## 2 Data Model  
A general illustration of the non-contiguous orthogonal frequency division multiplexing (NCOFDM) transmitter that utilizes the proposed method is shown in Figure 1.  
The input bit stream \(x_{n}\in\mathbf{x=}[x_{1},x_{2},...,x_{N}]^{T}\) is first modulated into symbols using Mary phase shift keying (PSK) or Mary quadrature amplitude modulation (QAM) \(s_{k}\in\mathbf{s=}[s_{1},s_{2},...,s_{N}]^{T}\). These modulated symbols are then divided into \(N\) parallel streams using serial to parallel (S/P) converter. An NCOFDM system has the ability to activate only those subcarriers that are located in vacant spectral bands detected by spectrum sensing techniques. These active subcarriers go through the inverse Fourier transform block following parallel to serial (P/S) converter. The cyclic prefix of length \(N_{p}\) is added to mitigate the impact of inter-symbol interference (ISI). The discrete time baseband transmitted NCOFDM signal that is to be transmitted in the time domain can be represented as [33, 34, 35]:  
Figure 1: NCOFDM transmitter. subtracted from the signal of the upper branch, resulting in the signal of NCOFDM having suppressed or zero sidelobes. To find the expressions for \(\mathbf{w}_{q}\), \(\mathbf{B}\) and \(\mathbf{w}_{a}\), consider the NCOFDM signal given in Equation (3), represented by \(M\) samples, which are collected in the vector \(\mathbf{u}\!=\! [\mathbf{u}_{1},\mathbf{u}_{2},\!...\mathbf{u}_{M}]^{T}\) with uncorrelated elements. After passing through GSC it is given as  
\[Y=\mathbf{w}^{H}\mathbf{u} \tag{4}\]  
where the superscript \(H\) represents Hermitian and \(\mathbf{w}\!=\! [\mathbf{w}_{0},\mathbf{w}_{1},\!...,\mathbf{w}_{M-1}]^{T}\) is a weight vector with dimension \(M\times I\). LCMV determines the optimal weight vector \(\mathbf{w}^{H}\) that minimizes the output power having multiple linear constraints. The optimization problem for LCMV is given by:  
\[\min_{\mathbf{w}}\mathbf{w}^{H}\mathbf{R}_{\mathbf{a}}\mathbf{w}\ \ s.t\ \mathbf{w}^{H}\mathbf{C}\!=\!\mathbf{g}^{H} \tag{5}\]  
On solving Equation (5), we obtain:  
\[\mathbf{w}_{o}^{H}=\mathbf{g}^{H}\big{(}\mathbf{C}^{H}\mathbf{R}_{\mathbf{a}} ^{-1}\mathbf{C}\big{)}^{-1}\mathbf{C}^{H}\mathbf{R}_{\mathbf{a}}^{-1} \tag{6}\]  
where \(\mathbf{R}_{\mathbf{a}}\!=\!E[\mathbf{u}\mathbf{u}^{H}]\!=\!\sigma^{2}\mathbf{ I}\) is the correlation matrix, with dimension \(M\times M\), \(\mathbf{I}\) is the identity matrix with dimension \(M\times M\), \(\sigma^{2}\) is the variance, while \(\mathbf{C}\) is the constraint matrix with dimension \(M\times N\), having \(N\) steering vectors given by Equation (7):  
\[\mathbf{C}\!=\![\mathbf{s}_{1},\mathbf{s}_{2},\!...,\mathbf{s}_{N}] \tag{7}\]  
where \(N\) represents the total number of frequencies in the desired portion of the signal, as shown in Figure 2; \(\mathbf{s}_{i}\!=\! [s_{1},s_{2},\!...s_{M}]^{T}\) is an \(i^{th}\) steering vector with dimension \(M\times I\) containing \(M\) samples of the \(i^{th}\) spectrum; and \(\mathbf{g}\!=\! [1,\!1,\!...,\!1]^{T}\) is called the gain vector with dimension \(N\times I\), which contains the desired gain related to each steering vector. Figure 3: Generalized sidelobe canceller (GSC): (**a**) block diagram of GSC; (**b**) equivalent diagram of GSC. The effective implementation of LCMV is the division of a field with dimension \(M\times M\) into the constraint subfield defined by columns of **C**, an \(M\times N\) matrix, and an orthogonal subfield defined by columns of **B** an \(M\times(M-N)\) matrix such that  
\[\textbf{C}^{H}\textbf{B}=\textbf{O} \tag{8}\]  
where **O** is a null matrix with dimension \(N\times(M-N)\) and **B** is a blocking matrix that blocks the desired portion of the NCOFDM signal.