# Compact Sub 6 GHz Slot Multiantenna System for 5G Laptops
## 4 Channel Capacity of a Six-Antenna Array  
The ergodic channel capacity can be evaluated by averaging the capacities observed in a multipath Rayleigh fading environment. When a transmitter does not have channel state information (CSI), and its power is equally divided by each transmit antenna unit, its ergodic channel capacity can be calculated with the following formula [2]:  
\[C=E\bigg{\{}\log_{2}\bigg{[}\text{det}\bigg{(}I_{n_{r}}+\frac{\Gamma}{n_{t}}HH^{ \dagger}\bigg{)}\bigg{]}\bigg{\}}, \tag{1}\]  
where \(\Gamma\) is the signal-to-noise ratio (SNR), \(H\) is the wireless channel matrix, ()+ is the Hermitian transpose of the matrix, \(n_{t}\) is the number of transmit antennas, \(n_{r}\) is the number of receive antennas, \(I_{n_{r}}\) is the identity matrix of size \(n_{r}\), det is the determinant, and \(E\) is the expectation concerning different channel realizations. In addition, the wireless channel matrix \(H\) in (1) is evaluated as follows [32]:  
\[H=\text{Y}^{1/2}H_{iid}, \tag{2}\]  
where \(H_{iid}\) is the independently and identically distributed (i.i.d.) wireless channel matrix in a Rayleigh fading environment. Considering the effects of antennas in the test, the receiver correlation matrix \(\text{Y}\) can be calculated by the following formula:  
\[\text{Y}=\Phi^{1/2}\overline{\text{Y}}\Phi^{1/2}. \tag{3}\]  
In (3), \(\Phi\) is the diagonal efficiency matrix, i.e.,  
\[\Phi=\text{diag}[\eta_{1},\eta_{2},\cdots,\eta_{n_{r}}], \tag{4}\]  
where \(\eta_{i}\) is the efficiency of antenna \(i\). The complex correlation coefficient matrix \(\overline{\text{Y}}\) is expressed as [33]:  
\[\overline{\text{Y}}=\left[\begin{array}{cccc}1&\rho_{12}&\cdots&\rho_{1n_{ r}}\\ \rho_{12}^{*}&1&\cdots&\rho_{2n_{r}}\\ \vdots&\vdots&\ddots&\vdots\\ \rho_{1n_{r}}^{*}&\rho_{2n_{r}}^{*}&\cdots&1\end{array}\right], \tag{5}\]  
where \(*\) is the conjugate operator, and \(\rho_{ij}\) is the complex correlation coefficient between antennas \(i\) and \(j\). The complex correlation coefficient can be expressed in terms of the S parameters of an antenna system as follows [34]:  
\[\rho_{ij}=\frac{-(S_{ii}^{*}S_{ij}+S_{ii}^{*}S_{jj})}{\sqrt{(1-\left|S_{ii} \right|^{2}-\left|S_{ji}\right|^{2})(1-\left|S_{jj}\right|^{2}-\left|S_{ij} \right|^{2})}}, \tag{6}\]  
where \(\left|\cdot\right|\) is the magnitude operator.  
The results of the calculated ergodic capacity with a 20-dB SNR are shown in Figure 10. The proposed 5G six-antenna array in free space was compared with the case of a six-antenna ideal 6 \(\times\) 6 MIMO system and the case of one ideal single-input, single-output (SISO) system. The ideal transmitting and receiving antennas have 100% efficiency, null ECC, and i.i.d. channels with a Rayleigh fading environment. At each frequency point, all results of capacity were calculated by averaging 1000 channel realizations to ensure that they converged to specific values. In the free space, the capacity in 3300-3600 MHz was between 27.9 and 29.6 bit/s/Hz, while the capacity in 4800-5000 MHz was from 26.8 to 27.8 bit/s/Hz. The highest capacity value in the band of 3300-3600 MHz was 3 bit/s/Hz lower than the capacity of 6 \(\times\) 6 MIMO i.i.d. channels. The proposed six-antenna array achieved capacity much larger than that of a SISO system; therefore, it is promising for future 5G laptop application.