# Interference Alignment in Multi-Hop Cognitive Radio Networks under Interference Leakage
## 2 Signal and System Model  
This paper considers a cognitive multi-hop relay-aided network with \(L\) PUs and two SUs in which the secondary source \(S\) wishes to communicate with the secondary destination \(D\) over \(K-1\) closely-located DF relays, as shown in Figure 1. We assume that all terminals are operating in a half-duplex fashion, and the direct path between \(S\) to \(D\) is not available due to heavy shadowing or large path loss effect. The uniformly-located relay terminals are clustered together and thus experiencing the same scale of fading even though the instantaneous SNR varies. In the SN, each node is equipped with \(M\) transmit/receive antennas applying maximal ratio transmission (MRT) and maximum ratio combining (MRC) techniques at the transmitter and receiver, respectively. In underlay CRNs, the transmit powers of the SUs are generally set to a predefined power level to meet the interference power constraints of the PUs [30]. However, in the proposed scheme shown in Figure 2, we adopt the linear IA method to mitigate the interference occurring at the SN without reducing the powers of the SUs. With the aid of precoding and linear suppression matrices, the single symbol detection at the \(i\)th hop (or \(i+1\)th relay) can be expressed as [31]:  
\[\mathbf{y}_{R_{i+1}}= \mathbf{U}_{R_{i+1}}^{\mathrm{H}}\mathbf{H}_{R_{i}\to R_{i+1}} \mathbf{V}_{R_{i}}\mathbf{x}_{s}+\sqrt{\alpha}\mathbf{U}_{R_{i+1}}^{\mathrm{H }}\sum_{j=1}^{L}\mathbf{H}_{P_{j}\to R_{i+1}}\mathbf{V}_{P_{j}}\mathbf{x}_{j}+ \mathbf{U}_{R_{i+1}}^{\mathrm{H}}\mathbf{n}_{R_{i+1}}, \tag{1}\]  
where \(\mathbf{x}_{s}\) is the source signal, \(\mathbf{x}_{j}\) is the information of the primary user, \(\mathbf{H}_{R_{i}\to R_{i+1}}\) denotes the channel information at the \(i\)th hop of the SN, \(\mathbf{H}_{P_{j}\to R_{i+1}}\) denotes the channel coefficients matrix between PN and SN, \(\mathbf{V}\) and \(\mathbf{U}\) denote the corresponding precoding and linear suppression matrices, \(\alpha\) gives the interference leakage coefficient varying between 0 and 1 [32], \(\mathbf{n}_{R_{i+1}}\) is the zero-mean unit-variance (\(\sigma_{\mathbf{n}_{R_{i+1}}}^{2}=\mathbf{I}\)) circularly symmetric additive white Gaussian noise (AWGN) vector, and \((.)^{\mathrm{H}}\) stands for the Hermitian operation. Note that all signal and system model parameters are listed in Table 1. The interference between PN and SN can be perfectly aligned if the following conditions are satisfied [31]:  
\[\mathbf{U}_{R_{i+1}}^{\mathrm{H}}\mathbf{H}_{\bar{\jmath}\to R_{i+1}} \mathbf{V}_{\bar{\jmath}}=0\] \[\mathrm{Rank}\Big{(}\mathbf{U}_{R_{i+1}}^{\mathrm{H}}\mathbf{H}_{ \bar{\jmath}\to R_{i+1}}\mathbf{V}_{\bar{\jmath}}\Big{)}=d, \tag{2}\]  
where \(d\) is the data stream transmitted by each user [33]. Using the ideal linear IA assumption, (1) can be expressed as:  
\[\mathbf{y}_{R_{i+1}}=\widehat{\mathbf{H}}_{\bar{\imath}_{i}\to R_{i+1}} \mathbf{x}_{\mathrm{s}}+\widehat{\mathbf{n}}_{R_{i+1}}, \tag{3}\]  
where \(\widehat{\mathbf{H}}_{\bar{\imath}_{i}\to R_{i+1}}\triangleq\mathbf{U}_{R_{i+1 }}^{\mathrm{H}}\mathbf{H}_{\bar{\imath}_{i}\to R_{i+1}}\mathbf{V}_{R_{i}}\) and \(\widehat{\mathbf{n}}_{R_{i+1}}\triangleq\mathbf{U}_{R_{i+1}}^{\mathrm{H}} \mathbf{n}_{R_{i+1}}\).  
Figure 1: Multi-hop underlay cognitive radio network.  
Figure 2: Block diagram of the system model.