## 2 System Model
Perfect channel state information is assumed to have been obtained through channel estimation [17] and effective feedback techniques [18] in this paper, so we can focus on the precoding design. Based on [PERSON]'s theorem, the system spectral efficiency can be expressed as:  
\[\rm{SE}=\rm{log_{2}}\left(\rm{\left|I_{N_{r}}+\frac{P}{N_{S}\sigma^{2}}HFBB^ {H}F^{H}H^{H}\right|}\right) \tag{5}\]  
where \(\rm{(\cdot)^{H}}\) represents the conjugate transpose of the matrix, and \(\rm{\left|\cdot\right|}\) represents the determinant of the matrix. We set \(\rm{B}=\rm{\left(F^{H}F\right)^{-\frac{1}{2}}\hat{B}}\), and Equation (5) can be equivalent to [15]:  
\[\rm{SE}=\rm{log_{2}}\left(\rm{\left|I_{N_{r}}+\frac{P}{N_{S}\sigma^{2}}H_{eff} \widehat{BB}^{H}H_{eff}^{H}\right|}\right) \tag{6}\]  
where \(\rm{H_{eff}}\) represents the effective channel, defined as:  
\[\rm{H_{eff}}=\rm{H}\overline{F} \tag{7}\]  
\[\rm{\overline{F}}=\rm{F}\rm{\left(F^{H}F\right)^{-\frac{1}{2}}} \tag{8}\]If the analog precoder F is given, the effective channel \(\text{H}_{\text{eff}}\) can be determined, and the digital precoder B can be obtained through traditional SVD decomposition. The design of the analog precoder is related to the precoding structure used. The dynamic sub-connected structure used in this article on the transmit side is shown in Figure 1. The dynamic connection network in the Figure 1 is the core of this structure, which is realized by a switch network. The switches dynamically allocate the transmit antennas connected to different phase shifters to the RF link. The switch has smaller power consumption than the phase shifter, although it would generate a certain insertion loss. However, reference [19] shows that the implementation of switch-based hybrid precoding is a promising solution after simulation verification. In the dynamic connection network, each RF chain in the analog precoding process is connected to a non-empty and non-repetitive subset of the transmitting antenna \(\left\{\text{S}_{\text{j}}\right\}_{\text{j=1}}^{\text{N}_{\text{rf}}^{\text{ tr}}}\text{S}_{\text{j}}\) represents the set of transmit antennas connected to the j-th RF chain. For example, if the i-th transmit antenna is connected to the j-th RF chain, then \(\text{i}\in\text{S}_{\text{j}}\). Therefore, the constraint for designing the analog precoding matrix can be written as [20]:  
\[\left|\text{F}_{\text{ij}}\right|=1_{\text{S}_{\text{j}}}(\text{i})\quad \forall(\text{i},\text{j}) \tag{9}\]  
where \(1_{\text{S}_{\text{j}}}(\text{i})\) can be defined as:  
\[1_{\text{S}_{\text{j}}}(\text{i})=\left\{ \begin{array}{ll}1&\text{if }\text{i}\in\text{S}_{\text{j}}\\ 0&\text{otherwise}\end{array} \right. \tag{10}\]  
the set of all non-empty subsets \(\text{S}_{\text{j}}\) is S, which is defined as follows:  
\[\text{S}=\left\{\left\{\text{S}_{\text{j}}\right\}_{\text{j=1}}^{\text{N}_{ \text{rf}}^{\text{tr}}}|\cup_{\text{j=1}}^{\text{N}_{\text{rf}}^{\text{tr}}} \text{S}_{\text{j}}=\left\{1,2,\ldots,\text{N}_{\text{t}}\right\},\text{S}_{ \text{j}}\cap\text{S}_{\text{k}}=\mathcal{O},\forall\text{j}\
eq\text{k}. \right\} \tag{11}\]  
and the problem of designing the analog precoding matrix F can be transformed into:  
\[\max_{\left\{\text{S}_{\text{j}}\right\}\in\text{S}}\text{SE}\big{(}\left\{ \text{S}_{\text{j}}\right\}\big{)} \tag{12}\]  
Equation (12) is a joint optimization problem, which requires exhaustive search for the combination of all non-empty and no internal repeating elements subset \(\left\{\text{S}_{\text{j}}\right\}_{\text{j=1}}^{\text{N}_{\text{rf}}^{\text{ tr}}}\) to find the optimal solution, and the total search volume is the second type Stirling number [21], as follows:  
\[\left|\text{S}\right|=\frac{1}{\text{N}_{\text{rf}}^{\text{t}}.\text{i}}\sum _{\text{k=0}}^{\text{N}_{\text{rf}}^{\text{t}}.\text{i}}(-1)^{\text{N}_{ \text{rf}}^{\text{tr}}-\text{k}}\left( \begin{array}{c}\text{N}\text{t}\\ \text{k}\end{array}