## 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
Then the receiver side signal model can be expressed by,  
\[\mathbf{r}_{\mathrm{NOMCR}}(\mathrm{n})=\mathbf{h}_{\mathrm{pu}}\otimes\mathbf{ s}^{\mathrm{pu}}(\mathrm{n})+\mathbf{h}_{\mathrm{su}}\mathbf{w}_{\mathrm{m}, \mathrm{LL}}^{\mathrm{su}}(\mathrm{n})+\mathbf{w}(\mathrm{n}), \tag{8}\]  
In (8), \(\mathbf{h}_{\mathrm{pu}}\) and \(\mathbf{h}_{\mathrm{su}}\) represent frequency selective channels corresponding to PU and SU transmissions respectively. Let the estimated PU-OFDM signal and its estimated channel be denotedby \(\hat{\mathbf{g}}^{\text{pu}}(\text{n})\) and \(\hat{\mathbf{h}}_{\text{pu}}\) respectively. Then the estimated high-power PU-OFDM signal can be treated as interference and cancelled from the received LDM signal. To obtain a reliable estimate of the higher power PU-OFDM signal, the received signal needs to undergo OFDM demodulation, channel estimation and equalization, de-interleaving, and channel decoding to obtain a reliable bit decision estimate for the PU-OFDM. The estimated bit sequence is re-encoded, re-interleaved and re-modulated to generate the reconstructed transmitted PU-OFDM signal. Although the reconstruction process incurs higher complexity, it provides a reliable estimate. **SU Signal Model with PU interference estimation and cancellation:**  
The estimate of the SU signal, corrupted by the wireless fading channel, can be obtained as  
\[ \begin{split}\mathbf{r}_{\text{su}}(\text{n})=\mathbf{h}_{\text{su }}\otimes\mathbf{w}_{\text{m,LL}}^{\text{su}}(\text{n})+\left(\mathbf{h}_{\text {pu}}\otimes\mathbf{s}^{\text{pu}}(\text{n})-\hat{\mathbf{h}}_{\text{pu}} \otimes\hat{\mathbf{g}}^{\text{pu}}(\text{n})\right)+\mathbf{w}(\text{n})\\ =\mathbf{h}_{\text{su}}\otimes\mathbf{w}_{\text{m,LL}}^{\text{su}} (\text{n})+\mathbf{i}^{\text{pu}}(\text{n})+\mathbf{w}(\text{n})\end{split}  \tag{9}\]  
In (9), \(\mathbf{i}^{\text{pu}}(\text{n})\) denotes the residual interference after estimated PU-OFDM signal cancellation. The channel equalization of \(\mathbf{r}_{\text{su}}(\text{n})\) can be performed with the help of the channelization weights of the fast convolution filter-bank structure. In other words, the sub-block based equalization is embedded within the matched filter. The frequency domain weights of the embedded \(\mathbf{R}_{\text{m}}\)th sub-block equalizer of the mth sub-band can be calculated by  
\[\hat{\mathbf{c}}_{\mathbf{R}_{\text{m},\text{n}}}=\frac{\left(\frac{\hat{ \mathbf{H}}_{\mathbf{R}_{\text{m},\text{n}}}^{\text{s}}\cdot\hat{\mathbf{c}}_{ \text{m},\text{n}}}{\left|\hat{\mathbf{H}}_{\text{R}_{\text{m},\text{n}}} \cdot\mathbf{c}_{\mathbf{R}_{\text{m},\text{n}}}\right|^{2}+\left|\hat{ \mathbf{H}}_{\text{R}_{\text{m},\text{n}}}\cdot\mathbf{c}_{\mathbf{R}_{\text{m },\text{n}}}\right|^{2}+\left(\frac{1}{8}\right)} \tag{10}\]  
where \(\hat{\mathbf{H}}_{\mathbf{R}_{\text{m},\text{n}}}\) denotes the estimated \(\mathbf{R}_{\text{m}}\)th sub-block frequency response at the nth FFT bin and \(\mathbf{c}_{\mathbf{R}_{\text{m},\text{n}}}\) denotes the RRC pulse shaping filter coefficients of the \(\mathbf{R}_{\text{m}}\)th sub-block, nth FFT bin. The channel estimation can be performed by any standard channel estimation method like the scattered pilot-based channel estimation [23; 24]. The demodulation of each sub-block is carried out by the matched filter  
\[\hat{\mathbf{c}}_{\mathbf{m},\mathbf{r}}^{\text{su}}=\mathbf{S}_{\text{L}_{ \text{m}}}^{\text{su}}\cdot(\mathbf{W}_{\text{L}_{\text{m}}}^{\text{su}})^{-1 }\cdot(\mathbf{P}_{\text{N}_{\text{sym}}}^{\text{su}})^{\frac{\text{N}_{ \text{sym}}^{\text{su}}}{2}}\cdot\mathbf{C}_{\text{m}}^{\text{su}}\cdot( \mathbf{M}_{\text{m},\text{r}}^{\text{su}})^{\text{T}}\cdot\mathbf{W}_{\text{ N}_{\text{sym}}^{\text{su}}}^{\text{su}} \tag{11}\]  
The RHS of (11) is almost similar to (4) except for the embedded sub-block equalizer matrix, \(\mathbf{C}_{\text{m}}^{\text{su}}\), containing the terms \(\hat{\mathbf{c}}_{\mathbf{R}_{\text{m},\text{n}}}\) along its main diagonal and zeros elsewhere.