# Efficient Preamble Design Technique for Millimeter-Wave Cellular Systems with Beamforming
## 2 Proposed Preamble Design Technique  
The two ZC sequences used to generate a preamble sequence in the proposed technique are given by [14]:  
\[Z_{s}^{r,d}\left(m\right)=Z^{r_{s}}\left(m+d_{s}\right)=e^{-j\pi r_{s}\left(m+d_ {s}\right)\left(m+d_{s}+1\right)/L},GCD\left(r_{s},L\right)=1 \tag{1}\]  
where \(m\in\left\{0,\ 1,\ \cdots,\ L-1\right\},\ s\in\left\{0,\ 1\right\}\). \(L\), \(s\), and \(GCD\) denote a prime number representing the length of the ZC sequence, a parameter indicating one of the two different ZC sequences, and the greatest common divisor, respectively. In addition, \(r_{s}\in\left\{1,\ \ \cdots,\ L-1\right\}\) and \(d_{s}\in\left\{0,\ 1,\ \ \cdots,\ L-1\right\}\) denote the root index and the cyclic shift of the \(s\)-th ZC sequence, respectively. The number of preamble sequences that can be generated by a combination of the root indices and the cyclic shifts of the two ZC sequences, is given by \(\left(L-1\right)^{2}L^{2}\). Therefore, the number of preamble sequences that can be generated by combining two ZC sequences is considerably larger than the ones generated by a single ZC sequence. For example, the number of available preamble sequences is 3660 for a single ZC sequence with a length of 61 and 864,900 for two ZC sequences with lengths of 31 each. Therefore, the number of preamble sequences that can be generated by combining two ZC sequences with length of 31 is 236 times greater than the ones generated by a single ZC sequence with a length of 61. The proposed preamble is generated by allocating two ZC sequences with lengths, \(L\), in the frequency domain, as follows:  
\[X^{r,d}\left(k\right)=\sum\limits_{s=0}^{1}\widetilde{X}_{s}^{r,d}\left(k \right)e^{-j2\pi ks/N} \tag{2}\]  
where \(\widetilde{X}_{s}^{r,d}\left(\left(k+N_{0}\right)_{\forall nN}\right)= \  
Here, % denotes the modulo operation. The parameter, \(q\), is used to distinguish the even samples (\(q=0\)) from the odd samples (\(q=1\)). As shown in Figure 1c, \(N_{0}\)-point FFTs of the even and odd samples of \(x^{\prime,d}\left(n\right)\) are given by \(\widetilde{X}_{0}^{\prime,d}\left(\left(k\right)_{\gamma_{n}N_{0}}\right)\) and \(\widetilde{X}_{1}^{\prime,d}\left(\left(k\right)_{\gamma_{n}N_{0}}\right)\), respectively. In other words, even and odd samples of \(x^{\prime,d}\left(n\right)\) consist of \(Z_{0}^{\prime,d}\left(m\right)\) and \(Z_{1}^{\prime,d}\left(m\right)\), respectively. Because of this property, the proposed preamble can have a low PAPR characteristic similar to the one obtained using a single ZC sequence. The correlation property for each sequence (composed of either odd or even samples) can be easily obtained from \(x^{\prime,d}(n)\)[14]. The upper bound for the sum of the correlation values of the two sequences, \(\widetilde{\mathcal{C}}^{\prime,\overline{d}}\), that will be used for the sequence detection in Section 3, can be derived using the triangular inequality as follows:  
\[\widetilde{\mathcal{C}}^{\overline{r},\overline{d}}=\left\{ \begin{array}{cc} 1/2\sum\limits_{s=0}^{1}\delta\left(\overline{d}_{s}\right),&if\,F=0\,\left( \text{Property}\,1\right)\\ 1/\left(2\sqrt{L}\right)\sum\limits_{s=0}^{1}\delta\left(\left\{\overline{d }_{s}\right\}_{\gamma_{n}\mathbb{I}}\right),&if\,P\
eq 0\,\left(\text{Property}\,2 \right)\\ 1/2\left(1/\sqrt{L}+\sum\limits_{s=0}^{1}\delta\left(\overline{d}_{s}+ \overline{r}_{s}\right)\right),&if\,F\
eq 0,\,P=0\,\left(\text{Property}\,3 \right)\end{array} \right.