# Positioning-Based Uplink Synchronization Method for NB-IoT in LEO Satellite Networks
## 3 Position-Aided Uplink Synchronization Method
### NPSS Detection
#### 3.1.1 Coarse Estimation of Time Delay Observations  
Due to the time-domain repetition characteristics of the NPSS, the traditional autocorrelation-based Doppler cancellation algorithm produces significant side peaks at positions corresponding to one OFDM symbol interval, which adversely affect detection performance. Therefore, when using conjugate multiplication of the signal to mitigate Doppler frequency offset, it is necessary to fully exploit the time-domain autocorrelation properties of the ZC sequence to design an appropriate metric function. We adopt the commonly used 1.92 MHz sampling rate for NB-IoT, which, combined with the 15 kHz subcarrier spacing, results in an FFT size of 128. The specific steps of the algorithm are described below \(y(n)\) is the received signal defined in Equation (2). For each sampling point \(n\), the subsequent signal segment of one NPSS duration is divided equally into 11 parts, yielding \(\mathbf{z_{1}},\mathbf{z_{2}},\mathbf{z_{3}}\cdot\ldots\ldots\mathbf{z_{11}}\):  
\[\mathbf{z_{m}}=y\{[n+N\cdot(m-1)]:[n+N\cdot m]\},m=1,2,\ldots\ldots,11; \tag{3}\]  
Figure 3: Flowchart of the MSA algorithm. As shown in Figure 4, each of the 11 signal segments is cross-correlated with the complex conjugates of the other 10 segments, resulting in 11 correlation values \(v_{m}\):  
\[ \begin{array}{ll}v_{m}&=\sum\limits_{i=1}^{11}\mathbf{z_{m}\cdot z^{*}}_{i}\\ &=\sum\limits_{i=1}^{11}\sum\limits_{d=0}^{N-1}y(n+d+N\cdot m-N)\cdot y(n+d+N \cdot i)^{*},\ i\
eq m\end{array}  \tag{4}\]  
Here, \(d\) is a counting variable introduced by the correlation operation. \(i\
eq m\) holds because the autocorrelation values of the individual segments are large. To prevent these from contributing to the final result and causing false peaks, they are excluded from the calculation. All frames are independent of each other and free from corruption. We ignore multipath and other interference. Substituting Equation (2) into the computation yields:  
\[v_{m}=\sum\limits_{i=1}^{11}\sum\limits_{d=0}^{N-1}x(n+d+N*m-N-\delta)\cdot x( n+d+N*i-\delta)\cdot e^{j2\pi(n+d+N*m-N)\epsilon/N}\cdot e^{-j2\pi(n+d+N*t) \epsilon/N} \tag{5}\]  
For each sampling point \(n\), 11 different correlation values \(v_{m}\) are obtained corresponding to different values of \(m\). The metric value \(p(n)\) at sampling point \(n\) is defined as the minimum among these 11 correlation values:  
\[p(n)=[\min(v_{1},v_{2},\ldots,v_{11})]^{2} \tag{6}\]  
According to the position of the sampling point \(n\), the sampling point distribution is classified into three categories in Figure 5: partially overlapping, fully aligned, and the start positions of adjacent symbols. Correlation between two identical ZC sequences produces a distinct peak, but in the presence of Doppler frequency offset, the peak may be reduced or shifted. 1. \(n\
eq\delta+a\cdot N,a=0,\pm 1,\pm 2\ldots\ldots\); When the current sampling point is located in the middle of an OFDM symbol or outside the NPSS, the result is small due to the correlation properties of the ZC sequence. 2. \(n=\delta\); When the current sampling point corresponds to the start position of the NPSS, a peak appears. Due to the time-domain repetition property of NPSS, the 11 OFDM symbols starting from the beginning are identical. Our algorithm fully exploits this  
Figure 4: Computation diagram for different m values. Figure 5: Illustrations of three sampling point distributions. characteristic, so we can get \(x(d+N*m-N)=x(d+N\cdot i)=x(d)\). The resulting metric \(v(m)\) is calculated as follows:  
\[ \begin{array}{ll}v(m)&=\sum\limits_{i=1}^{11}e^{j2\pi\epsilon(m-i-1)}\sum \limits_{d=0}^{N-1}x(d+N*m-N)\cdot x(d+N\cdot i)\\ &=\sum\limits_{i=1}^{11}e^{j2\pi\epsilon(m-i-1)}\sum\limits_{d=0}^{N-1}|x(d) |^{2}\\ &=A\sum\limits_{i=1}^{11}e^{j2\pi\epsilon(m-i-1)}\end{array}  \tag{7}\]  
In the final result, \(A=\sum\limits_{d=0}^{N-1}|x(d)|^{2}\) is a fixed constant, and \(\sum\limits_{i=1}^{11}e^{j2\pi\epsilon(m-i-1)}\) is unaffected by carrier frequency offset \(\epsilon\).