# Effective PSCCH Searching for 5G-NR V2X Sidelink Communications
## 3 Proposed Effective PSCCH Searching
### Proposed Effective PSCCH Searching Based on Correlation  
Based on the observations of the previous section, we now design a new efficient algorithm, where all correlations are firstly completed, without performing all the remaining process. After that, the processing for SCI/PSCCH search and decoding are completed for the pair \((n,i)\), where the correlation is stronger and it continues in descending order. The pseudo-code of the Proposed algorithm (Algorithm 2) is presented following. ```
1 For \(n=0,\ \ldots,\ N_{subCh}^{SL}-1\) do
2 For \(i=1,2,3\) do
3 \(\mathbf{c}_{n,i}=\sum\limits_{p=1}^{N_{qr}}\mathrm{corr}(\mathbf{s}_{r,x,p}, \mathbf{d}_{PSCCH}^{n,i})\)
4 \(\mathbf{c}^{\max}(3n+i-1)=\max\big{(}\mathbf{c}_{n,i}\big{)}\)
5 End
6 End
7 \((\mathbf{c}^{sorted},\mathbf{c}^{sorted}_{index})=\mathrm{sort}_{dsc}( \mathbf{c}^{\max})\)
8 For \(p=0,\ldots,3N_{subCh}^{SL}-1\) do
9 \(n=\mathbf{c}^{sorted}_{index}(p)/3\)
10 \(i=\mathrm{rem}\big{(}\mathbf{c}^{sorted}_{index}(p),3\big{)}-1\)
11  Compute \(f_{offset}(q_{\max})\), where \(q_{\max}=\underset{q\in\{0,\ldots,Q_{\max}-1\}}{\mathbf{c}_{n,i}(q)}\)
12 \(\mathbf{s}^{sync}_{rx,p}(q)=\mathbf{s}_{rx,p}(q+f_{offset})\), \(q=0,\ldots,Q_{\max}-1-f_{offset}\)
13  Perform the CP-OFDM demodulation
14  Extract the PSCCH candidate

Figure 4: Correlation results: (**a**) Trying different subchannels with \(i=1\); (**b**) Trying in subchannel \(n=1\) for indexes \(i=0,1\) and \(2\). In the Proposed algorithm, first we compute for all subchannels (line 1) and all indexes \(i\) (line 2), the correlations \(\text{corr}(\mathbf{s}_{rx,\varphi},\mathbf{s}_{PSCCH}^{H})\) (line 3), and store the value of main peak associated to each pair \((n,i)\) in \(\mathbf{c}^{\text{max}}\in\mathbb{R}^{3N_{subCh}^{SL}}\) (line 4). After that, we sort the vector \(\mathbf{c}^{\text{max}}\) in descending order and store the indexes of original positions in \(\mathbf{c}_{indexs}^{sorted}\in\mathbb{Z}^{3N_{subCh}^{SL}}\) (line 7). Then, for each element of \(\mathbf{c}_{indexs}^{sorted}\), we get the pair \((n,i)\) (lines 9-10), and we apply the remain processing (lines 11-21), similar to the Blind algorithm. ### Theoretical Runtime Analysis

In this section, a theoretical runtime analysis is completed, in order to compare the performances of the two presented algorithms, i.e., the conventional Blind algorithm, and the Proposed algorithm. For that, first let us define the processing time of each part of the algorithms. These definitions are presented in Table 3.

Let \(E_{1}\) and \(E_{2}\) be the expected number of attempts associated to Blind and Proposed algorithms, respectively. Therefore, the total runtime needed to perform the search and decoding procedure of an SCI/PSCCH, associated to Blind and Proposed algorithms, are given, respectively, by

\[T_{1}=E_{1}(T_{corr}+T_{rem}), \tag{5}\]

\[T_{2}=3N_{subCh}^{SL}\Big{(}T_{corr}+T_{fmax}\Big{)}+T_{sort}+E_{2}\big{(}T_{ gpair}+T_{rem}\big{)}. \tag{6}\]

For a low SNR scenario, \(E_{1}=E_{2}=3N_{subCh}^{SL}\), and from Equations (5) and (6) we obtain

\[T_{1}=3N_{subCh}^{SL}(T_{corr}+T_{rem}), \tag{7}\]

\[T_{2}=3N_{subCh}^{SL}\Big{(}T_{corr}+T_{fmax}+T_{gpair}+T_{rem}\Big{)}+T_{sort}. \tag{8}\]

Since \(T_{fmax}\), \(T_{gpair}\) and \(T_{sort}\) are much lower than \(T_{corr}\) or \(T_{rem}\), for a low SNR scenario, we conclude that \(T_{1}\approx T_{2}\). For a high SNR scenario, \(E_{1}=\frac{1}{2}\big{(}3N_{subCh}^{SL}+1\big{)}\) and \(E_{2}=1\). Then, from Equations (5) and (6) we obtain

\[T_{1}=1.5N_{subCh}^{SL}T_{corr}+1.5N_{subCh}^{SL}T_{rem}+0.5T_{corr}+0.5T_{rem}, \tag{9}\]

\[T_{2}=3N_{subCh}^{SL}T_{corr}+T_{rem}+3N_{subCh}^{SL}T_{fmax}+T_{sort}+T_{gpair}. \tag{10}\]