Abstract
In this paper, we consider a two-way communication system using the unmanned aerial vehicle (UAV) as a relay (UAV-aided). This system eliminates impulse interference through an adaptive filter based on the least mean square (LMS) and uses the received signal transmitted by the UAVs to construct a parallel factor (PARAFAC) model. Based on the identifiability condition of the PARAFAC model, a pulse interference cancellation orthogonal pilot tensor (PIC-OPT) receiver without iteration is proposed. Our algorithm is also used in millimeter-wave to achieve the acquisition of channel information. Compared with the least squares method, the simulation results demonstrate the superiority of the proposed semiblind receiver in terms of the relative mean square error and bit error rate.
1. Introduction
With the rapid development of the growing demand for mobile Internet services, two-way communication technology is now widely used in smart metering, security, and load management [1] and has become a research hotspot in the field of modern communications. The use of unmanned aerial vehicles (UAVs) as relays (UAV-aided) for information transmission in the two-way system has gradually come into our field of vision, for their ability to improve spectral efficiency and reduce end-to-end delay and cost [2, 3]. The reliability of signal detection in the two-way UAV-aided system strongly depends on the accuracy of the channel state information (CSI) for all the links involved in the communication process [4].
In two-way communication systems, the main methods to obtain CSI include the autoregressive method [5], the pilot-based channel estimation method [6], and the tensor-based method [7]. In [6], two-channel estimation algorithms, namely the superimposed channel training scheme and the two-stage channel estimation algorithm, were developed for two-way multiple-input multiple-output (MIMO) relay communication systems. In [7], to obtain accurate channel state information, a tensor-based channel estimation algorithm was proposed for two-way MIMO relay systems.
In recent years, tensor-based semiblind receivers have been proposed to solve channel estimation problems with no or only a small number of pilot sequences. The successful application of tensor methods in two-way cooperative systems was proposed in [8–10] using parallel factor (PARAFAC) models. In view of the advantages of tensors in constructing receivers, they can be used to solve the problems of signal and CSI acquisition in two-way UAV-aided communication systems.
In this paper, a two-way UAV-aided cooperative communication system with pulse interference cancellation (PIC) is considered. The signals are transmitted in two stages: first, the two users send signals to the UAVs as relays, and second, the received signals are amplified and forwarded to both users, and the two users still send information to the relay. Pulse interference is introduced in the transmission and needs to be eliminated first. Then, a PIC orthogonal pilot tensor (PIC-OPT) receiver is proposed to be used to obtain accurate CSI. Compared with other iterative receivers, such as alternating least squares (ALS) receivers and minimum mean square error (MMSE) receivers, the proposed receiver has lower complexity and requires no iteration. Simulation results show the superiority of the PIC-OPT receiver in terms of the relative mean square error (rMSE).
1.1. Notations
Column vectors, matrices, and tensors are denoted by boldface lower-case , bold-face capital , and calligraphic letters, respectively. , , and are the transpose, the pseudoinverse, and the inverse, respectively. represents the estimation of . The operator forms a diagonal matrix by putting the vector on its main diagonal. , and denote the outer, the Kronecker, and the Khatri–Rao matrix products, respectively.
2. Two-Way UAV-Aided PIC Cooperative Communication System
A two-way UAV-aided cooperative communication system is considered, as illustrated in Figure 1. In this system, two sensors and UAVs are assumed to be users and relays, respectively, with , , and antennas. The signals may be attenuated in the channel, and other disturbances may be introduced. The entire transmission process can be divided into two phases. In the first phase, the users need to send their own information to the relays. In the second phase, the relays amplify and forward the information received in the first phase to the users. The information sent to relays is represented by matrices and where is the pilots’ length. The channel matrices between the users and the UAVs are represented by and , respectively. The matrix is the signal received by the relays, with the additive noise vectors at the relay station. The signals received by the relays can be written as follows:
In the second phase, the relays amplify and transmit signals to the users. It is assumed that the channel has reciprocity in this system. Therefore, the signals received by the two users can be expressed as follows:where and are the pulse interference, with the noise and , and is the coding matrix. In this situation, an adaptive filter is proposed to eliminate pulse interference, as shown in Figure 2. The least mean square (LMS) is the most common adaptive filtering algorithm in real life. The principle of LMS is to minimize the system output by constantly adjusting the weight of the finite impulse respond (FIR) filter. This method can eliminate the correlation linear weight of the received signal and the reference signal. The structure of the adaptive filter is shown in Figure 3 and the specific steps of the LMS algorithm are shown in Table 1. For example, our input signal is , which is the sum of the transmitted signal and the pulse interference . , the interference sampling of interference source, is another pulse interference related to . The adaptive filter adjusts its own parameters so that the output at a time is the best estimate of . represents the filter weighting coefficient obtained by the adaptive algorithm at time . The error signal is the difference between and the input signal, which means is the best estimate of the useful signal. Therefore, the output of the system is the useful signal obtained by filtering. In Table 1, = , …, , and = , …, are the actual input vector and filter weight coefficient vector, respectively. is the filter length, is the fixed step. The convergence condition of the algorithm is that the step factor satisfies the following:where is the maximum eigenvalue of the autocorrelation matrix of the input signal of the adaptive filter. The adaptive filter can achieve a cancellation amplitude of 20 dB [11, 12]. The hardware circuit design is divided into two parts. In the first part, the transmitter sends the baseband signal through the digital-to-analog converter (DAC) and the digital up-conversion, and in the second part, the receiver converts the received signal through the digital down-conversion and the analog-to-digital converter (ADC) into the baseband signal. Because it is disturbed by pulse interference at the receiving end, the adaptive filter is used to eliminate the interference. After interference processing, the received signals can be simplified as follows:
In order to obtain the channel information at the two user terminals, we make the following definitions:
3. The PIC-OPT Receiver
In this part, the technology of multidimensional matrices is used to solve the channel estimation problem. Through this solution, estimates of and can be calculated from training data. First, for the convenience of calculation, we only write the equation without the presence of noise, which achieves an approximate result. In addition, we first obtain the solution for the first user and the solution for the second user is similar.
Initially, a multidimensional matrix [13, 14] is considered, with rank . By using the parallel factorization method, can be decomposed as follows:where is the multidimensional matrix of size and the matrices are the factor matrices of this decomposition. In order to reduce the calculation difficulty, we select to deduce the design rules for these factor matrices from the derivation of the channel estimation algorithm instead of the multidimensional matrix . Inserting (7) into (4), we can get the following:
According to the slice of the multidimensional matrix [15], the 3-mode unfolding of this matrix can be written as follows:
Here, we use elementary properties of the n mode determinant of linear algebra. The Khatri-Rao product can be isolated with the inverted operation of . Therefore, the orthogonal matrix can be used to isolate the Khatri–Rao product as follows:
Since is the orthogonal matrix, the Moore–Penrose pseudo inverse-operation causes the Khatri-Rao product in (9) to be measured in reverse per column. Therefore, there must exist matrices and with conditions as shown in [16].
The proposed PIC-OPT receiver as described in Table 2 includes the PIC using an adaptive filter and the singular value decomposition (SVD) of the channel matrices recovery. The computation complexity of the proposed receiver is .
We design a pilot matrix with orthogonal rows, with the restrictive condition . Additionally, we assume that the pilots transmitted by two users are orthogonal to each other, which can be written as follows:
In consideration of the orthogonality constraint of , substitute (15) in (13) and we can obtain the following:
The channel and can be estimated as follows:
According to the uniqueness theorem of PARAFAC model decomposition, (8) is decomposed to obtain unique and , then the Kruskal rank of the three-factor matrices in the PARAFAC model satisfies the following [17]:where is the Kruskal rank of the matrix (similarly to and . In addition, we design the pilot matrix with orthogonal rows, with the restrictive condition . It is assumed that satisfies k rank, and and are random matrices, so the factor matrices of the PARAFAC model all satisfy K rank. (20) that can be expressed as follows:
Then, the loading matrices , and are essentially unique up to the permutation and scalar ambiguity, meaning that any other triple is related to viawhere is the permutation matrix, and is the diagonal scaling matrices satisfying . Considering that the system is a semiblind receiver, the encoding matrix is known and full rank, so the permutation ambiguity is not considered.
4. Extension in Millimeter Wave MIMO Systems
In the following, we show that the proposed algorithm can also be used to obtain the channel information in millimeter wave MIMO systems. In a downlink millimeter wave MIMO system, the channel matrix in the time domain can be written as follows:
This mmWave channel model with scatterers between the user and the base station. , , and represent the time delay, angles of arrival, and departure of a scatter, respectively. is the delta function, is the complex path gain of the -th path, and and are the antenna array response vectors at the transmitter and the base station. Because of the random distribution of scatterers in space, we assume that different scatterers have different , and , as well as . Therefore, we can write the channel matrix with the kth subcarrier as follows:where is the sampling rate. In order to obtain from the received signals, we can construct multidimensional matrices by assuming the digital precoding matrices and the pilot symbols remain the same in different subcarriers, which means
Therefore, the received signals at the kth subcarrier are as follows:where is comprised of the radio frequency (RF) combining vector used at the kth subcarrier, where with , and is a common RF precoder for all subcarriers. In this way, can be deduced as follows:
This can be written as a style of multidimensional matrix comprised of vectors.and can be written in the matrix form as follows:where are the factor matrices of this decomposition. Considering the sparse scattering nature of the mmWave channel, is usually smaller than the dimensions of , which means the multidimensional matrix has a low-rank structure, which ensures that the canonical decomposition [16] is unique with scaling and permutation ambiguities. Therefore, we can obtain the estimation parameters , which construct the mmWave channels by performing a multidimensional matrix decomposition of the received signal .
5. Simulation Results
In this section, 5000 Monte Carlo simulations are used to verify the performance of the proposed semiblind receiver. The rMSEs of channel matrices are used to characterize the estimation performance [18, 19].
, , and are set as discrete Fourier transform (DFT) matrices. For the first case in Figure 4, the number of antennas is set as , and , which satisfies the first case with . The factor matrices are , , and , where is the DFT matrix. As for the second case, the number of antennas is set as , , , and , , and , where is the DFT matrix. In Figures 4(a) and 4(b), we show the estimation accuracy of the channel and symbol in both cases. It is easy to see that the proposed method has lower rMSE and bit error rate (BER) compared with the least squares (LS) algorithm and MMSE algorithm under both conditions, and as the signal-to-noise ratio (SNR) value increases, the reductions in the rMSE and BER are observed.
(a)
(b)
Figures 5(a) and 5(b) show the influence when the number of antennas at the relays and the users is different. For the case in Figure 5(a), the number of relay antennas is set as , which satisfies . For the case in Figure 5(b), the number of relay antennas is set as , which satisfies . With the increasing number of users antennas, the spatial diversity gain increases, which improves the estimation accuracy and the rMSE performance of the algorithm gradually improve.
(a)
(b)
Figure 6 plots the complementary cumulative distribution function (CCDF) of the rMSE for a fixed SNR of 30 dB and randomly drawn channel realizations. The rMSE value of the proposed method is concentrated in the lower range than that of the LS algorithm and the MMSE algorithm.
6. Conclusion
In this paper, a PIC-OPT receiver based on the PARAFAC model of a two-way MIMO UAV-aided cooperative communication system is proposed. The simulation results show that compared with the traditional LS receiver, the proposed PIC-OPT receiver has better rMSE performance and BER performance. Our numerical results confirm the effectiveness of the proposed receiver. In addition, the application of this method in millimeter-wave MIMO systems is a prospect of this paper. Our work considers the PIC-OPT receiver in stationary scenarios, and the extended work including channel modeling and estimation in some high-mobility scenarios can be considered [20, 21].
Data Availability
The data used in this paper are provided by simulations and that the material used to support the findings of this study is available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the Key Research and Development Program (No. 2020YFC0811004), NSFC under Grant 62001008, Beijing Municipal Natural Science Foundation under Grants 4212002 and L192034, Beijing Education Committee Project under Grant KM201910009011, Key Laboratory of Universal Wireless Communications Ministry of Education under Grant KFKT-2020104, and NCUT Startup Project.