# Low-Complexity Synchronization Scheme with Low-Resolution ADCs
## 3 Analysis of Synchronization Algorithm
### Feedback-Based Synchronization Scheme with Low-Resolution ADCs  
Assuming the received sequence is \(R[n]\), then \(R[n]\) can be represented as  
\[R[n]=R_{B}[n]+N_{0}, \tag{13}\]  
where \(R_{B}[n]\) is a transmitted sequence, and \(N_{0}\) donates a white complex Gaussian noise signal with zero mean and variance \(\sigma^{2}\). In general, we make the sampling clock error \(\Delta t_{s}=0\), so we can respectively express the two training sequences (beacon) contained in the \(R[n]\) as  
\[r_{B1}[n]=x_{B1}[n]\exp[-j(2\pi\Delta f_{c}nT_{s}+\Delta\varphi)]+\eta_{1}[nT _{s}], \tag{14}\]  
\[r_{B2}[n]=x_{B2}[n]\exp\{-j[2\pi\Delta f_{c}(n+N_{B})T_{s}+\Delta\varphi]\}+ \eta_{2}[nT_{s}], \tag{15}\]  
where \(\Delta f_{c}\) is the carrier frequency offset, \(T_{s}\) is the sampling period, \(\Delta\varphi\) denotes the phase delay, \(N_{B}\) is the length of the training sequence, \(n=0,1,2\cdots N_{B}-1\). \(\eta_{1}\) and \(\eta_{2}\) are Gaussian noise, and \(x_{B1}[n]\) and \(x_{B2}[n]\) represent the first half and the second half of the sequence (preamble), respectively. Therefore, we can use a \(2N_{B}\)-point sliding window to sample the received signal, and then the similarity between the first \(N_{B}\)-point samples and the last \(N_{B}\)-point samples can be computed by an autocorrelation property. The similarity is maximized when the sliding window moves to the starting point of the OFDM symbol. The experimental results in Figure 5 show that the autocorrelation of the PN sequence is not changed by low-resolution quantization. Thus, we first consider the symbol-timing synchronization with one-bit ADCs. In our system, we can define a function of the one-bit ADCs operator as  
\[\mathbf{r}=\mathrm{sgn}(R[n])=\mathrm{sgn}(R_{B}[n]+N_{0}), \tag{16}\]  
where \(\mathrm{sgn}(\cdot)\) is the signum function applied separately to the real and imaginary parts, \(N_{0}\) is a statistically independent Gaussian noise, and \(\mathbf{r}_{i}\in\{1+j,1-j,-1+j,-1-j\}\), \(0\leq i\leq N_{B}-1\). Since \(x_{B1}\) is identical to \(x_{B2}\), the relationship between the autocorrelation and the location of the starting point can be expressed as  
\[ \begin{array}{ll}\hat{d}&=\underset{d}{\operatorname{argmax}}\left\{\sum \limits_{n=0}^{N_{B}-1}\operatorname{sgn}(R[n+d])\operatorname{sgn}(R^{*}[n+d+ N_{B}])\right\}\\ &=\underset{d}{\operatorname{argmax}}\left\{\sum\limits_{n=0}^{N_{B}-1} \left|\operatorname{sgn}(x_{B1}[n])\right|^{2}\right\}\end{array}  \tag{17}\]  
where \(\hat{d}\) is the starting point of the OFDM symbol. Considering the case of multi-bit quantization, we normalize the received sequence \(R[n]\) and pass the normalized sequence through the multi-bit ADCs. In the sliding window, since the phase difference between the first \(N_{B}\)-point samples and the corresponding last \(N_{B}\)-point samples caused by CFO can be computed by an autocorrelation property, we can get the autocorrelation at the starting point \(\hat{d}\) as follows:  
\[B\exp\left(j\phi^{\prime}\right)\big{|}_{\hat{d}=\hat{d}}=\big{|}\hat{\rho}^{ 2}\cdot\sum\limits_{n=0}^{N_{B}-1}y^{\prime}_{B1}[n]y^{\prime}_{B2}*[n+N_{B}] \bigg{|}_{\hat{d}=\hat{d}}, \tag{18}\]  
where \(y^{\prime}_{B1}[n]\) and \(y^{\prime}_{B2}[n]\) are elements of the quantized output in the case of multi-bit quantization, \(\phi^{\prime}\) is the phase difference, and \(\rho\) is a normalization factor. For \(\phi^{\prime}=2\pi\Delta f_{c}{{}^{\prime}}N_{B}T_{s}\), the expression of frequency offset estimation module under low-resolution quantization can be given by  
\[\Delta f_{c}{{}^{\prime}}=\frac{\operatorname{arg}\big{[}\left.B\exp(j\phi^{ \prime})\right|_{\hat{d}=\hat{d}}\big{]}}{2\pi}\cdot\frac{B_{s}}{N_{B}}, \tag{19}\]  
where \(\operatorname{arg}(\cdot)\) represents the angle function, \(B_{s}\) is the bandwidth of OFDM signal, and \(B_{s}=1/T_{s}\).