## 4 Suggested Categorization Approach  
As was formerly demonstrated, the spike positions of the correlation function \(U^{(a)}(\tau)\) vary between STBC types. This trend acts as a distinctive characteristic, and a test is used to determine whether \(U^{(a)}(\tau)\) has spikes or not. The assessment of the following technical issues is essential to the categorization procedure.  
* Considering the realistic limitations of the surveillance range, the issue of how to calculate the correction function, \(U^{(a)}(\tau)\), emerges.
* Accordingly, the presence of non-zero values at positions where \(U^{(a)}(\tau)\) should be zeros results from an estimation error of \(U^{(a)}(\tau)\). Because of this, the judgment about the existence of the peaks of \(U^{(a)}(\tau)\) is influenced. Therefore, in the event of a non-ideal estimator, even in the absence of background understanding of channel information, the start of STBC blocks, and noise characteristics, a judgment criterion must be constructed.  
To counteract these challenges, we provide FFT-based spike detection techniques. A series of samples \(r_{\mathcal{D}}^{(i)}(m)\) of length \(M\) where \(m=0\), \(1\),\(\cdots\), \(M-1\) and \(r_{\mathcal{D}}^{(i)}(m)=y_{\mathcal{D}}^{(i)}(m+m_{0})\) are taken and analyzed. Here we denote \(m_{0}\) as an unknown propagation delay to emphasize the absence of details regarding the beginning of STBC blocks. It is important to note that the strategy that has been given does not need the knowledge of the unknown propagation delay, as will be detailed further on. Define the sequence  
\[\mathcal{X}=g_{\mathcal{D}}(m+\tau)=r_{\mathcal{D}}^{(1)}(m)r_{\mathcal{D}}^{(2 )}(m+\tau),m=0,1,\cdots,M-1. \tag{15}\]  
\(\alpha=c_{3}\) show spikes when \(\tau=\pm 1\) and \(\tau=\pm 3\) at the FFT bins 0, \(M/8-1\), \(M/4-1\), \(3M/8-1\), \(M/2-1\), \(5M/8-1\), \(3M/4-1\), and \(7M/8-1\). We propose two algorithms for categorizing STBC signals relying on these properties of the FFT outputs. We define the following three sequences.  
\[\begin{array}{l}\mathcal{X}_{1}=r_{\mathcal{D}}^{(1)}(m)r_{\mathcal{D}}^{(2 )}(m+1),m=0,1,\cdots,M-1\\ \mathcal{X}_{2}=r_{\mathcal{D}}^{(1)}(m)r_{\mathcal{D}}^{(2)}(m+2),m=0,1, \cdots,M-1\\ \mathcal{X}_{3}=r_{\mathcal{D}}^{(1)}(m)r_{\mathcal{D}}^{(2)}(m+3),m=0,1, \cdots,M-1.\end{array} \tag{25}\]  
The FFT outputs of these sequences are designated as \(Q_{1}=\mathrm{FFT}(\mathcal{X}_{1})\), \(Q_{2}=\mathrm{FFT}(\mathcal{X}_{2})\), and \(Q_{3}=\mathrm{FFT}(\mathcal{X}_{3})\). We define \(q_{1}^{(1)}\) and \(q_{2}^{(1)}\) as the locations of the first two maximum values of sequence \(Q_{1}\). The first four greatest values of sequence \(Q_{2}\) are identified by the coordinates \(q_{1}^{(2)}\), \(q_{2}^{(2)}\), \(q_{3}^{(2)}\), and \(q_{4}^{(2)}\), respectively. The first eight greatest values of sequence \(Q_{3}\) are identified by their positions \(q_{1}^{(1)},\cdots,q_{8}^{(3)}\).  
Figure 4: The absolute value of the FFT output of sequence \(\mathcal{X}\) at different values of \(\tau\) for \(\alpha=c_{2}\).  
Figure 5: The absolute value of the FFT output of sequence \(\mathcal{X}\) at different values of \(\tau\) for \(\alpha=c_{3}\).  
Figure 3: The absolute value of the FFT output of sequence \(\mathcal{X}\) at different values of \(\tau\) for \(\alpha=c_{1}\).