# Efficient Preamble Design Technique for Millimeter-Wave Cellular Systems with Beamforming
## 3 Efficient CBID Detection Technique for mmWave Cellular Systems
\(r_{s}^{g}\), \(d_{s}^{cb}\), and \(O_{s}^{v}\) denote the root index for the GID (\(g\)), the cyclic shift corresponding to the CID (\(c\)) and BID (\(b\)), and the phase rotation offset for the SID (\(v\)), respectively, for the \(s\)-th ZC sequence. In Equation (6), \(d_{s}^{c,b}\) is selected to satisfy \(\left(r_{s}^{g}d_{s}^{c,b}\right)_{\gamma_{0}L}=\widehat{d}_{s}^{v,b}\) so that the phase rotation in \(P_{s}^{v,b}\left(m\right)\) does not depend on \(r_{s}^{g}\). The GID (\(g\)) is mapped to the root index pair of the two ZC sequences, \(\left\{r_{0}^{g},r_{1}^{g}\right\}\). The mapping rule for the GID is given by:  
\[\left\{r_{0}^{g},r_{1}^{g}\right\}=\left\{\left(\widetilde{g}^{\prime}\right) _{\gamma_{0}\left(N_{r}+1\right)}+1,\left(\widetilde{g}^{\prime}+\left| \widetilde{g}^{\prime}/\left(N_{r}+1\right)\right|\right)_{\gamma_{0}\left(N_{ r}+1\right)}+1\right\} \tag{7}\]  
where \(\widetilde{g}^{\prime}=g+\widetilde{g}/\left(\widetilde{g}+1\right)/2\)  
\[\widetilde{g}=\left[\left(g+\left|g/N_{r}\right|\right)/\left(\left|g/N_{r} \right|+1\right)/2\right)/N_{r}\right]\]  
\[r_{s}^{g}\in\left\{1,2,\cdots,N_{r}\right\},\left(N_{r}\leq L-1\right) \&\left(N_{r}\leq N_{g}\right)\]  
Figure 3: Preamble structure in the proposed technique. Here, \(N_{r}\) denotes the number of root indices used in the preamble design. An example of Equation (7) when \(N_{g}=12\) and (\(N_{r}=12\), \(N_{r}=7\), \(N_{r}=5\)) is given in Table 1. In this table, the root index pairs corresponding to the GIDs are listed. For example, 12 root indices are used when \(N_{r}=12\), whereas five root indices are used when \(N_{r}=5\). As given in Equation (6), \(v\) is mapped to a pair of phase rotation offsets, \(\{O_{0}^{v},O_{1}^{v}\}\). Therefore, \(c\) is mapped to a combination of \(\left\{r_{0}^{g},r_{1}^{g}\right\}\) and \(\{O_{0}^{v},O_{1}^{v}\}\). When \(w\) is defined as \(\left\lceil N_{g}/N_{r}\right\rceil\), \(N_{C}\) can be expressed by \(w\left(N_{r}-\left(w-1\right)/2\right)N_{v}\). The maximum value of \(N_{C}\) is obtained when \(N_{r}=L-1\) and is given by \(w\left(L-\left(w+1\right)/2\right)N_{v}\). Therefore, the proposed technique can increase the number of available CIDs compared to the previous technique with a single ZC sequence. For example, when the length of a ZC sequence is 1021, the number of available CIDs is 1020 in the previous technique and 6084 in the proposed technique, when \(L\), \(w\), and \(N_{v}\) are set to 509, 3, and 4, respectively. The number of CIDs in the proposed technique is six times larger than the one in the previous technique. The values of \(\widehat{d}_{s}^{v,b}\) are selected such that the conditions \(\widehat{d}_{s}^{v^{\prime},b^{\prime}}\
eq\widehat{d}_{s}^{v^{\prime},b^{ \prime\prime}}\
eq^{b^{\prime}}\) and \(\widehat{d}_{s}^{v^{\prime},b^{\prime}}\
eq\widehat{d}_{s}^{v^{\prime\prime} \
eq v^{\prime},b^{\prime\prime}}\) are satisfied for two different preambles with \(\left(v^{\prime},b^{\prime}\right)\) and \(\left(v^{\prime\prime},b^{\prime\prime}\right)\). Here, \(0\leq v^{\prime},v^{\prime\prime}\leq N_{p}-1\) and \(0\leq b^{\prime},b^{\prime\prime}\leq N_{B}-1\). This condition is required to remove the poor correlation properties (Property 1 and 3) in Equation (5). Note that the correlation property of the proposed preamble can be changed depending on the value of \(N_{r}\). As \(N_{r}\) decreases, the number of side peaks satisfying the condition in Property 3 increases. The maximum number of side peaks is given by \(2\left(w-1\right)\). In the example for the GID mapping rule in Table 1(b/c), the maximum number of side peaks satisfying the condition in Property 3 is two/four because the value of \(w\) is two/three. For example, the root index 2/3 appears twice/four times for different GIDs, producing high correlation values (side peaks). Thus, as \(N_{r}\) decreases, the number of side peaks producing high correlation values at incorrect positions can increase.