## 2 Non-Orthogonally Multiplexed Cognitive Radio (NOMCR) System
### Fast Convolution Filtered OFDM Based-NOMCR System
#### 2.2.1 Fast Convolution Filtered OFDM (FC-F-OFDM) Transceiver  
Let \(\mathrm{N}^{\mathrm{su}}\mathrm{{}_{data}}\) be the number of subcarriers in the SU- FC-F-OFDM symbol and let \(\mathrm{N}^{\mathrm{su}}\mathrm{{}_{cp}}\) be its cyclic prefix. Then, the length of the SU- FC-F-OFDM symbol in time samples is given by, \(\mathrm{N}^{\mathrm{su}}\mathrm{{}_{sym}}=\mathrm{N}^{\mathrm{su}}\mathrm{{}_ {data}}+\mathrm{N}^{\mathrm{su}}\mathrm{{}_{cp}}\). At the transmitter side, FC-F-OFDM synthesis transform operates on the CP-OFDM pre-processed SU data on a sub-band basis to generate the final wide-band signal. The high rate \(\mathrm{mth}\) sub-band waveform, \(\mathbf{w}^{\mathrm{su}}\mathrm{{}_{m}}\), of the FC-F-OFDM based SU system can be generated as  
\[\mathbf{w}^{\mathrm{su}}_{\mathrm{m}}=\mathbf{F}^{\mathrm{su}}_{\mathrm{m}} \cdot\mathbf{x}^{\mathrm{su}}_{\mathrm{m}}=\big{[}\mathrm{diag}\big{(}\mathbf{ F}^{\mathrm{su}}_{\mathrm{m},0},\mathbf{F}^{\mathrm{su}}_{\mathrm{m},1},\ldots, \mathbf{F}^{\mathrm{su}}_{\mathrm{m},\mathrm{R}_{\mathrm{m}}-1}\big{)}\big{]} \cdot\mathbf{x}^{\mathrm{su}}_{\mathrm{m}} \tag{1}\]  
where \(\mathbf{F}^{\mathrm{su}}_{\mathrm{m}}\) is the block diagonal SFB transform matrix on the narrow-band, low-rate \(\mathrm{mth}\) sub-band signal \(\mathbf{x}^{\mathrm{su}}\mathrm{{}_{m}}\) and \(\mathrm{R}_{\mathrm{m}}\) is the total number of sub-blocks of \(\mathbf{x}^{\mathrm{su}}\mathrm{{}_{m}}\). The SFB operations of the \(\mathrm{rth}\) sub-block of \(\mathbf{x}^{\mathrm{su}}\mathrm{{}_{m}}\) are expressed mathematically as  
\[\mathbf{F}^{\mathrm{su}}_{\mathrm{m},\mathrm{r}}=\mathbf{S}^{\mathrm{su}}_{ \mathrm{N}^{\mathrm{su}}\mathrm{{}_{sym}}}\cdot(\mathbf{W}^{\mathrm{su}}_{ \mathrm{N}^{\mathrm{su}}\mathrm{{}_{sym}}})^{-1}\cdot\mathbf{M}^{\mathrm{su}} _{\mathrm{m},\mathrm{r}}\cdot\mathbf{D}^{\mathrm{su}}_{\mathrm{m}}\cdot\big{(} \mathbf{P}^{\mathrm{su}}_{\mathrm{L}_{\mathrm{m}}}\big{)}\cdot\mathbf{W}^{ \mathrm{su}}_{\mathrm{L}_{\mathrm{m}}} \tag{2}\]  
where \(\mathbf{W}^{\mathrm{su}}_{\mathrm{L}_{\mathrm{m}}}\) is the \(\mathrm{L}_{\mathrm{m}}\)-by-\(\mathrm{L}_{\mathrm{m}}\) DFT matrix, \((\mathbf{W}^{\mathrm{su}}_{\mathrm{N}^{\mathrm{su}}\mathrm{{}_{sym}}})^{-1}\) is the \(\mathrm{N}^{\mathrm{su}}_{\mathrm{sym}}\)-by-\(\mathrm{N}^{\mathrm{su}}_{\mathrm{sym}}\) IDFT matrix, \(\left(\mathbf{P}^{\mathrm{su}}_{\mathrm{L}_{\mathrm{m}}}\right)^{(\mathrm{L}_{ \mathrm{m}}/2)}\) is the circulant permutation matrix which cyclically left shifts \(\mathrm{L}_{\mathrm{m}}\)-by-\(\mathrm{L}_{\mathrm{m}}\) identity matrix by \(\mathrm{L}_{\mathrm{m}}/2\) positions, \(\mathbf{M}^{\mathrm{su}}_{\mathrm{m},\mathrm{r}}\) is the \(\mathrm{N}^{\mathrm{su}}_{\mathrm{sym}}\)-by-\(\mathrm{L}_{\mathrm{m}}\) frequency domain mapping matrix which maps the \(\mathrm{L}_{\mathrm{m}}\) incoming samples of the sub-block to the respective center frequency in the sub-band and \(\mathbf{D}^{\mathrm{su}}_{\mathrm{m}}\) is the \(\mathrm{L}_{\mathrm{m}}\)-by-\(\mathrm{L}_{\mathrm{m}}\) diagonal matrix with frequency domain window weights of the \(\mathrm{mth}\) sub-band on its main diagonal. The frequency domain diagonal matrix of filter coefficients can be expressed as  
\[\mathbf{D}^{\mathrm{su}}_{\mathrm{m}}=\mathrm{diag}( \begin{bmatrix}\mathbf{ 0}^{\mathrm{su}}_{([\frac{1}{2}\mathrm{m-L}\mathrm{extm}]}-\mathrm{L}_{ \mathrm{TBW},\mathrm{m})-\mathrm{by}-1}\\ \mathbf{d}^{\mathrm{su}}_{\mathrm{0},\mathrm{m}}\\ \cdots\\ \mathbf{d}^{\mathrm{su}}_{\mathrm{L}_{\mathrm{TBW}}-1,\mathrm{m}}\\ \mathbf{1}_{\mathrm{L}_{\mathrm{ACT}},\mathrm{m}-\mathrm{by}-1}\\ \mathbf{d}^{\mathrm{su}}_{\mathrm{TBW}-1,\mathrm{m}}\\ \cdots\\ \mathbf{d}^{\mathrm{su}}_{0,\mathrm{m}}\\ ([\frac{|\mathrm{L}_{\mathrm{m}-\mathrm{L}\mathrm{extm}}|}{2}|-\mathrm{L}_{ \mathrm{TBW},\mathrm{m})-\mathrm{by}-1}\end{bmatrix}