# Efficient Preamble Design Technique for Millimeter-Wave Cellular Systems with Beamforming
## 3 Efficient CBID Detection Technique for mmWave Cellular Systems
However, the computation complexity for the CBID detection is reduced as \(N_{r}\) decreases because the number of available root indices decreases. In the CBID detection, it is assumed that \(N_{0}\)-point FFT is used. However, because the length of the ZC sequence, \(L\), is usually different from the FFT size, \(N_{0}\), \(P_{\mathrm{s}}^{v,b}\left(m\right)\) in Equation (6) can be rewritten as:  
\[Z_{s}^{c,b}\left(m\right)=Z_{s}^{g}\left(m+d_{s}^{c,b}\right)=Z_{s}^{g}\left(m \right)P_{s}^{v,b}\left(m\right)S_{s}^{c,b} \tag{6}\]0  
where \(\rho\) denotes the ratio of \(N_{0}\) to \(L\) and is a rational number. The non-integer factor, \(\rho\), usually degrades the performance of the CBID detection. In order to compensate the effect of the fractional part in \(\rho\) on the CBID detection, we multiply \(Z_{s}^{c,b}\left(m\right)\) by a polyphase sequence, \(D_{s}^{v,b}\left(m\right)\), as follows:  
\[ \begin{array}{l}\widehat{Z}_{s}^{c,b}\left(m\right)=Z_{s}^{c,b}\left(m\right) D_{s}^{v,b}\left(m\right)\\ =S_{s}^{c,b}Z_{s}^{\mathrm{S}}\left(m\right)e^{-i\pi 2\widehat{d}_{s}^{v,b}m/N_{0}} \end{array}  \tag{9}\]  
where \(D_{s}^{v,b}\left(m\right)=e^{-j\pi 2\widehat{d}_{s}^{v,b}m/N_{0}}\), \(\Delta_{s,b}^{v}=\left(1-\rho\right)\widehat{d}_{s}^{v,b}\). In the following, \(Z_{s}^{r,d}\), \(\widehat{X}_{s}^{r,d}\), \(X^{r,d}\), and \(x^{r,d}\), used in Equations (2)-(4), are replaced by \(\widehat{Z}_{s}^{c,b}\), \(\widehat{X}_{s}^{c,b}\), \(X^{c,b}\), and \(x^{c,b}\), respectively, to formulate the equations for the CBID detection in mmWave cellular systems. In addition, there may exist a residual STO even after the synchronization procedure has been completed with the SP. Because the residual STO causes a phase rotation in the frequency domain, the BP needs to be designed to be robust to the residual STO [16]. Next, we analyze the effect of the residual STO on the CBID detection. When the residual STO (\(\chi\)) exists, the decimated version (even and odd samples) of \(x^{c,b}\left(\left(n-\chi\right)_{\infty_{\mathrm{N}}}\right)\) with \(q\in\left\{0,\ 1\right\}\) is given by:  
\[ \begin{array}{l}x^{c,b}\left(\left(2n+q-\chi\right)_{\infty_{\mathrm{N}}} \right)\\ =2/N\sum\limits_{k=0}^{N_{0}-1}\left(\widehat{X}_{s=\left(q+l\right)_{\infty 2}}^{c,b}\left(k\right)e^{-j2\pi\left(u_{l}+l \left(q+l\right)_{\infty 2}\right)k/N_{0}}\right)e^{j2\pi\chi u/N_{0}}\\ =2/NS_{s=\left(q+l\right)_{\infty 2}}^{c,b}\sum\limits_{m=0}^{L-1}\left(Z_{s= \left(q+l\right)_{\infty 2}}^{g}\left(m\right)e^{-j\pi 2\left(\widehat{d}_{s}^{v,b} \right)_{\infty 2}+u_{l}+l\left(q+l\right)_{\infty 2}\right)\widehat{m}/N_{0}} \end{array}  \tag{10}\]  
where \(\widehat{m}=N_{0}/2-\left[L/2\right]+m\), \(\chi=2\mu_{l}+l\)  
\[U_{l}=\left\{u_{l}\right|\ 0\leqslant u_{l}\leqslant\left\lfloor\chi_{Max}/2 \right\rfloor-l\left(\chi_{Max}+1\right)_{\forall\varsigma 2}\right\}\]  
\[l=\left(\chi\right)_{\forall\varsigma 2},\ \chi\in\left\{0,1,\cdots,\chi_{Max}\right\}\]  
Here, \(l\) denotes a parameter indicating that \(\chi\) is an even number (\(l=0\)) or an odd number (\(l=1\)). It can be seen from Equation (1) that \(s=\left(q+l\right)_{\infty 2}\) changes depending upon the value of the STO for the same \(q\). When \(\chi\) is an even number, the root index of the ZC sequence of a decimated version with \(q=0\) in Equation (10) corresponds to the case, \(s=0\). However, when \(\chi\) is an odd number, the root index of a ZC sequence of a decimated version with \(q=0\) corresponds to the case, \(s=1\). Hence, an ambiguity in the GID detection may occur because the order of the indices in a root pair corresponding to the GID can be reversed depending upon the value of the STO in the CBID detection.