1. Two types of channel capacity classification

The wireless transmission environment is time-varying and affected by multipath effects. Since the channel changes in real time, the channel capacity of the MIMO system also changes randomly and is not quantitative. Therefore, in order to achieve an accurate description of the channel, the performance of the channel is determined by using the average capacity of the channel and the outage capacity of the channel.

1.1 Average Capacity

Due to the time-varying nature of the channel, the average capacity can be used to judge the overall channel capacity performance of the MIMO system.

1.2 Interrupt capacity

Outage capacity requires channel capacity to ensure that MIMO systems can reliably transmit signals. Under certain probability conditions, the MIMO system can ensure that the rate of information transmission is reliable. The outage probability of a signal can be expressed as follows:

$P_{out}=P_r\lbrace C\leq C_{out}\rbrace$

In the above formula $C_{out}$ is the interrupt capacity of the channel.

Thus, the probability that a channel can reliably transmit information can be defined as follows:

$(100-P_{out})%$

Since the time-varying channel state information is unknown when the transmitting end of the MIMO system sends signals, when the wireless channel is a quasi-static random channel, the performance of the channel can be judged by defining the transmission rate of the interrupt information. When the sender sends information, because it does not know the channel state information of the wireless channel, the information rate sent is greater than the channel capacity, and the information cannot be transmitted normally, causing communication interruption. A relationship between outage probability and outage capacity can also be implied here.

2. When the CSI is unknown at the sender, the channel capacity of the MIMO system

If the CSI can be known at the transmitting end, the transmitted signal can be processed, and the spatial sub-channels with better channel conditions will be allocated more power, and the spatial sub-channels with poor channel conditions will be allocated less or no power, so as to achieve the maximum channel capacity. value.

In the process of wireless communication, the MIMO system cannot determine the time-varying complex channel state in more cases when sending signals, and the CSI is unknown when sending signals, but the CSI during signal transmission can be analyzed at the receiving end. In order to make the channel capacity of the MIMO system better, it is very important to study the channel capacity under the condition of unknown CSI.

Under the condition of unknown CSI, the transmitting end distributes the transmission power equally, and the receiving end can obtain CSI. The spatial sub-channels are independent in the Rayleigh fading environment. Therefore, the channel capacity of the MIMO system can be expressed as follows:
$C=log_2[det(I_{min}+\frac{\rho}{N_T}HH^H)]$

In the above formula, $I_{min}$ it is the identity matrix of min✖️min order, $N_T$ which is the number of antennas at the transmitting end, det represents the determinant of the matrix, ρ represents the average signal-to-noise ratio of the receiving end, and H represents the channel matrix.

If a singular value decomposition is performed on the channel matrix $HH^T$ , the channel capacity can be changed as follows:

$C=\sum_{i=1}^m log_2(1+\frac{\rho}{N_R}\lambda_i)$

In the above formula, $N_R$ is the number of antennas at the receiving end, $\lambda_i$ and is the eigenvalue of the channel matrix.

2.1 The number of receiving and transmitting antennas is equal

If coherent detection and combining technology is used at the receiving end to perform the same-frequency and same-phase processing on each antenna, that is, the channel matrix of the MIMO system is all 1, then the MIMO system with multiple transmission and multiple reception can be equivalent to using or $N_T$ a $N_R$ SISO System, through each equivalent SISO system that exists independently, the diversity gain obtained is $N_TN_R$ 2 times, and the further channel capacity is as follows:

$C=log_2(1+N_T N_R\rho)$

2.2 Non-coherent detection combination at the receiving end

If the receiving end uses non-coherent detection and combining technology for processing, due to the interference and noise in the channel, the signals of the spatial sub-channels will be different, and the signal-to-noise ratio obtained by each receiving antenna at the receiving end is still ρ, and the total received SNR The ratio is $N_R\rho$ , compared with the equivalent SISO system under the coherent detection combining technology in 2.1, the diversity gain of the equivalent MIMO system at this time is $N_R$ 1 times, so the channel capacity under the non-coherent detection is expressed as follows:

$C=log_2(1+N_R\rho)$

2.3 Summary of general formula

Because the MIMO technology adopts space diversity, the parallel space sub-channels that can be formed by the MIMO system can be transmitted in the channel orthogonal to each other without interfering with each other. If the number of antennas at the transmitting end and the receiving end are equal, both are L. The channel matrix is expressed as the identity matrix of L✖️L $I_L$ , and the channel capacity can be obtained as follows:

$C=log_2[det(I_L+\frac{\rho}{L}HH^H)]=Llog_2[1+\rho]$

It can be seen from the above formula that when the number of antennas at the transmitting end and the receiving end are equal, the MIMO system can obtain a gain of L times compared with the SISO system.

CSI channel state information (English: channel state information, abbreviated as CSI)

SISO single input single output (English: simple input simple output, abbreviated as SISO)

Channel Capacity Analysis of MIMO System