Research on Signal Detection Algorithm for Vertically Layered Space-Time Code System

O Introduction
In a non-line-of-sight mobile communication environment, it is a challenge for designers to design a communication technology that has a very high transmission rate, good service quality and a large service range. Multiple-input multiple-output MIMO (Multiple Input Multiple-Output) technology refers to the technology of using multiple transmitting antennas and multiple receiving antennas for wireless transmission. The wireless communication system using this technology is a MIMO system. MIMO technology can multiply the capacity and spectrum utilization of the communication system without increasing the bandwidth, so its research has become a frontier field of communication theory research.
The essence of MIMO technology is to provide spatial multiplexing gain and spatial diversity gain for wireless systems. Spatial multiplexing technology can greatly improve channel capacity, while spatial diversity technology can improve channel reliability and reduce channel bit error rate.
Vertical Layered Space-Time Code System (V-BLAST) is a space-time code system based on multiple-input multiple-output (MIMO) transmission mode proposed by Bell Labs. It is a representative of spatial multiplexing technology.
The V-BLAST structure decomposes the data stream to be transmitted into multiple parallel sub-data streams, independently encodes, modulates and maps each data stream to its corresponding transmitting antenna, and separates the multiple sub-data streams at the receiving end using a detection algorithm combined with interference elimination and other technologies.
Generally speaking, the V-BLAST system exchanges high-frequency band utilization at the expense of partial diversity gain. Since V-BLAST cannot obtain the maximum diversity gain, the detection algorithm selected by the receiving end when detecting the signal is crucial to improving the performance of the entire system. This paper conducts an in-depth study of the two detection algorithms in the V-BLAST system, analyzes their performance through simulation results, and compares the applicability of the two algorithms.

1 Traditional Receiver Detection Algorithm
There are many algorithms for MIMO signal detection technology. The best algorithm is the maximum likelihood (ML) decoding algorithm. However, the complexity of the ML algorithm increases exponentially with the increase of the number of antennas and the modulation order, which is not practical. Therefore, various simplified algorithms have been proposed. Among them, the commonly used detection algorithms include the zero forcing (ZF) linear algorithm and the minimum mean square error (MMSE) linear algorithm.
Assuming that the MIMO channel is flat fading, the signal vector received by the receiver at time t is expressed as:

where rt represents the nR×1 received signal vector, H is the nR×nr-dimensional channel response matrix, xt is the nT×l transmitted signal vector, and nt is the nR×1 AWGN noise vector, where each component is an independent normally distributed random variable with a mean of 0 and a variance of σ2.
1.1 ZF algorithm
ZF is the simplest linear detection algorithm, which uses the linear transformation matrix G to multiply the received vector rt on the left to completely or partially eliminate interference from other antennas.

G is the Penrose-Moore inverse (also called generalized inverse) of H. Assuming that the channel matrix is ​​invertible, the received signal vector estimate is:

In order to ensure the existence of the generalized inverse, nT must be less than or equal to nR, otherwise HHH is a singular matrix and its inverse does not exist.
Although the ZF algorithm can make the interference of other antennas zero, it has the disadvantage of amplifying noise, so a detection method based on the MMSE criterion is proposed.
1.2 MMSE algorithm
The MMSE algorithm selects the matrix wMMSE (wMMSE is the linear combination coefficient matrix of nT×nR) based on the received vector rt to minimize the mean square error, that is:

According to the orthogonal principle, the optimal solution is:

The algorithm can minimize the error caused by noise and interference from other antennas. The received signal vector estimate can be expressed as:

Among them, σ2 is the variance of AWGN, and InT is the unit matrix of σ2→0.
It can be seen from the ZF algorithm and the MMSE algorithm that although the ZF algorithm can make the interference of other antennas zero, the noise is amplified because the noise is multiplied by the factor G before the noise, so the detection performance is relatively poor. The MMSE algorithm does not completely eliminate the interference of other antennas, but it makes a compromise between reducing the interference of other antennas and enhancing the noise, so that the total error rate is minimized. If the signal-to-noise ratio is very high, that is, σ2→0, the MMSE algorithm can be simplified to the ZF algorithm.
Whether it is the ZF algorithm or the MMSE algorithm, its essence is based on the method of inverting the channel matrix. In order to make the inversion of the channel matrix have a unique solution, it is necessary to require that the number of receiving antennas is greater than or equal to the number of transmitting antennas.

2 Serial Interference Cancellation Detection Algorithm
The serial interference cancellation detection algorithm borrows the idea of ​​serial interference cancellation (SIC) in multi-user detection. The idea of ​​SIC is to first detect a user's signal, and then restore it to the transmitted signal multiplied by the channel parameters as interference to other users, and subtract it from the received signal, so as to reduce the interference to other users' signals. There will be errors in this process. If the user signal with the highest reliability is deleted first, the probability of error transmission will be reduced. Therefore, the detection and deletion can be carried out in the order of the signal-to-noise ratio of each user signal from high to low. In the MIM0 system, the SIC idea is combined with the traditional detection algorithm, which is the decoding algorithm of the V-BLAST system. In this paper, SIC is combined with the ZF algorithm and the MMSE algorithm for research.
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3 Analysis of simulation results
3.1 Simulation
MATLAB software was used for simulation, using 2-transmit 2-receive and 4-transmit 4-receive antenna models, and the channel was a narrow-band Rayleigh fast fading channel. It was assumed that the receiving end had an ideal channel estimation and the modulation method was BPSK. The ZF algorithm, MMSE algorithm, ZF-SIC algorithm, and MMSE-SIC algorithm were simulated respectively, and the results are shown in Figures 1 and 2.

3.2 Comparison of performance of various algorithms
1) From the simulation results of Figure 1 and Figure 2, it can be seen that when the number of transmitting and receiving antennas is the same, the detection performance of the traditional ZF algorithm is the worst, and the performance of the traditional MMSE algorithm and the ZF-SIC algorithm of V-BLAST is similar. However, as the signal-to-noise ratio increases, the performance of the ZF-SIC algorithm is better. The detection performance of the MMSE-SIC of V-BLAST is the best, and its diversity gain becomes more obvious as the signal-to-noise ratio increases.
2) By comparing the simulation results of Figure 1 and Figure 2, it can be seen that the performance advantage of the V-BLAST detection algorithm used in the 4-transmitter 4-receiver model is more obvious than that of the V-BLAST detection algorithm used in the 2-transmitter 2-receiver model.
3) Although the detection performance of the MMSE-SIC algorithm is the best, its complexity is higher than that of the other algorithms, followed by the ZF-SIC algorithm. The performance of the MMSE algorithm is moderate, and its structure is simple and the complexity is low. Therefore, in actual application, the performance and complexity of the balanced detection algorithm should be considered comprehensively according to the specific situation.

4 Conclusion
This paper introduces two traditional detection algorithms for MIMO systems and the V-BLAST detection algorithm that incorporates the idea of ​​serial interference cancellation based on these two algorithms, and compares and analyzes the performance and complexity of various detection algorithms. MIMO technology has been widely studied and applied in wireless communication systems, especially the V-BLAST space-time code system based on MIMO systems. How to further improve the performance of the detection algorithm will be an inevitable trend to significantly improve system performance.