## 2 Non-Orthogonally Multiplexed Cognitive Radio (NOMCR) System
### Fast Convolution Filtered OFDM Based-NOMCR System
#### 2.2.2 FC-F-OFDM Based NOMCR System Model  
Let \(\mathrm{N}^{\mathrm{pu}}_{\mathrm{data}}\) be the number of subcarriers in the PU-OFDM symbol and \(\mathrm{N}^{\mathrm{pu}}_{\mathrm{cp}}\) be its cyclic prefix. Then, the length of the PU-OFDM symbol in time samples is given by, \(\mathrm{N}^{\mathrm{pu}}_{\mathrm{sym}}=\mathrm{N}^{\mathrm{pu}}_{\mathrm{data} }+\mathrm{N}^{\mathrm{pu}}_{\mathrm{cp}}\). The transmitter side baseband PU-OFDM signal in time domain can be represented by the following mathematical equation  
\[\mathbf{s}^{\mathrm{pu}}(\mathrm{n})=\left(\frac{1}{\sqrt{\mathrm{N}_{\mathrm{ data}}^{\mathrm{pu}}}}\right)\sum_{\mathrm{m}=-\infty}^{\infty}\sum_{\mathrm{k}=0}^{ \mathrm{N}_{\mathrm{data}}^{\mathrm{pu}}-1}\mathbf{b}_{\mathrm{m},\mathrm{k}}^ {\mathrm{pu}}\cdot\mathbf{p}\left(\mathrm{n}-\mathrm{m}\cdot\mathrm{N}_{ \mathrm{sym}}^{\mathrm{pu}}\right)\cdot\mathrm{e}^{\mathrm{j}\cdot 2\pi\mathrm{k}\cdot \frac{\mathrm{n}-\mathrm{m}\cdot\mathrm{N}_{\mathrm{sym}}^{\mathrm{pu}}}{N_{ \mathrm{data}}^{\mathrm{pu}}}},\,\mathrm{n}=0,1,\ldots,\,\mathrm{N}_{\mathrm{ sym}}^{\mathrm{pu}}-1, \tag{5}\]  
where \(\mathbf{p}(\mathrm{n})\) is the transmitter side pulse shaping filter, \(\mathbf{b}^{\mathrm{pu}}_{\mathrm{m},\mathrm{k}}\) is the data symbol at the kth subcarrier of the mth OFDM symbol and \(\mathbf{s}^{\mathrm{pu}}(\mathrm{n})\) is the transmit side PU-OFDM signal. In matrix format, the PU-OFDM signal can be expressed mathematically as  
\[\mathbf{s}^{\mathrm{pu}}(\mathrm{n})=\mathbf{F}_{\mathrm{cp}}^{\mathrm{pu}} \cdot\mathbf{W}^{\mathrm{H}}\cdot\mathbf{P}\cdot\mathbf{b}^{\mathrm{pu}},\, \,\mathrm{where}\,\,\mathbf{F}_{\mathrm{cp}}^{\mathrm{pu}}=\left[ \begin{array}[ ]{cc}\mathbf{0}&\mathbf{I}_{\mathrm{N}_{\mathrm{cp}}^{\mathrm{pu}}}\\ \mathbf{I}_{\mathrm{N}_{\mathrm{data}}^{\mathrm{pu}}-\mathrm{N}_{\mathrm{cp}}^{ \mathrm{pu}}}&\mathbf{0}\\ \mathbf{0}&\mathbf{I}_{\mathrm{N}_{\mathrm{cp}}^{\mathrm{pu}}}\end{array} \right] \tag{6}\]  
where \(\mathbf{F}^{\mathrm{pu}}_{\mathrm{cp}}\) is the cyclic prefix (CP) matrix, \(\mathbf{W}\) is the DFT matrix, \(\mathbf{P}\) is the diagonal matrix of frequency domain pulse shaping filter coefficients and \(\mathbf{b}^{\mathrm{pu}}\) is the column vector of \(\mathrm{N}^{\mathrm{pu}}_{\mathrm{data}}\)-by-1 PU data samples. As evident from the formulation of the PU-OFDM signal in Equations (5) and (6), the total number of SU data samples, \(\mathrm{N}^{\mathrm{su}}_{\mathrm{data}}\), within each SU sub-band is a subset of the total number of PU-OFDM data samples within that band. If there are M total number of sub-bands belonging to the SU, then the FC-F-OFDM based SU signal component for the entire band can be mathematically represented by  
\[\mathbf{w}_{\mathrm{m},\mathrm{LL}}^{\mathrm{su}}(\mathrm{n})=\mathbf{F}_{ \mathrm{m}}^{\mathrm{su}}\cdot(\mathrm{g}\cdot\mathbf{x}_{\mathrm{m}}^{ \mathrm{su}}(\mathrm{n})) \tag{7}\]  
In Equation (7), the term 'g' represents the power level of the SU signal relative to the PU-OFDM signal. Let the total transmission power of the NOMCR signal be \(\mathrm{P}_{\mathrm{s}}\). The injection level for the FC-F-OFDM based SU signal is denoted by \(\rho\) dB, where \(\rho=-20\times\log_{10}\mathrm{g}\). As evident from Equation (7), to decode the SU signal, the receiver is required to first cancel the estimate of the PU-OFDM signal from the received NOMCR signal. Then the residual SU signal undergoes matched filtering per sub-band (i.e., inverse of the transmit side overlap-and-save fast convolution based digital filtering) to recover the sub-band CP-OFDM signal. **Received NOMCR Signal Model:**  
Let \(\mathbf{r}_{\mathrm{NOMCR}}(\mathrm{n})\) be the received NOMCR signal, corrupted by AWGN \(\mathbf{w}\) (\(\mathrm{n}\)).