out, and its parameters are recognized as the final estimated \(\tau\) and \(f_{d}\). The computation overhead of selecting the largest \(\xi^{(2)}_{ij}\)(\(\tau_{i}\)_f\(j\)) among M test variables is small. The state diagram of the proposed method is demonstrated in Figure 1. Its detection probability \(P_{D}\) and false alarm probability \(P_{F}\) are obtained as follows:  
\[P_{D}=P_{D}^{1}P_{D}^{2}P_{F}=\left(1-P_{D}^{1}\right)+P_{D}^{1}\left(1-P_{D} ^{2}\right), \tag{3}\]  
where \(P_{D}^{1}\) is the probability that the M test variables include the signal test variable in the first dwell, and \(P_{D}^{2}\) is the probability that detects the signal test variable in the second dwell. For simplicity, the signal test variable is regarded as existing in one of the searching cells randomly, and the noise test variable in spare ones. Hypothesis testing is used where the alternative hypothesis (\(H_{1}\)) means that the signal test variable exists and the null hypothesis (\(H_{0}\)) means the noise test variable exists. The distribution of the signal test variable is in connection with the energy detector pattern. Without loss of generality, coherent and non-coherent integration algorithms are adopted here. Thus, signal test variable and noise test variable are regarded as non-central chi-square distribution \(f_{H_{1}}\) and central chi-square distribution \(f_{H_{0}}\)[18]. If \(M\) is large enough to include the signal test variable in the whole \(M\) selected test variables, the minimum test variable \(V_{th}\) can be expressed as:  
\[M=\int_{V_{th}}^{\infty}f_{H_{1}}(x)dx+(N-1)\int_{V_{th}}^{\infty}f_{H_{0}}(x)dx. \tag{4}\]  
Thus, the equivalent threshold \(V_{th}\)(\(M\),\(N\),\(\sigma\)) which picks up the largest \(M\) test variables including the signal test variable from all \(N\) variables can be deduced as:  
\[V_{th}(M,N,\sigma)=\big{[}F_{H_{1}}(x)+(N-1)F_{H_{0}}(x)\big{]}^{-1}\big{(}F_ {H_{1}}(\infty)+(N-1)F_{H_{0}}(\infty)-M\big{)}, \tag{5}\]  
where \(F_{H0}\) and \(F_{H1}\) are the probability distribution functions of signal and noise test variables, respectively, which are determined by SNR. The SNRs of the signal test variables are related to signal power level and integration time as:  
Figure 1: The state diagram of the proposed multi-peak double-dwell acquisition method.