CN113642181B - MIMO radar waveform optimization method for rapid manifold modeling - Google Patents

Abstract

The application discloses a MIMO radar waveform optimization method for rapid manifold modeling, relates to the technical field of radars, and solves the problem of detection performance degradation when the radars are subjected to strong sidelobe interference. The method comprises the steps of carrying out joint optimization on signal to noise ratio under constant modulus constraint, adding weighting factors, combining into a new objective function for optimizing SINR, converting constraint problems in Euclidean space in the objective function into unconstrained optimization problems based on Riemann manifold, optimizing by adopting a Riemann manifold BFGS algorithm limiting memory, obtaining an optimized MIMO radar receiver waveform matrix, introducing the waveform matrix into MIMO radar data, and carrying out radar signal transceiving through an antenna. The method and the device optimize the waveforms of all the snapshots of the radar respectively, so that the performance of the MFL of the radar is obviously improved.

Description

MIMO radar waveform optimization method for rapid manifold modeling

Technical Field

The application relates to the technical field of radars, in particular to a MIMO radar waveform optimization method for rapid manifold modeling.

Background

MIMO (Multiple-Input Multiple-Output) radar has attracted considerable attention in recent years due to its superiority in target detection and parameter estimation performance. The MIMO radar with densely distributed antennas can provide more flexibility for the design of the transmitting directional diagram by transmitting different waveforms on each antenna, and effectively improve the signal-to-interference-and-noise ratio. Therefore, waveform design is an extremely important topic for MIMO radar, and attracts a wide range of attention.

Recently, waveforms based on maximized SINR (signal to interference plus noise ratio) are designed as a research hotspot, and the methods mainly include two types: the first is to jointly optimize the waveform of the transmitting end and the filter weight vector of the receiving end. Such as the SDP (semidefinite programming, semi-definite programming) -random optimization method as described in the documents "G.Cui, H.Li, and M.Rangaswamy, MIMO radar waveform design with constant modulus and similarity constraints," IEEE Trans.Signal Process, vol.62, no.2, pp.343-353, jan.2014 ", and the SQR (successive quadratically constrained quadratic programing refinement, successive quadratic programming iterations) method as described in the documents" O.Aldayel, V.Monga and M.Rangaswamy, successive QCQP Refinement for MIMO Radar Waveform Design Under Practical Constraints, "IEEE Trans.Signal Process, vol.64, no.14, pp.3760-3774, july 2016. The second type of waveform optimization method is to optimize only the transmit waveform. Of these, the IA-CPC (iterative algorithm for continuous phase case, sequential phase iterative algorithm) method proposed in the literature G.Cui, X.Yu, G.Foglia, Y.Huang and j.li, quadratic optimization with similarity constraint for unimodular sequence synthesis, "IEEE trans.signal process," vol.65, no.18, pp.4756-4769, sep.2017, "X.Yu, G.Cui, L.Kong, J.Li and g.gui, constrained Waveform Design for Colocated MIMO Radar With Uncertain Steering Matrices," IEEE trans.aerosp.electron. Syst., vol.55, no.1, pp.356-370, feb.2019, "Dinkel-CD (Dinkelbachs coordinate decent, dinkel-descent algorithm) method, and the CR (convex relaxation, convex relaxation algorithm) method proposed in the literature M.Bolhasani, E.Mehrshahi, and s.ghorashi," Waveform covariance matrix design for robust signal-dependent interference suppression in colocated MIMO radars, "Signal Processing, vol 152,2018, pp.311-319 (hereinafter abbreviated as literature 1) are more typical.

In a practical scenario, since the radar transmitting end always operates in a saturated state in order to avoid waveform distortion, the constant modulus constraint is a constraint that must be added in waveform design. Due to the constraint of constant modulus, the SINR optimization problem is always NP-hard type and thus difficult to solve. Existing studies have generally adopted phase-only waveform optimization methods and optimized the overall waveform for all snapshots.

There are two limitations in this case, firstly, since the radar inevitably receives environmental disturbance in actual operation, the waveform design of the phase only is not well satisfied in practice. Second, the overall design of waveforms for all snapshots can make a large difference between waveform performance for different snapshots, which can affect the MFL (matched filtering loss, matched filter loss) performance of the radar.

Disclosure of Invention

The technical problems to be solved by the application are as follows: and when the radar is interfered by strong sidelobes, the detection performance is reduced. The present application provides a MIMO radar waveform optimization method for fast manifold modeling to maximize signal-to-interference-and-noise ratio, by designing a MIMO radar waveform with signal-dependent interference and gaussian white noise and using the output SINR as a design indicator for existing prior information about target direction angle and signal interference direction angle, which can be obtained from previous radar beam scans by existing target detection methods. By converting the optimized objective function to the Riemann complex manifold for searching and optimizing, the interference existing at the notch can be obviously restrained, and the signal-to-interference-and-noise ratio can be improved. Meanwhile, the waveforms of all the snapshots of the radar are optimized respectively, so that the performance of the MFL of the radar is obviously improved.

The application is realized by the following technical scheme:

a MIMO radar waveform optimization method for rapid manifold modeling comprises the following steps:

step A: constructing a MIMO radar model to obtain a waveform matrix;

and (B) step (B): adding interference data obtained by target radar beam scanning into the MIMO radar model, wherein the interference data comprises additive noise and a plurality of interference sources with independent signals, and calculating complex vectors of a radar antenna receiving end in the interference direction of the interference sources;

step C: calculating the signal-to-noise ratio of the output end of the receiver comprising a plurality of variables, carrying out joint optimization on the signal-to-noise ratio under the constraint of a constant modulus, adding a weighting factor, combining the signal-to-noise ratio into a new objective function for optimizing SINR, converting the constraint problem in the Euclidean space in the objective function into an unconstrained optimization problem based on the Riemann manifold, and obtaining a cost function for optimizing the SINR objective function by calculating the unconstrained quadratic optimization problem of the complex circular manifold which extends the Euclidean space to the Riemann manifold;

step D: optimizing a radar model, a cost function and a Riemann manifold framework, adopting a Riemann manifold BFGS algorithm limiting a memory to calculate the iteration direction of BFGS, performing linear search, continuously updating the step length, calculating the step length coefficient of BFGS, searching a convergence value conforming to the Wolff criterion, calculating the cost function value and the Riemann gradient at the moment, and iterating under the preset iteration times and iteration stopping conditions of the BFGS algorithm;

step E: the iteration is terminated to obtain an optimized MIMO radar receiver waveform matrix, the waveform matrix is led into MIMO radar data, radar signal receiving and transmitting are carried out through an antenna, a directional diagram obtained by using the waveform to carry out wave beam formation can generate a notch in the interference direction, so that the interference in the direction is greatly inhibited, and the radar detection performance is improved.

The MIMO radar is provided with M _t Root transmit antenna and M _r The method comprises the steps of co-arranging MIMO radars with receiving antennas, wherein all the transmitting and receiving antennas are isotropic, the transmitting and receiving antennas are uniform linear arrays, the distance between adjacent array elements is half wavelength, during the radar detection process, the radar adjusts the used waveforms according to the interference condition in a scene, a pattern notch is generated at any position where interference exists, and interference in the direction is restrained.

The waveform sequence transmitted by the mth antenna isWherein, m=1, M _t L is the number of samples per waveform, the waveform matrix is +.>

Wherein, the target direction of the MIMO radar is theta ₀ The directions of the independent interference sources of the Q signals are theta _i I=1, …, Q, M for the far-field target, the reception at the receiving end at time index l _r The x 1 complex vector is:

y(n)＝α ₀ A(θ ₀ )x _n +d(n)+v(n) (1)

wherein,

1)α ₀ representing complex amplitude of targeta _t (θ) and a _r (θ) are transmit and receive steering vectors, respectively, expressed as:

2)interference signal representing scattering interference of independent uncorrelated point sources of Z signals, the Z-th interference source being located at theta _z Z=1,..z, d (n) represents the interference signal superposition of Z interference sources:

α _z representing the complex amplitude of the Z-th interferer;

3)the additive noise is represented as gaussian noise:

the signal-to-noise ratio of the output end of the radar receiver obtained through the complex vector of the receiving end is as follows:

wherein the omega of the numerator is an information matrix containing the target direction, and the Y of the denominator is a matrix containing the interference direction and the noiseThe acoustic information matrix, the numerator is the signal power and the denominator is the interference and noise power;the vec (·) operator represents the reconstruction of the matrix into a column vector, where the specific expression of the other symbols is as follows: />

The problem of optimizing SINR under constant modulus constraint is described as:

maximizing x under constant modulus constraint ^H While Ω x, minimize x ^H Yx is multiplied by a weighting factor in each term, and the objective function is generated in a convex combination manner as follows:

wherein, xi= (1-gamma) Y-gamma omega and gamma epsilon [0,1]

Solving equation (5) in Riemann space.

The complex circular manifold is calculated by expressing the Riemann manifold as the product of complex Euclidean space and complex circle as:

converting equation (5) into an unconstrained quadratic optimization problem on the complex circular manifold, the unconstrained quadratic optimization problem being expressed as:

obtaining a cost function (7), projecting the gradient and Hessian matrix under the complex Euclidean space into a tangent plane of the Riemann manifold, calculating the Riemann gradient and the Riemann Hessian matrix, performing gradient descent search in the tangent plane, and retracting the searched points into the manifold by using a defined retraction function;

and stopping obtaining the waveform function when the algorithm meets the condition.

The application has the following advantages and beneficial effects:

according to the application, the waveform design problem conforming to the constant modulus constraint is quickly modeled and solved by utilizing the Riemann manifold algorithm, the constraint problem is converted into the unconstrained optimization problem on the Riemann complex manifold, the problem is directly optimized, the approximation error is reduced, and a deeper notch can be obtained on the directional diagram. Meanwhile, the waveforms of all the snapshots of the radar are optimized respectively, so that the performance of the MFL of the radar is obviously improved. In addition, the application can obtain higher signal-to-noise ratio under the condition of lower signal-to-noise ratio (SNR).

Drawings

The accompanying drawings, which are included to provide a further understanding of embodiments of the application and are incorporated in and constitute a part of this specification, illustrate embodiments of the application and together with the description serve to explain the principles of the application. In the drawings:

fig. 1 is a graph showing the comparison of the performance of the emission pattern in the example.

Fig. 2 is a graph showing comparison of snr performance in the embodiment.

Detailed Description

Before any embodiments of the application are explained in detail, it is to be understood that the application is not limited in its application to the details of construction set forth in the following description or illustrated in the drawings. The application is capable of other embodiments and of being practiced or of being carried out in various ways. All other embodiments, which can be made by those skilled in the art based on the embodiments of the application without making any inventive improvements, are intended to fall within the scope of the application.

Examples:

co-located MIMO radar with M _t Root transmit antenna and M _r And a root receiving antenna. All transmit and receive antennas are isotropic. The transmitting antenna and the receiving antenna are uniform linear arrays, and the distance between adjacent array elements is half wavelength. The waveform sequence transmitted by the mth antenna can be expressed asWherein, m=1, M _t L is the number of samples per waveform. Then the waveform matrix can be expressed as +.>

In this embodiment, the radar detects an aircraft target with a target direction θ ₀ Q strong scatterers or artificial interference sources appear in the airspace, target echoes are submerged in interference, so that the detection of the radar is greatly influenced, and the directions of the Q independent interference sources are theta _i I=1, …, Q. The case where the target and the disturbance are in the same range means that the detection performance is poor because the target delays through the range gate do not contribute to the improvement of the detection performance. However, for some far-field target, the receiving end receives M _r The x 1 complex vector (at time index l) is expressed as

y(n)＝α ₀ A(θ ₀ )x _n +d(n)+v(n) (1)

Wherein,

1)α ₀ representing complex amplitude of targeta _t (θ) and a _r (θ) are transmit and receive steering vectors, respectively, which can be expressed as follows:

2)interference representing point source scattering interference of independent uncorrelated Z signalsA signal. In the present application the Z-th interference source is located at θ _z Z=1,..z, d (n) represents the interference signal superposition of Z interference sources:

α _z representing the complex amplitude of the Z-th interferer.

3)Representing additive noise, here taken as gaussian noise:

after the radar model is established, the cost function of the optimization algorithm is set forth next.

When definingThe signal to noise ratio at the output of the radar receiver can then be expressed as follows:

wherein,the vec (·) operator represents the reconstruction of the matrix into a column vector, where the specific expression of the other symbols is as follows: />

The goal is to optimize SINR under constant modulus constraint (CM constraint), the problem can be expressed by the following equation:

equation 4 essentially seeks to maximize x under constant modulus constraints ^H While Ω x, minimize x ^H Yx, the best possible joint optimization of the two equations, grasping this essence, multiplies each term by a weighting factor, and forms the objective function in a convex combination, as can be expressed by the following equation:

wherein, xi= (1-gamma) Y-gamma omega and gamma e [0,1].

Unlike the existing search algorithm in euclidean space, in the method, equation (5) is solved in the euclidean space, the euclidean manifold is the product of the complex euclidean space and the complex circle, and based on the euclidean manifold, the constraint problem in euclidean space can be converted into the unconstrained optimization problem, so that a large number of excellent algorithms based on the gradient descent concept in euclidean space can be popularized to the euclidean manifold. The expression of the complex circular manifold is as follows:

thus, the problem of converting equation (5) to unconstrained quadratic optimization on complex circular manifolds can be expressed as

So far, the cost function to be optimized is obtained, and the definition of Guan Liman manifold in the algorithm is further described below.

Under the Riemann manifold, the display of the optimization algorithm is derived as follows: 1. the Riemann gradient and the Riemann Hessian matrix may be calculated by projecting the gradient and Hessian matrix in Euclidean space into the tangent plane of the Riemann manifold. 2. Gradient descent search is performed in the tangential plane. 3. The searched points are retracted into the manifold using a defined retraction function.

The Riemann gradient of a function f (x) can be represented by the following equation

Wherein,represents R from ⁿ To T _x S ^n-1 Orthogonal projection of +.>Representing the euclidean gradient of the function f (x).

It is also necessary to select the Riemann metric, which is generally selected as follows

g _x (ε,η)＝<ε,η>＝Re{ε ^H η} (9)

The withdraw operator (recovery) is defined as follows

Based on the signal model, cost function and Riemann manifold framework described above, the proposed fast modeling optimization algorithm, the Riemann manifold BFGS algorithm with memory limitation (Limited-memory Riemannian BFGS method), is described next.

The BFGS algorithm was proposed in 1970 by Broyden, fletcher, goldfar and Shanno and is therefore abbreviated as BFGS, which uses a matrix B without second derivatives _(k+1) Approximating the Hesse matrix. The BFGS algorithm is widely regarded as the best quasi-newton algorithm in euclidean space, and the results obtained in numerical experiments by the BFGS algorithm are generally superior to the DFP (Davidon-Fletcher-Powell) algorithm, another common quasi-newton algorithm. In recent years, the Riemann manifold has attracted considerable attention because of its excellent performance in several areas. The Riemann BFGS algorithm is the popularization of the BFGS algorithm on the Riemann manifold and is derived from the BFGS algorithm in the algorithm ideaThe method is also a method for carrying out descending search on the Riemann manifold by applying an inverse updating formula.

However, when the optimization scale is large, the dimension of the matrix in the BFGS algorithm is quite striking, so that the algorithm is difficult to realize, and the idea of the L-BFGS is to only store the information of the last m iterations, so that the storage scale of a computer can be greatly reduced, and the realization of the algorithm is possible.

The Riemannian BFGS algorithm has similar defects as the BFGS algorithm, so the Riemannian L-BFGS should make similar improvement to the L-BFGS algorithm, namely the latest s is saved _k And y _k Generating vectors in new tangent space using these data to avoid storing entire bulky H _k Matrix such that the algorithm can be implemented on a computer.

The Riemannian L-BFGS algorithm is specifically described as follows:

the update formula of RBFGS can be expressed as

Wherein,representing function composition (.) ^-1 Representing an inverse function; />

Assume that the latest l-1 s are stored _k And y _k Will beIterative to (11) to obtain

Wherein, representing the initial Hessian approximation in the next iteration. />Represented by s _i Converted lie->Upper tangent vector.

The stopping conditions of the algorithm are as follows: 1. the cost function value is smaller than the threshold value |f (x _k )|＜ε ₀ The method comprises the steps of carrying out a first treatment on the surface of the 2. The modulus of the Riemann gradient is less than the threshold gradf (x _k )||＜ε ₁ The method comprises the steps of carrying out a first treatment on the surface of the 3. The iteration number has reached a preset maximum value k < k _max 。

The following specific steps of the algorithm are as follows:

initializing: definition of Riemann manifoldAnd a corresponding Riemann metric g (α, η); withdrawal R (&) is in accordance with-> Is equal to the equidistant transmission vector T of (1) _s In->An upper smooth function f, an initial search point on manifold,/>Setting k=0, epsilon > 0,0 < c ₁ ＜0.5＜c ₂ ＜1，γ ₀ ＝1，l＝0；

0. When any stopping condition is reached, stopping the algorithm, outputting a result, otherwise executing the following steps:

1.obtaining +.>

2.q＝gradf(x _k )；

3. Cycle 1 is performed with the cycle variable i=k-1, k-2

4.

5.

6. When i=l, cycle 1 ends.

7.

8. Cycle 2 is performed with the cycle variable i=l, l+1

9.

10.

11. When i=k-1, cycle 2 ends.

12. Let eta _k ＝-r；

13. Finding coefficients alpha meeting Wolfe criteria _k ；

Wherein the Wallf criterion is:

f(x _k+1 )≤f(x _k )+c ₁ α _k g(gardf(x _k ),η _k )

14. order the

15. Order the

16. Let l=max { k-l,0}; will beAnd ρ _k Added to the memory storage. If k > l, deleting vector pairsAnd expanding ρ from memory storage _l-1 The method comprises the steps of carrying out a first treatment on the surface of the Will->And->By T _s From->Switch to->Obtain->And->

17. Let k=k+1, return to step 0;

outputting the searched waveform when the algorithm meets the condition and stopsThe method is the method.

For a given angle θ, the transmission pattern may be expressed as follows:

wherein,representing a waveform matrix +_>Representing the covariance matrix of the waveform, the SINR may be given by:

wherein the method comprises the steps of

Wherein,represents SNR, & gt>INR representing the qth interferer.

In the above embodiment, a method for optimizing a MIMO radar waveform by fast manifold modeling includes the steps of:

step 1: several parameters required for the construction of the algorithm: number of transmitting array elements M _T Target location θ ₀ ，N _J Angle theta of the interference ₁ ,θ ₂ …Sequence length M, weight factor γ. Minimum step size minstepsize, maximum iteration number maximation, minimum gradient min grad (x) _k )||。

Step 2: calculating BFGS direction: according to ρ _i 、Search direction q from last iteration _n-1 Obtaining the search direction q of the current iteration _n 。

Step 3: linear search is performed: last iteration point x _k Searching a step length along the gradient descent direction to obtain a new point x _k+1 Updating the step size.

Step 4: calculating BFGS step length coefficient and finding alpha meeting Wolfe criteria _k 。

Step 5: calculating a new candidate point x _k+1 Cost function value and Riemann gradient q=gradf (x _k )。

Step 6: obtaining new variables for this iterationρ _k 、γ _k+1 And beta _k 。

Step 7: if any termination condition is not triggered, returning to the step 2, and terminating the algorithm to obtain a waveform when the algorithm meets any stop condition

In this embodiment 2, consider a device having a transmitting antenna M _T A uniform linear array of=16, the number of snapshots being m=128. The target direction is theta ₀ =0°. The total of five interference sources are called interference 1, interference 2, interference 3, interference 4 and interference 5 for convenience of definition. The three directions of interference are respectively theta ₁ ＝-80°,θ ₂ ＝-45°,θ ₃ ＝-15°，θ ₁ ＝30°，θ ₁ =60°. The simulation scenario is the same as the implementation scenario in document 1. The parameters of the Riemannian L-BFGS algorithm are set to the weight factor γ=10 ^-6 The method comprises the steps of carrying out a first treatment on the surface of the Minimum step size stepsize=10 ^-10 Maximum iteration number maximum=5000, minimum gradient grad (x _k )||＝10 ^-6 。

Fig. 1 shows that the transmission pattern produced by the proposed method produces five notches of depth-156.3 dB, -137.9dB, -143.9dB, -155.7dB, -143.8dB, respectively, which are improved by 48.8dB,30.3dB,36.4dB,48.2dB,36.3dB, respectively, over the three notches of depth-107.5 dB, -107.6dB, -107.5dB, respectively, of the prior art. The proposed algorithm can achieve deeper notches in the interference direction and about 30-50dB deeper than the notches of prior art scheme 1.

Fig. 2 shows the variation of the SINR of the output with the SNR of the input. It can be seen that the proposed algorithm results in an SINR that is 0.5dB higher than that of the existing scheme 1. Therefore, the waveform obtained by the present application has better SINR performance than the waveform obtained by the prior art scheme 1.

The foregoing description of the embodiments has been provided for the purpose of illustrating the general principles of the application, and is not meant to limit the scope of the application, but to limit the application to the particular embodiments, and any modifications, equivalents, improvements, etc. that fall within the spirit and principles of the application are intended to be included within the scope of the application.

Claims (3)

1. The MIMO radar waveform optimization method for the rapid manifold modeling is characterized by comprising the following steps of:

step A: constructing a MIMO radar model to obtain a waveform matrix;

and (B) step (B): adding interference data obtained by target radar beam scanning into the MIMO radar model, wherein the interference data comprises additive noise and a plurality of interference sources with independent signals, and calculating complex vectors of a radar antenna receiving end in the interference direction of the interference sources;

step C: calculating the signal-to-noise ratio of the output end of the receiver comprising a plurality of variables, carrying out joint optimization on the signal-to-noise ratio under the constraint of a constant modulus, adding a weighting factor, combining the signal-to-noise ratio into a new objective function for optimizing SINR, converting the constraint problem in the Euclidean space in the objective function into an unconstrained optimization problem based on the Riemann manifold, and obtaining a cost function for optimizing the SINR objective function by calculating the unconstrained quadratic optimization problem of the complex circular manifold which extends the Euclidean space to the Riemann manifold;

step D: optimizing a radar model, a cost function and a Riemann manifold framework, adopting a Riemann manifold BFGS algorithm limiting a memory to calculate the iteration direction of BFGS, performing linear search, continuously updating the step length, calculating the step length coefficient of BFGS, searching a convergence value conforming to the Wolff criterion, calculating the cost function value and the Riemann gradient at the moment, and iterating under the preset iteration times and iteration stopping conditions of the BFGS algorithm;

step E: the iteration is terminated to obtain an optimized MIMO radar receiver waveform matrix, the waveform matrix is imported into MIMO radar data, and radar signal receiving and transmitting are carried out through an antenna;

wherein the MIMO radar is provided with M _t Root transmit antenna and M _r Co-antenna for root receptionArranging the MIMO radar, wherein all transmitting and receiving antennas are isotropic, the transmitting and receiving antennas are uniform linear arrays, and the distance between adjacent array elements is half wavelength;

the waveform sequence transmitted by the mth antenna isWherein, m=1, M _t L is the number of samples per waveform, the waveform matrix is +.>

The target direction of the MIMO radar is theta ₀ The directions of the independent interference sources of the Q signals are theta _i I=1, …, Q, M for the far-field target, the reception at the receiving end at time index l _r The x 1 complex vector is:

y(n)＝α ₀ A(θ ₀ )x _n +d(n)+v(n) (1)

wherein,

1)α ₀ representing complex amplitude of targeta _t (θ) and a _r (θ) are transmit and receive steering vectors, respectively, expressed as:

transmitting a steering vector:

receiving a steering vector:

2)representing independent uncorrelation of Z signalsThe point source scatters the interfering signal of the interference, and the Z-th interfering source is positioned at theta _z Z=1,..z, d (n) represents the interference signal superposition of Z interference sources:

α _z representing the complex amplitude of the Z-th interferer;

3)the additive noise is represented as gaussian noise:

the signal-to-noise ratio of the output end of the radar receiver obtained through the complex vector of the receiving end is as follows:

wherein, the omega of the numerator is an information matrix containing the target direction, the Y of the denominator is an information matrix containing the interference direction and the noise, the numerator is the signal power, and the denominator is the interference and the noise power;the vec (·) operator represents the reconstruction of the matrix into a column vector, where the specific expression of the other symbols is as follows: />

2. The method for optimizing MIMO radar waveforms for fast manifold modeling according to claim 1, wherein the problem of optimizing SINR under constant modulus constraint is described as:

maximizing x under constant modulus constraint ^H While Ω x, minimize x ^H Yx is multiplied by a weighting factor in each term, and the objective function is generated in a convex combination manner as follows:

wherein, xi= (1-gamma) Y-gamma omega and gamma epsilon [0,1]

Solving equation (5) in Riemann space.

3. A method of optimizing MIMO radar waveforms for fast manifold modeling according to claim 2, wherein the complex circular manifold is calculated by expressing the risman manifold as the product of complex euclidean space and complex circle as:

converting equation (5) into an unconstrained quadratic optimization problem on the complex circular manifold, the unconstrained quadratic optimization problem being expressed as:

obtaining a cost function (7), projecting the gradient and Hessian matrix under the complex Euclidean space into a tangent plane of the Riemann manifold, calculating the Riemann gradient and the Riemann Hessian matrix, performing gradient descent search in the tangent plane, and retracting the searched points into the manifold by using a defined retraction function;

and stopping obtaining the waveform function when the algorithm meets the condition.