Implementation method of broadband MIMO array transmitting beam based on manifold optimization with embedded momentum

Abstract

The present invention discloses a method for implementing a transmit beam of a broadband MIMO array based on manifold optimization with embedded momentum, belonging to the technical field of MIMO array beamforming. Since the problem of broadband waveform design for a MIMO array is an unconstrained quadratic problem on a complex circular manifold. Based on this feature, the present invention realizes the design of the transmit beam of the MIMO array based on a manifold optimization embedding method without relaxation of momentum information flow, that is, first converts the problem into an unconstrained quadratic polynomial on the CCM, and then uses a manifold optimization method algorithm with embedded momentum information to solve the problem by adaptively updating the descent direction and step size based on momentum to complete the implementation of the transmit beam of the MIMO array. Compared with the existing methods, the present invention has a faster convergence speed and better performance.

Description

Technical Field

[0001] The present invention belongs to the technical field of MIMO array beamforming shaping, and particularly relates to a method for realizing a broadband MIMO array transmitting beam based on manifold optimization with embedded momentum. Background Art

[0002] The waveform design of a constant modulus constrained transmitting broadband beam pattern is a key technology in a multiple-input multiple-output (MIMO) radar system. First, the literature "E. Raei, S. Sedighi, M. Alaee-Kerahroodi, and M. B. Shankar, 'Mimo radar transmit beampattern shaping for spectrally dense environments,' IEEE Transactions on Aerospace and Electronic Systems, pp. 1–13, 2022" shows that appropriate transmit beam pattern design can improve the detection probability of the MIMO array system and effectively suppress clutter interference. In addition, the literature "A. Aubry, A. De Maio, M. A. Govoni, and L. Martino, 'On the design of multi-spectrally constrained constant modulus radar signals,' IEEE Transactions on Signal Processing, vol. 68, pp. 2231–2243, 2020" shows that constant modulus constraints (CMC) are important in waveform design because the MIMO array system uses non-linear power amplifiers. Therefore, the use of CMC for transmit beam pattern design has been widely studied.

[0003] Generally speaking, the existing methods can be divided into two categories: narrowband MIMO array beam pattern design and broadband MIMO array beam pattern design.

[0004] In the literature "A. Aubry, A. De Maio, and Y. Huang, 'Mimo radar beampattern design via sl / isl optimization,' IEEE Transactions on Signal Processing, vol. 64, no. 15, pp. 3955–3967, 2016", for narrowband beam pattern design, two different optimization strategies are widely used. The first type of method uses a two-step approach to synthesize the transmit beam pattern. More precisely, they first optimize the waveform covariance matrix based on the desired beam pattern, and then synthesize the waveform based on the optimized covariance matrix. However, relaxing the objective function or CMC will lead to relaxation errors. To avoid relaxation errors, the second type of method directly designs the waveform. Usually, in the literature "K. Xu, P. Gong, Y. Wu, J. Li, and X. Deng, 'Power constraint waveform design based on admm algorithm for mimoradar transmit beampattern,' Digital Signal Processing, vol. 130, p. 103713, 2022", an alternating direction multiplier (ADMM) method is proposed by slightly relaxing the CMC. However, due to obtaining an approximate constant modulus solution, the practical value of this method is limited.

[0005] In the literature "H. He, P. Stoica, and J. Li, 'Wideband mimo systems: Signal design for transmit beampattern synthesis,' IEEE Transactions on Signal Processing, vol. 59, no. 2, pp. 618–628, 2011", for wideband beam pattern design, it is usually proposed to form a wideband beam pattern (WBFIT) through iterative techniques by relaxing the CMC to a peak-to-average power ratio (PAPR) constraint, which reduces the design accuracy. To improve the design accuracy, a successive closed-form (SCF) method by relaxing the objective function was proposed in the literature "O. Aldayel, V. Monga, and M. Rangaswamy, 'Tractable transmit mimo beam pattern design under a constant modulus constraint,' IEEE Transactions on Signal Processing, vol. 65, no. 10, pp. 2588–2599, 2017". However, due to the relaxation, its performance degrades. To address this issue, a projection, descent, and retraction (PDR) method was proposed in the literature "K. Alhujaili, V. Monga, and M. Rangaswamy, 'Transmit mimo radar beam pattern design via optimization on the complex circle manifold,' IEEE Transactions on Signal Processing, vol. 67, no. 13, pp. 3561–3575, 2019" by updating the step size using a line search strategy. However, balancing performance and complexity is a challenge.To avoid this drawback, a method based on Majorization-Minimization (MM) is proposed in the literature "W.Fan, J.Liang, G.Lu, X.Fan, and H.C.So, “Spectrally-agile wave-form design for wideband mimo radar transmit beampattern synthesis via majorization-admm,” IEEE Transactions on Signal Processing, vol.69, pp.1563–1578, 2021". By finding a surrogate function, the performance is reduced because it is very difficult to derive a tractable surrogate function that follows the shape of the original function. Summary of the Invention

[0006] Aiming at the waveform implementation problem of the constant modulus constrained transmit wideband beam pattern of the MIMO array system, the present invention proposes a method for implementing the transmit beam of a wideband MIMO array based on manifold optimization with embedded momentum to reduce the computational complexity and improve the XX performance.

[0007] The technical solution adopted by the present invention is as follows:

[0008] Step 1, construct an optimization model for wideband waveform design of the MIMO array:

[0009]

[0010] Among them, f(x) represents the objective function, and the complex circular manifold x represents the waveform, M represents the number of transmit antennas of the MIMO array, and N represents the number of sampling points of the waveform;

[0011] The matrix P, the vector q, and the parameter r are respectively set as follows:

[0012]

[0013]

[0014]

[0015] Intermediate quantity A p 、 and d p are respectively set as:

[0016]

[0017] Among them, the weighting factor ρ ∈ [0, 1];

[0018] Step 2: Iteratively solve the optimization model constructed in Step 1 based on the manifold with embedded momentum:

[0019] Step 201: Calculate the search direction η at the current iteration j :

[0020]

[0021] where represents the complex circular manifold gradient of the objective function, β represents the step factor in the momentum direction, Trans(·) represents the transformation operator from the manifold space to the tangent space, and η j-1 represents the search direction obtained in the previous iteration, with the initial value being a preset value;

[0022] Complex circular manifold gradient is: where real(·) represents the real part operator, conj(·) represents the conjugate operator,.* represents element-wise multiplication, represents the Euler gradient, and

[0023] Step 202: Judge whether the update condition is satisfied based on the current value of the search step α. If so, update the search step α to γ K α; otherwise, keep the search step α unchanged; and execute Step 203 based on the current search step;

[0024] where the initial value of the search step α is a preset value, x j , x j-1 represent the feasible solutions of the waveforms obtained in the last two iterations, γ is a coefficient greater than 0, and K represents the preset number of search times (preferably K < 10);

[0025] Step 203: Calculate the feasible solution x j :

[0026] Calculate the solution in the tangent space based on the current search step α and search direction η j Calculate the solution in the tangent space

[0027] Based on the solution in the tangent space obtain the feasible solution back to the manifold: where represents the element-wise division symbol, and |·| represents the element-wise modulus symbol;

[0028] Step 204: Judge whether the preset iteration convergence condition is satisfied. If so, obtain the MIMO array transmission beam based on the recently obtained feasible solution x j ; otherwise, return to Step 201.

[0029] Further, in step 204, the iteration convergence condition is that the number of iterations reaches a preset maximum number of iterations, or the iteration convergence condition is set to: |f(x j ) - f(x j-1 )| ≤ ε, where ε represents a preset iteration convergence threshold.

[0030] Further, in step 201, Trans(η j-1 ) = η j-1 - real(η j-1 .* conj(x)).* x.

[0031] Further, the step size factor β is specifically set as:

[0032]

[0033] where the gradient difference

[0034] To further accelerate convergence, in step 202, when the update condition of the search step size α is satisfied, different update strategies are adopted according to the value of the search number K:

[0035] If K = 1, then the search step size α is updated to: ξ 1 γ K α, where the preset parameter ξ 1 > 1;

[0036] If K = 2, then the search step size α is updated to: γ K α;

[0037] If K > 2, then the search step size α is updated to: ξ 2 γ K α, where the preset parameter ξ 2 > ξ 1 .

[0038] The technical solution provided by the present invention at least brings the following beneficial effects:

[0039] The present invention aims at the wideband beam pattern waveform design under constant modulus constraints, and proposes a non-convex quadratic optimization problem with constant modulus constraints. Existing methods mainly solve the problem by relaxing the objective function, resulting in performance degradation. Different from these methods, the present invention notices that the original problem is a quadratic unconstrained problem on the complex circular manifold, and based on this feature, proposes a method for realizing the wideband MIMO array transmit beam based on manifold optimization with embedded momentum. This method adaptively updates the step size and the descent direction based on the momentum on the complex circular manifold. On this basis, the problem is solved using this method. Compared with the existing methods, this method has a faster convergence speed and better performance. Description of the Drawings

[0040] To more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the accompanying drawings required for the description of the embodiments. Obviously, the accompanying drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other accompanying drawings can be obtained based on these drawings.

[0041] Figure 1 is a comparison diagram of the convergence speed of the method proposed in the embodiment of the present invention and the existing method;

[0042] Figure 2 is the spatial-frequency nulling transmit beam pattern of the method proposed in the embodiment of the present invention;

[0043] Figure 3 is the spatial-frequency nulling transmit beam pattern of the existing method MM;

[0044] Figure 4 The spatial-frequency nulling transmit beam pattern of the existing method PDR. Specific implementation manners

[0045] To make the objectives, technical solutions, and advantages of the present invention clearer, the following will further describe the embodiments of the present invention in detail with reference to the accompanying drawings.

[0046] The inventors of the embodiments of the present invention noticed during implementation that the problem of MIMO array broadband waveform design is an unconstrained quadratic problem on the complex circle manifolds (CCM). Based on this feature, the embodiments of the present invention propose a method for designing the transmit beam of the MIMO array based on an unrelaxed momentum information manifold optimization embedding method. That is, first, the problem is transformed into an unconstrained quadratic polynomial on the CCM, and then the manifold optimization embedding with momentum information (MOEMI) algorithm is used to solve the problem by adaptively updating the descent direction and step size based on momentum to complete the implementation of the MIMO array transmit beam.

[0047] As a possible implementation, the specific implementation process of the embodiments of the present invention is as follows:

[0048] First, establish a system model. Consider a centralized broadband MIMO array system with M transmit antennas and a uniform linear array (ULA) configuration. The bandpass signal transmitted by the m-th antenna can be expressed as:

[0049]

[0050] where t is the sampling time, fc is the carrier frequency, and x m (t) is the baseband signal.

[0051] Assume that the spectrum of x m (t) is within the interval [-B / 2, B / 2], where B is the bandwidth in Hz. Using digital signal processing techniques, the baseband signal transmitted by the m-th antenna after sampling can be expressed as:

[0052]

[0053] where N is the number of sampling points, n is the sampling point number, and T s = 1 / B is the sampling rate.

[0054] Then, the discrete Fourier transform (DFT) of x m (n) is expressed as:

[0055]

[0056] where p represents the normalized frequency.

[0057] The beam pattern with respect to the angle θ and the normalized frequency p is given by:

[0058] P(θ,p) = |a H (θ,p)y p | 2 (4)

[0059] where is the steering vector at the angle θ and the frequency at , and its expression is:

[0060]

[0061] where c represents the speed of light. y p represents the vector composed of the discrete Fourier transforms of each point of the signal x m (n), and y p = [y 0 (p)y 1 (p)...y M-1 (p)].

[0062] In addition, the spatial angle θ can be obtained by dividing [0°, 180°] into K subintervals, and the steering vector a(θ,p) can be further written as:

[0063] a kp = a(θ k ,p), k = 1,..., K (5)

[0064] In this sense, the beam pattern in Equation (4) can be redefined as:

[0065]

[0066] where, from and x m = [x m (0)x m (1)...x m (N - 1)] is obtained. The matrix F p , e p are respectively defined as I M represents the identity matrix of dimension M.

[0067] Denote the desired beam pattern as d kp , and the beam pattern design problem of CMC is formulated as:

[0068]

[0069] where, φ k,p is the phase variable.

[0070] In practical applications, spatial frequency nulls are required to suppress strong unwanted echoes and ensure spectral coexistence with communication users. Inspired by this, the null beam pattern design problem is:

[0071]

[0072] where, ρ = [0, 1] is the weighting factor, Θ 1 is the region of interest for beam pattern matching, Θ 2 is null shaping, Θ = Θ 1 ∪Θ 2

[0073] For the form in (8), the problem will be carried out in two minimization stages: in the first stage, φ k,p is fixed with respect to x, and in the second stage, φ k,p is fixed with respect to x.

[0074] The φ k,p in the first minimization stage is obtained by:

[0075]

[0076] It should be noted that the phase variable φ k,p is not inherent to the problem, but is introduced to make the problem tractable. Therefore, in this work, optimizing f(x) under CMC is being sought, where φ k,pis fixed. The problem in (9) can be rewritten as:

[0077]

[0078] where

[0079]

[0080]

[0081]

[0082] Note that the original problem is an unconstrained quadratic problem on the CCM, expressed as:

[0083]

[0084] where M = {x ∈ C MN×1 : |x(n)| = 1, n = 1,..., MN}.

[0085] First, the momentum-based descent direction update strategy is derived, and then the descent step size is adaptively updated, that is, the optimization principle on the manifold is used to ensure the monotonicity and convergence of the objective function. Finally, the solution is updated through the manifold backtracking operation.

[0086] Compared with the disadvantage that the traditional gradient descent method is prone to falling into local optima, the momentum-based descent method in the embodiments of the present invention better matches the problem model and improves the ability of the algorithm to jump out of local optima. The traditional gradient descent method does not consider the influence of momentum and fails to truly match the problem model. The feasible solution point can be regarded as a moving mass point, and the optimization process is equivalent to the mass point updating the feasible solution on a non-convex high-dimensional plane according to the set descent direction and step size. Since the moving mass point has an initial momentum (velocity), the descent direction of the mass point is the combined direction of the mass point gradient descent direction and the initial momentum direction.

[0087] The momentum-based descent direction can be expressed as:

[0088]

[0089] where is the complex circular manifold gradient of the objective function, β is the step size factor in the momentum direction, Trans(·) is the conversion operator that switches the manifold space to the tangent space, and η j-1 is the initial movement direction.

[0090] The complex circular manifold gradient can be expressed as:

[0091]

[0092] where real(·) is the real part operator and conj(·) is the conjugate operator. is the Euler gradient, and.* represents element-wise multiplication, which is expressed as:

[0093]

[0094] The conversion operation of the initial motion direction can be expressed as:

[0095] Trans(η j-1 ) = η j-1 -real(η j-1 .*conj(x)).*x (15)

[0096] The value of the step size factor β in the momentum direction depends on the gradient difference, which is expressed as:

[0097]

[0098] where diff is the gradient difference, which is expressed as

[0099] Next, the step size is updated using an adaptive backward search strategy, which can adaptively adjust the search step size according to the change of momentum, ensure the monotonicity of the algorithm, and improve the convergence speed of the algorithm.

[0100] The adaptive update strategy can be expressed as

[0101]

[0102] where α is the initial step size, γ is a coefficient greater than 0, and K is the number of searches (K < 10).

[0103] After K searches, the obtained adaptive step size is:

[0104] α new = γ K *α (18)

[0105] That is, only when the formula (17) is satisfied, the update method given by the formula (18) is executed. At the same time, to further accelerate the convergence, the initial step size can be appropriately adjusted according to the change of K. If K = 1, it indicates that the initial step size is selected too conservatively, and the initial step size for the next iteration should be increased to α = ξ 1 α new , ξ 1 > 1; if K = 2, it indicates that the initial step size is selected reasonably, and the step size should be maintained as α = α new in the next iteration; if K > 2, it indicates that the number of searches is too large, resulting in an overly small adaptive step size, and the initial step size for the next iteration should be increased to α = ξ 2 αnew , ξ 2 > 1.

[0106] Given the descent direction η on the tangent space of the manifold j (13) and the search step size α new (19), the solution of the tangent space can be expressed as

[0107]

[0108] Then, the feasible solution back to the manifold can be expressed as

[0109]

[0110] where is the element-wise division symbol, and |·| is the element-wise modulo symbol.

[0111] Based on the above analysis and discussion, the specific solution steps for solving the optimization problem corresponding to formula (11) include:

[0112] Step 1: Calculate the search direction η according to formula (12) j ;

[0113] Step 2: Calculate the step size α using the adaptive search strategy (i.e., formula (18)) new ;

[0114] Step 3: Calculate the feasible solution x according to formulas (19) and (20) j ;

[0115] Step 4: If |f(x j ) - f(x j-1 )| ≤ ε, output x = x j . Otherwise, return to Step 1; where ε represents the preset comparison threshold.

[0116] Embodiment

[0117] To verify the performance of the method proposed in the embodiment of the present invention, it is simulated and compared with the existing method.

[0118] Consider a MIMO array system with the number of transmit antennas M = 10, the weighting coefficient ρ = 0.85, the carrier frequency f c being 1 GHz, the bandwidth B = 200 MHz, the element spacing d = c / 2(f c + B / 2), the number of samples N = 32, and the angle region of interest being [0°, 180°] and uniformly discretized with a step size of Δθ = 1°. The spatial frequency domain of interest is defined as Θ ∈ {(θ, f)|θ ∈ [0°, 180°], f ∈ [f c - B / 2, f c+B / 2], and the null region is Θ 2 ∈{(θ,f)|θ∈[15°,45°], f∈[fc,fc + B / 2]}, and the desired transmit beam pattern Θ 1 is

[0119] This embodiment is mainly compared with two existing methods (PDR method, MM method).

[0120] Among them, for the existing method PDR method, specific references can be found in the literature "K. Alhujaili, V. Monga, and M. Rangaswamy, "Transmit mimo radar beampattern design via optimization on the complex circle manifold," IEEE Transactions on Signal Processing, vol. 67, no. 13, pp. 3561–3575, 2019" and "K. Alhujaili, X. Yu, G. Cui, and V. Monga, "Spectrally compatible mimo radar beampattern design under constant modulus constraints," IEEE Transactions on Aerospace and Electronic Systems, vol. 56, no. 6, pp. 4749–4766, 2020".

[0121] For the existing method MM method, specific references can be found in the following literature:

[0122] Literature 1: "W. Fan, J. Liang, G. Lu, X. Fan, and H. C. So, "Spectrally-agile waveform design for wideband mimo radar transmit beampattern synthesis via majorization-admm," IEEE Transactions on Signal Processing, vol. 69, pp. 1563–1578, 2021";

[0123] Reference 2: 《R. Liu, M. Li, Q. Liu, and A. L. Swindlehurst, “Dual-functional radar-communication waveform design: A symbol-level precoding approach,” IEEE Journal of Selected Topics in Signal Processing, vol. 15, no. 6, pp. 1316–1331, 2021》;

[0124] Reference 3: 《W. Fan, J. Liang, G. Yu, H. C. So, and G. Lu, “Mimo radar waveform design for quasi-equiripple transmit beampattern synthesis via weighted lp-minimization,” IEEE Transactions on Signal Processing, vol. 67, no. 13, pp. 3397–3411, 2019》;

[0125] Reference 4: [Z. Huang, B. Tang, H. Wang, and L. Qin, “Wideband mimo radar transmit beampattern synthesis via majorization–minimization,” Circuits, Systems, and Signal Processing, vol. 40, no. 11, pp. 5594–5615, 2021];

[0126] Reference 5: [L. Wu, P. Babu, and D. P. Palomar, “Transmit waveform / receive filter design for mimo radar with multiple waveform constraints,” IEEE Transactions on Signal Processing, vol. 66, no. 6, pp. 1526–1540, 2018].

[0127] Figure 1 The comparison of the convergence speeds between the proposed method and the existing methods is given. From this, it can be concluded that the method of the embodiment of the present invention is superior to the existing two algorithms in terms of convergence speed.

[0128] Figure 2 and 3 Figs. 4 presents the nulling transmit beam patterns of the proposed method and the existing method. It can be seen from them that the performance of the method according to the embodiments of the present invention is better than that of the existing methods.

[0129] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features. And these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

[0130] The above are only some embodiments of the present invention. For those of ordinary skill in the art, without departing from the inventive concept of the present invention, several deformations and improvements can still be made, and these all belong to the protection scope of the present invention.

Claims

1. Method for implementing wideband MIMO array transmitting beam based on embedded momentum manifold optimization, Characterized in that, Comprising the following steps: Step 1, constructing an optimization model for wideband waveform design of MIMO array: where f(x) represents the objective function, and the complex circular manifold x represents the waveform, M represents the number of transmit antennas of the MIMO array, and N represents the number of sampling points of the waveform; Matrix P, vector q and parameter r are respectively set as follows: Intermediate quantity A p , and d p are respectively set to: Wherein, the weighting factor ρ ∈ [0, 1]; Step 2, iteratively solving the optimization model constructed in Step 1 based on the embedded momentum manifold; Step 201, calculate the search direction η at the current iteration number j : η j = -▽ M f(x) + βTrans(η j-1 ); Among them, represents the complex circular manifold gradient of the objective function, β represents the step factor in the momentum direction, Trans(·) represents the conversion operator for switching from the manifold space to the tangent space, and η j-1 represents the search direction obtained in the previous iteration, and the initial value is a preset value; Complex circular manifold gradient is: where real(·) represents the real part operator, conj(·) represents the conjugate operator,.* represents element-wise multiplication, ▽f(x) represents the Euler gradient, and ▽f(x) = 2Px - 2q; Step 202, determine whether the update condition f(x j ) > f(x j-1 ) + γ K * α * (▽ M f(x) H * η i ) is satisfied. If so, update the search step α to γ K α; otherwise, keep the search step α unchanged; and execute Step 203 based on the current search step; Among them, the initial value of the search step size α is a preset value, and x j , x j-1 represent the feasible solutions of the waveforms obtained in the last two iterations, γ is a coefficient greater than 0, and K represents the preset number of search times; Step 203, calculate the feasible solution x j : Based on the current search step size α and search direction η j Calculate the solution of the tangent space Solution based on the tangent space Obtain a feasible solution that traces back to the manifold: where represents the element-wise division symbol, and |·| represents the element-wise modulus symbol; Step 204: Determine whether the preset iterative convergence condition is satisfied. If so, obtain the MIMO array transmission beam based on the most recently obtained feasible solution x j ; otherwise, return to Step 201.

2. The method according to claim 1, Characterized in that, In step 204, the iteration convergence condition is that the number of iterations reaches the preset maximum number of iterations, or the iteration convergence condition is set to: |f(x j ) - f(x j-1 )| ≤ ε, where ε represents the preset iteration convergence threshold.

3. The method according to claim 1, Characterized in that, In step 201, the transformation operator Trans(η j-1 ) = η j-1 - real(η j-1 .* conj(x)).* x.

4. The method according to claim 1, Characterized in that, In Step 201, the step size factor β is specifically set as: β = (▽ M f(x)) H *diff / real(diff H *Trans(η j-1 )) Among them, the gradient difference 5. The method according to claim 1, Characterized in that, In Step 202, when the update condition of the search step α is satisfied, different update strategies are adopted according to the value of the search times K: If K = 1, update the search step length α to: ξ 1 γ K α, Among them, the preset parameter ξ 1 > 1; If K = 2, update the cable step length α to: γ K α; If K > 2, then update the search step size α to: ξ 2 γ K α, where the preset parameter ξ 2 > ξ 1 .

6. The method according to any one of claims 1 to 5, Characterized in that, The value of the search times K is less than 10.

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