## 4 Numerical Results
### Degrees of Freedom  
The information rate contained in symbol \(s^{[i,i]}\) is defined as \(R_{i,j}\). Assuming that the noise random variable is a complex Gaussian independent identical distribution that satisfies the zero mean and unit variance, the signal-to-noise ratio of each transmission channel is \(P\). We define the degrees of freedom (DoF) of a channel as the dimension of the received signal space. The degrees of freedom from user \(i\) to user \(j\) can be defined as  
\[d_{i,j}\triangleq\lim\limits_{SNR\rightarrow\infty}\frac{R_{i,j}(SNR)}{\log( SNR)}=\lim\limits_{P\rightarrow\infty}\frac{R_{i,j}(P)}{\log(P)} \tag{33}\]  
In the relay cooperative communication network in this paper, the total DoF of the system is \(d_{total}=\sum\limits_{i=1}^{K}\sum\limits_{j\in S}d_{i,j}\). The DoF upper bound of the user \(i\) is  
\[\sum\limits_{j\in S}^{K}d_{i,j}\leq d_{i}^{upper}=\min\left\{\underbrace{\min \{M,N\}}_{A},\underbrace{\min\{(K-1)M,N\}}_{B}\right\}=\min\{M,N\} \tag{34}\]  
where \(A\) and \(B\) are the boundaries of the cut from the user \(i\) and \(j\) to the relay. Additionally,  
\[d_{total}\leq\sum\limits_{i=1}^{K}d_{i}^{upper}=\min\{KM,KN\} \tag{35}\]  
From the perspective of MAC and BC transmission process, the cut from all \(K\) source nodes to relay nodes is defined as \(C\), and the cut from relay nodes to all \(K\) nodes is defined as \(D\). We can therefore obtain  
\[d_{total}\leq\underbrace{\min\{KM,N\}}_{C}+\underbrace{\min\{KM,N\}}_{D}=\min \{2 KM,2N\} \tag{36}\]  
Through comprehensive consideration of Formulas (35) and (36), the upper bound of system freedom is  
\[d_{total}\leq\min\{KM,KN,2 KM,2N\}=\min\{KM,2N\} \tag{37}\]  
It can be seen from Formula (37) that if the number of node antennas is determined as discussed at the end of Section 3.3, the degrees of freedom are only determined by the  
Figure 10: Performance comparison of sum rate under different methods with [25; 30]. interactive degrees of freedom \(d\) and the number of users \(K\). The upper bound of the maximum degrees of freedom that the system can achieve is only related to the number of node antennas from the cut set theorem. Table 3 lists the research results on the upper bound of degrees of freedom in recent years. The table lists the range of the upper bound of degrees of freedom of the classical channel model and the proposed scheme in this paper. In order to enable information interaction between each user pair, the number of antennas between the user and the relay must meet the constraint \(N\leq\text{min}\Big{\{}2M-d_{(i,j)}\Big{\}}\), \(\forall d_{(i,j)}>0\). The following results can be obtained:  
\[\frac{N}{M}\leq 2-\frac{d_{(i,j)}}{M} \tag{38}\]  
\(M\) must satisfy \(M\geq\frac{[N+d]}{2}\). When \(N\) and \(M\) respectively take the lower bound \(\frac{K(K-1)d}{2}\) and \(\frac{K(K-1)d}{2}+d/2\), constraint (38) can be rewritten as  
\[\frac{N}{M}\leq 2-\frac{2d}{\frac{K(K-1)d}{2}+d}=\frac{2K^{2}-2K}{K^{2}-K+2} \tag{39}\]  
It can be seen from Figure 11 that in [40], the channel diagonalization technique can be used to achieve the upper bound \(2N\) of system degrees of freedom at \(\frac{N}{M}\leq 1\). In this paper, by using the heterogeneous transmission of system degrees of freedom, the algorithm can make the upper bound \(2N\) of system degrees of freedom realizable when \(\frac{N}{M}\leq\frac{2K^{2}-2K}{K^{2}-K+2}\). When \(K\geq 3\), the realizable range of the upper bound \(2N\) of the system degrees of freedom is extended. When \(\frac{N}{M}\geq K\), due to sufficient relay resources, the upper bound \(KM\) can be achieved due to the number of user antennas. The algorithm proposed by the author in [39] can achieve the upper bound of channel degrees of freedom more quickly due to full connection communication. Figure 11: \(K\) User can achieve the upper bound of degrees of freedom with [39; 40].