# Efficient Preamble Design Technique for Millimeter-Wave Cellular Systems with Beamforming
## 3 Efficient CBID Detection Technique for mmWave Cellular Systems
For example, \(N_{0}\)-point FFTs of the even and odd samples of \(x^{c,b}\left(\left(n-\chi\right)_{\infty_{\mathrm{N}}}\right)\) are expressed as \(\widehat{X}_{0}^{c,b}\left(\left(k\right)_{\infty_{\mathrm{N}}}\right)e^{-j2 \pi n_{0}k/N_{0}}\) and \(\widehat{X}_{1}^{c,b}\left(\left(k\right)_{\infty_{\mathrm{N}}}\right)e^{-j2 \pi n_{0}k/N_{0}}\), when \(\chi\) is an even number, but as \(\widehat{X}_{1}^{c,b}\left(\left(k\right)_{\infty_{\mathrm{N}}}\right)e^{-j2 \pi\left(u_{1}+1\right)k/N_{0}}\) and \(\widehat{X}_{0}^{c,b}\left(\left(k\right)_{\infty_{\mathrm{N}}}\right)e^{-j2 \pi u_{1}k/N_{0}}\), when \(\chi\) is an odd number. In order to avoid ambiguity, the value of \(\widehat{d}_{0}^{v,b}\) needs to be selected such that it is not equal to \(\widehat{d}_{1}^{v,b}\) in the preamble design. For example, if the value of \(\widehat{d}_{0}^{v,b}\) is an odd number, then the value of \(\widehat{d}_{1}^{v,b}\) needs to be an even number. Besides the ambiguity in the GID detection, ambiguities in detecting the SID and BID may occur because the effect of the STO is exhibited by a phase rotation in the frequency domain. For example, an ambiguity in the BID detection occurs when \((\widehat{d}_{s}^{v^{\prime},b^{\prime}}+u_{0})_{\infty_{\mathrm{N}}}\) or \(\left(\widehat{d}_{s}^{v^{\prime},b^{\prime}}+u_{1}\right)_{\infty_{\mathrm{N}}}\) is equal to \(\left(\widehat{d}_{s}^{v^{\prime},b^{\prime\prime}}+u^{b}\right)_{\infty_{ \mathrm{N}}}\); an ambiguity in the SID detection occurs when \(\left(\widehat{d}_{s}^{v^{\prime},b^{\prime}}+u_{0}\right)_{\infty_{\mathrm{N}}}\) or \(\left(\widehat{d}_{s}^{v^{\prime},b^{\prime\prime}}+u^{b}\right)_{\infty_{ \mathrm{N}}}\) is equal to \(\left(\widehat{d}_{s}^{v^{\prime\prime},b^{\prime\prime}}+u^{b}\right)_{\infty_ {\mathrm{N}}}\). Therefore, the value of \(\widehat{d}_{s}^{v,b}\)should be selected such that \(\left(\widehat{d}_{s}^{y^{\prime},b^{\prime}}+u_{0}\right)_{\forall\text{N}_{0}} \
eq\left(\widehat{d}_{s}^{y^{\prime},b^{\prime\prime}}\
eq b^{\prime}\right)_{ \forall\text{N}_{0}}\) and \(\left(\widehat{d}_{s}^{y^{\prime},b^{\prime}}+u_{0}\right)_{\forall\text{N}_{ 0}}\
eq\left(\widehat{d}_{s}^{y^{\prime\prime},b^{\prime\prime}}\
eq\omega^{ \prime},b^{\prime\prime}\right)_{\forall\text{N}_{0}}\) because \(U_{1}\subseteq U_{0}\). Next, we consider an OFDM-based mmWave cellular system with a Tx-Rx beamforming. When the proposed preamble with the CID, \(c\), and the BID, \(b\), is transmitted from the \(b\)-th Tx beam of the BS, the received signal at the \(i\)-th Rx beam in the MS is given in the frequency domain as follows:  
\[Y_{i}(k)=\sum_{c=0}^{N_{\text{C}}-1}\sum_{b=0}^{N_{\text{B}}-1}G_{i}^{c,b}\left( k\right)\text{ }X^{c,b}\left(k\right)e^{-j2\pi\chi^{c,b}k/N}+W_{i}\left(k\right) \tag{11}\]  
where \(G_{i}^{c,b}(k)=\eta^{c,b}\mu^{c,b}H_{i}^{c,b}(k)\). Here, it is assumed that the initial STO and CFO synchronizations have been completed and only the residual STO exists. The parameters \(\eta\), \(\mu\), \(H\), \(G\), and \(W\) denote the Tx array gain, Rx array gain, channel frequency response, total gain including the Tx and Rx array gains, and the additive white Gaussian noise (AWGN), respectively.