# Interspectral Error Tracking and Compensation of DSDT in Satellite Internet of Things
## 2 System Model  
DSDT utilizes filter bank technology to divide the signal into multiple sub-spectra, inserts them into frequency-offset holes for transmission, and recovers the original signal through receiver-side combination filters. The schematic diagram of this transmission system is shown in Figure 1. A, B and C are the sub-spectra divided by the filter.  
In practical satellite communication, not all frequency holes appear densely, and some frequency gaps have large spacing, thus requiring the use of different transceivers for transmission. The larger the spacing between channel frequency bands, the greater the  
Figure 1: Spectrum division transmission system.  
difference in channel characteristics. This means that using spectrum-based transmission technology across large frequency bands or even on multiple frequency transceivers will inevitably face the problem of inconsistent amplitude, frequency, and phase between sub-spectra, as shown in Figure 2.  
Since the actual frequency hole occurs in multiple frequency bands, these sub-spectrums will experience independent fading channels, thus denoting the fading coefficient of the \(i\)th sub-spectrum channel as \(A_{i}e^{i\phi_{i}}\). Meanwhile, considering a relative movement between satellite and ground station, received sub-spectrum signals in different frequency bands face different Doppler shifts, where the Doppler offset in the \(i\)th channel is denoted as \(e^{i2\pi f_{di}}\). Therefore, the expression of the received signal across the \(i\)th combination filter can be written as follows:  
\[s_{ri}(t)=s_{i}(t)\times A_{i}e^{i(2\pi f_{di}t+\phi_{i})}+n_{i}(t), \tag{1}\]  
where \(n_{i}(t)\) is noise in the \(i\)th channel, and \(S_{i}(t)\) represents the \(i\)th sub-spectrum of the original signal \(s_{o}(t)\) across division filter \(hd_{i}\) and combination filter \(hc_{i}\)[8] (\"\" is the convolution operation).  
\[s_{i}(t)=hc_{i}*hd_{i}*s_{o}(t)\]  
As mentioned in [9], the original signal \(s_{o}(t)\) could be MASK, MPSK, or MQAM and so on. Then, pilot synchronization is operated for estimations of the amplitude \(\hat{A}_{i}\), frequency\(\hat{f}_{i}\), and phase \(\hat{\phi_{i}}\). After compensation, \(s_{ri}(t)\) turns into \(s^{\prime}_{ri}(t)\):  
\[s^{\prime}_{ri}(t)=s_{i}(t)\times A^{\prime}_{i}e^{i(2\pi\Delta f_{i}t+\Delta \phi_{i})}+n_{i}(t), \tag{2}\]  
where residual synchronization errors are  
\[A^{\prime}_{i}=A_{i}/\hat{A}_{i}\]  
\[\Delta f_{i}=f_{di}-\hat{f}_{i}\]  
\[\Delta\phi_{i}=\phi_{i}-\hat{\phi}_{i}\]  
Figure 2: Transmission across large frequency domain sub–spectra under multiple transponders.  
Finally, the combined signal with \(N\) sub-spectrums can be expressed as  
\[s_{r}(t)=\sum_{i=1}^{N}s^{\prime}_{ri}(t), \tag{3}\]  
When a residual synchronization error exists, the spectrum structure of Equation (3) is shown as the combined signal in Figure 2. It can be seen that the combined signal appears distorted, thus resulting in a loss of bit error rate (BER) performance. In this case, a free-pilot synchronization scheme become a good choose for reducing the synchronization error. In addition, a schematic diagram of the signal processing flow for DSDT with a free-pilot module is shown in Figure 3.