## 4 Proposed Method  
The suggested method takes use of the existence of spikes in the previously described cross-correlation functions for modulation awareness. For the sake of demonstration, we take a look at the set of modulation schemes {QPSK, 16 \(-\) PSK, 64 \(-\) QAM}. When \(F_{2}(k_{1},k_{2})\) has spiked, the set of {QPSK, 64 \(-\) QAM} is said to possess; otherwise, the 16 \(-\) PSK modulation scheme is selected. Finally, we examine the spikes of \(F_{4}(k_{1},k_{2})\) to distinguish between 64 \(-\) QAM and QPSK; if they exist, QPSK is reported present; otherwise, 64 \(-\) QAM is accepted. For spike detection, the accompanying fundamental considerations must be addressed. * In reality, the observation duration is constrained, which prompts the question of how to calculate the cross-correction functions. Figure 5: \(|F_{4}(k_{1},k_{2})|\) for QPSK and 64 \(-\) QAM modulations. Figure 6: The architecture of a cross-correlation function when there are spikes. * Given the preceding fact, an estimating error of \(F_{i}(k_{1},k_{2})\), \(i=1,\cdots,4\), outcomes a non-zero level when \(F_{i}(k_{1},k_{2})\) should remain zero. This has a negative impact on the conclusion of whether or not statistical peaks of \(F_{i}(k_{1},k_{2})\) exist. Due to this, judgment criteria have to be established in situations where a non-ideal predictor is present, even if previous information of the estimate error statistics is not available. * It is also assumed that channel information and noise power are unknown. This assumption constitutes an extra obstacle in the way of successfully completing the spike detection task. To determine the most probable assigned modulation scheme, we offer a spike-detection method bound by the chance of false alarm while dealing with the aforementioned constraints. The fundamental concept is to calculate a test statistic and establish a barrier. If the test statistic exceeds the barrier, the method asserts that spikes exist; otherwise, no spikes are discovered. The explanation that follows gives a more explicit mathematical analysis of the suggested method. Let us examine a number of successive samples, \(K\), from each of the two vectors \(z_{D}^{(1)}(k)\) and \(z_{D}^{(2)}(k)\), \(k=0,\cdots,K-1\). When spikes appear in a correlation function, they tend to cluster around the diagonal, as demonstrated in Figures 2-6. The test statistic \(\mu\) is defined as  
\[\mu=\frac{1}{K}\biggl{|}\sum_{k=0}^{K-1}f\Bigl{(}z_{D}^{(1)}(k),z_{D}^{(2)}(k) \Bigr{)}\biggr{|}, \tag{13}\]  
where  
\[f\Bigl{(}z_{D}^{(1)}(k),z_{D}^{(2)}(k)\Bigr{)}= \begin{cases}z_{D}^{(1)}(k)z_{ D}^{(2)}(k)&\text{if $F_{1}$ is considered}\\ z_{D}^{(1)}(k)\Bigl{(}z_{D}^{(2)}(k)\Bigr{)}^{3}&\text{if $F_{2}$ is considered}\\ z_{D}^{(1)}(k)\Bigl{(}z_{D}^{(2)}(k)\Bigr{)}^{7}&\text{if $F_{3}$ is considered}\\ \Bigl{(}z_{D}^{(1)}(k)\Bigr{)}^{4}\Bigl{(}z_{D}^{(2)*}(k)\Bigr{)}^{2}&\text{if $F_{4}$ is considered}.\end{cases}  \tag{14}\]  
There are two hypotheses to investigate in the spike detection challenge:  
\[\mathcal{H}_{0}:\ \mu = e,\] \[\mathcal{H}_{1}:\ \mu = S+e, \tag{15}\]  
where \(\mathcal{H}_{1}\) and \(\mathcal{H}_{0}\) are the hypotheses related to whether or not a spike exists. Here, \(S\) represents the spike's amplitude and \(e\) denotes the impact of the AWGN and estimate error due to the usage of a restricted number of samples. Using hypothesis \(\mathcal{H}_{0}\) and the assumption that \(e\) is a Rayleigh random variable, with mean \((\pi\sigma^{2})/2\) and cumulative distribution function (CDF) supplied by  
\[g(a)=1-\exp\Bigl{(}-a^{2}/2\sigma^{2}\Bigr{)},\ a\geq 0. \tag{16}\]  
The method's primary idea is to create a barrier \(\lambda\), and the spike is stated to appear if \(\mu>\lambda\); alternatively, there would be no spike. Our aim is to achieve a given level of false alarm probability \(P_{f}\), which is described as the possibility of incorrectly reporting the presence of a spike.