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Received 19 July 2023, accepted 2 August 2023, date of publication 9 August 2023, date of current version 7 September 2023. Digital Object Identifier 10.1109/ACCESS.2023.3303814 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO With OTFS Modulation GUO-RUI WANG AND ZHONG-XUN WANG School of Physics and Electronic Information, Yantai University, Yantai 264005, China Corresponding author: Zhong-Xun Wang (ytdxwzx@163.com) This work was supported in part by the Graduate Innovation Foundation, Yantai University, under Grant KGIFYTU2306. ABSTRACT In this paper, we propose a channel estimation method for massive multi-input multioutput (MIMO) with orthogonal time frequency space (OTFS) modulation. Our method aims to reduce inter-symbol interference (ISI) when the subtraction of the two delay paths is smaller than the resolution of the system. We propose a 3D inner product proportion reduce difference structured orthogonal matching pursuit (3D-IPRDSOMP) algorithm that combines the characteristics of 3D structured sparse channels. With knowledge of the number of dominant paths, we use the number of local optimums to match the number of distinguishable dominant paths with backtracking. The algorithm approximates similar and indistinguishable dominant paths as identical ones. The estimated number of distinguishable paths as the iteration’s termination condition increases the channel estimation’s accuracy. Simulation results show better performance than the traditional algorithm under different conditions. INDEX TERMS Massive MIMO, OTFS, OMP, compressed sensing, channel estimation. I. INTRODUCTION In the future, more wireless communication scenarios will be considered, particularly wireless communication in high-mobility situations become more important [1], [2]. Motivated by the demand, the amount of related research is increasing. However, orthogonal frequency division multiplexing (OFDM) modulation is sensitive to Doppler shift, significantly affecting communication efficiency [3]. The two modulations are related in the mathematical construction, and OTFS modulation is considered a precoding scheme based on OFDM [4]. To solve the problem, an OTFS modulation scheme was proposed in 2D modulation, which provides both time and frequency diversity gain, transforming the time-varying channel into a quasi-time-invariant channel [5]. The efforts have addressed the formulation, equalization, estimation, and detection of the input-output relations in vectorization. OTFS massive MIMO allows for higher spectrum efficiency and robustness in rapidly changing channels [6], [7], [8]. A framework for channel estimation schemes in OTFS massive MIMO systems was proposed in [7]. There is sparThe associate editor coordinating the review of this manuscript and approving it for publication was Li Zhang. VOLUME 11, 2023 sity in the delay and Doppler dimension. This property is exploited to reduce the pilot overhead and complexity channel estimation. Massive MIMO systems offer substantial benefits in enhancing spectrum efficiency. However, the downlink channel ought to be estimated to obtain channel state information (CSI). The uplink and downlink channel state information for time division duplexing (TDD) is complex conjugate and interconverted without transmitting CSI information [9]. In [10], the authors proposed methods for the semi-blind estimation of sparse channels in MIMO systems. This approach reduces the computational complexity by converting the semi-blind estimation problem into a reduced-rank filtering problem through a novel formulation of the sparse matrix. In [11], the authors estimated channels in MIMO systems with discrete priors. Channel estimation is an essential component of signal demodulation, and the channel estimation algorithms in the delay-Doppler domain are described in [12] and [13]. Similarly, obtaining CSI for OTFS massive MIMO systems is essential, and accurately estimating the CSI of the channel is a challenging topic. In [14] and [15], the authors proposed a sparse reconstruction algorithm for block sparse signals, using the prior information of block sparsity. This approach enables the recovery of the estimated sparse block signals at low sampling rates. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/ 94679 G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO TABLE 1. Summary of literature survey. In OFDM massive MIMO systems, [6] proposed a sparse reconstruction channel estimation scheme. With the idea, we apply it to OTFS massive MIMO channel estimation problem, where we reconstruct the channel matrix using fewer pilots because of the prior information that the channel is known to be sparse. In [20], the authors proposed a compressed sensing channel estimation model for traditional multipath channels. The application of compressive sensing in the channel and a novel method for estimating sparse multipath channels are proposed in [21]. The input-output models of OTFS single-input singleoutput (SISO) systems use a method that transmits the received signal as a two-dimensional (2D) convolution in the delay-Doppler domain and a pulse-based channel estimation technique. A pseudo-random noise (PN) sequence was proposed by the pilots in the delay-Doppler domain, and they proposed that a long PN sequence achieved better performance [22]. In [7], the authors proposed an estimation channel model with compressed sensing for OTFS massive MIMO systems. They compared the traditional orthogonal matching pursuit (OMP) algorithm with the proposed 3D structured orthogonal matching pursuit (3D-SOMP) algorithm. The 3D-SOMP channel estimation algorithm increases the performance with the number of dominant paths as the prior information. However, this is not a practical sparse channel recovery algorithm. There may be similar delay paths in the channel model, for example, the spatial channel model (SCM) channel [23]. If the interval of the dominant paths is below the delay resolution of OTFS systems, it is considered a single path, and the estimation based on the number of dominant paths necessarily introduces error paths, leading to a degradation of the estimation performance. To solve this problem, we analyze the reasons for the poor performance of the traditional OMP algorithm. We optimize the 3D-SOMP algorithm by backtracking. We work backward from the number of dominant paths to match the local optimum near the number of dominant paths. We approximate that the solution is the number of distinguishable dominant paths. The proposed algorithm effectively reduces the selection of extra atoms because of coherent inter-symbol interference and increases channel estimation accuracy. The specific contributions are summarized as follows. 94680 1) We formulate the channel estimation problem as a sparse signal recovery problem based on the previously demonstrated characteristics of the sparsity of the OTFS massive MIMO channel structure. Then, we analyze the reason for the poor performance of the traditional OMP algorithm for sparse channel reconstruction. The traditional iterative approach gets stuck in the local optimum because of a lack of error correction mechanisms, resulting in poor channel reconstruction. Although this deficiency is compensated by setting different steps, they are required to be different in different cases, which does not fundamentally solve the problem. 2) The 3D-SOMP algorithm is designed to exploit sparsity in the time domain, block sparsity in the Doppler domain, and burst sparsity in the angular domain. It combines different methods for different dimensions. We propose the 3D-IPRDSOMP algorithm by exploiting the property that the inner product descent ratio difference of the OMP algorithm has a maximum value near sparsity. We match the index with a maximum value smaller than the number of dominant paths and use it as the number of distinguishable paths and one of the conditions for iteration termination. The rest of the paper is organized as follows. In Section II, we introduce the structural framework for OTFS SISO systems, OTFS massive MIMO systems, and discrete inputoutput relations. In Section III, the reason why the number of distinguishable dominant paths is smaller than the number of dominant paths is presented, as is the core principle of the proposed algorithm. We introduce the 3D sparsity of the OTFS massive MIMO downlink, the traditional OMP algorithm, and illustrate the principle of the 3D-IPRDSOMP algorithm. In Section IV, we present the simulation results. In Section V, we present the conclusions of the paper. II. SYSTEM MODEL In Fig. 1, the OTFS modulation is located before OFDM modulation, and the demodulation part is located after OFDM modulation, essentially the precoding operation of OFDM systems. The time-varying channel is converted into a time-invariant channel by performing a fast Fourier transform (FFT) and an inverse fast Fourier transform (IFFT) on the two dimensions of OFDM, respectively, thus reducing the performance degradation caused by the time-varying channel. VOLUME 11, 2023 G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO FIGURE 1. OTFS SISO architecture. OTFS modulation is seen as a combination of OFDM modulation and processing. A. OTFS SISO MODULATION The SISO model of the OTFS systems consists of two components: modulation and demodulation. OTFS modulation is a 2D data manipulation of the time and frequency dimensions of the OFDM channel, transforming the symbols from the time-frequency dimension to the delay-Doppler dimension. The modulation takes the M × N 2D constellation symbols (after QAM, QPSK, etc. mapping) in the delay-Doppler domain XDD [l, k] and performs an inverse symplectic finite Fourier transform (ISFFT) to transform the symbols from the delay-Doppler domain to the frequency-time domain, which are represented as XFT [f , t]. The ISFFT is divided into two parts. First, the IFFT modulation is performed on the time-delay dimension symbols in the delay-Doppler domain, transforming the symbols from the delay dimension to the frequency dimension. Then, the FFT is performed on the Doppler dimension symbols, transforming the symbols from the Doppler dimension to the time dimension. Where the mathematical expression of ISFFT is XISFFT [m, n] = √ 1 N −1 M −1 X X MN k=0 l=0 j2π XDD [l, k]e nk ml N −M , (1) where XDD [l, k] is the transmit symbol in the delay-Doppler domain, and XISFFT [m, n] is the transmit symbol in the frequency-time domain, and l = 0, 1, · · ·, M − 1, k = 0, 1, · · ·, N − 1, m = 0, 1, · · ·, M − 1, n = 0, 1, · · ·, N − 1 [19]. We convert the mathematical expression (1) into a matrix expression [7] as XISFFT = FM XDD FH N, (2) where FM ∈ CM ×M is the discrete Fourier transform (DFT) N ×N is the inverse discrete Fourier matrix and FH N ∈ C transform (IDFT) matrix, and XDD is a matrix consisting of XDD [l, k]. The symbol XISFFT needs to be added to the transmit windowing Wtx ∈ CM ×N after ISFFT, and the transmit windowing symbol obtains the frequency-time domain symbol XFT by performing Hadamard product operation with XISFFT , which XFT can be expressed as XFT = XISFFT ⊙ Wtx . VOLUME 11, 2023 (3) In this paper, we use the ideal transmit windowing and the ideal receive windowing. A transmit windowing Wtx is an allone matrix, so the time-frequency symbols in the transmitter are approximated by the symbols after ISFFT. Therefore, XFT is similar to XISFFT as XFT [f , t] = XISFFT [m, n] , (4) where f = 0, 1, · · ·, M − 1, t = 0, 1, · · ·, N − 1. The signal is converted from the frequency-time domain XFT [f , t] to the delay-time domain XIFFT [τ, t] by performing an IFFT on the frequency dimension in the 2D frequency domain symbol XFT [f , t] as M −1 1 X j2π fMτ , XFT [f , t]e XIFFT [τ, t] = √ M m=0 (5) where XFT [f , t] is the transmit symbol in the frequency-time domain, XIFFT [τ, t] is the transmit symbol in the frequency domain, and τ = 0, 1, · · ·, M − 1, t = 0, 1, · · ·, N − 1 [19]. The corresponding matrix representation as FT XIFFT = FH MX , (6) M ×M denotes the IDFT matrix, XIFFT ∈ where FH M ∈ C M ×N C is a matrix representation of XIFFT [τ, t]. For convenience of representation, the 2D matrix S ∈ CM ×N is used to denote the time-delay domain matrix XIFFT . The IFFT modulation is similar to the traditional OFDM modulation in that both convert the frequency dimension symbols into the time dimension, enabling channel transmission. The integration of the equations (2)(6) is obtained as DD H S = FH FN = XDD FH M FM X N, (7) where S = [s1 , s2 , · · · , sN ] is a M × N matrix, each vector si is an OFDM symbol frame, the bandwidth occupied by each OFDM symbol frame is M 1f , the duration is T , and the subcarrier interval is 1f . Then we transform the matrix S into a one-dimensional time vector by vectorization. To avoid inter-symbol interference, we add a cyclic prefix (CP) to the modulation by multiplying the CP addition matrix ACP ∈ C(M +NCP )×M with 94681 G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO the matrix S. The specific formula is s = vec {ACP S} , (8) where vec {} denotes the vectorizing of a matrix [7]. The resulting vector s ∈ C(M +NCP )N ×1 is the vector that is transformed into a signal that is transmitted in the time dimension after adding cyclic prefixes and vectorizing. B. OTFS SISO DEMODULATIO In this section, we describe the demodulation of OTFS SISO systems. OTFS demodulation is the modulation inverse at the receiver. First, the received time series is the vector r, which results from the vector through the channel response. The received vector r needs to be matrixed as R = invec {r} , (9) where invec {} denotes the invectorizing of a vector and R is the result of matrixing the receiver vector r, which contains the CP. Correspondingly, we use the CP deletion matrix at the receiver. In contrast to the modulation, the signal is restored to the delay-Doppler domain after FFT and SFFT. Then, the matrix R after removing CP, which is abbreviated as RDCP ∈ CM ×N , needs to be FFT in the time dimension to transform the signal from the delay-time domain to the frequency-time domain, thus obtaining the frequency-time domain receive matrix. The corresponding matrix-form transformation equation is YFT = FM RDCP , (10) where YFT ∈ CM ×N is the frequency-time domain receive matrix. The corresponding mathematical expression for the FFT is YFFT M −1 1 X j2π τMf , RDCP [τ, t]e [f , t] = √ M m=0 (12) (13) where the mathematical expression of YDD is denoted by YFT ,W [m, n], where m = 0, 1, · · ·, M − 1, n = 0, 1, · · ·, N − 94682 YDD [l, k] = √ MN N −1 M −1 X X j2π YFT ,W [m, n]e kn lm N −M , k=0 l=0 (14) where YDD [l, k] is the received symbol in the matrix YDD . Combining equation (14), the total matrix equation is obtained as YDD = FH M FM RDCP FN , (15) YDD where is the matrix in the delay-Doppler domain, corresponding to the delay-Doppler domain matrix XDD at the transmitter. The relationship between the receive matrix YDD and the transmit matrix XDD is a 2D convolution concerning the delay-Doppler channel HDD ∈ CM ×N . C. OTFS MASSIVE MIMO Previously, we introduced the discrete input-output relation for the OTFS SISO systems. To improve spectrum efficiency, we are increasingly adopting massive MIMO [24]. The next step is to extend the discrete input-output relationship from the OTFS SISO system to the OTFS massive MIMO system. The symbol Nt is the number of antennas on the transmitter and Nr is the number of antennas on the receiver. Each antenna uses OTFS modulation. In general, the rectangular windowing Wtx [n, m], Wrx [n, m] is used to match the filters at the transmitter and receiver. For every single transmitting and receiving antenna, we consider that there are P paths (taps). The index i represents the i-th delay tap τi , the Doppler shift tap νi and the channel gain hi . The impulse response in the delay-Doppler domain is h(τ, ν) = The received matrix YFT,W is subjected to an M -point IFFT that converts the symbols from the frequency dimension to the delay dimension and an N -point FFT that converts the symbols from the time dimension to the Doppler dimension. The matrix of the delay-Doppler domain signal transformation is obtained as FT,W YDD = FH FN , MY 1 (11) where RDCP [τ, t] is receive matrix after removing CP. After obtaining the frequency-time domain signal YFT (consisting of YFFT [f , t]), it needs to be transformed into the delayDoppler domain. The matrix YFT,W ∈ CM ×N is obtained by Hadamard product with the corresponding receive windowing matrix Wrx ∈ CM ×N in the previous section, where the corresponding receive windowing matrix is also the all-one matrix. The matrix form of the operation is YFT,W = YFT ⊙ Wrx . 1 and YFT ,W [m, n] is the mathematical expression consists of YFT,W . The corresponding mathematical expression is P X hi δ (τ − τi ) δ (ν − νi ) , (16) i=1 where δ (τ − τi ) and δ (ν − νi ) are the Dirac delta function with τi and νi shift, separately. Considering the number of taps present between the p-th transmitter and the q-th receiver, the impulse response in the delay-Doppler domain is hqp (τ, ν) = P X hqpi δ (τ − τi ) δ (ν − νi ) , (17) i=1 where p = 1, 2, · · ·, Nt , q = 1, 2, · · ·, Nr . In the following, the input-output relation between the transmitter and receiver antennas will be described. βi i , νi = NT where ai For SISO systems, we define τi = Mα1f and βi denote the delay taps and Doppler taps, respectively. Among them, ai and βi are considered integer parts, although fractional delay and fractional Doppler effects will be present in practical situations. The authors generalized this from SISO to the MIMO modulation in [16]. From the MIMO channel input-output relationship, we deduce that y = Hx + v, (18) VOLUME 11, 2023 G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO where the vectors x, y, v ∈ CNM ×1 , H ∈ CNM ×NM [24]. Then, we derive the discrete input-output relation in OTFS massive MIMO. With the equations of the SISO modulation, we deduce the input-output relation model in OTFS massive MIMO is y1 = H11 x1 +H12 x2 + · · · + H1Nt xNr + v1 y2 = H21 x1 +H22 x2 + · · · + H2Nt xNr + v2 (19) .. .. .. .. .. . . . . . yNr = HNr 1 x1 +HNr 2 x2 + · · · + HNr Nt xNr + vNr where Hqp denotes the equivalent channel matrix between the p-th transmit antenna and the q-th receive antenna and (19) represents a linear model of the input-output relation, which is transformed into a matrix H11 H12 ... H1Nr H21 H22 ... H2Nr HMIMO = . (20) . .. . . .. .. .. . HNt 1 HNt 2 . . . HNt Nr After further simplifying the matrix HMIMO , we obtain yMIMO = HMIMO xMIMO + vMIMO , wherevMIMO ∈ Cnr NM ×1 xMIMO ∈ Cnt NM ×1 , HMIMO ∈ Cnt NM ×na NM , which denotes the OTFS massive MIMO discrete input-output matrix. III. 3D-IPRDSOMP BASED CHANNEL ESTIMATION IN OTFS MASSIVE MIMO SYSTEMS In this section, we analyze different channel dimensions to transform a channel estimation problem into a sparse signal recovery problem. We performed an analysis of estimation errors in the OMP algorithm with different steps. Using the observation that the inner product in sparse signal recovery continuously decreases with the number of iterations and that the difference in decrease ratio is the largest value after all paths are selected. We proposed the 3D-IPRDSOMP algorithm to achieve better performance. A. DELAY PATHS AND SYSTEM RESOLVABLE DELAY PATHS Simulated channels typically include the common Rayleigh and Rice fading channels. We use the 3GPP SCM channel model based on geometric random variables to characterize the communication channel more accurately. This model includes three types: urban macro-cell, urban micro-cell, and suburban macro-cell. In the case of the urban macro as an example, the number of delay paths is six, and the delay correlation coefficient is determined by τn′ = −rDS σDS ln zn , n = 1, . . . , N , (21) where N represents the number of delayed paths, zn (n = 1, 2, · · ·, N ) obeys the uniform distribution within (0, 1), rDS is the scale factor of the delay distribution, σDS is the delay spread. The delay values of the N paths satisfy ′ > τ′ ′ the descending order τ(N ) (N −1) > τ(1) . The specific value of the delay tap τn is determined by ( ′ ) ′ τ(n) − τ(1) Tc floor , n = 1, 2, · · · , N , (22) τn = 16 Tc /16 VOLUME 11, 2023 where the function floor is obtained by rounding the nearest ′ − τ ′ = 0, integer to negative infinity [25]. Because of τ(1) (1) τ1 = 0. It represents the main path. The minimum spacing between different path delays is 1τ = Tc /16. For OTFS or OFDM systems, the matrix estimated by the channel is discrete and does not distinguish all the delay paths, which is related to the sampling rate of the system. We consider the minimum delay 1τ that is distinguished by the OTFS systems. When ∃τi − τi−1 < 1τ, i = 2, 3, · · ·, N , if the minimum interval in N delay paths is less than the resolution of the OTFS systems, the number of distinguishable delay paths of the system K < N . For the whole OTFS system, the number of delayed paths of the channel is K , not N . It is found that the relationship between inner product summation and residuals is more significant with the number of iterations in the compressed sensing channel estimation. When the dominant path N is six, the number of distinguishable delay paths of the system K ≤ 6. The most likely to occur are K = 4, 5, 6. (n+1) uτ , (23) (n) uτ where uτ represent the inner product after selecting dominant paths. It represents the correlation between the received signal and the sparse basis vector. It represents the proportional decrease of the inner product. The ∇u denotes the division between inner products. To equalize the inner product decline ratio using the residual, the formula as n ∇u n ω =1 , (24) γn ∇un = where γ represents the residual in the reconstruction process, and the confidence of the reconstruction vector increases gradually as γ decreases. After all distinguishable delay paths are selected, the decrease in the inner product significantly reduces, and the residuals become smaller and stabilize. In this case, the difference between the inner product uτ and the residual vector γ is obtained by taking the difference between the inner product and the residual vector. There is a local maximum ωNp around the number of distinguishable paths. As shown in Fig. 2, the number of iterations at the local maximum value approximates the number of distinguishable dominant paths in OTFS systems. Therefore, the number of distinguishable dominant paths is estimated in channel reconstruction with compressive sensing, reducing selected wrong dominant paths. B. 3D STRUCTURED SPARSITY CHANNEL AND TRAINING PILOTS Multiple antennas in massive MIMO systems allow the signal to be studied in spatial dimensions, increasing the information dimension. We consider a base station with Nt antennas at the base station and K single-antenna users at the receiver. The users are assumed to have some degree of concentration, but the direction between them and the base station is random. 94683 G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO FIGURE 2. Variation of differential values with the number of iterations in the channel reconstruction process in different distinguishable delay paths. A uniform linear array (ULA) of antennas is used, where the spatial angle ψsi is defined as d sin θsi , (25) λ where d is the antenna spacing and λ is the wavelength of the carrier frequency [26]. The model assumes that the system has multiple dominant paths, each with multiple sub-paths, and ignores the difference in delay in the sub-paths. The channel response in the delay-Doppler-space domain is obtained by the discrete Fourier transform (DFT) of the equation ψsi = DDS Hℓ,k,p = N X recovery problem while reducing the pilot overhead [7]. However, the sparse characteristics of the three dimensions are different, and with prior information, we adopted different schemes to recover the channel information in different dimensions. The figure above shows the location of the pilots in the delay-Doppler domain. The guard intervals are left around the pilots to prevent mutual interference between the pilot response and the data symbol response. This interference affects subsequent channel estimation and symbol detection. To reduce the correlation between pilots and increase channel recovery performance, the PN sequence is used for the pilots. Different pilots need to be adopted for different antennas, and the orthogonalized pilots distinguish users at the receiver end and reduce inter-user interference. The pilots are placed in the p-th antenna as follows. The delay domain is placed as ℓ = 0, 1, · · · , Mτ − 1, the Doppler dimension is placed as k = −Nν /2, · · · , 0, · · · , Nν /2−1, and the spatial dimension is placed on the antenna p = 0, 1, · · · , Nt − 1 for the pilots. The desired signals and the pilots on the Nt antennas of the transmitter transmit simultaneously, and the received signals are separated from the desired signals and the pilots. The received pilots in the delay-Doppler domain at the user side can be expressed as yℓ,k = g −1 NX t −1 M X Ng 2 −1 e p=0 ℓ′ =0 k ′ =− Ng 2 (M +NCP ) H DDS ℓ′ ,k ′ ,p xℓ−ℓ′ ,k−k ′ ,p + vℓ,k . (29) k h(n−1)(M +NCP )+1,(ℓ)M ,p e−j2π(n−1) N , (26) n=1 DDS channel is where the delay-Doppler-space-domain Hℓ,k,p DDA as transformed to Hℓ,k,r DDA Hℓ,k,r NX t −1 To simplify the equation (29), denote ϕℓ−ℓ′ (ℓ−ℓ′ )k ′ j2π e N (M +NCP ) . By combining (27) and (29), yℓ,k = rp DDS j2π Nt , Hℓ,k,p e (27) and −1 Mg −1 2 −1 X X X ′ r=− N2t ℓ =0 × DDS Hℓ,k,p NtX /2−1 r=−Nt /2 DDA Hℓ,k,r e pr −j2π N t = Ng Nt 2 p=0 NX t −1 k ′ =− −j2π Nrp e t ϕℓ−ℓ′ HℓDDA ′ ,k ′ ,r Ng 2 xℓ−ℓ′ ,k−k ′ ,p + vℓ,k , (30) p=0 , (28) DDS , H DDA ∈ CM ×N ×Nt represent the channel where Hℓ,k,p ℓ,k,r response in the delay-Doppler-space and delay-Dopplerangle dimension separately [7]. In OTFS massive MIMO systems, considering the case of finite channel dominant paths, the locations of the delay taps are related to the dominant channel and are relatively random. Here, the delay-time domain in the channel is considered to be absolutely sparse. Ideally, the value of the delay index is zero except for the delay tap location. In summary, the sparsity properties of channel models in three dimensions have been demonstrated, and we transformed the channel estimation problem into a sparse signal 94684 (ℓ−ℓ′ )k ′ j2π N X where k = −Nν /2, · · · , 0, · · · , Nν /2 − 1, ℓ = 0, 1, · · · , Mτ − 1, k ′ = −Ng /2, · · · , 0, · · · , Ng /2 − 1, ℓ′ = 0, 1, · · · , Mg − 1 [7]. The authors proposed the relationship between the response of the pilots after passing the system and the input signals, thus establishing a model of the DDA provide theoretical support for input-output relation to Hℓ,k,r subsequent channel estimation. The derivation process is omitted, and the final expression that conforms to the standard form of compressive sensing is obtained as y = Zc,W h + v, (31) where Zc,W = [W ⊙ Zc,−Nt /2 , . . . , W ⊙ Zc,0 , . . . , W ⊙ Zc,Nt /2−1 ] ∈ CMτ Nν ×Mg Ng Nt , Zc,r ∈ CMτ Nν ×Mg Ng , where r = VOLUME 11, 2023 G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO −Nt /2, · · · , 0, · · · , Nt /2−1, the (ℓNν +k +1+Nν /2, ℓ′ Ng + P t −1 −j2π rp Nt x k ′ + 1+ Ng /2)-th element equal to N ℓ−ℓ′ ,k−k ′ ,p p=0 e in Zc,r , where k = −Nν /2, · · · , 0, · · · , Nν /2 − 1, ℓ = 0, 1, · · · , Mτ − 1, k ′ = −Ng /2, · · · , 0, · · · , Ng /2 − 1, and ℓ′ = 0, 1, · · · , Mg − 1. Where W ∈ CMτ Nν ×Mg Ng , the (ℓNν +k +1+Nν /2, ℓ′ Ng + k ′ +1+Ng /2)-th element equal to iT h ϕℓ−ℓ′ in W, and h = hT−Nt /2 , · · · , hT0 , · · · , hTNt /2−1 − 1 ∈ CMg Ng Nt ×1 , where hr ∈ CMg Ng ×1 from HℓDDA ′ ,k ′ ,r , it correspondence the (ℓNg + k + 1 + Ng /2)-th element in hr . By simplifying the complex formula to the above form, the sensing matrix in compressive sensing is represented as Zc,W , and the channel vector is transformed into a 3D delay-Doppler-angle domain tensor. The received vector is known, and the sensing matrix (obtained from the receiver) is used to estimate the channel vectors through the sparse reconstruction principle. In this paper, the OMP and 3DSOMP algorithms are used for sparse channel reconstruction, and we analyze the reasons for the poor performance of the OMP algorithm and propose a 3D-IPRDSOMP algorithm. FIGURE 3. An OTFS frame in the delay-Doppler domain with pilots and guard intervals. C. OMP ALGORITHM ANALYSIS AND 3D-IPRDSOMP ALGORITHM PRINCIPLE The traditional OMP algorithm, as well as the 3D-SOMP algorithm, rely on sparsity as prior knowledge. However, sparsity is unknown in wireless channels, making these algorithms impractical. To solve the problem, we proposed an algorithm to estimate the number of dominant paths in the channel. It is based on the decreasing differential ratio between inner products and is applied to sparse channel reconstruction. We analyze the poor performance of the traditional OMP algorithm and propose a solution to the problem. The traditional OMP algorithm [27] recovers signals from random measurements via orthogonal matching pursuit. As one of the earliest and most widely used iterative algorithms, the matching pursuit (MP) algorithm [28] is known as the base algorithm of the greedy algorithm. For channel estimation in massive MIMO systems, the compressive sensing formulation is y = 9h + v, (32) where 9 ∈ CMτ Nν ×Mg Ng Nt is the sensing matrix, the noise vector v ∈ CMg Ng Nt ×1 [21]. The channel vector h ∈ CMg Ng Nt ×1 of massive MIMO systems is converted from the channel matrix in the delay-Doppler-space domain. The core of the OMP algorithm is to select the column with the highest correlation with the current residual vector, perform Schmitt orthogonalization and normalization on the selected atom, and update the residuals. This process ensures that the residuals are orthogonal to the previously selected atoms. In each iteration, the algorithm updates the observation matrix by removing the selected columns, which ensures that the atoms are not repeatedly selected and the residuals are rapidly reduced. By iteratively selecting the atoms with the highest correlation with the residuals and updating the residuals and VOLUME 11, 2023 FIGURE 4. The normalized mean square error (NMSE) performance curves of OMP algorithm with different step sizes. observation matrix, the OMP algorithm effectively estimates the sparse channel vector. The main steps of the OMP algorithm are as follows. First, it only selects one atom at each iteration, and once an atom is selected, it cannot be changed in the next iteration. Therefore, the larger the length of the vector h, the greater the number of atoms to be selected, and the larger the number of iterations, the greater the probability of selecting the error atom in the same case, with a certain sparsity. When only one atom is selected at each iteration, it is easy to introduce the wrong atoms and get stuck in the local optimum that does not reconstruct the channel well in subsequent iterations. Fig. 4 shows the simulation results of the OMP algorithm for different step sizes. The channel reconstruction performance increases as the step size increases and more atoms are selected at each iteration. The improvement results from the reduced probability of getting stuck in the local optimum. 94685 G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO However, excessive steps also lead to the wrong selection of atoms, significantly degrading the performance of channel reconstruction. Therefore, selecting only one atom at each iteration with the traditional OMP algorithm results in poor channel reconstruction. To solve this problem, the 3D-SOMP algorithm was proposed to increase the atom selection efficiency based on the structured characteristics of the channel and the correlation between the indexes of the vector [7]. For the delay-Dopplerangle domain, with a fixed number of dominant paths, ideally, there are non-zero values only at the delay index of the channel and zero values for the other delay indexes. The decreased ratio of the inner product of the dominant path changes drastically when the orthogonal matching pursuit algorithm selects the wrong atoms. To mitigate the effect of these extreme values in the early stage of reconstruction, we proposed using each iteration’s residual as the confidence level of the channel reconstruction effect. Specifically, we increase the estimation performance of distinguishable dominant paths by dividing the differential between indicators of the inner product decrease ratio by the residual. Then, we backtrack to find the local optimum index. The index is used as the number of distinguishable dominant paths, and the corresponding reconstructed channel vector is used as the estimated channel vector. The approach increases the reconstruction accuracy, especially for scenarios with many dominant paths and significant channel sparsity. Algorithm 1 Proposed 3D-IPRDSOMP Algorithm Input: Vector y, Sensing matrix 9 Output: Recovered channel vector b h = h(Pos) 1 Initialization: 2 i ← 0, j ← 0, max_ω = 0 (i) ← 0, r(i) = y − 9h(i) , γ i = r(i) 3 ← ∅, h 2 4 for i ≤ Np do 5 i = i + 1; 6 e = 9 H r; 7 E = invec{e}; 8 eτ (m) = E(1) (m, :) ; 9 uτ (i) = max(eτ (m)); (i) 10 mτ = arg max m eτ (m); 11 12 13 14 15 16 17 18 19 D. 3D-IPRDSOMP ALGORITHM The proposed 3D-IPRDSOMP algorithm is presented in Algorithm 1. The proposed algorithm exploits the structured characteristics of the channel and incorporates the fundamental concept of the OMP algorithm to increase channel estimation accuracy. Furthermore, it exploits the differential between mean inner products during the iterative process to reach the local maximum when searching from the known number of dominant paths to estimate the number of distinguishable dominant paths. The proposed method enables more efficient and accurate channel estimation. The traditional OMP algorithm considers that the vector h is sparse. However, it is known from [7] that the channel has different sparse characteristics in the delay-Doppler-angle domain, which is exploited by the 3D-SOMP algorithm. It has different characteristics for different algorithms, but the number of dominant paths is assumed to be known. In a wireless channel, if the delay intervals between dominant paths are smaller than the resolution of the system, it causes selected wrong paths and overestimates the channel. Therefore, the number of distinguishable dominant paths is used as an approximate number of dominant paths. The inner product vector between the residuals and the observation matrix is solved by e = 9 H r, 94686 (33) 20 21 (i) eν (n) = E mτ , n, : ; N N (i) nν = arg minn eν 2g − n : 2g + n − 1 , s.t. eν N2t − n : N2t + n − 1 ≥ ϵ ∥eν ∥; n o N N N (i) (i) (i) 3ν = 2g − nν , · · · , 2g , · · · , 2g + nν − 1 ; (i) (i) eθ (r) =?E mτ , 3ν , r ; dθ = LH eθ ; gθ (r) = ∥Dθ (r, :)∥; ps = arg maxr gθ (r); (i) 3θ = {ps, ps + 1, · · · , p s + D − 1}; (i) (i) (i) = ∪ mτ , 3ν , 3θ ; † h(i) = 9 y, h(i) r(i) = y − 9h(i) ; † = 9c y; 22 23 24 25 26 27 28 29 end for j = 0 to Np − 1 do ∇uNp −j = (Np −j+1) uτ ωNp −j = 1 (Np −j) uτ ; ∇uNp −j ; γ Np −j N −j ω p then if max_ω < max_ω = ωNp −j ; else max_index = Np − j + 1; break; 30 end 31 32 end Pos = max_index; where r ∈ CMτ Nν ×1 is the residual vector and e ∈ CMg Ng Nt ×1 is the correlation of the observation matrix with the channel residual vector. The structural characteristics of the channel are known to provide prior information. When one of the dominant path indexes is determined, the indexes of the other Doppler-angle VOLUME 11, 2023 G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO where the equation is found in the maximum gθ index, it is regarded as the starting location of the burst sparsity. Then we find the angle-dimension support with ps and D as (i) 3θ = {ps , ps + 1, · · · , ps + D − 1}, (39) (i) where 3θ selects the angle-dimension indexes with starting location ps and totaling D, representing the total number of indexes in the angle-dimension. Then we select all the indexes of the delay-Doppler-angle domain by (i) (i) = ∪ m(i) , (40) τ , Av , Aθ FIGURE 5. Extracting the same delay indexes in the channel vector. (i) domains corresponding are also non-zero, as well as the delay indexes that are not selected. This will greatly reduce the number of atom selections and thus avoid getting stuck in the local optimum, which greatly increases the accuracy of channel reconstruction and reduces the probability of selecting the wrong atoms. Then, we use the structural characteristics of the channel to find the indexes in the same dominant path, as shown in Fig. 5. The same color indexes in the figure indicate that they have the same delay index, representing that they have the dominant delay indexes. The specific equation is (i) (i) where mτ , Av , Aθ representing delay, Doppler, and angle domains, respectively. After the above algorithm steps, the maximum inner prod(i) uct uτ corresponding to Np delayed dominant paths and the L2-norm γ i of the residuals are obtained by ri = y − 9h(i) , (41) where the residual vector ri after the i-th iteration is obtained from (41). The residual vector represents the similarity between the estimated and wireless channels. Smaller residuals represent more accurate estimates. The proposed equation (Np −j+1) E = invec{e}. (34) To simplify the summation, the (34) is used to organize indexes e into a tensor E ∈ CMg ×Ng ×Nt . The tensor E is rearranged to obtain E(1) ∈ CMg ×Ng Nt , and calculate the corresponding residual eτ (m) = E(1) (m, :) , (35) where E(1) (m, :) is the inner products of all indexes of the same dominant path, we obtain the L2-norm eτ (m) of the total inner product of each delay domain, which represents the total inner product of the τ -th delay, and find the index with the largest total inner product among them by m(i) τ = arg max eτ (m). m (36) With the 3D-IPRDSOMP algorithm, the index of the inner product and the dominant path corresponding to the maximum inner product after summation is obtained by (36). The arg max denotes the index corresponding to finding the (i) maximum value. Where mτ denotes the index of the dominant delay of the sum of the maximum inner product. The (i) corresponding inner product eτ (mτ ) = max eτ (m) is also recorded. In [7], step 11-step 15 has been explained. gθ (r) = ||Dθ (r, :)||, (37) CNt ×D where matrix Dθ ∈ obtained from, then, obtain the L2-norm gθ (r) of Dθ (r, :). To record the starting location of the burst sparse, we find the maximum value of gθ as ps = arg maxr gθ (r), VOLUME 11, 2023 (38) ∇uNp −j = uτ (Np −j) , (42) uτ represents the proportion of the maximum inner product uτ for each iteration The division between ∇u and the residual γ as Np −j+1 ∇u , (43) ωNp −j = 1 γ Np −j where the symbol denotes the difference between them. We obtain the value ω of variation in the scaled decrease of the maximum inner product. The corresponding values ω change from fast to slow when the overestimating channel and selection of wrong atoms result in a maximum inner product and slow variations after max_ω < ωNp −j . Then, starting from the Np -th value, the local maximum max_ω and the corresponding index max_index are obtained by calculating ωNp −j and comparing them with the maximum with backtracking. The corresponding index Pos is marked as the corresponding index vector h| = h(Pos) . IV. SIMULATION RESULTS In this section, we compared the performance of four different algorithms: the traditional least squares (LS) algorithm, the traditional OMP algorithm, the 3D-SOMP algorithm, and the proposed 3D-IPRDSOMP algorithm. We use 3GPP Standardized Channel Model (SCM)to create the channel. We use the normalized mean square error (NMSE), which represents the error of channel estimation. The NMSE of the proposed 94687 G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO TABLE 2. Simulation parameters. FIGURE 7. The NMSE performance comparison against the pilot overhead ratio η. FIGURE 6. The NMSE performance comparison against the SNR. 3D-IPRDSOMP based channel estimation technique is computed as k= N2 ℓ=M −1 P P NMSE = k=− N2 ℓ=0 DDA − H DDA Ĥℓ,k,r ℓ,k,r k= N2 ℓ=M −1 P P k=− N2 ℓ=0 2 , DDA Hℓ,k,r (44) 2 DDA ∈ CMg ×Ng ×Nt , H DDA ∈ CMg ×Ng ×Nt are where Ĥℓ,k,r ℓ,k,r obtained by matrixing the channel vectors h and h. The effectiveness of channel reconstruction is proven by evaluating the channel estimation effect. We construct the 3GPP SCM link channel. The channel model is configured with six dominant paths and twenty sub-paths for each dominant path. However, the path delay does not have a one-to-one correspondence with the OTFS modulation because of the resolution. The simulation parameters are summarized in Table 2. 94688 Fig. 6 shows the NMSE performance under different signal-to-noise ratios (SNR)s. The number of antennas is 32, the user velocity is 100 m/s, and the pilot overhead ratio is 0.5. Notably, the proposed 3D-IPRDSOMP algorithm outperforms the traditional 3D-SOMP algorithm by approximately 2 dB. We observe that the NMSE performance of channel estimation increases as the SNR increases. The performance in the channel is compromised because of the low accuracy of the OTFS massive MIMO delay dimension. Specifically, only one dominant path exists on the integer delay index, despite the presence of multiple paths in the channel. Consequently, the number of distinguishable dominant paths is smaller than the number of dominant channel paths. The proposed algorithm matches the number of approximate primary paths to achieve better performance. In this case, the traditional algorithm estimates the atom selection based on the number of dominant paths, resulting in wrong atom selection and the consequent degradation of the NMSE performance. Fig. 7 shows the NMSE performance under different pilot overhead ratios. The number of antennas is 16, the user velocity is 100 m/s, and the SNR is 5 dB. The proposed 3DIPRDSOMP algorithm outperforms the traditional algorithm under the same pilot overhead ratio. We observe that the NMSE performance of channel estimation increases as the pilot overhead ratio increases. It is because the solvability of the matrix is enhanced as the number of pilot overheads increases, leading to an approximate solution closer to the optimal solution. Furthermore, the 3D-IPRDSOMP algorithm exploits the structural characteristics of the channel, where the indexes between different dimensions are correlated. It reduces the number of atom choices without increasing the step, resulting in better channel reconstruction performance. Fig. 8 shows the performance of NMSE with different numbers of BS antennas. The pilot overhead ratio is 0.5, the user velocity is 100 m/s, and the SNR is 5dB. The proposed 3D-IPRDSOMP algorithm outperforms traditional algorithms with the same number of antennas. We observe VOLUME 11, 2023 G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO dimension increase. The larger Doppler shift for the same number of samples, the smaller the resolution of the Doppler dimension, and hence the accuracy of the channel estimation decreases. Notably, the proposed 3D-IPRDSOMP algorithm outperforms the traditional 3D-SOMP algorithm, especially when the user velocity is low. V. CONCLUSION FIGURE 8. The NMSE performance comparison against the number of BS antennas. FIGURE 9. The NMSE performance comparison against the number of user velocities. In this paper, we studied the number of distinguishable dominant path problems in channel estimation in OTFS massive MIMO systems. First, we analyze the reasons for the poor performance of the traditional OMP algorithm. We adjust the step size of the OMP algorithm to escape the local optimum and be closer to the global optimum. Then, using residuals and the inner product to characterize the completion of reconstruction, we proposed a method that approximates the number of distinguishable dominant paths. It is used as an iterative termination condition to select atoms, thus avoiding performance degradation because of the introduction of erroneous atoms. The 3D-IPRDSOMP algorithm increases the accuracy of channel estimation by exploiting the 3D structured characteristics of the channel and reducing inter-symbol interference. The 3D-IPRDSOMP algorithm has better performance than traditional algorithms. In the future, we aim to enhance the accuracy of the dominant path estimation and eliminate inter-path interference. In addition, we plan to increase the performance of estimating the number of dominant paths in a channel without any prior information. 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Marzetta, ‘‘Massive MIMO for next generation wireless systems,’’ IEEE Commun. Mag., vol. 52, no. 2, pp. 186–195, Feb. 2014. [25] J. Salo, G. D. Galdo, J. Salmi, P. Kyösti, M. Milojevic, D. Laselva, and C. Schneider, MATLAB Implementation of the 3GPP Spatial Channel Model, document 3GPP TR 25.996, 2005. [26] F. Hlawatsch and G. Matz, Wireless Communications Over Rapidly TimeVarying Channels. New York, NY, USA: Academic, 2011. [27] S. K. Sahoo and A. Makur, ‘‘Signal recovery from random measurements via extended orthogonal matching pursuit,’’ IEEE Trans. Signal Process., vol. 63, no. 10, pp. 2572–2581, May 2015. [28] S. G. Mallat and Z. Zhang, ‘‘Matching pursuits with time-frequency dictionaries,’’ IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3397–3415, Dec. 1993. GUO-RUI WANG was born in Dongying, Shandong, in 1999. He received the B.S. degree in electronic information engineering from Yantai University, in 2021, where he is currently pursuing the M.S. degree in electronic science and technology. ZHONG-XUN WANG was born in Yantai, Shandong, in 1964. He received the B.S. degree from the Physics Department, Shandong Normal University, in 1986, the M.S. degree from the Department of Hydroacoustic, Harbin Engineering University, in 2008, and the Ph.D. degree from the Naval Aeronautical Engineering Institute, in 2009. His current research interest includes source channel coding in wireless communication. VOLUME 11, 2023