# Cognitive Interference Alignment Schemes for IoT Oriented Heterogeneous Two-Tier Networks
## 4 Performance Analyses of the Cognitive IA Schemes
### Proof of Convergence  
In this subsection, we illustrate how the proposed iterative algorithms are proved to be convergent. Without a loss of generality, we assume the transmit power of each transmitter is normalized. In each iteration of the interference nulling based cognitive IA scheme, the decoder, \(\mathbf{U}_{k}\overset{\leftarrow}{\mathbf{U}}_{k}\), is designed to nullify the original/reciprocal interference power imposed on each IoT node, which is associated with the covariance matrices in (21) and (22). More specifically, in the original channel, the decoder, \(\mathbf{U}_{k}\), is designed as (23) to minimize the overall interference power, \(P_{I,k}\). Since the decoders, \(\mathbf{U}_{k},\forall k\in[1,K]\), are uncoupled from each other and the sum interference power is given by \(P_{I}=\sum_{k=1}^{K}P_{I,k}\), the objective function can be decoupled as:\[\min_{\mathbf{U}_{1},\mathbf{U}_{2},\cdots,\mathbf{U}_{K}}P_{I}=\min_{\mathbf{U}_{1 },\mathbf{U}_{2},\cdots,\mathbf{U}_{K}}\sum_{k=1}^{K}P_{I,k}=\sum_{k=1}^{K}\left\{ \min_{\mathbf{U}_{k}}P_{I,k}\right\}, \tag{30}\]  
where:  
\[P_{I,k}=\sum_{i=1,j\
eq k}^{K}\text{tr}\Big{\{}\mathbf{U}_{k}^{H}\widetilde{ \mathbf{H}}_{k,i}\widetilde{\mathbf{V}}_{i}\widetilde{\mathbf{V}}_{i}^{H} \widetilde{\mathbf{H}}_{k,i}^{H}\mathbf{U}_{k}\Big{\}}+\text{tr}\Big{\{} \mathbf{U}_{k}^{H}g_{k,p}g_{k,p}^{H}\mathbf{U}_{k}\Big{\}}. \tag{31}\]  
We know that, in each iteration of the original channel, it always chooses the optimal \(\{\mathbf{U}_{1},\ \cdots,\ \mathbf{U}_{K}\}\) to minimize the sum interference power in (30) and reduce the value of \(P_{I}\).  
In the reciprocal channel, the reversed decoder, \(\overset{\leftarrow}{\mathbf{U}}_{k}\), associated with the \(k\)-th IoT node is designed without considering the cross-tier interference from the macrocell. Hence, the reciprocal objective function of the \(k\)-th IoT node is given as:  
\[\overset{\leftarrow}{P}_{I,k}=\sum_{i=1,j\
eq k}^{K}\text{tr}\bigg{\{} \overset{\leftarrow}{\mathbf{U}}_{k}^{H}\overset{\leftarrow}{\mathbf{H}}_{ k,k}\overset{\leftarrow}{\mathbf{V}}_{i}\widetilde{\mathbf{V}}_{i}^{H} \overset{\leftarrow H}{\mathbf{H}}_{i,k}\widetilde{\mathbf{U}}_{k}\bigg{\}}. \tag{32}\]  
By substituting \(\overset{\leftarrow}{\mathbf{U}}_{k}=\widetilde{\mathbf{V}}_{k}\), \(\overset{\leftarrow}{\mathbf{V}}_{k}=\mathbf{U}_{k}\) and \(\overset{\leftarrow}{\mathbf{H}}_{i,k}=\widetilde{\mathbf{H}}_{k,i}^{H}\) to (32), we get:  
\[\overset{\leftarrow}{P}_{I,k}=\sum_{i=1,j\
eq k}^{K}\text{tr}\bigg{\{} \overset{\leftarrow}{\mathbf{U}}_{k}^{H}\overset{\leftarrow}{\mathbf{H}}_{ i,k}\widetilde{\mathbf{V}}_{i}\widetilde{\mathbf{V}}_{i}^{H}\overset{ \leftarrow H}{\mathbf{H}}_{i,k}\widetilde{\mathbf{U}}_{k}\bigg{\}}. \tag{33}\]  
As can be seen from (31) and (33), the first term of \(P_{I,k}\) is equal to the reciprocal objective function, \(\overset{\leftarrow}{P}_{I,k}\), while the second term of \(P_{I,k}\) is not affected by the reversed precoder, \(\overset{\leftarrow}{\mathbf{U}}_{k}=\widetilde{\mathbf{V}}_{k}\). It means that the optimal decoder that ensures a receiver suffers the least co-tier interference also causes the least interference to others in the reciprocal channel. Therefore, when the value of \(\overset{\leftarrow}{\mathbf{U}}_{k}\) is chosen to minimize \(\overset{\leftarrow}{P}_{I,k}\), it also reduces the value of \(P_{I,k}\). Since the value of the interference power decreases monotonically in every iteration and is lower bounded by zero, hence, convergence of the algorithm can be guaranteed. The convergence of the partial cognitive IA scheme can also be discussed in a similar way, and a semi-definite programming (SDP) problem generally requires \(\mathcal{O}\Big{[}\sqrt{Kd}\log(1/\varepsilon)\Big{]}\) iterations to converge with \(\varepsilon\) as the predefined solution accuracy.