At present, the traditional channel estimation algorithms have the disadvantages of over-reliance on initial conditions and high complexity. The bacterial foraging optimization (BFO)-based algorithm has been applied in wireless communication and signal processing because of its simple operation and strong self-organization ability. But the BFO-based algorithm is easy to fall into local optimum. Therefore, this paper proposes the quantum bacterial foraging optimization (QBFO)-binary orthogonal matching pursuit (BOMP) channel estimation algorithm to the problem of local optimization. Firstly, the binary matrix is constructed according to whether atoms are selected or not. And the support set of the sparse signal is recovered according to the BOMP-based algorithm. Then, the QBFO-based algorithm is used to obtain the estimated channel matrix. The optimization function of the least squares method is taken as the fitness function. Based on the communication between the quantum bacteria and the fitness function value, chemotaxis, reproduction and dispersion operations are carried out to update the bacteria position. Simulation results show that compared with other algorithms, the estimation mechanism based on QBFO-BOMP algorithm can effectively improve the channel estimation performance of millimeter wave (mmWave) massive multiple input multiple output (MIMO) systems. Meanwhile, the analysis of the time ratio shows that the quantization of the bacteria does not significantly increase the complexity.

The main expectation of the next generation wireless communication system is to obtain higher data rates and better user service quality, so as to meet the needs of national economic and social development. The communication system has gradually expanded to more and more application scenarios, such as intelligent transportation, cloud computing, medical treatment, automatic driving and other new fields. Therefore, data transmission with low latency, high throughput and high rate has become the requirement of communication development. In mmWave massive MIMO systems, the acquisition of channel state information is of great significance to the reliability of communication. Among them, the optimization and research of channel estimation for mmWave massive MIMO systems is a popular direction of research in the evolution of communication systems [

In mmWave massive MIMO systems, due to the sparsity of mmWave, the combination of channel estimation and compressed sensing (CS) technology is a research hotspot [

Traditional channel estimation algorithms are easy to fall into the local optimization and depend on initial conditions. However, the quantum optimization algorithms have the advantages of independent of initial conditions, independent of the solution space, and well-stabilized. In the quantum optimization algorithms, there are interaction between personals and between personal and environment, and thus it has good self-organization ability [

Dasgupta et al. [

Inspired by the above works, the QBFO-BOMP-based algorithm is proposed for mmWave massive MIMO in this paper. When the measured value of signal is corrupted by noisy, the support set of the sparse signal can be recovered via the BOMP-based algorithm. The binary matrix is constructed depending on whether the atom is selected, and then the received signal is projected onto the linear subspace of the selected column to update the residual. In particular, the QBFO-based algorithm is used to obtain the channel matrix to be estimated. The optimization function of the least squares method is used as the fitness function. The bacterial population is measured to collapse into a classical state. The bacterial location is then updated by chemotaxis, reproduction, and dispersal operations. Finally, the optimal solution is obtained.

The remainder of this paper is summarized as follows.

MmWave massive MIMO systems is considered. The numbers of transmitter and receiver are

For the uniform linear arrays (ULA) adopted in this paper,

Therefore,

Assuming that the channel remains stationary for the duration of the transmission unit training symbol, where a pilot transmitted signal is written as

In this paper, a QBFO-BOMP-based algorithm is proposed for the channel eatimation.

Firstly,

The grid, which quantizes the angular parameters, is defined as follows:

The corresponding rows of the array response matrix are orthogonal and can be represented as follows:

Thus,

The key point of the selection atom step is to find the atom that is highly correlated with the received signal. Although

The signal of the

In this paper, the residual error is updated in the following ways:

The specific processes of the BOMP-based algorithm are summarized in Algorithm 1. Noting, the QBFO-based algorithm is used to obtain the estimated channel matrix in Algorithm 1. The corresponding specific steps are described in

In this section, QBFO is applied to mmWave massive MIMO, and the fitness corresponding function is:

Firstly, the quantum state bacteria is expressed as the possible value of the channel matrix by using binary coding. Each quantum state is then measured and collapsed to the binary of classical state. Finally, it is decoded into the decimal system, such that each bacterium represents the value of the channel matrix. In addition, the number of bacterial population

Therefore, the corresponding value of the channel matrix in

In quantum theory, the quantum state can collapse by measurement. Specifically, the initial quantum state collapses to the state given by the measurement. The

In the QBFO-based algorithm, the position information of the bacteria is converted into multiple qubits via quantum states. So every bacteria is in a superposition state. However, the quantum state of each bacteria is determined by the probability amplitude. According to

The bacterial population composed of

The chemotactic operation refers to the optimization process of the entire bacterial population. The quantum rotating gate is used to update the quantum state of the bacteria in this paper.

The direction of rotation can be positive or negative when

This paper uses the fitness function

It is determined that the number of bacteria entering the new population is one.

A new population is get by repeating

After the reproduction operation is completed, it enters the dispersion stage. At this stage, each bacteria corresponds to a random number. Given a dispersion probability

The value of

The flow chart and steps of the QBFO-based algorithm are shown in

The proposed QBFO-based algorithm does not have the process of derivation and inversion. Instead, the possible values of the channel matrix are directly given in terms of the actual situation. That is to get the minimum value of the fitness function. Therefore, the proposed QBFO-based algorithm is simple to operate and does not depend on the solution range space of the function.

Noting, the expression of the initial state of each quantum bacteria is [

It can be shown from

This paper selects the traditional OMP-based algorithm [

The parameters are set to:

This paper focuses on the channel estimation of mmWave massive MIMO systems and proposes the QBFO-BOMP-based algorithm. The quantum transmission technique is introduced to estimate the channel matrix. The proposed QBFO-BOMP-based algorithm mainly consists of two main components. The first part is the BOMP-based algorithm. Firstly, the BOMP-based algorithm is used to recover the support set of the sparse signal. The binary matrix is constructed according to whether the atom is chosen, and then the residual is updated by projecting the received signal onto the linear subspace of the selected column. The second part is to use the QBFO-based algorithm to get the estimated channel matrix. According to the communication between quantum bacteria and the fitness function value, chemotaxis, reproduction and dispersion operations are carried out to update the bacterial position. Simulation analysis shows that the proposed QBFO-BOMP-based algorithm has better estimation performance than other algorithms.

The authors express their gratitude to the editor and referees for their valuable time and efforts on our manuscript.

This work was supported by the National Natural Science Foundation of China (Nos. 61861015, 62061013 and 61961013), Key Research and Development Program of Hainan Province (No. ZDYF2019011), National Key Research and Development Program of China (No. 2019CXTD400), Young Elite Scientists Sponsorship Program by CAST (No. 2018QNRC001), the Scientific Research Setup Fund of Hainan University (No. KYQD(ZR) 1731) and the Natural Science Foundation High-Level Talent Project of Hainan Province (No. 622RC619).

The authors declare that they have no conflicts of interest to report regarding the present study.

_{1/2}-SVD based channel estimation for mmWave massive MIMO