# NC-OFDM Satellite Communication Based on Compressed Spectrum Sensing
## 3 Compressed Spectrum Sensing  
Currently, the frequency range used by one Ku-band communication satellite is about 500 MHz. According to the monitoring results of actual satellite communication frequency occupation, frequency bandwidths of spectrum holes are in the range of about 10 MHz to 100 MHz. Considering the system control complexity brought by the increasing number of subcarriers and the full utilization of spectrum holes, we set the bandwidth of each NC-OFDM subcarrier to 4.125 MHz. So, 500 MHz of bandwidth can be allocated to 122 subcarriers, which include 100 data subcarriers, 12 pilot subcarriers, and 10 protection subcarriers, as shown in Table 1.  
Traditional spectrum sensing methods used for cognitive radio include classic energy detection methods [16], cyclic spectrum estimation [23], etc., but these methods are many used for narrowband communication. On the task of simultaneously monitoring all the subchannels among the 500 MHz bandwidth, several challenges arise such as power consumption and cost of high-speed analog-to-digital converters (ADCs). In recent years, compressed sensing has been proposed for broadband spectrum signal monitoring and multiband signal acquisition [24; 25; 26]. With a periodic non-uniform sampling framework working far below the Nyquist rate, the signal processing platform based on a software-defined radio can realize real-time, full-band spectrum monitoring within 2 GHz bandwidth, and the spectrum monitoring results can be directly used for spectrum holes extraction and subcarriers allocation. We hereby propose to integrate the sub-Nyquist sampling and spectrum sensing scheme into the NC-OFDM satellite communication. We adopt a period non-uniform sampling (PNS, also known as Multicoset sampling) architecture. In the PNS, the baseband signal \(x(t)\) is sampled by an interleaved ADC array composed of \(p\) low-speed ADCs working under the sampling rate of \(f_{Nyq/L}\), where \(f_{Nyq}\) is the Nyquist frequency of the signal and \(p<L\). By setting the analog delay in the \(i\)-th way as \(c_{i}/f_{Nyq}\), where \(c_{i}\) is a non-negative integer smaller than \(L\), the sampling sequence in the \(i\)-th way can be expressed by Equation:  
\[x_{i}[n]=x\biggl{(}\frac{nL}{f_{Nyq}}+\frac{c_{i}}{f_{Nyq}}\biggr{)},\ n\ =\ 1,2,\ldots. \tag{3}\]  
The \(i\)-th way of PNS gives the following measurement:  
\[Y_{i}(f)=\frac{L}{f_{Nyq}}\ exp\Bigl{(}-j2\pi\frac{c_{i}}{L}f\Bigr{)}X_{i} \biggl{(}\ exp\biggl{(}j2\pi\ f\frac{L}{f_{Nyq}}\biggr{)}\biggr{)}, \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Subcarrier Division Parameters** & **Definition** \\ \hline number of data subcarriers & 100 \\ \hline number of pilot subcarriers & 12 \\ \hline number of guard subcarriers & 10 \\ \hline total number of subcarriers & 122 \\ \hline subcarrier frequency spacing & 4.125 MHz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Subcarriers of Ku satellite with 500 MHz frequency band. where \(X_{i}\Big{(}\exp\Big{(}j2\pi\,f\frac{L}{f_{Nqq}}\Big{)}\Big{)}\) is the discrete time Fourier transform (DTFT) of \(x_{i}[n]\). A linear relationship between the measurement and the original spectrum \(X(f)\) is given by Equation (5):  
\[y(f)=\big{[}Y_{1}(f)Y_{2}(f)\dots Y_{p}(f)\big{]}^{T}, \tag{5}\]  
where:  
\[x(f)=\bigg{[}X\bigg{(}f+\frac{(-L/2)f_{Nqq}}{L}\bigg{)}X\bigg{(}f+\frac{(-L/2+ 1)f_{Nqq}}{L}\bigg{)}\dots\bigg{(}f+\frac{(L/2-1)f_{Nqq}}{L}\bigg{)}\bigg{]}^{T} \tag{6}\]  
and **A** is a \(p\times L\) matrix with its \((i,j)\)-th element being \(a_{i,l}=e^{j2\pi\frac{c_{i}}{L}(-2/L+l-1)}\). The above equation is underdetermined and has infinite solutions.