The Internet of Vehicles (IoV) generally requires short communication delays, limiting the communication distance between terminals and base stations. As a result, the distribution density of base stations in the IoV has increased in comparison to the traditional mobile communication network. Due to the increasing shortage of low-frequency spectrum resources, the IoV must rely on higher-frequency resources such as millimetre waves and terahertz. However, the higher-frequency electromagnetic waves have the characteristics of weak diffraction and short effective communication distance. It requires the construction of more base stations, which leads to a further increase in base station density. The construction of denser base stations requires more energy. To meet the purpose of energy saving, low-order pure phase modulations (such as 2PSK) are adopted. Under the same conditions, this single-information modulation mode can save approximately 75% of the energy compared with the quadrature amplitude modulation (QAM) mode.

If the transmitter sends a pure phase signal, the received signal is still a complex signal due to the attenuation and phase shift. Then, the receiver needs to extract phase information from the complex signal. In order to demodulate the information data contained in the received signal, the receiver also needs to estimate the channel state information in advance. Usually, a certain number of pilot signals are embedded in the information signal to form the actual transmitted signal in each coherence period of the channel. The means to estimate the channel of the IoV communication system and decode the received data has become one of the research hotspots of IoV communication technology [1–6]. The channel estimation problem or information decoding problem belongs to the category of signal estimation and detection. In terms of channel estimation, Appaiah et al. [7] use linear minimum mean square error (LMMSE) estimator to perform channel estimation through inter-cell cooperation. Abdallah and Darya [8] use the expectation maximisation (EM) algorithm to estimate the channel. The studies of Liu et al. [9] and Liu and Lau [10] utilise the sparsity of the channel to improve the estimation accuracy. Wang et al. [11] utilise the approximate message passing (AMP) algorithm and channel statistics to improve the accuracy of sparse channel estimation. These research results try to utilise the statistical correlation characteristics of channel elements to improve the accuracy of channel estimation based on compressed sensing.

Signal detection algorithms are often used, including linear zero-forcing algorithm [12], minimum mean square error algorithm [13], nonlinear spherical decoding [14], AMP algorithm [15], and maximum likelihood algorithm [16]. In general, nonlinear algorithms have a high complexity but a low error rate, whereas linear algorithms often have a low complexity but a high error rate. In the nonlinear IoV, linear detection techniques cannot be applied due to the nonlinear distortion of the signal. Further, some nonlinear detection algorithms have been studied and applied to channel estimation. Wang et al. [17] use Bayesian theory to transform the nonlinear detection problem into a linear mixed problem and solve it through the generalised AMP (GAMP) algorithm. However, it does not perform well in terms of algorithm convergence. Wang et al. [18] solve the channel estimation and multiuser detection problems with a non-convex optimisation method. The task of channel estimation is completed by calculating the eigenvector corresponding to the smallest eigenvalue. Unfortunately, it also does not perform well in terms of algorithm convergence.

If the receiver can only detect phase information from the received complex signal, full information including channel amplitude attenuation, frequency offset and phase offset cannot be obtained. In order to obtain full information, a new scheme is considered, which proposes the introduction of an additional full-information channel including amplitude, frequency and phase between the transmitter and receiver. The difficulty of the new scheme lies in the need for formulation of a method to make full use of the added full-information channel to help estimate the full state information of the nonlinear channel. Experiments have proved that the wireless channel has sparse characteristics in many cases. The channel estimation procedure can make use of these sparse characteristics to increase channel estimation accuracy. The estimated channel is then utilised to decode the multiuser information data after the channel state information has been estimated.

In order to meet the high reliability and green energy-saving requirements of the IoV, low-order pure phase modulation mode is usually adopted. The receiver can obtain phase information from the received signal. In the scenario where a large amount of information needs to be transmitted, solely relying on phase to carry information cannot meet the requirement of high data rate. Hence, it is necessary to consider the QAM modulation mode in which both the amplitude and the phase carry information. However, since the receiver can only detect phase information, the means to estimate channel state information with both amplitude and phase information poses a challenge. An additional channel that includes amplitude and phase information can be added between the transmitter and receiver. Different from single-information channels, the receiver of this full-information channel can decode both amplitude and phase information. This full-information channel can be formed by directly superimposing all single-antenna channels together in a multi-input multi-output (MIMO) system. The block diagram of the communication system with the full-information channel is shown in Figure 1. Each receiving antenna first allocates the power of the received signal. The power of the α ratio is allocated to the full-information channel, and the power of the 1-α ratio is allocated to the phase detector, i.e. the single-information channel. The increased energy consumption and processing overhead can be ignored if the number of single-information channels is significantly greater than that of full-information channels. The estimator of the single-information channel relies on the detector of phase to obtain the phase information of the channel. The estimator of the full-information channel relies on the detector of amplitude and phase to estimate the full information of the channel. Information concerning the two parts of the estimated value are then exchanged to obtain more accurate channel estimation. Finally, the information data symbols sent with the pilot symbols are decoded according to the estimated channel.

In the future, the number of terminals of the IoV will become greater. More data will need to be transmitted by the terminals. The problem of channel estimate gets more challenging as additional antennas are mapped to the full-information channel. Additionally, multiuser detection becomes a more challenging process. To address this issue, the single-antenna channels can be divided into several groups. The single-antenna channels in each group are superimposed together to form a full-information channel. Each full-information channel is first estimated. Then, the estimation results of different full-information channels are combined to form the estimated value of the entire multi-antenna communication channel. Such a structure is equivalent to a simple repetition of multiple full-information channel communication systems. In order to increase the redundancy of the full-information channels, a channel corresponding to one antenna can be mapped to multiple full-information channels. When multiuser detection is performed, multiple full-information channels are jointly estimated to obtain better estimation performance. The block diagram of such a communication system is shown in Figure 2.

It is shown in the Figure 2 that each full-information channel is formed by the superposition of several single-antenna channels. These single-antenna channels corresponding to a full-information channel are selected by a random selector. The random selector completes the mapping task through a pseudo-random sequence. Each symbol of the pseudo-random sequence corresponds to a single-antenna channel. The single-antenna channel participates in the full-information channel when the symbol ‘1’ is used, and does not participate in it when the symbol ‘0’ is used. According to the received pilot signal, both the detector of phase in single-information channel and the detector of amplitude and phase in full-information channel need to estimate the channel. What poses a challenge is the formulation of a method by which the two detectors can be arranged in such a way that they cooperate to obtain more accurate channel estimation with full information. It should be emphasised that the state information includes both amplitude information and phase information.

Each receive antenna gets signals from all transmit antennas. Directly employing orthogonal pilot sequences will result in a higher proportion of pilot signals within the channel coherence time and significantly reduce transmission efficiency. Therefore, it is necessary to use non-orthogonal pilot sequences. These pilot signals are superposed together. Given that the number of antennas in the base station and the number of terminals are both large, what poses a challenge is the development of a method of detecting the desired pilot signal from the superposed signals belonging to different terminals.

In the IoV system’s uplink, the terminal sends the information data signal and the pilot signal to the base station. Prior to decoding the information data, the base station performs channel estimation based on the received pilot signal. In order to improve the transmission efficiency, it is ensured that the length of the pilot sequence is less than the number of sub-channels. The length of the pilot sequence and the number of sub-channels are denoted as

The pilot signal received by an antenna in base station can be expressed as:
_{K} is a constant with value <1,

In the system shown in Figure 1, the estimation result of the full-information channel is actually the superposition of multiple single-antenna channels with full-information. Each channel state information of single-antenna needs to be separated from the total state information. The successive interference cancellation algorithm can be used to accomplish this task. Since the detection of the first single-antenna channel is greatly interfered by other single-antenna channels, the detection accuracy is poor. As a result, it degrades the accuracy of estimation for subsequent single-antenna channels. In order to improve the channel estimation accuracy, it has to be ensured that the number of single-antenna channels superimposed in the full-information channel cannot be too large. Apart from this, a single-antenna channel can be mapped to different full-information channels. After estimating the state information of the same single-antenna channel in different full-information channels, this information can be exchanged with each other. Then each estimator estimates the channel repeatedly to obtain a more accurate estimation. The estimation process will be terminated when the maximum number of iterations is reached or when the estimation result converges. We use circles to represent single-antenna channels and squares to represent full-information channels. If a single-antenna channel b0 is mapped to the full-information channel c0, a line segment is drawn between the circle b0 and the square c0. It indicates that there is a mapping relationship between the two channels. For example, there are seven single-antenna channels and three full-information channels in Figure 3.

Assuming that there are three full-information channels, the flow chart of the estimation process of all single-antenna channels is shown in Figure 4.

The estimation process of the channel is described as follows. Firstly, the terminal transmits the pilot signal. Each single antenna of the base station estimates the phase information of the channel based on the received pilot signal. This phase information is used to create the initial value of each single-antenna channel with the full information. Then the full-information channel 1 estimates those single-antenna channels that have a mapping relationship with it. The method of successive interference cancellation is used to separate the independent single-antenna channel from the superposition of all channel state information. This step is equivalent to updating the state information of these single-antenna channels. The updated channel state information of single-antenna is sent to the estimator of full-information channel 2. The full-information channel 2 estimates those single-antenna channels that have a mapping relationship with it based on the received pilot signal and updated channel information. The updated channel state information of single-antenna is sent to the estimator of full-information channel 3. A similar process applies to full-information channel 3. The updated single-antenna channel state information is passed to the full-information channel 1 for channel estimation update for next iteration. The above process is performed iteratively until the maximum number of iterations is reached or the estimation result is converging. The phase and amplitude information of all single-antenna channels is finally obtained.

The EM algorithm [8] is used in the full-information channel estimation process. First, the channel parameters are estimated according to the pilot response signal. Then, the missing data are estimated under the parameter model. Next, the channel parameters are re-estimated according to the estimated missing data and the pilot response signal. The above process is repeated iteratively until the maximum number of iterations is reached. The minimum mean square error algorithm is used for the channel estimation in each iteration. Since the single-information channel directly discards the amplitude information, it is very suitable for the EM algorithm to estimate the full information of the channel based on phase information.

In order to measure the accuracy of the channel estimation, the normalised mean square error (NMSE) is adopted, which is calculated by the following formula:

The comparison of the estimation accuracy of the full-information channel with mapping different numbers of single-antenna channels to the same full-information channel is shown in Figure 5. In the simulation, the base station has a total of eight antennas and one full-information channel. The full-information channel is variously mapped with one, two, four, six and eight single-antenna channels. It can be seen from the simulation results that, with the increase of the number of single-antenna channels, the channel estimation accuracy is also improved. However, the performance improvement becomes smaller and smaller. This is owing to the larger number of single-antenna channels giving the greater mutual interference, which degrades the improvement of the estimation accuracy of each single-antenna channel.

The comparison of the estimation accuracy of the full-information channel with different mapping methods of single-antenna channels mapping to full-information channels is shown in Figure 6. Assume that there are three full-information channels in the simulation. The base station has a total of seven antennas, and the mapping schemes are shown variously in Figures 3(A)–3(D). It can be seen from the simulation results that the channel estimation accuracy of schemes (A) and (D) is better. The channel estimation accuracy of schemes (B) and (C) is poor. The reason for poor performance in scheme (B) is that the messages are repeatedly transmitted among the nodes b1, b3, c1, and c2. The channel information cannot be effectively updated. In scheme (C), the single-antenna channels are divided into two independent groups. The channel information cannot be exchanged between the two groups. The channel information cannot be effectively passed, which results in the decrease in the estimation accuracy.

The comparison of the estimation accuracy with different iterations is shown in Figure 7. The number of iterations is variously two, four, six, and eight. The base station has a total of eight antennas, to which two full-information channels need to be mapped. The mapping method adopts the random mapping method. The comparison result shows that the channel estimation accuracy gradually increases and then tends to be stable as the number of iterations increases. It indicates that the benefits obtained by increasing the number of iterations approach to a negligible degree as this number increases. It shows the exchange of information has been fully utilised.

The comparison result is listed in Table 1 with the information transmission rate, channel estimation accuracy, bit error rate and power consumption. There are one, two and three full-information channels in schemes 1, 2 and 3. Assume that the baud rates are 1M symbols per second. The single-antenna channel adopts 2PSK modulation, and the full-antenna channel adopts 64 QAM modulation. The power consumption of the single-antenna link is 0.25 W and the power consumption of the full information link is 1 W. The signal-to-noise ratio in the channels is 10 dB. The comparison results show that the proposed scheme can obtain the benefit of 6 times the information rate at the expense of 2.5 times the power consumption.

Performance comparison of different schemes

Scheme | Baud rates (Mbps) | NMSE | Power (W) | BER |
---|---|---|---|---|

Without full-information channel | 1 | 0.23 | 2 | 4.9 × 10^{–4} |

1 full-information channel | 6 | 0.52 | 3 | 4.1 × 10^{–3} |

2 full-information channels | 6 | 0.44 | 4 | 1.9 × 10^{–3} |

3 full-information channels | 6 | 0.37 | 5 | 8.8 × 10^{–4} |

In order to meet the requirement of high-speed data rate in IoV, the paper proposes a scheme of adding a small number of full-information channels based on the single-information channel. These two types of channels constitute a nonlinear channel. Aiming at the channel estimation problem of the nonlinear channel, the paper proposes a compressed sensing based channel estimation algorithm. By designing the mapping relationship between the single-information channel and the full-information channel, the channel information is exchanged between different full-information channels. Then, the channel estimator performs channel estimation again to obtain a more accurate channel estimation. The paper also simulates the accuracy of channel estimation under different conditions and analyses the hidden reasons behind the phenomenon. Simulation results show that the proposed scheme can achieve a substantial improvement in the data rate at the expense of a little power consumption.

#### Performance comparison of different schemes

Scheme | Baud rates (Mbps) | NMSE | Power (W) | BER |
---|---|---|---|---|

Without full-information channel | 1 | 0.23 | 2 | 4.9 × 10^{–4} |

1 full-information channel | 6 | 0.52 | 3 | 4.1 × 10^{–3} |

2 full-information channels | 6 | 0.44 | 4 | 1.9 × 10^{–3} |

3 full-information channels | 6 | 0.37 | 5 | 8.8 × 10^{–4} |

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prediction of e-commerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Evaluation and prediction of regional human capital based on optimised BP neural network Study on inefficient land use determination method for cities and towns from a city examination perspective A study of local smoothness-informed convolutional neural network models for image inpainting Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network