# Interspectral Error Tracking and Compensation of DSDT in Satellite Internet of Things
## 3 Adaptive Error Tracking and Equalization
### Sub-Spectrum Error Synchronization Tracking and Adaptive Compensation  
The second scheme involves adjusting the tap weight coefficients of each sub-spectrum separately using decision feedback to achieve adaptive compensation for the combined recovered signal. Although this processing is done in the time domain, it effectively modifies the frequency components of the combined recovered signal to adaptively compensate for errors. Therefore, it can also be considered as frequency domain equalization. The objective function is as follows:  
\[\min_{g\in S}E\left[\sum_{l=0}^{n}\lambda^{n-1}\left|\hat{s}_{l}-\sum_{i=1}^{N} \hat{s}_{i}y_{l,i}\right|^{2}\right], \tag{6}\]  
where \(y_{l,i}\) is \(l\)th symbol of \(i\)th sub-spectrum after the downsampling operation  
\[y_{l,i}=s^{\prime}_{ri}(lT_{s}).\]  
In Equation (6), \(\lambda\) represents the forgetting factor, \(\widehat{s}_{l}\) is the normalized ideal constellation value of the lth symbol after combination,\(g_{i}\) represents the equalizer weights of the \((i+1)\)th sub-spectrum, and \(y_{l,i}\) represents the sampling point of the lth symbol in the \((i+1)\)th sub-spectrum under symbol synchronization. When compensating for sub-spectrum errors, \(\sum_{i=1}^{N}g_{i}y_{l,i}\) approximates the estimated ideal constellation value under symbol estimation. Therefore, the mean square error can be used as the objective function to modify the equalizer coefficients. Compared to the recovered signal after combination, the pre-combination sub-spectra retain more synchronization error information. Sub-spectrum level attenuation compensation can theoretically achieve better combination performance. To address the drawback of the decision feedback algorithms relying on the previous symbol's hard decision result, a forgetting factor is introduced to enable rapid convergence of the error function and to improve the phase tracking capability, thereby reducing the risk of symbol misjudgment. Figure 4 illustrates the iterative process of the tap coefficient \(\mathbf{g}(l)\) of the algorithm. \(\mathbf{g}(l)\) represents the tap coefficient value corresponding to each sub-spectrum under the \((l+1)\)th symbol, and \(\overline{e(I)}\) is the difference between the conjugate operation and the ideal normalized constellation. In the Recursive Least Squares (RLS) algorithm, the update step for the equalizer tap coefficients is determined by calculating a weighted sum of the mean square error indices in the least squares sense. \(\mathbf{k}(l)\) is the step size for updating the tap coefficients of the equalizer. The updating process is as follows:  
1. Calculate the combined signal and make hard decision \(\hat{s}\) :\[\mathfrak{s}(l)=\mathcal{D}[\sum_{i=1}^{N}y_{i,i}\mathfrak{s}_{l-1,i}] \tag{7}\]
2. Calculate the data autocorrelation term \(\mathbf{R}(l)\) and cross-correlation term \(\mathbf{p}(l)\): \[\mathbf{R}(l)=\sum_{l=0}^{N}\lambda^{n-l}\mathbf{y}(l)\mathbf{y}(l)^{\mathsf{ T}},\] (8) \[\mathbf{p}(l)=\sum_{l=0}^{N}\lambda^{n-l}\mathfrak{s}_{l}^{n} \mathbf{y}(l),\] (9)
3. Calculate the tap coefficient increment \(\mathbf{k}(l)\): \[\mathbf{T}(l)=\mathbf{R}^{-1}(l),\] (10) \[\mathbf{k}(l)=\frac{\mathbf{T}(l-1)\mathbf{y}(l)}{\lambda+\mathbf{y}^{ \mathsf{T}}(l)\mathbf{T}(l-1)\mathbf{y}(l)},\] (11)
4. Obtain the tap coefficient: \[\mathbf{g}(l)=\mathbf{g}(l-1)+\mathbf{k}(l)\tilde{e}(l),\] (12) where \(\tilde{e}(l)=\mathfrak{s}(l)-\sum_{i=1}^{N}y_{i,i}\mathfrak{s}_{l-1,i}\). Assuming the decisions in the iteration are correct, the lower bound of the convergence error for the iteration can be obtained. According to the recursive least squares theory [19], our mean square error is obtained as below  
\[D(l)\approx\frac{1-\lambda}{2}\sigma_{n}^{2}tr[\mathbf{R}_{y}^{-1}]+\frac{1}{ 2(1-\lambda)tr[\mathbf{R}_{e}]},\quad\text{when}\quad l>>1 \tag{13}\]  
where \(\mathbf{R}_{y}\) is the covariance matrix of vector \(\mathbf{y}\), and \(\mathbf{y}=[y_{l,1},y_{l,2},y_{l,3},\ldots,y_{l,N}]\).