# Interference Alignment in Multi-Hop Cognitive Radio Networks under Interference Leakage
## 2 Signal and System Model
### Interference Alignment Approach and End-to-End SNR Analysis  
Throughout the paper, we apply the linear IA scheme to align the interference occurring between the PN and the SN. In order to suppress the interference signal, we use a minimum mean squared error (MMSE)-based decoder, which aims to maximize the capacity at the receiver part. The size of precoding matrix at each transmitter node \(\mathbf{V}_{c}\), \((c\in P_{j},R_{i})\) and the suppression matrix of relay receiver at the \(i\)th hop \(\mathbf{U}_{R_{i+1}}\) are \(M\times\dfrac{M}{2}\) and \(\dfrac{M}{2}\times M\), respectively, where \(M\) is a positive even number. We assume that \(\mathbf{V}_{c}=\sqrt{\mathcal{P}_{c}}\mathbf{Q}_{c}\mathbf{X}_{c}\), where \(\mathbf{Q}_{c}\) is an \(M\times\dfrac{M}{2}\) matrix, whose columns form the orthonormal basis for \(\mathbf{V}_{c}\) and \(\mathbf{X}_{c}\). \(\mathbf{X}_{c}\) is an \(\dfrac{M}{2}\times\dfrac{M}{2}\) unitary matrix, which is obtained by using QR decomposition of the precoding matrix \(\mathbf{V}_{c}\). Besides, \(\mathcal{P}_{c}\) is the average power of each stream as \(\mathcal{P}_{c}=2P_{c}/M\) where \(\mathcal{P}_{c}\), \((c\in P_{j},R_{i})\) is the total transmit power at the transmitter side. The suppression matrix at the \(i\)th hop can be written as \(\mathbf{U}_{R_{i+1}}=\tilde{\mathbf{U}}_{R_{i+1}}\tilde{\mathbf{U}}_{R_{i+1}}\)[34]. When all CSI is known at each receiver node, the first term of the suppression matrix is obtained as:  
\[\tilde{\mathbf{U}}_{R_{i+1}}=\mathbf{H}_{ef,R_{i}\to R_{i+1}}^{\mathrm{H}} \left(\sum_{j=1}^{L}\dfrac{\mathcal{P}_{P_{j}}}{\mathcal{P}_{R_{i}}}\mathbf{H }_{ef,P_{j}\to R_{i+1}}\mathbf{H}_{ef,P_{j}\to R_{i+1}}^{\mathrm{H}}+ \dfrac{\sigma_{n}^{2}}{\mathcal{P}_{R_{i}}}\mathbf{I}\right)^{-1}, \tag{4}\]  
where \(\mathbf{H}_{ef,c\to R_{i+1}}=\mathbf{H}_{c\to R_{i+1}}\mathbf{Q}_{c}\), \((c\in P_{j},R_{i})\) denotes the effective channel matrix and \(\sigma_{n}^{2}\) is the noise variance. Then, applying the Cholesky factorization as:  
\[\tilde{\mathbf{U}}_{R_{i+1}}\left(\sum_{j=1}^{L}\mathcal{P}_{P_{j}}\mathbf{H }_{ef,P_{j}\to R_{i+1}}\mathbf{H}_{ef,P_{j}\to R_{i+1}}^{\mathrm{H}}+ \sigma_{n}^{2}\mathbf{I}\right)\tilde{\mathbf{U}}_{R_{i+1}}^{\mathrm{H}}= \boldsymbol{\zeta}_{R_{i+1}}\xi_{R_{i+1}}^{\mathrm{H}}, \tag{5}\]  
\(\boldsymbol{\zeta}_{R_{i+1}}\) can be obtained. The second term of the suppression matrix is calculated as \(\tilde{\mathbf{U}}_{R_{i+1}}=\mathbf{S}_{R_{i+1}}^{H}\boldsymbol{\zeta}_{R_{i +1}}^{-1}\), where \(\mathbf{S}_{R_{i+1}}\) is obtained by using singular-value decomposition (SVD) as \(\mathbf{H}_{R_{i+1}}=\mathbf{S}_{R_{i+1}}\mathbf{\Lambda}_{R_{i+1}}\mathbf{D}_{ R_{i+1}}^{\mathrm{H}}\) and \(\mathbf{H}_{R_{i+1}}\) denotes the \(\frac{M}{2}\times\frac{M}{2}\) block channel matrix \(\mathbf{H}_{R_{i+1}}=\boldsymbol{\zeta}_{R_{i+1}}^{-1}\tilde{\mathbf{U}}_{R_{i +1}}\mathbf{H}_{ef,R_{i}\to R_{i+1}}\). Finally, the suppression matrix can be written as a multiplication of these two terms:  
\[\mathbf{U}_{R_{i+1}}=\mathbf{S}_{R_{i+1}}^{\mathrm{H}}\boldsymbol{\zeta}_{R_{ i+1}}^{-1}\tilde{\mathbf{U}}_{R_{i+1}}. \tag{6}\]  
Interested readers are referred to [34] and the references therein for a review of decoding matrix design. As \(\mathbf{U}_{R_{i+1}}(\sum_{j=1}^{L}\mathcal{P}_{R_{j}}\mathbf{H}_{ef,R_{i} \to R_{i+1}}\mathbf{H}_{ef,R_{i}\to R_{i+1}}^{\mathrm{H}}+\sigma_{n}^{2}\mathbf{ I})\mathbf{U}_{R_{i+1}}^{\mathrm{H}}=\mathbf{I}\), the interference can be aligned. With the aid of the proposed IA approach, the interference can be perfectly aligned in the CR network. By doing so, the hop-by-hop transmission can be accomplished via the single-input