Low-Complexity Joint Estimation Algorithm for Carrier Frequency Offset and Sampling Frequency Offset in 5G-NTN Low Earth Orbit Satellite Communications
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摘要: 在5G非地面网络(5G-NTN)低轨卫星通信系统中，多普勒效应会带来载波频偏(CFO)、采样频偏(SFO)以及子载波间频偏(ISFO)。研究发现，当正交频分复用(OFDM)信号子载波数量较大且采用高阶调制时，ISFO会成为制约接收机性能的关键因素。现有算法多针对CFO和SFO的估计及补偿展开研究，极少考虑ISFO的影响。另外，采用传统的最大似然估计算法对CFO和SFO进行联合估计时，需要进行一维或二维网格搜索，计算复杂度非常高。针对上述问题，该文利用5G-NTN中导频信号的分布特点，提出两种低复杂度的CFO和SFO联合估计算法。首先，利用5G-NTN中解调参考信号的互相关向量在主瓣内的单峰特性，设计一种基于二分搜索的联合估计算法，可以实现快速收敛。然后，设计基于观测量自相关的L&R估计算法，推导了待估参数的近似闭式解。典型实例分析和仿真表明，两种算法性能接近采用一维或二维搜索的最大似然估计算法，且所提二分搜索估计算法运算量仅为二维搜索最大似然估计算法的4%、一维搜索最大似然估计算法的44%。Abstract:
Objective The Doppler effect is a major impairment in Low Earth Orbit (LEO) satellite communications within 5G Non-Terrestrial Networks (5G-NTN). It introduces Carrier Frequency Offset (CFO), Sampling Frequency Offset (SFO), and Inter-Subcarrier Frequency Offset (ISFO) across subcarriers. Although existing estimation algorithms focus mainly on CFO and SFO, the effect of ISFO is insufficiently addressed. ISFO becomes highly detrimental to receiver performance when Orthogonal Frequency-Division Multiplexing (OFDM) systems use a large number of subcarriers and high-order modulation. Moreover, under joint CFO and SFO conditions, conventional Maximum Likelihood Estimation (MLE) methods often require one- or two-dimensional grid searches. This results in high computational cost. To reduce this cost, two joint estimation algorithms for CFO and SFO are proposed. Methods The influence of non-ideal factors at the transmitter, receiver, and channel, such as local oscillator offset, SFO in Digital-to-Analog Converters (DACs) and Analog-to-Digital Converters (ADCs), and the Doppler effect, is analyzed. A mathematical model for the received OFDM signal is developed, and the mechanism through which SFO and ISFO distort the phase of frequency-domain subcarriers is derived. Leveraging the pilot structure of 5G-NTN, two joint CFO and SFO estimation algorithms are proposed. (1) Algorithm 1 uses the sequence correlation between the received frequency-domain Demodulation Reference Signal (DMRS) vectors. After phase pre-compensation is applied, the normalized cross-correlation vector is computed. An objective function is constructed from this vector, and its unimodal behavior in the main lobe is used to estimate the parameters through a bisection search. (2) Algorithm 2 treats the estimation parameter as analogous to a CFO in single-carrier systems and adopts an L&R-based autocorrelation method to derive approximate closed-form expressions. Results and Discussions A computational complexity analysis compares the proposed algorithms with one-dimensional (1D-ML) and two-dimensional (2D-ML) grid-search MLE methods. Numerical results show that Algorithm 1 reduces complexity substantially. The number of complex multiplications, which represent the main computational cost, is 4% of that of the 2D-ML method, 8% of that of Algorithm 2, and 44% of that of the 1D-ML method. Although Algorithm 2 is more computationally demanding, it yields a closed-form estimation expression. The performance of each algorithm is evaluated through the Mean Square Error (MSE) of the estimated parameters. Simulations show that for a subcarrier number of 3072, the 1D-ML algorithm performs slightly better than the others at Signal-to-Noise Ratios (SNRs) below 5 dB. However, because robust modulation schemes such as BPSK and QPSK typically used at low SNRs tolerate larger offsets, the medium-to-high SNR range is of greater practical relevance. In this range, all four algorithms demonstrate comparable estimation performance. Conclusions This study addresses the effect of Doppler in 5G-NTN LEO satellite communications by analyzing the mechanism and influence of ISFO and by proposing two joint estimation algorithms for CFO and SFO. First, a mathematical model of the received signal is established considering non-ideal factors such as CFO, SFO, and ISFO. The combined effect of SFO and ISFO on OFDM signals is derived to be equivalent to their linear superposition, which expands the range of the equivalent SFO. Second, the objective function is defined using the cross-correlation vector of two DMRS sequences. By using its unimodal behavior within the main lobe, a binary search enables fast convergence. Subsequently, the parameter determined by SFO and ISFO is then treated as analogous to the CFO in single-carrier systems, allowing an approximate closed-form estimation solution to be obtained through the L&R method. Finally, complexity analysis and performance simulations show that the proposed algorithms provide significant computational savings and strong estimation performance. These results can support the development of 5G-NTN LEO satellite payloads and terminal products. -
1 二分搜索估计算法
输入：$ \boldsymbol{Y}_{\mathrm{M1}} $，$ \boldsymbol{Y}_{\mathrm{M2}} $，$ \rho $，$ {k}_{0} $，$ d $，$ I $，$ {\alpha }_{\max } $，$ N{}_{\text{fi}} $ 1：if $ \rho dI{\alpha }_{\max } \gt 2\text{π} $ then 2： 确定粗搜索次数：$ {N}_{\text{co}}=2\times \left\lceil \rho dI{\alpha }_{\max }/(2\text{π} )\right\rceil $，并设置
$ \boldsymbol{Z}={[Z(1),Z(2),\cdots ,Z({{N}_{\text{co}}})]}^{\mathrm{T}} $；3： for $ l=1,2,\cdots,N_{\text{co}} $ do 4： 获取粗搜索点：$ \beta =2\text{π} (l-({N}_{\text{co}}+1)/2)/(\rho dI) $； 5： 根据式(10)—式(12)依次计算$ {\mathbf{\tilde{\boldsymbol{Y}}}}_{\text{M2}} $、$ \boldsymbol{R}(\beta ) $和$ f(\beta ) $，
并将$ f(\beta ) $存入$ Z(l) $中；6： end 7： 搜索得到$ \boldsymbol{Z} $中最大值的序号$ {l}_{\max } $，计算粗估计值
$ {\overline{\alpha }}_{{\mathrm{mid}}}=2\text{π} ({l}_{\max }-({N}_{\text{co}}+1)/2)/(\rho dI) $；8：else 9： 粗估计值$ \overline{\alpha}_{\mathrm{mid}}=0 $； 10：end 11：确定精搜索的初始基准点$ \gamma=\overline{\alpha}_{\mathrm{mid}} $及步进$ \Delta =\text{π} /(\rho dI) $； 12：for $ n=1,2,\cdots,N_{\text{fi}} $ do 13： if $ f(\gamma +\Delta ) \gt f(\gamma -\Delta ) $ then 14： 调整基准点和步进：$ \gamma =\gamma +\Delta $，$ \Delta =\Delta /2 $； 15： else 16： 调整基准点和步进：$ \gamma =\gamma -\Delta $，$ \Delta =\Delta /2 $； 17： end 18：end 19：计算多普勒参数估计值：$ \overline{\alpha }=\gamma $，
$ {\overline{\varepsilon }}_{0}=\arctan (\displaystyle\sum\nolimits_{i=0}^{I-1}{R}_{i}(\overline{\alpha }))/\rho $；输出：$ \overline{\alpha } $，$ {\overline{\varepsilon }}_{0} $ 
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表 1 算法复杂度比较
算法名称 复数乘 实数乘 复数加 本文算法1 73728 36864 36864 本文算法2 893568 4608 768 2D-ML算法 1658880 0 829440 1D-ML算法 165888 0 82944
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表 2 仿真系统参数
参数名称 参数值 子载波数量($ K $) 768(64RB), 3072 (256RB)导频子载波间隔($ d $) 2 DMRS信号序号 M1=2, M2=6 SFO($ \eta $) 30 ppm CFO($ {\varepsilon }_{0} $) 0.02 ISFO($ \Delta \varepsilon $) 24 ppm 信道类型 AWGN 编码调制方式 MCS10(BPSK, 码率308/ 1024 ),
MCS25(16QSK, 码率873/1024 )天线配置 SISO
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