OTFS MIMO system sparse channel estimation method based on 3D-SOMP algorithm

Abstract

The invention discloses an OTFS MIMO system sparse channel estimation method based on a 3D-SOMP algorithm. The OTFS MIMO system sparse channel estimation method comprises the following steps: S1, channel estimation in an OTFS SISO system; s2, analyzing the three-dimensional structure sparsity of a channel in the OTFS MIMO system; s3, expressing a downlink channel estimation problem as a sparse signal recovery problem; and S4, solving the sparse signal recovery problem of the sensing matrix by using a 3D-SOMP algorithm. The invention proves that the OTFS MIMO channel has 3D structure sparsity: normal sparsity along the delay dimension, block sparsity along the Doppler dimension and burst sparsity along the angle dimension. Simulation results show that the algorithm provided by the invention can obtain accurate channel state information with relatively low pilot frequency overhead.

Description

Technical Field

[0001] The present invention belongs to the technical field of wireless communications, and in particular relates to a sparse channel estimation method for an OTFS MIMO system based on a 3D structured orthogonal matching pursuit algorithm. Background Art

[0002] One of the goals of future wireless communications (emerging 5G or above) is to support reliable communications in high-mobility scenarios, such as high-speed railways with speeds of up to 500 km / h or vehicles with speeds of up to 300 km / h. Currently, the main modulation technology for 4G and emerging 5G is orthogonal frequency division multiplexing (OFDM). For high-mobility scenarios, OFDM may encounter significant inter-carrier interference (ICI). ICI is caused by Doppler spreading of time-varying channels (also called dual-selective or dual-dispersion channels). When using traditional transceivers, ICI will severely degrade the performance of OFDM systems.

[0003] In order to deal with ICI, some improvements have been made to traditional OFDM, but at the cost of making the transceiver design more complex. Relevant technicians have proposed linear equalization and nonlinear equalization to eliminate ICI at the receiving end. Some transmitter processing methods to mitigate ICI have also been proposed, including polynomial cancellation coding and pulse shaping. Using the transmitting and receiving end processing, a multiple access scheme based on channel-independent block extension is proposed for multi-user scenarios, which can eliminate ICI and multi-user interference. For time-varying channels, some modulation schemes that use transmit diversity to improve system performance are proposed instead of trying to eliminate ICI. For example: using the frequency oversampling technology of the zero-padding OFDM system, frequency diversity is achieved through the transmission signal design, and vector OFDM transmits multiple groups of linear precoded symbols on the channel subcarriers to provide frequency diversity; using Doppler-resilient orthogonal signal division multiplexing technology, it multiplexes multiple data vectors and a pilot vector into a data stream to fully utilize the frequency-time diversity in the time-varying channel.

[0004] Orthogonal Time-Frequency-Space (OTFS) is an alternative to OFDM for addressing time-varying channels. Using the Basis-Spread Model (BEM) of the channel, OTFS transforms the time-varying channel into a time-independent channel in the Delay-Doppler domain, so that the information-bearing data is multiplexed into a roughly constant channel in the Delay-Doppler domain. OTFS differs from previous work in that it multiplexes data in the Delay-Doppler domain.

[0005] Like OFDM multi-antenna systems, OTFS with massive multiple-input multiple-output (MIMO) can further improve spectral efficiency. These benefits require the transmitter to know the channel state information (CSI) to design the transmit beamforming vector. When the OTFS massive MIMO system operates in frequency division duplex (FDD) mode, downlink channel estimation is necessary due to the lack of channel reciprocity. In OTFS massive MIMO systems, downlink channel estimation is challenging due to the large number of antennas configured at the base station (BS).

[0006] Prior art proposes time-varying channel estimation schemes for massive MIMO systems. The time-varying MIMO channel is modeled based on several joint sparse time-independent coefficients of boundary elements. These joint sparse coefficients can be estimated with high accuracy by a distributed compressed sensing algorithm. In order to further reduce the effective dimension of massive MIMO time-varying channels, a spatial domain boundary element method is proposed to reduce the training overhead of the downlink. In addition, there is a general framework for compressed channel sensing. Based on the sparse multipath structure of massive MIMO time-varying channels, compressed channel sensing can achieve the target estimation error with less overhead.

[0007] In order to reduce the computational complexity of channel estimation, the Heisenberg-Weil signature sequence is designed to make the signature method of Doppler delay channel estimation have lower complexity. The above channel estimation techniques are proposed for OFDM massive MIMO systems. They are not directly applicable to OTFS massive MIMO systems. This is because in OTFS systems, the data carrying information is multiplexed in the delay Doppler domain, rather than in the frequency-time domain as in OFDM systems.

[0008] For the OTFS system, the related art proposes a pulse-based channel estimation technique for the OTFS single-input single-output (SISO) architecture. The BS acts as a training pilot to send pulse signals in the delay-Doppler domain. The received signal in the delay-Doppler domain can be regarded as a two-dimensional periodic convolution of the transmitted pulse and the delay-Doppler channel. The delay-Doppler channel can then be estimated from the received signal using a threshold method. For the OTFS SISO system, another method using a pseudo-random noise (PN) sequence as a training pilot in the delay-Doppler domain is proposed. In this method, channel estimation is done in the discrete domain, where three quantities of interest, namely delay shift, Doppler shift, and fading coefficient, are estimated. The delay-Doppler channel is then calculated. The pulse-based scheme is extended to the OTFS MIMO system by sending multiple pulses with appropriate protection between two adjacent pulses to distinguish different BS antennas. Existing channel estimation techniques cannot be directly extended to OTFS massive MIMO because a large number of antennas need to be distinguished by transmitting orthogonal pilots, which will result in high pilot overhead.

[0009] To address this problem, we propose a discrete-time formulation for the OTFS system and prove that the OTFS massive MIMO channel has a delay-Doppler angle 3D structured sparsity. The 3D channel is sparse along the delay dimension due to the limited number of dominant propagation paths. The 3D channel is block-sparse along the Doppler dimension because the Doppler frequency of the paths is usually much smaller compared to the system bandwidth. The only non-zero block is concentrated around zero, but the length of the non-zero block is unknown. The 3D channel is burst-sparse along the angle dimension because the angle of departure (AOD) spread of the paths at the BS is usually small. The length of the non-zero burst can be regarded as a constant, but the starting position of each non-zero burst is unknown. Based on the 3D structured sparse channel, we formulate the downlink channel estimation problem for the OTFS massive MIMO system as a sparse signal recovery problem. The proposed estimator exploits the training pilots transmitted in the delay-Doppler domain. We propose that the pilots for different antennas are independent complex Gaussian random sequences that overlap to reduce the overall pilot overhead. The received pilot can be represented as a phase-compensated 2D periodic convolution of the transmitted pilot with the delay-Doppler channel by inserting a guard interval between the pilot and the data. By decomposing the channel according to its structure, we formulate the downlink channel estimation problem as a sparse signal recovery problem and propose a 3D-SOMP algorithm to solve the formal channel estimation problem. Summary of the invention

[0010] In order to solve the above problems, the present invention proposes a sparse channel estimation method for an OTFS MIMO system based on a 3D-SOMP algorithm.

[0011] The sparse channel estimation method of the OTFS MIMO system based on the 3D-SOMP algorithm of the present invention comprises the following steps:

[0012] S1. Channel estimation in OTFS SISO system;

[0013] S2. Analyze the three-dimensional structure sparsity of the channel in the OTFS MIMO system;

[0014] S3. The downlink channel estimation problem is formulated as a sparse signal recovery problem;

[0015] S4. Use 3D-SOMP algorithm to solve the sparse signal recovery problem of the perception matrix.

[0016] The S1 includes OTFS SISO modulation and OTFS SISO demodulation, which are achieved by adding a preprocessing block before the frequency domain of the traditional modulator at the transmitting end and adding a corresponding post-processing block after the frequency domain of the traditional demodulator at the receiving end. The time-varying channel is converted into a time-varying channel in the delay Doppler domain through the preprocessing and post-processing modules.

[0017] The pre-processing block maps the two-dimensional data block in the delayed Doppler domain to the two-dimensional block in the frequency-time domain, and then converts the two-dimensional block in the frequency-time domain into a one-dimensional transmission signal through a traditional frequency-time modulator; the frequency-time demodulator at the receiving end transforms the received signal into a two-dimensional block in the frequency-time domain, and the post-processing block transforms the two-dimensional block into a two-dimensional data block in the delayed Doppler domain; the two-dimensional data block received in the delayed Doppler domain is given by the two-dimensional periodic convolution of the transmitted two-dimensional data block and the delayed Doppler channel impulse response in the delayed Doppler domain with phase compensation.

[0018] The OTFS SISO modulation and OTFS SISO demodulation are formulated in the discrete time domain, and the biorthogonal characteristics of the transmit waveform and the receive waveform are completely maintained along the time dimension, but only approximately maintained along the frequency dimension.

[0019] The periodic convolution is expressed as

[0020]

[0021] in, is the additive noise in the delay-Doppler domain, is the delay Doppler CIR H DD The (l+1, k+1+N / 2) elements of

[0022]

[0023] Where, (l) M is the remainder after dividing l by M, and

[0024] The three-dimensional sparse structure of the channel in the OTFS MIMO system in S2 includes the sparsity of the delay-Doppler channel H along the delay dimension i, the sparsity of the delay-Doppler angle channel H along the Doppler dimension k and the sparsity of the delay-Doppler angle channel H along the angle dimension r. We denote the delay-Doppler channel Expressed as

[0025]

[0026] Will Arrange the layers into a 3D tensor in is H's (l+1,k+N / 2+1,R+N t / 2+1) elements, (l=0,1,...,M-1,k=-N / 2,...,0,...,N / 2-1) elements and (r=-N T / 2,...,0,...,N t / 2-1) elements;

[0027] Function γ N(x) has the following characteristics: when |x|>>1, |γ N (x)|≈0

[44] , so Only when k ≈ v si NT,l≈τ i Mτ max and r≈Ψ si N t There is a dominant element only when ; the sparsity of the delay-Doppler channel H along the delay dimension i is judged according to the number of dominant paths.

[0028] The method for determining the sparsity of the delay Doppler angle channel H along the Doppler dimension k is as follows: i The Doppler frequency of the ith subpath is expressed as Where v is the user's moving speed, φ si is the angle between the user's moving direction and the arrival direction of the i-th subpath, and the maximum Doppler of the subpath is The sparsity of the delay-Doppler angle channel H along the Doppler dimension k is determined according to the maximum Doppler of the subpath.

[0029] The method for determining the sparsity of the delay Doppler angle channel H along the angle dimension r is as follows: Since the angular spread of the dominant path is very small, in (-1 / 2, 1 / 2), φ si Distribution is N P Block, therefore, the delay-Doppler angle channel H is burst-sparse along the angle dimension r.

[0030] In S3, we formulate the OTFS channel estimation problem as a sparse signal recovery problem of the perception matrix y=ψh+v.

[0031] The beneficial effects of the present invention are:

[0032] This paper proves that the OTFS MIMO channel has 3D structural sparsity: normal sparsity along the delay dimension, block sparsity along the Doppler dimension, and burst sparsity along the angle dimension. Based on the three-dimensional channel sparsity, we formulate the downlink channel estimation problem as a sparse signal recovery problem and solve it using the proposed 3D-SOMP algorithm. Simulation results show that the algorithm can obtain accurate channel state information with low pilot overhead, proving the superiority of this method. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 It is the OTFS SISO architecture.

[0034] Figure 2 It is the OTFS massive MIMO architecture.

[0035] Figure 3 It is a delay-Doppler angle three-dimensional channel.

[0036] Figure 4 is an OTFS frame with a delay-Doppler domain size of M×N.

[0037] Figure 5 It is the comparison of NMSE performance and pilot overhead ratio η.

[0038] Figure 6 is the comparison of NMSE performance with the number of BS antennas.

[0039] Figure 7 is the performance comparison of NMSE and signal-to-noise ratio.

[0040] Figure 8 It is the comparison of NMSE performance and user speed.

[0041] Fig. 9 It is a performance comparison of bit error rate and signal-to-noise ratio. DETAILED DESCRIPTION

[0042] Embodiments of the present invention are described in detail below, and examples of the embodiments are shown in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and intended to be used to explain the present invention, but should not be understood as limiting the present invention.

[0043] The sparse channel estimation method of the OTFS MIMO system based on the 3D-SOMP algorithm of the present invention comprises the following steps:

[0044] S1. Channel estimation in OTFS SISO system;

[0045] OTFS of SISO system, including OTFS modulation and discrete time formula of OTFS demodulation.

[0046] Figure 1 The OTFS SISO architecture is shown. OTFS is a modulation / demodulation technology. It can be implemented by adding a pre-processing block before the frequency domain of the traditional modulator (such as OFDM modulator) at the transmitting end, and adding a corresponding post-processing block after the frequency domain of the traditional demodulator (such as OFDM demodulator) at the receiving end. Through the pre-processing and post-processing modules, the time-varying channel is converted into a time-varying channel in the delay-Doppler domain. Therefore, the data carrying information can be multiplexed in a roughly constant delay-Doppler channel. At the same time, the transmission data of the OTFS system can make full use of the diversity characteristics of the frequency-time channel. In this way, OTFS improves the system performance over OFDM in high mobility scenarios.

[0047] S101.OTFS SISO modulation

[0048] First, the quadrature amplitude modulation (QAM) data sequence of length MN is rearranged into a two-dimensional data block. This is called the delay-Doppler domain X DD ∈C M×N The 2D OTFS frame in the delay-Doppler domain is represented by M and N, where M and N are the number of resource units along the delay dimension and the Doppler dimension. The OTFS modulation at the transmitter consists of a preprocessing block and a conventional frequency-time modulator such as OFDM or filter bank multicarrier (FBMC). The preprocessing block transforms the two-dimensional number X in the delay-Doppler domain into DD The data block is mapped to a two-dimensional block X in the frequency and time domain FT ∈C M×N It is achieved by using the Inverse Symplectic Finite Fourier Transform (ISFFT) and a transmission window function. DD The ISFFT of

[0049]

[0050] Among them, F M ∈C M×N and F N ∈C N×N is the discrete Fourier transform (DFT) matrix. The transmission window matrix W tx ∈C M×N X ISFFT Multiply element-wise to generate a 2D block in the frequency domain X FT for

[0051] X FT =X ISFFE ☉W tx (2)

[0052] The window matrix has several uses. For example, a windowing matrix can be designed to randomize the phase of the transmitted symbols to eliminate inter-element interference. For simple representation, we assume a trivial window at the transmitter, W t x is a matrix of all 1s. Then, the two-dimensional block X in the frequency-time domain is converted to FT Converted into a one-dimensional transmitted signal s. Assuming there is an OFDM modulator, for X FT Perform m-point inverse DFT (IDFT) on each column of to obtain a two-dimensional transmitted signal block S∈C M ×N ,Right now

[0053]

[0054] where S = [s1, s2, s3, ..., s N ], each column vector s of S i ∈C M×1 It can be regarded as an OFDM symbol. Note that N OFDM symbols occupies bandwidth MΔf and has duration NT, where Δf and T are the subcarrier spacing and symbol duration. Combining (1)-(3),

[0055]

[0056] In order to avoid inter-symbol interference between blocks, OFDM modulators usually use CP plus matrix Add a cyclic prefix (CP) to each OFDM symbol si, where N CP is the length of CP. By reading the two-dimensional transmission signal block by column, the one-dimensional transmission signal is obtained.

[0057] s=vec{A CP S} (5)

[0058] S102.OTFS SISO demodulation

[0059] Receiving Signals Through the time-varying channel h l,k The k bits after the last one are L+1 in length. k Expressed as

[0060]

[0061] where v k is the additive noise at the receiver. The OTFS demodulation at the receiving end consists of a traditional frequency-time demodulator (such as OFDM or FBMC demodulator) and a post-processing module, such as Figure 1 The frequency-time demodulator transforms the received signal r into the frequency-time domain Y FT ∈C M×N Specifically, assuming an OFDM demodulator, the received signal r is first rearranged into a two-dimensional block of size (M+N CP )×M matrix r, that is:

[0062] R = invec{r}(7)

[0063] Each column vector of R can be regarded as a received OFDM symbol containing CP. Then, the OFDM demodulator multiplies R by the CP removal matrix Remove CP and get OFDM symbol R without CP CP R. For each OFDM symbol without CP (i.e. R CP Applying the point DFT to each column vector of R, we get the received two-dimensional block Y in the frequency domain FT for

[0064] Y FT =F M RCP R(8)

[0065] In the post-processing block, Y FT Transformed into a two-dimensional data block Y in the delay-Doppler domain DD ∈C M×N It receives the window matrix W rx ∈C M×N And SFFT implementation. The receiving window matrix W rx With element Y FT Multiply, that is

[0066] Y FT，W =Y FT ⊙W rx (9)

[0067] Then, for Y FT ,W performs SFFT processing to obtain a two-dimensional data block Y in the delay Doppler domain DD for

[0068]

[0069] As with the transmitter, we consider a trivial window on the receiver for a simple expression, namely, W rx is a matrix of all 1s. Combining equations (8) to (10), we can get

[0070] Y DD =R CP RF N (11)

[0071] Received 2D data block Y in the delay-Doppler domain DD The 2D data block X sent by DD and the delayed Doppler channel impulse response (CIR) H DD ∈C M×N The two-dimensional periodic convolution for phase compensation in the delay-Doppler domain is given as shown in Lemma 1 below. Note that by applying OTFS pre-processing / post-processing on the OFDM modulator / demodulator (with CP), the OTFS modulation / demodulation is formulated in the discrete time domain. In this case, the biorthogonal property of the transmit / receive waveform is fully maintained along the time dimension, but only approximately maintained along the frequency dimension. This is different from the case of ideal waveforms and rectangular waveforms.

[0072] Lemma 1: We denote X DD and Y DD The (l+1,k+1,+N / 2) element is and Where l = 0, 1, ..., M-1 and k = -N / 2, ..., 0, ..., N / 2-1. Then Can be expressed

[0073]

[0074] in, is the additive noise in the delay-Doppler domain. is the delay Doppler CIR, H DD and the elements of (l+1,k+1,+N / 2)

[0075]

[0076] Where (l) M It is the remainder after dividing l by M. Note Therefore (12) can be regarded as a periodic convolution.

[0077] Proof: See Appendix.

[0078] It can be seen from equation (12) that due to the delay Doppler CIR, H DD is time-independent (H DD Don't follow h k,l The variable k changes), so in the delay Doppler domain, the transmitted data Approximately constant channel H DD Moreover, since each transmitted data in the delay-Doppler domain is extended to the entire frequency-time domain, as shown in (1) and (2), it can utilize the full diversity of the frequency-time channel. Therefore, OTFS outperforms traditional OFDM in performance, especially in high mobility scenarios.

[0079] In order to eliminate the interference between symbols, equalization is required, because each time data is transmitted in (12) Available, and there is also inter-symbol interference Available, Available, k'≠k. In order to eliminate the interference between symbols through equalization, it is necessary to use the delay Doppler CIR H obtained by downlink channel estimation DD .

[0080] S103.OTFS Massive MIMO

[0081] Figure 2 The OTFS massive MIMO architecture is shown. The BS is equipped with antennas that can serve U single-antenna users simultaneously. Downlink precoding is performed to eliminate inter-user interference.

[0082] Downlink precoding requires downlink channel CSI, which is obtained from the uplink channel feedback of the FDD system after precoding, and the transmission data block X in the delayed Doppler domain DD Through OTFS modulation in N tAt the user end, the received signal is first demodulated by OTFS demodulation to obtain the received data block Y in the delay Doppler domain. DD In order to eliminate inter-symbol interference, equalization is performed based on the downlink CSI.

[0083] Next, we will focus on downlink channel estimation in OTFS SISO / massive MIMO systems. Note that due to the signal processing time and feedback time, the delay-Doppler CIR H DD However, it can still be used for downlink precoding because the channel coherence time in OTFS systems is increased by several orders of magnitude compared to OFDM systems.

[0084] S2. Analyze the three-dimensional structure sparsity of the channel in the OTFS MIMO system;

[0085] The goal of channel estimation is to obtain the delayed Doppler data block Y received from (12) DD Delay Doppler CIRH DD . Estimation H DD An intuitive approach is to transmit pulses in the delay-Doppler domain as a training exercise for pilots.

[0086] The transmitted pulse is expressed as

[0087]

[0088] According to equation (12), the received signal in the delayed Doppler domain can be expressed as

[0089]

[0090] Delay-Doppler It can be estimated by the least squares (LS) estimator or the minimum mean square error (MMSE) estimator. It is estimated from (15). Note that only The non-zero part of needs to be estimated due to its finite support, which will be explained later.

[0091] However, this pulse-based channel estimation technique is not suitable for large-scale MIMO systems due to the huge pilot overhead required. t The delay-Doppler channel associated with the BS antennas requires N transmissions. t pulses. We assume that N t Delay-Doppler There is limited support in the delay dimension [0:M max -1], in the Doppler dimension In order to avoid interference between multiple antennas, the guard interval between two adjacent pulses is not less than N in the Doppler dimension. max , not less than N in the delay dimension max Therefore, the transmission N in the OTFS massive MIMO system t The pilot length of the pulse should be ∝N t N max M max With a large number of BS antennas, the pilot's overhead would be overwhelming.

[0092] To solve this problem, we propose a channel estimation technique based on the 3D-SOMP algorithm, which can obtain accurate CSI while greatly reducing the pilot overhead.

[0093] We consider a scenario with N at the BS and U single-antenna users. t The downlink channel estimation is the same for U users. Therefore, we focus on a certain user and omit the user's subscript without losing generality. We consider a downlink time-varying channel consisting of Np dominant propagation paths. Each main path consists of n subpaths. The i-th subpath in the i-th main path has a complex path gain α si and Doppler frequency v si The delay of all sub-paths on the ith main path can be regarded as the same τ i We will i The physical AoD of a sub-path is expressed as When considering a typical uniform linear array (ULA) antenna, The relevant spatial angle is defined as Where d is the antenna spacing and λ is the wavelength of the carrier frequency. Usually, d = λ / 2 and therefore The time-varying channel associated with the (p+1) antenna (p=0,1,...,N t -1) can be expressed as

[0094]

[0095] Among them, p rc (τ) is the value between τ and The band-limited pulse shaping filter response at , MΔf is the system sampling interval. Based on equation (13), we express the delay-Doppler CIR of the (p+1) antenna (called the delay-Doppler space).

[0096] OTFS in massive MIMO systems Where l, k and p correspond to delay, Doppler and spatial index respectively, as follows

[0097]

[0098] In order to study the three-dimensional structure sparsity of the channel in the OTFS massive MIMO system, we Apply the inverse DFT along the spatial p dimension to define the delay-Doppler angle channel

[0099]

[0100] In the formula is the angular index. Then, substituting (17) into (18), we can write the delay-Doppler channel Expressed as

[0101]

[0102] we will Arranged into a three-dimensional tensor in is the (l+1,k+N / 2+1,r+Nt / 2+1) element of H, (l=0,1,...,M-1,k=-N / 2,...,0,...,N / 2-1,

[0103] r=-NT / 2,...,0,...,Nt / 2-1).

[0104] Function γ N (x) has the following characteristics: when |x|>>1, |γ N (x)|≈0. Therefore, Only when k ≈ v si NT,l≈τ i Mτ max and r≈ψ si N t Only then will there be an advantage.

[0105] like Figure 3 As shown in the figure, since the number of advantage paths is small, such as Np = 6, then the N s The delay-Doppler channel H is sparse along the delay dimension i. Assume that the maximum path delay is τ max , then H has finite support in the delay dimension τ [0:M max -1], where M max ≈τ max MΔf.

[0106] In addition, the Doppler frequency of the si-th subpath in the ith subpath can be expressed as Where v is the user's moving speed, is the angle between the user's moving direction and the arrival direction of the i-th subpath. Therefore, the maximum Doppler of the subpath is Due to φ si Distributed in [-π / 2,π / 2], Distributed in Therefore, H has finite support along the Doppler dimension k. Where N max ≈v max For example, for a typical subcarrier spacing Δf = 15kHz, a carrier frequency of 2.15GHz, and a user speed of 180km / h, the maximum Doppler therefore, There are only about 5% dominant elements in the Doppler dimension. That is, the delay-Doppler angle channel H is block-sparse along the Doppler dimension k, where the only non-zero block is centered at k=0, but the length of the non-zero block is unknown.

[0107] Finally, for the angular dimension r, since the angular extension of the dominant path is small, in [-1 / 2,1 / 2], is distributed into Np blocks. Therefore, the delay-Doppler angle channel H is burst-resolved along the angle dimension r. There are Np non-zero blocks, but the starting position of each block is unknown because the path may arrive from any direction. Note that the difference between burst sparsity and traditional block sparsity is that the starting position of the non-zero burst is not necessarily {1,1+D,1+2D,...}, where D is the length of the non-zero block.

[0108] In summary, we decompose the multipath of a time-varying channel and show its structured sparsity along the delay dimension, Doppler dimension, and angle dimension, as Figure 3 As shown in Figure 2, the three-dimensional channel tensor H is sparse along the delay dimension, block sparse along the Doppler dimension, and burst sparse along the angle dimension. This three-dimensional structure sparsity can be used for CSI estimation with low pilot overhead.

[0109] S3. The downlink channel estimation problem is formulated as a sparse signal recovery problem;

[0110] Figure 4 The OTFS frame with a delay-Doppler domain size of M×N is shown. The lengths of the pilots along the Doppler dimension and the delay dimension are N v and M τ We recommend using complex Gaussian random sequences as training pilots. In order to avoid interference of two-dimensional periodic convolution in the delay-Doppler domain on the pilot and data, a guard interval needs to be set.

[0111] Note that the delay-Doppler angle channel H in the OTFS massive MIMO system has limited support in the Doppler dimension and [0:M max-1] along the delay dimension. The length of the guard is Figure 4 As shown, the interval along the Doppler dimension is Along the delay dimension is M g ≥M max -1. In order to reduce the overall pilot overhead of the OTFS massive MIMO system, we propose a non-orthogonal pilot pattern, that is, the transmitted pilots of different antennas completely overlap in the delay-Doppler domain, while the complex Gaussian random sequences (pilots) of different antennas are independent.

[0112] The training pilot in the delay-Doppler domain at the (p+1) antenna is denoted by x l,k,p , where l=0,1,...,M τ -1, In N t The OTFS frame on the antenna will be modulated and transmitted simultaneously. After passing through the channel, the received signal is demodulated and then the guard interval is discarded. From equation (12), it can be seen that the pilot received in the delayed Doppler domain at the user side can be expressed as

[0113]

[0114] in, The compensation phase Delay Doppler spatial channel It can be expressed as a delay-Doppler angle channel Based on the DFT of formula (18),

[0115]

[0116] Note that in actual operation, the parameter N max and M max should be estimated carefully. Uncertainty in parameter estimates can lead to interference between the pilot and the data.

[0117] Substituting (21) into (20), we get

[0118]

[0119] have

[0120]

[0121] To simplify the expression, we rewrite (22) into a vector-matrix form. Convert to column vector Where (lN v +k+N v / 2+1) is equal to y l,k We will also Convert to column vector The fourth element of hr (l'Ng +k'+N g / 2+1) is equal to the abbreviation Therefore, (22) can be written in vector matrix form as

[0122]

[0123] The two-dimensional periodic convolution matrix is ​​(lN v +k+N v / 2+1,l'N g +k'+N g / 2+1) c,r , which is equivalent to is a matrix (lN v +k+N v / 2+1,l'N g +k'+N g / 2+1) element W l-l',k' .

[0124] By indicating and

[0125] It can be expressed as

[0126] y=Z c,W h+v. (24)

[0127] Note that h can be devectorized to obtain the truncated delay-Doppler angle channel That is H g = invec{h}, which consists of the non-zero parts of H, where In this way, we formulate the OTFS channel estimation problem as a sparse signal recovery problem for the sensing matrix y = ψh + v.

[0128] y=ψh+v (25)

[0129] S4. Use 3D-SOMP algorithm to solve the sparse signal recovery problem of the perception matrix.

[0130] Algorithm 1 gives the 3D-SOMP algorithm. We borrow the main idea of ​​OMP to obtain the sensor matrix ψ and the residual measurement of each column r = y-ψh (0) The correlation vector between (0) =0.

[0131]

[0132] As the initial channel estimate

[0133] e=ψ H r (26)

[0134] For the traditional OMP algorithm, the support of the sparse channel vector h can be identified by finding the columns in Ψ that are most correlated with the residual measure r. Different from OMP, in order to utilize h (or H g ), we rearrange the relevant vector e into a tensor in step 6

[0135] ε=invec{e} (27)

[0136] For convenience, we first introduce N-dimensional (N ≥ 3) tensor

[0137] By fixing all indices except the nth index of M, that is, M(i1,i2,...,:,:,i n+2 ,...,i N ), we get n-type fiber, by fixing all indices except the two indices of M, namely M(i1,i2,...,:,:,i n+2 ,...,i N ) to obtain slices.

[0138] Finally, the expansion operation converts the n-dimensional tensor into a two-dimensional matrix. All mode-n fibers are treated as M (n) The columns of the mode-n expansion matrix can be obtained

[0139] Our proposed 3D-SOMP algorithm identifies the 3D support of each dominant path in a one-by-one manner. For each dominant path, the algorithm first obtains the type 1 expansion matrix By calculating E (1) The l2 norm of the row vector of , and the correlation vector with the mth element along the delay dimension is obtained

[0140] e r (m)=||E (1) (m,:)|| (28)

[0141] Therefore, by finding e τ The largest element of The delay dimension index of the i-th advantage path can be obtained Then, the user fixes the delay dimension metric Focus on the slice Identify the Doppler and angular supports. By calculating the thin slice The l2 norm of the row vector of , and the correlation vector with the nth element along the Doppler dimension is obtained

[0142]

[0143] Since the truncated 3D channel is block sparse along the Hg Doppler dimension and has only one non-zero block centered at v = 0, the length of the only non-zero block is unknown. It can be estimated by finding the smallest block in the Doppler dimension correlation vector, where the norm of the block is the same as ||e v The ratio between || should be greater than the threshold ∈, that is,

[0144]

[0145] Therefore, the Doppler support of the ith dominant path obtained in step 11 is

[0146] Finally, we focus on To obtain the angular support of the ith dominant path. Similarly, by calculating The l2 norm of the column vector of , we get the angular dimension correlation vector of the rth element

[0147]

[0148] Truncated 3D channel H g It is burst sparse along the angular dimension. Assuming the length of the non-zero burst is d, the user needs to estimate the starting position of the non-zero burst, which is related to the AoD of the i-th dominant path. The user first converts the burst sparsity into traditional block sparsity by the following lifting transformation method. In this method, a block of size N is transformed into t ×1 burst sparse vectors by lifting the matrix With higher dimensions N t The starting positions of non-zero bursts in the burst sparse vector are related to the supporting points of non-zero blocks in the high-dimensional block sparse vector.

[0149] The ((i-1)D+j)th column, L(i=1,2,...,N t and j=1,2,...,D) at position There is only one non-zero element 1, where

[0150]

[0151] In order to truncate the three-dimensional channel H g The burst sparsity along the angular dimension is transformed into the traditional block sparsity by using the lifting matrix L to transform the diagonal correlation vector e θ Make corrections

[0152] d θ =L He θ (33)

[0153] Then Rearrange to N t ×D matrix. By calculating D θ The l2 norm of the row vector of , we get in step 14 Therefore, the starting position of the non-zero burst is p s By finding g θ Therefore, the angular support associated with the i-th dominant path is

[0154] So far, the delay-Doppler angle three-dimensional support obtained in the i-th iteration is Users can estimate the channel through the LS part: Where Ω c represents the complementary set of Ω. Then, by subtracting h in the i-th iteration in step 19 (i) The contribution of N is used to calculate the residual measurement value. p After iterations, the complete channel estimate is

[0155] Performance Comparison

[0156] For traditional impulse-based channel estimation techniques (extended to OTFS massive MIMO systems), the pilot overhead is ∝N t N max M ma x. According to CS theory, in our proposed channel estimation technique, the pilot overhead (i.e., the length of the measurement) is ∝Slog(L), where S and L are the sparsity level and length of the sparse vector h. And S = N max N p D and L = N g M g N t Therefore, the pilot overhead of our proposed channel estimation technique is ∝N max N p Dlog(N g M g N t ) 5 .

[0157] Note that the number of dominant paths is usually small, such as N p = 6. Since the angular spread of the dominant path is usually small, the non-zero block length along the angular dimension D is usually much smaller than the number of BS antennas N. t , such as D≈N t / 10. Guard interval length N g and M g Can be set to Nmax and M max Therefore, the pilot overhead of the proposed 3D-SOMP based channel estimation is much lower than that of the previously proposed impulse based channel estimation.

[0158] Simulation Results

[0159] We study the performance of the proposed 3D-SOMP based channel estimation technique from the following aspects.

[0160] Table 1 System simulation parameters

[0161] Parameter Values Carrier frequency(GHz) 2.15 Duplex mode FDD Subearrier spacing(kHz) 15 Cyclic prefix duration(us) 16.6 FFT size 1024 Transmission bandwidth(#of resource blocks) 50 Size of a OTFS frame (M, N) (600,12) #of BS antennas 1～64 #of user antennas 1 Channel model:3GPP standardized channel model Urban macro cell #of dominant channel paths 6 #of sub-paths per dominant path 20 User velocity(m / s) 10～160

[0162] Normalized mean square error (NMSE) of channel estimation. Using the traditional pulse-based channel estimation technique as a benchmark, the LS estimator is used to estimate the delay-Doppler channel. and the antennas in (15) for

[0163]

[0164] The traditional NMSE calculation based on impulse channel estimation technology is:

[0165]

[0166] Will be in N t The channel vector h in formula (25) can be estimated by 3D-SOMP algorithm as Then, Rearranged to M g ×N g ×N t Tensor H g =invec{h}. Therefore, the delay-Doppler angle channel can be estimated as: Where Γ is The NMSE of the proposed 3D-SOMP channel estimation technique is calculated as

[0167]

[0168] We also give the NMSE of the traditional OMP-based channel estimation technique to compare with formula (25) when h is recovered using the traditional OMP algorithm. We simulate the 3GPP standardized spatial channel model considering an urban macrocell environment. The detailed system parameters are shown in Table 1. Note that due to limited computing power, the simulated N cannot be set too large. When N exceeds 10, the error of the channel estimation formula (25) is not obvious due to the limitation of N. We define the pilot overhead ratio η as the ratio of the number of resource units transmitted by the pilot to the total number of resource units in the delay-Doppler domain. Considering the total number, since the sparsity of the three-dimensional channel along the Doppler dimension is not obvious when N is not very large, we set N v = N. We will compare the NMSE performance of the proposed 3D-SOMP based channel estimation technique, the conventional impulse based channel estimation technique, and the conventional OMP based channel estimation technique, as well as the pilot overhead ratio, the number of BS antennas, the signal-to-noise ratio (SNR), and the user speed.

[0169] exist Figure 5 In this paper, we show the performance comparison of NMSE and pilot overhead ratio η. The number of BS antennas is 16, the signal-to-noise ratio is 5db, and the user speed is 100m / s. We observe that when considering the same pilot overhead ratio, the proposed 3D-SOMP based channel estimation technique outperforms the traditional pulse based channel estimation technique. When the pilot overhead ratio is small, that is, the interval between two adjacent pulses is less than N in the Doppler dimension, the proposed 3D-SOMP based channel estimation technique outperforms the traditional pulse based channel estimation technique. max and or less than the delay M max When the number of channels is ∝N, the conventional pulse-based techniques perform poorly due to insufficient pilot overhead. Therefore, interference from adjacent pulses degrades the NMSE performance of conventional pulse-based channel estimation techniques. In contrast, non-orthogonal pilots are used in the proposed 3D-SOMP based channel estimation technique. The required pilot overhead is ∝N max N p Dlog(N g M g N t ), which is much smaller than the traditional pulse-based channel estimation technique. For example, in order to achieve an NMSE of 0.03, the proposed 3D-SOMP-based channel estimation technique only requires a 32% pilot overhead ratio. For the traditional pulse-based channel estimation technique, a 60% pilot overhead ratio is required to achieve an NMSE of 0.3. In addition, the 3D-SOMP-based channel estimation technique proposed in this paper has better performance than the traditional OMP-based channel estimation technique with larger pilot overhead because it utilizes the three-dimensional structure sparsity of the delay-Doppler angle channel in the OTFS massive MIMO system.

[0170] exist Figure 6In this paper, we present a comparison of NMSE performance with the number of BS antennas. The pilot overhead ratio is set to 50%, the signal-to-noise ratio is 5dB, and the user speed is 100m / s. We observe that the performance of traditional NMSE pulse-based channel estimation techniques is severely degraded when the number of BS antennas exceeds 8 (NMSE is greater than 10 -1 ). This is because when the number of BS antennas is large and the pilot overhead ratio is constant, the interval between two adjacent pulses is insufficient. In contrast, the 3D-SOMP-based channel estimation technique proposed in this paper works well with a large number of BS antennas. The performance gap between the 3D-SOMP algorithm and the traditional OMP algorithm decreases with the increase of. This is because the 3D-SOMP algorithm, due to the use of the structured sparsity of the 3D channel, will lead to performance degradation in the case of insufficient pilots. This problem can be solved by increasing the pilot overhead or the signal-to-noise ratio.

[0171] exist Figure 7 In , we show the performance comparison of NMSE and SNR. The number of antennas is set to 32, the pilot overhead ratio is 50%, and the user speed is 100m / s.

[0172] We observe that the proposed 3D-SOMP based channel estimation technique outperforms the conventional impulse based channel estimation technique. The conventional impulse based channel estimation technique suffers from the NMSE layer pilot overhead due to the insufficient interference between multiple antennas. For the proposed 3D-SOMP based technique, the NMSE performance improves with the increase of SNR. Moreover, the proposed 3D-SOMP based technique outperforms the conventional OMP based technique by about 6dB.

[0173] exist Figure 8 In , we show the comparison of NMSE performance with user speed v. The pilot overhead ratio is η = 50% and the signal-to-noise ratio is 5dB. We observe that the NMSE performance of the channel estimation decreases as the user speed increases. This is because the Doppler spread As the user speed ν increases, and the channel supports along the Doppler dimension. Therefore, the required pilot overhead is increased to ensure a constant NMSE. In our simulations, when the pilot overhead ratio η = 50%, the NMSE performance of the channel estimation decreases as the user speed increases.

[0174] exist Fig. 9In this paper, we show the comparison of bit error rate (BER) and signal-to-noise ratio, where OFDM and OTFS systems are considered. The user speed is 100m / s. In this high mobility case, the ICI caused by large Doppler spread degrades the BER performance of the OFDM system, which is represented by the curve “OFDM under ICI”. With the knowledge of perfect CSI, we overcome the ICI using MMSE detection, as shown in the curve “OFDM with perfect CSI”, which is better than the “OFDM under ICI” case. In the OTFS system, the perfect CSI, the CSI estimated based on the traditional pulse method, and the CSI estimated based on the 3D-SOMP algorithm are used for OTFS signal detection by delay-Doppler 2D deconvolution. We observe that the OTFS system outperforms the OFDM system with the knowledge of perfect CSI under high SNR conditions. This is because the transmitted data in the OTFS system can fully utilize the diversity of the frequency-time channel. We also observe that the proposed 3D-SOMP algorithm can achieve satisfactory BER performance, which is very close to the case with perfect CSI in the OTFS system. The CSI obtained by the traditional pulse-based channel estimation method is inaccurate, resulting in poor bit error rate.

[0175] Proof of Lemma 1

[0176] Proof: Based on OTFS modulation, each column vector s of S i ∈C M×1 is an OFDM symbol (without CP)

[0177]

[0178] Where X DD is the two-dimensional data block in the delay-Doppler domain, f i ∈C N×1 is the DFT matrix F N Then, CP is added to each OFDM symbol, and these OFDM symbols with CP are transmitted through the BS. After passing through the channel, the received OFDM symbols are removed by CP and arranged into a matrix Z = [z1,z2,...,z N ]∈C M×N ,

[0179] Z=R CP R(38)

[0180] In order to avoid inter-symbol interference, CPN CP The length of is usually greater than the channel length L, that is, N CP > L. Therefore, the received i-th OFDM symbol (without CP) is z i ∈C M×1Given by the circular convolution of the i-th transmitted OFDM symbol (without CP) has a time-varying channel, i.e.

[0181]

[0182] In the formula is a circular convolution matrix, where the (a, b) element can be expressed as Among them (ab) M It is the remainder after ab is divided by M. i is the additive noise vector. Substituting (37) into (39),

[0183]

[0184] Transform the received OFDM symbol Z without CP into the delay-Doppler domain Y DD The two-dimensional data block is as shown in formula 11, that is,

[0185] Y DD =ZF N =[z1,z2,...,z N ]F N (41)

[0186] We write (41) as

[0187]

[0188] Substituting (40) into (42)

[0189]

[0190] Where V DD =[v1,v2,...,v N ]F N We denote Y DD and X DD The (l+1,k+1+N / 2) element of and where l = 0, 1, ..., M-1 and k = -N / 2, ..., 0, ..., N / 2-1. Expand to (43) Depend on

[0191]

[0192] We define The focus is on calculation We first expand the time-varying channel h based on the Fourier series l,k

[0193]

[0194] Where P is the number of frequency components of the time-varying channel. fp,l is the p-frequency component of the first channel tap. ω p,l for Corresponding non-zero coefficients. Based on (45) Expressed as

[0195]

[0196] We define a function Then (46) can be rewritten as

[0197]

[0198] Now we can define Delay Doppler and k=-N / 2,...,0,...,N / 2-1) is

[0199]

[0200] in, Now let's prove it

[0201]

[0202] Specifically, we first calculate

[0203]

[0204] It can be noted that the function γ N (x) has the following characteristics: when |x|>>1, |γ N (x)|→0. It follows that there are p explicit terms in (50), which are Since 0≤1≤M-1, we have

[0205]

[0206] Therefore, combining equations (49), (50), and (51),

[0207]

[0208] Finally, substitute equation (52) into equation (44) to prove

[0209]

[0210] Although the above embodiments have been shown and described, it is to be understood that the above embodiments are exemplary and are not to be construed as limitations on the present invention. Changes, modifications, substitutions and variations of the above embodiments by those of ordinary skill in the art are all within the scope of protection of the present invention.

Claims

1. A sparse channel estimation method for OTFS MIMO system based on 3D-SOMP algorithm, characterized in that: The following steps are involved: S1. Channel estimation in OTFS SISO system; S2. Analyze the three-dimensional structure sparsity of the channel in the OTFS MIMO system; S3. The downlink channel estimation problem is formulated as a sparse signal recovery problem; S4. Use 3D-SOMP algorithm to solve the sparse signal recovery problem of the perception matrix.

2. The sparse channel estimation method for OTFS MIMO system based on 3D-SOMP algorithm according to claim 1, characterized in that: The S1 includes OTFS SISO modulation and OTFS SISO demodulation, which are achieved by adding a preprocessing block before the frequency domain of the traditional modulator at the transmitting end and adding a corresponding post-processing block after the frequency domain of the traditional demodulator at the receiving end. The time-varying channel is converted into a time-varying channel in the delay Doppler domain through the preprocessing and post-processing modules.

3. The sparse channel estimation method for OTFS MIMO system based on 3D-SOMP algorithm according to claim 2, characterized in that: The pre-processing block maps the two-dimensional data block in the delayed Doppler domain to the two-dimensional block in the frequency-time domain, and then converts the two-dimensional block in the frequency-time domain into a one-dimensional transmission signal through a traditional frequency-time modulator; the frequency-time demodulator at the receiving end transforms the received signal into a two-dimensional block in the frequency-time domain, and the post-processing block transforms the two-dimensional block into a two-dimensional data block in the delayed Doppler domain; the two-dimensional data block received in the delayed Doppler domain is given by the two-dimensional periodic convolution of the transmitted two-dimensional data block and the delayed Doppler channel impulse response in the delayed Doppler domain with phase compensation.

4. The sparse channel estimation method for OTFS MIMO system based on 3D-SOMP algorithm according to claim 3, characterized in that: The OTFS SISO modulation and OTFS SISO demodulation are formulated in the discrete time domain, and the biorthogonal characteristics of the transmit waveform and the receive waveform are completely maintained along the time dimension, but only approximately maintained along the frequency dimension.

5. The sparse channel estimation method for OTFS MIMO system based on 3D-SOMP algorithm according to claim 1, characterized in that: The three-dimensional structure sparsity of the channel in the OTFS MIMO system in S2 includes the sparsity of the delay-Doppler channel H along the delay dimension i, the sparsity of the delay-Doppler angle channel H along the Doppler dimension k and the sparsity of the delay-Doppler angle channel H along the angle dimension r. Only when k ≈ v si NT,l≈τ i Mτ max and r≈Ψ si N t There is a dominant element only when ; the sparsity of the delay-Doppler channel H along the delay dimension i is judged according to the number of dominant paths.

6. The sparse channel estimation method for OTFS MIMO system based on 3D-SOMP algorithm according to claim 5, characterized in that: The method for determining the sparsity of the delay Doppler angle channel H along the Doppler dimension k is as follows: i The Doppler frequency of the ith subpath is expressed as Where v is the user's moving speed, φ si is the angle between the user's moving direction and the arrival direction of the i-th subpath, and the maximum Doppler of the subpath is The sparsity of the delay-Doppler angle channel H along the Doppler dimension k is determined according to the maximum Doppler of the subpath.

7. The sparse channel estimation method for OTFS MIMO system based on 3D-SOMP algorithm according to claim 5, characterized in that: The method for determining the sparsity of the delay Doppler angle channel H along the angle dimension r is as follows: Since the angular spread of the dominant path is very small, in (-1 / 2, 1 / 2), φ si Distribution is N P Block, therefore, the delay-Doppler angle channel H is burst-sparse along the angle dimension r.

8. The sparse channel estimation method for OTFS MIMO system based on 3D-SOMP algorithm according to claim 1, characterized in that: In S3, we formulate the OTFS channel estimation problem as a sparse signal recovery problem of the perception matrix y=ψh+v.

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