# A Semiblind Receiver for a Two-Way UAV-Aided PIC System Based on the PARAFAC Model

Authors: Xi Han; Yichen Liu; Qin Liu; Lijun Wang; Zhengping Li
Journal: International Journal of Antennas and Propagation
 (2022)

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## 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.


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## Body

## 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-casex, bold-face capital X, and calligraphic X letters, respectively. XT, X†, and X−1 are the transpose, the pseudoinverse, and the inverse, respectively. A^ represents the estimation of A. The diaga operator forms a diagonal matrix by putting the vector a on its main diagonal. °, ⊗, and ⊙ denote the outer, the Kronecker, and the Khatri–Rao matrix products, respectively.


## 1.1. Notations

Column vectors, matrices, and tensors are denoted by boldface lower-casex, bold-face capital X, and calligraphic X letters, respectively. XT, X†, and X−1 are the transpose, the pseudoinverse, and the inverse, respectively. A^ represents the estimation of A. The diaga operator forms a diagonal matrix by putting the vector a 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 Figure1. In this system, two sensors and R UAVs are assumed to be users and relays, respectively, with M1, M2, and Mr 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 X1∈CM1×Np and X2∈CM2×Np where Np is the pilots’ length. The channel matrices between the users and the UAVs are represented by H1∈CMr×M1 and H2∈CMr×M2, respectively. The matrix r∈CMr×Np is the signal received by the relays, with the additive noise vectors nR∈CMr×1 at the relay station. The signals received by the relays can be written as follows:(1)r=H1X1+H2X2+nR.Figure 1
A UAV-aided PIC system.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:(2)Y1w=C×1H1T×2r+W1+N1,Y2w=C×1H2T×2r+W2+N2,where W1 and W2 are the pulse interference, with the noise N1 and N2, and C 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 Y1w, which is the sum of the transmitted signal Y1 and the pulse interference W1. Wr, the interference sampling of interference source, is another pulse interference related to W1. The adaptive filter adjusts its own parameters so that the output yn at a time n is the best estimate of W1. wn represents the filter weighting coefficient obtained by the adaptive algorithm at time n. The error signal en is the difference between yn and the input signal, which means en is the best estimate of the useful signal. Therefore, the output of the system is the useful signal obtained by filtering. In Table 1, Xn = xn,xn−1, …, xn−L+1, and Wn = wn,wn−1, …, wn−L+1 are the actual input vector and filter weight coefficient vector, respectively. L is the filter length, u is the fixed step. The convergence condition of the algorithm is that the step factor u satisfies the following:(3)0<u<1λmax,where λmax 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:(4)Y1=C×1H1T×2r+N1,(5)Y2=C×1H2T×2r+N2.Figure 2
Diagram of pulse interference cancellation.Figure 3
The structure of the adaptive filter.Table 1
The LMS algorithm.
Step 1: initialize parameters and the initial value of the adaptive filter weighting coefficientW0. The order and step value are set as M and u, respectivelyStep 2: calculate the output result of the filter estimationyn=WTnXnStep 3: calculate the error of the filteren=Y1w−ynStep 4: the filter weighting coefficient is updated town+1=wn+2uenXnStep 5: repeat steps 2–4 above until the weight coefficient of the filter approaches the best filterIn order to obtain the channel information at the two user terminals, we make the following definitions:(6)H=H1,H2∈CMr×M1+M2,X=X1X2∈CM1+M2×Np.


## 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 ofH1 and H2 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] C is considered, with rank Mr. By using the parallel factorization method, C can be decomposed as follows:(7)C=I3,Mr×1C1×2C2×3C3,where I3,Mr is the multidimensional matrix of size Mr×Mr×Mr and the matrices C1,C2,C3∈CMr×Mr 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 C. Inserting (7) into (4), we can get the following:(8)Y1=I3,Mr×1H1TC1×2XTHTC2×3C3.According to the slice of the multidimensional matrix [15], the 3-mode unfolding of this matrix can be written as follows:(9)Y13=C3H1TC1⊙XTHTC2T.Here, we use elementary properties of then mode determinant of linear algebra. The Khatri-Rao product ⊙ can be isolated with the inverted operation of C3. Therefore, the orthogonal matrix C3 can be used to isolate the Khatri–Rao product as follows:(10)C3†Y13T=H1TC1⊙XTHTC2.SinceC3 is the orthogonal matrix, the Moore–Penrose pseudo inverse-operation C3† causes the Khatri-Rao product in (9) to be measured in reverse per column. Therefore, there must exist matrices E1∈CM1×Mr and E2∈CM2×Mr with conditions as shown in [16].(11)E1=H1TC1,(12)E2=XTHTC2.The proposed PIC-OPT receiver as described in Table2 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 M12Np+M1Mr3+Mr3.Table 2
The PIC-OPT receiver.
Step 1: use the LMS adaptive filter to eliminate the pulse interference, which could eliminate the correlation linear weight of the signalStep 2: joint estimation ofE1 and E22.1 according to (11)–(13), we set Ψ≈E1⊙E2; the mth column can be obtained as Ψm≈e1,m⊗e2,m2.2 recast themth column vector Ψ as the matrix Ψ˜m≈e2,me1,mT∈CNp×M12.3 setm=12.4m←m+12.5 compute the SVD asΨ˜m=UmΣmVmHThe best rank-one approximation ofΨ˜m in the Fibonacci norm is obtained by e^1,m=σ1v1,m∗,e^2,m=σ1u1,mWhereσ1 is the largest singular value, and u1,m and v1,m are the first columns of Um and Vm, respectively2.6 go to 2.4 untilm=MrStep 3: calculate the estimates ofH1 and H2We design a pilot matrix with orthogonal rows, with the restrictive conditionNp⩾M1+M2. Additionally, we assume that the pilots transmitted by two users are orthogonal to each other, which can be written as follows:(13)X1T†XT=IM1,0M1×M2,(14)X2T†XT=0M2×M1,IM2.In consideration of the orthogonality constraint ofX, substitute (15) in (13) and we can obtain the following:(15)E˜2=X1T†E2=H2TC2.The channelH1 and H2 can be estimated as follows:(16)E1C1†=H1T,E˜2C2†=H2T.According to the uniqueness theorem of PARAFAC model decomposition, (8) is decomposed to obtain unique E1 and E2, then the Kruskal rank of the three-factor matrices in the PARAFAC model satisfies the following [17]:(17)kE1+kE2+kC3⩾2Mr+2,where kE1 is the Kruskal rank of the matrix E1 (similarly to E2 and C3. In addition, we design the pilot matrix X with orthogonal rows, with the restrictive condition Np⩾M1+M2. It is assumed that C3 satisfies k rank, and H1 and H2 are random matrices, so the factor matrices of the PARAFAC model all satisfy K rank. (20) that can be expressed as follows:(18)Mr+minM1,Mr+minNp,Mr⩾2Mr+2.Then, the loading matricesE1, E2, and C3 are essentially unique up to the permutation and scalar ambiguity, meaning that any other triple E^1,E^2,C^3 is related to E1,E2,C3 via(19)E1=E^1Π1Δ1,E2=E^2Π2Δ2,C3=C^3Π3Δ3,where Π∈CMr×Mr is the permutation matrix, and Δi∈CMr×Mri=1,2,3 is the diagonal scaling matrices satisfying Δ1Δ2Δ3=IMr. Considering that the system is a semiblind receiver, the encoding matrix C3 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:(20)Hτ=∑l=1LαlαMSθlαBSTφlδτ−τl.This mmWave channel model withL scatterers between the user and the base station. τl, φl, and θl∈0,2π represent the time delay, angles of arrival, and departure of a scatter, respectively. δ⋅ is the delta function, αl is the complex path gain of the l-th path, and αMSθl and αBSφl 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 τl, and φl, as well as θl∈0,2π. Therefore, we can write the channel matrix Hk with the kth subcarrier as follows:(21)Hk=∑l=1Lαlexp−j2πτlfsk/K¯αMSθlαBSTφl,where fs is the sampling rate. In order to obtain Hk from the received signals, we can construct multidimensional matrices by assuming the digital precoding matrices Fkt and the pilot symbols Skt remain the same in different subcarriers, which means(22)Fkt=Ft,Skt=St,k=1,…,K.Therefore, the received signals at the kth subcarrier are as follows:(23)Yk=QkTHkP+Wk,k=1,…,K,where Qk is comprised of the radio frequency (RF) combining vector pt used at the kth subcarrier, where pt=FRFFtSt with t=1,2,…,T, and FRF is a common RF precoder for all subcarriers. In this way, Yk can be deduced as follows:(24)Yk=∑l=1Lα˜l,kQkTαMSθlαBSTφlP+Wk=∑l=1Lα˜l,kα˜MSθlα˜BSTφl+Wk.This can be written as a style of multidimensional matrix comprised of vectors.(25)Y=∑l=1Lα˜MSθl°α˜BSφl°el+W,and can be written in the matrix form as follows:(26)Y=I×1E×2B×3A+W,where E,B,andA are the factor matrices of this decomposition. Considering the sparse scattering nature of the mmWave channel, L is usually smaller than the dimensions of Y, which means the multidimensional matrix Y 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 τl,θl,φl,αl, which construct the mmWave channels by performing a multidimensional matrix decomposition of the received signal Y.


## 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].X, C1, and C3 are set as discrete Fourier transform (DFT) matrices. For the first case in Figure 4, the number of antennas is set as M1=M2=4, and Mr=3, which satisfies the first case with X∈C8×8. The factor matrices are C1=D3, C2=I3, and C3=D3, where D3∈C3×3 is the DFT matrix. As for the second case, the number of antennas is set as M1=M2=5, Mr=8, X∈C10×10, and C1=D8, C2=I8, and C3=D8, where D8∈C8×8 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.Figure 4
The rMSE and BER performances of the PIC-OPT algorithm compared with those of the LS algorithm and the MMSE algorithm.
(a)(b)Figures5(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 Mr=4, which satisfies M1,M2⩾Mr. For the case in Figure 5(b), the number of relay antennas is set as Mr=8, which satisfies M1,M2<Mr. 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.Figure 5
The rMSE performance with a different number of antennas. (a)Mr=4. (b) Mr=8.
(a)(b)Figure6 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.Figure 6
The CCDF performance of the PIC-OPT algorithm compared with 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].



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