A multi-objective time delay sparse reconstruction estimation method based on exponential filter

Abstract

The present invention provides a multi-target time-delay sparse reconstruction estimation method based on an exponential filter, belonging to the field of communication technologies. The method includes constructing an exponential filter H of an optimal order p by using a radiation source reference signal p ; inputting the received multi-target received signal into the exponential filter H p , obtaining the exponential-order cross-correlation function of the received signal; performing wavelet soft threshold denoising processing on the exponential-order cross-correlation function; constructing a multi-target time-delay parameter sparse reconstruction optimization model by using the denoised exponential-order cross-correlation function; and using an l1-norm sparse reconstruction algorithm to solve the multi-target time-delay parameter sparse reconstruction optimization model, so as to obtain a multi-target time-delay estimation value. The present invention solves the problems of insufficient time-delay estimation accuracy when multiple targets are relatively close, unstable algorithm and high algorithm complexity at low signal-to-noise ratio in the prior art

Description

Technical Field

[0001] The present invention belongs to the field of communication technology, and in particular relates to a multi-objective time delay sparse reconstruction estimation method based on an exponential filter. Background Art

[0002] Time delay estimation plays a vital role in many fields such as radar detection, sonar positioning, wireless communication and satellite communication. The most classic method for estimating multi-target time delay is the correlation function method based on matched filters. The correlation function method has low computational complexity and has the highest output signal-to-noise ratio in the case of a single target. However, when multiple targets are close to each other, the output of the matched filter, i.e., the correlation function, often has a fatter main peak and higher side lobes. When multiple targets are mixed together, peak overlap and the side lobes of large targets blocking the main peaks of small targets will occur, making it impossible to accurately obtain the time delay estimates of multiple targets. Therefore, the correlation function method based on matched filters is often limited by resolution and cannot meet the high-precision requirements of multi-target time delay estimation.

[0003] With the continuous advancement of information technology and the increasing requirements for delay accuracy in the field of signal processing, super-resolution multi-target delay estimation methods have gradually become a hot topic in delay estimation research. Many new theories and methods have been applied to super-resolution multi-target delay estimation problems in different environments and have achieved ideal results. Some scholars have proposed some improved mismatched filters and corresponding delay estimation algorithms for matched filters. For example, a controllable exponent p (p∈[-1,1]) is introduced into the frequency response function of the matched filter to obtain a mismatched filter, the exponential filter. It can be proved that the exponential filter with p=1 is the classic matched filter, and the exponential filter with an exponent less than 1 has a higher resolution than the matched filter (p=1). Its output (called the exponential order correlation function) has a sharper main lobe and lower side lobes than the traditional correlation function, and the smaller the exponent, the higher the resolution. However, the improvement of the resolution of the exponential filter is still at the expense of the loss of the output signal-to-noise ratio. This contradiction between the multi-target resolution and the output signal-to-noise ratio also limits the delay estimation accuracy of the improved delay estimation method based on the exponential filter.

[0004] The other two classic super-resolution delay estimation methods are maximum likelihood methods and subspace methods. As the theoretically optimal algorithm, the maximum likelihood method can approach the Cramer-Rao bound in delay estimation performance at low signal-to-noise ratios, but this method requires a large-scale grid search and therefore has a high computational complexity. The subspace method decomposes the received signal into two mutually orthogonal subspaces, the signal subspace and the noise subspace, and obtains multi-target delay estimates through pseudo-spectral peak searches. However, its performance will still be greatly reduced under the conditions of small samples and low signal-to-noise ratios, and cannot meet the increasing multi-target super-resolution requirements.

[0005] At the same time, compressed sensing sparse reconstruction, as an emerging theory, has attracted a lot of attention from academia and industry in the past 10 years, and has been successfully applied in many fields such as signal processing, image science, machine learning, statistical modeling, and genomics data analysis. At present, some scholars have applied sparse reconstruction theory to multi-target delay estimation. The multi-target delay estimation algorithm combining the classic matched filter and sparse reconstruction method converts the traditional delay estimation problem into a sparsely reconstructed linear observation model and solves it, which greatly improves the delay estimation resolution of the classic matched filter-based correlation function algorithm, and is still applicable in small sample conditions. However, since this type of algorithm uses the traditional correlation function as a template to construct a sparsely reconstructed linear observation model, its resolution is actually constrained by the resolution of the correlation function, and cannot meet the high-precision requirements of multi-target delay estimation when the multiple targets are very close. There is also a method that obtains a multi-target time delay parameter model based on the cross-correlation frequency domain form. According to the covariance fitting criterion, a sparse iterative algorithm is used to estimate the delay parameters for the covariance matrix corresponding to the time delay parameter model. However, the introduction of the covariance matrix iterative operation indirectly increases the computational complexity, and it also requires the number of multiple targets to be predicted in advance. Therefore, the scope of application of this method is limited to a certain extent. Summary of the invention

[0006] In view of the above-mentioned deficiencies in the prior art, the present invention provides a multi-target time delay sparse reconstruction estimation method based on an exponential filter, which solves the problems of insufficient delay estimation accuracy when multiple targets are closely spaced, algorithm instability and high algorithm complexity at low signal-to-noise ratio.

[0007] In order to achieve the above purpose, the technical solution adopted by the present invention is:

[0008] This scheme provides a multi-objective time delay sparse reconstruction estimation method based on exponential filter, including the following steps:

[0009] S1. Use the radiation source reference signal to construct an exponential filter H with the optimal order p p ;

[0010] S2, input the received multi-target reception signal into the exponential filter H p , get the exponential order cross-correlation function of the received signal;

[0011] S3, performing wavelet soft threshold denoising on the exponential order cross-correlation function;

[0012] S4. Construct a multi-objective delay parameter sparse reconstruction optimization model using the denoised exponential order cross-correlation function;

[0013] S5. Use l 1The norm sparse reconstruction algorithm solves the multi-objective delay parameter sparse reconstruction optimization model, obtains the multi-objective delay estimation value, and completes the sparse reconstruction estimation of the multi-objective delay.

[0014] Furthermore, the expression of the exponential order cross-correlation function is as follows:

[0015]

[0016] Among them, r p [m] represents the mth sample value of the p-order exponential cross-correlation function of the reference signal r(t), h p [m] and h p [k] represents the m-th sampling value and the k-th sampling value of the unit impulse response of the p-order exponential filter, M represents the total number of sampling values, r represents the discrete sampling signal of the reference signal r(t), * represents the discrete convolution operation, and r[m] represents the m-th sampling value of the reference signal r(t).

[0017] Furthermore, the expression of the multi-objective delay parameter sparse reconstruction optimization model is as follows:

[0018]

[0019]

[0020]

[0021]

[0022]

[0023] in, represents the multi-objective delay parameter sparse reconstruction optimization model, y p represents the linear observation sparse model of the received signal on the exponential correlation domain, |||| 1 ,|||| 2 Respectively represent the vector l 1 and l 2 norm, λ represents a hyperparameter, represents the observation matrix, It represents the N-dimensional target amplitude vector after zero-filling expansion, and its nth component is n=1,2,...,N, where N represents The total number of components, i.e., the dimension, express The Nth component of p Represents the noise vector, whose mth component is the mth sampling value of the noise Indicates w pThe Mth sample value of represents the p-th exponential cross-correlation function of the reference signal r(t). sampling values, T represents the vector transpose operation.

[0024] Furthermore, the use of 1 The norm sparse reconstruction algorithm solves the multi-objective delay parameter sparse reconstruction optimization model, which is specifically:

[0025] A1. Use l 1 The norm sparse reconstruction algorithm transforms the multi-objective delay parameter sparse reconstruction optimization model into the following convex quadratic optimization problem with linear inequality constraints:

[0026]

[0027] So that:

[0028] Among them, u n Indicates the N-dimensional target magnitude vector to be sought The nth component of The constraint boundary of

[0029] A2. From the initial value t = 1 / λ and u = [1,...,1] T ∈R N Start the iteration with Iterative sequence approximation solution Get the multi-target delay estimation value, where R N represents N-dimensional Euclidean space, and u represents the constraint boundary vector.

[0030] Furthermore, the Iterative sequence approximation solution Specifically:

[0031] B1. Use the conjugate gradient iterative algorithm to solve the following linear equations to give the target amplitude vector And the iteration direction vector Δa, Δu of each step of the constraint boundary vector u:

[0032]

[0033]

[0034]

[0035] Among them, t represents the time, the initial value is given in the first iteration, represents the transpose of the observation matrix, D 1 , D 2 , g1 and g 2 represents the intermediate variable, Δa, Δu represent the iteration vector u is the iteration direction of each step, diag[·] represents the diagonal matrix, Represents the iteration vector The nth component, n=1,2,...,N, where N represents The total number of components, i.e., the dimension;

[0036] B2. Calculate s = β ρ The value of is used to obtain the step size of each iteration, where α and β represent preset constants, ρ represents the smallest positive integer that satisfies the following inequality, and s represents the calculated intermediate variable:

[0037]

[0038] in,

[0039]

[0040]

[0041] Among them, φ t (·) represents a multivariate function, Δa, Δu represent vectors respectively u is the iteration direction of each step, represents the nth component of the iteration direction vector Δa, Δu n represents the nth component of the iteration direction vector Δu, n=1,2,...,N, N is and the total number of components of u, i.e., the dimension, β ρ represents β to the power of ρ;

[0042] B3, according to s and Iterate once to update and u:

[0043]

[0044] in, represents the column vector of Δa and Δu, Δa and Δu represent vectors respectively uThe iteration direction of each step;

[0045] B4. If the iteration error ξ is less than the preset threshold ε rel , then output the approximate solution Get the multi-target delay estimation value, if the iteration error ξ is greater than or equal to the preset threshold ε rel , then update t according to the following formula and return to step B1 to continue calculating iterations:

[0046]

[0047]

[0048]

[0049] Among them, μ and s min Both represent preset hyperparameters, N represents the total number of components, η and v both represent intermediate calculation variables, and v T represents the transpose of v, Representation vector The mth component of p ) (m) Represents vector y p The mth component of , where M represents the total number of components.

[0050] Beneficial effects of the present invention:

[0051] (1) In the present invention, an exponential filter with a controllable order p is used to obtain the exponential order cross-correlation function of the target received signal and the reference signal. A sparse reconstruction linear observation model for delay estimation of multiple targets is established based on the exponential order cross-correlation function. 1 The norm sparse reconstruction convex optimization algorithm is used to solve the model and obtain the final multi-objective delay estimation. The algorithm has low computational complexity.

[0052] (2) When multiple targets are closely spaced or there is clutter interference, the classic matched filter delay estimation method based on the cross-correlation function often has the problem of target main lobe aliasing and cannot accurately estimate the delays of all targets; the exponential filter based on the exponential order correlation function has a higher resolution and can more accurately distinguish each target, but in the case of low signal-to-noise ratio, the exponential filter is more susceptible to noise interference, so there is still a problem of insufficient delay estimation accuracy. The present invention establishes a sparse reconstruction model based on the exponential order cross-correlation function (after wavelet filtering), which can effectively improve the multi-target resolution of the estimation algorithm. At the same time, based on l 1 The norm sparse reconstruction optimization algorithm solves the problem and removes the solution components below the threshold, which can filter out noise and clutter to a certain extent, and ultimately improve the stability of the algorithm in a low signal-to-noise ratio environment.

[0053] (3) The present invention is based on l 1 When the norm sparse reconstruction optimization algorithm solves the sparse reconstruction model of the corresponding multi-objective time delay, there is no need to predict the number of multi-objectives, so it can be applied to more application scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 The figure is a flow chart of the method of the present invention.

[0055] Figure 2This is a comparison diagram of the output of the wavelet post-exponential filter used in the present invention and the output of the matched filter and exponential filter of the existing time delay estimation method.

[0056] Figure 3 The figure is a comparison diagram of the delay estimation results of the present invention and the existing delay estimation technology.

[0057] Figure 4 This is a comparison chart of the mean square error of delay estimation between the present invention and the existing delay estimation technology. DETAILED DESCRIPTION

[0058] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0059] Example

[0060] In the prior art, assuming that the reference signal is r(t), the noisy multi-target received signal containing K targets can be expressed as:

[0061]

[0062] Among them, a i and τ i Represent the amplitude and delay of the i-th target respectively. The most classic method for estimating multi-target delay is the correlation function method based on matched filter. Its technical solution is: first, a matched filter is constructed from the reference signal r(t). Assume that the Fourier transform of the reference signal r(t)r(t) is R(ω), and the frequency domain response function of the matched filter is R * (ω); then the multi-target received signals are input into the matched filter. When the targets are far apart, the output of the matched filter (i.e., the cross-correlation function of the reference signal and the received signal) will obtain peak values ​​at each time delay; therefore, the multi-target time delay estimation value can be obtained by searching its output peaks one by one.

[0063] Introducing a controllable exponent p into the frequency response function of the matched filter yields an exponential filter, whose frequency domain response function is:

[0064] H p (ω)=|R(ω)| 1+p R -1 (ω) (2)

[0065] When the controllable index p = 1, the exponential filter is a classic matched filter. The output of the received signal after the exponential filter is called the exponential order cross-correlation function. It can be proved that this type of exponential order cross-correlation function still obtains the extreme value at the target delay. Therefore, searching for its output peaks one by one can also obtain the multi-target delay estimation value. At the same time, when the order p < 1, the exponential order cross-correlation function has higher resolution than the output of the matched filter, that is, the classic correlation function.

[0066] The above-mentioned prior art solution is based on the correlation function method of the matched filter, which has a low computational complexity and a very high output signal-to-noise ratio in the case of a single target. Therefore, when multiple targets are far apart, the multi-target delay estimation can be achieved more accurately. However, when the targets are close together, the cross-correlation function output by a single target often has a fatter main peak and higher side lobes. When multiple targets are mixed together, peak overlap will occur, and the side lobes of large targets will block the main peaks of small targets, making it impossible to accurately obtain the multi-target delay estimation value. Therefore, this type of algorithm is limited by resolution and cannot meet the high-precision requirements of multi-target delay estimation.

[0067] The improved delay estimation algorithm based on the exponential filter improves the resolution of the output correlation function by introducing a controllable parameter p: the exponential correlation function with order p<1 has a sharper main lobe and lower side lobes than the traditional correlation function (p=1), and therefore has a higher multi-target resolution. However, the improvement in resolution comes at the expense of the output signal-to-noise ratio. It can be proved that the smaller the order p, the greater the output multi-target resolution, but the lower the output signal-to-noise ratio. Therefore, this contradiction between multi-target resolution and output signal-to-noise ratio also makes it impossible to select the order p of the exponential filter too small, which actually limits the delay estimation accuracy of the improved delay estimation algorithm based on the exponential filter.

[0068] In the prior art, with the rise of sparse reconstruction methods, some people have proposed a multi-objective delay estimation algorithm that combines the classic matched filter and sparse reconstruction methods. This method first inputs the received signal into a matched filter, whose output is the traditional cross-correlation function. The sparse reconstruction linear equation model of the delay parameters is constructed by the cross-correlation function (or its frequency domain form), and the sparse iterative algorithm of sparse reconstruction is used to solve this linear equation system to estimate the delay parameters.

[0069] Specifically, assume that the reference signal r(t) is a discrete sampling signal with a sampling interval Δt, r[m] = r(mΔt), m = 1, 2, ..., M. At the same time, assume that all target delays τ i are all at the sampling point, that is, τ i =d i Δt,i=1,2,...,K,d iis an integer from 0 to M, then the discrete form of the time domain multi-target received signal in equation (1) can be written as:

[0070]

[0071] The discrete received signal (3) is input into a matched filter (with the impulse response set to h[m], m = 1, 2, ..., M), and its output is:

[0072]

[0073] Among them, R[m] and W[m] are the outputs of the reference signal r(t) and noise n(t) after the matched filter respectively.

[0074] Then, the above observation model (4) is expanded into a sparse reconstruction linear equation model. a ,T b ) is set to a discrete delay grid with dimension N as: in, are all integers, i = 1,..., N. At the same time, it is assumed that the discrete values ​​of all target delays are in the grid, that is, Let the target amplitude vector a = (a 1 ,a 2 ,...,a N ) T The zero-filled extended form is If and only if (target real delay discrete value), Has non-zero component values (target true range), others When the total number of delayed grids N is much larger than the number of real targets K, is a sparse vector with sparsity K. Finally, the (overcomplete) linear observation sparse model of the received signal in the correlation domain is obtained as follows:

[0075]

[0076] The above formula can be regarded as The sparse reconstruction linear equations are: T , noise vector w=(W[1],W[2],...,W[M]) T , the completed observation matrix is:

[0077]

[0078] Finally, the multi-objective delay estimation algorithm combining the classic matched filter and sparse reconstruction method uses the sparse reconstruction algorithm to solve the The sparse reconstruction linear equations model (5) is used to obtain the target amplitude vector The estimated value of , where its non-zero elements are located, is the corresponding multi-target delay estimation value.

[0079] The above-mentioned prior art combines the classic matching filter and sparse reconstruction method of the multi-target delay estimation algorithm to convert the traditional delay estimation problem into a sparsely reconstructed linear observation model and solve it, which greatly improves the delay estimation resolution of the classic correlation function algorithm, and is still applicable in the case of small samples. However, since this type of algorithm uses the traditional cross-correlation function as a template to construct a sparsely reconstructed linear observation model, its resolution is actually constrained by the resolution of the cross-correlation function, and cannot meet the high-precision requirements of multi-target delay estimation when the multiple targets are very close. In addition, when this method uses the sparse reconstruction optimization method to solve the multi-target delay, it often needs to predict the number of multiple targets, which also limits the scope of application of this method to a certain extent.

[0080] The above-mentioned prior art has the disadvantages of insufficient delay estimation accuracy when multiple targets are close to each other, unstable algorithm and high algorithm complexity when the signal-to-noise ratio is low, such as Figure 1 As shown, the present invention provides a method of combining an exponential filter and a 1 The multi-objective delay estimation method of the norm sparse reconstruction algorithm is implemented as follows:

[0081] S1. Use the radiation source reference signal to construct an exponential filter H with the optimal order p p ;

[0082] S2, input the received multi-target reception signal into the exponential filter H p , get the exponential order cross-correlation function of the received signal;

[0083] S3, performing wavelet soft threshold denoising on the exponential order cross-correlation function;

[0084] S4. Construct a multi-objective delay parameter sparse reconstruction optimization model using the denoised exponential order cross-correlation function;

[0085] S5. Use l 1 The norm sparse reconstruction algorithm solves the multi-objective delay parameter sparse reconstruction optimization model, obtains the multi-objective delay estimation value, and completes the sparse reconstruction estimation of the multi-objective delay.

[0086] In this embodiment, the noisy multi-target received signal is assumed to be as shown in equation (1), where the additive noise n(t) is assumed to have zero mean and variance σ 2Gaussian white noise. The reference signal r(t) is a discrete sampling signal with a sampling interval of Δt, r[m] = r(mΔt), m = 1, 2, ..., M. The exponential filter H with an order of p is constructed by the reference signal r(t). p , its frequency response function is H p (ω)=|R(ω)| 1+p R -1 (ω), its impulse response function h p (t) is H p The reference signal r(t) is passed through an exponential filter H with an order of p. p The output is the p-order exponential autocorrelation function:

[0087]

[0088] Among them, r p [m] represents the mth sample value of the p-order exponential cross-correlation function of the reference signal r(t), h p [m] and h p [k] represents the m-th sampling value and the k-th sampling value of the unit impulse response of the p-order exponential filter, M represents the total number of sampling values, r represents the discrete sampling signal of the reference signal r(t), * represents the discrete convolution operation, and r[m] represents the m-th sampling value of the reference signal r(t).

[0089] Assume that all delays τ i are all at the sampling point, that is, τ i =d i Δt,i=1,2,...,K,d i is an integer from 0 to M, then the discrete signal of the above time domain received signal is:

[0090]

[0091] The discrete received signal is input into the exponential filter H p , the output is:

[0092]

[0093] Among them, n p [m] = n*h p [m] is the output of the noise n(t) after passing through the exponential filter.

[0094] In this embodiment, in order to further improve the output signal-to-noise ratio of the exponential filter, the output of the exponential filter is denoised by a wavelet soft threshold filtering algorithm, and the result is recorded as The result of noise after wavelet processing is recorded as Let the observation vector The target amplitude vector is a=(a 1 ,a 2 ,...,a K ) T , the noise vector is Then the above discrete received signal (8) can be rewritten as the following linear observation linear equation model:

[0095] y p =A p a+w p (10)

[0096] The observation matrix is:

[0097]

[0098] The above linear observation model (10) is not a sparse form. Now it is expanded into a sparse reconstruction model. In a certain delay range (T a ,T b ) is set to a discrete delay grid with dimension N as: in are all integers, i = 1,..., N. At the same time, it is assumed that the discrete values ​​of all target delays are in the grid, that is, Assume that the zero-filled extended form of the target amplitude vector is If and only if (target real delay discrete value), Has non-zero component values (target true range), others When the total number of delayed grids N is much larger than the number of real targets K, is a sparse vector with sparsity K.

[0099] At the same time, the observation matrix A p The expanded form is:

[0100]

[0101] In summary, the (overcomplete) linear observation sparse model of the received signal in the exponential correlation domain is:

[0102]

[0103] Can be seen as about The sparse reconstruction linear equations are underdetermined when the number of observations is small, and the traditional least squares method is often difficult to solve. The present invention considers converting it into a linear equation based on l 1 Sparse reconstruction optimization model of norm:

[0104]

[0105] in, represents the multi-objective delay parameter sparse reconstruction optimization model, y p represents the linear observation sparse model of the received signal on the exponential correlation domain, |||| 1 ,|||| 2 Respectively represent the vector l 1 and l 2 norm, λ represents a hyperparameter, represents the observation matrix, It represents the N-dimensional target amplitude vector after zero-filling expansion, and its nth component is N stands for The total number of components, i.e., the dimension, express The Nth component of p Represents the noise vector, whose mth component is the mth sampling value of the noise Indicates w p The Mth sample value of represents the p-th exponential cross-correlation function of the reference signal r(t). sampling values, T represents the vector transpose operation.

[0106] Utilize l 1 The norm sparse reconstruction algorithm solves the multi-objective delay parameter sparse reconstruction optimization model, which is specifically:

[0107] A1. Use l 1 The norm sparse reconstruction algorithm transforms the multi-objective delay parameter sparse reconstruction optimization model into the following convex quadratic optimization problem with linear inequality constraints:

[0108]

[0109] So that:

[0110] Among them, u n Indicates the N-dimensional target magnitude vector to be sought The nth component of The constraint boundary of

[0111] A2. From the initial value t = 1 / λ and u = [1,...,1] T ∈R N Start the iteration with Iterative sequence approximation solution Get the multi-target delay estimation value, where R N represents N-dimensional Euclidean space, and u represents the constraint boundary vector:

[0112] Said Iterative sequence approximation solution Specifically:

[0113] B1. Use the conjugate gradient iterative algorithm to solve the following linear equations to give the target amplitude vector And the iteration direction vector Δa, Δu of each step of the constraint boundary vector u:

[0114]

[0115]

[0116]

[0117] Among them, t represents the time, the initial value is given in the first iteration, represents the transpose of the observation matrix, D 1 , D 2 , g 1 and g 2 represents the intermediate variable, Δa, Δu represent the iteration vector u is the iteration direction of each step, diag[·] represents the diagonal matrix, Represents the iteration vector The nth component, n=1,2,...,N, where N represents The total number of components, i.e., the dimension;

[0118] B2. Calculate s = β ρ The value of is used to obtain the step size of each iteration, where α and β represent preset constants, ρ represents the smallest positive integer that satisfies the following inequality, and s represents the calculated intermediate variable:

[0119]

[0120] in,

[0121]

[0122]

[0123] Among them, φ t (·) represents a multivariate function, Δa, Δu represent vectors respectively u is the iteration direction of each step, represents the nth component of the iteration direction vector Δa, Δu n represents the nth component of the iteration direction vector Δu, n=1,2,...,N, N is and the total number of components of u, i.e., the dimension, β ρ represents β to the power of ρ;

[0124] B3, according to s and Iterate once to update and u:

[0125]

[0126] in, represents the column vector of Δa and Δu, Δa and Δu represent vectors respectively uThe iteration direction of each step;

[0127] B4. If the iterative error ξ=η / G(v) is less than the preset threshold ε rel , then output the approximate solution Get the multi-objective delay estimate:

[0128]

[0129]

[0130] On the contrary, when the error ξ ≥ ε rel When , update t according to the following formula and return to step B1 to continue calculating iterations:

[0131]

[0132] In this embodiment, s min and μ are both set constants.

[0133] The present invention will be further described below.

[0134] Simulation Example 1

[0135] The reference signal is a linear frequency modulation signal with a bandwidth of 10MHz and a starting carrier frequency of 3000KHz. The simulation experiment is carried out under the condition of small sample (i.e. small snapshot) and low signal-to-noise ratio. The noisy received signal contains 3 targets, the input signal-to-noise ratio is 0dB, and the actual delays of the 3 targets are 118, 120 and 130 (in the number of sampling points). The total number of snapshots of the received signal used for sparse reconstruction is M=200. Figure 2 The output of the received signal after the matched filter, the output of the exponential filter with the optimal order of p = -0.3, and the output of the exponential filter with the order of p = -0.3 after the wavelet filter are given respectively. The peak values ​​of the three outputs correspond to the positions of the estimated delay parameters. It can be seen that the present invention has a stronger resolution for multiple targets based on the output of the wavelet soft threshold filter and the exponential filter, and also has a good anti-noise ability. The corresponding multi-target delay estimation result is more accurate. Here, the input signal-to-noise ratio is defined as

[0136] Simulation Example 2

[0137] The same linear frequency modulation signal as in simulation example 1 is used as the reference signal. Consider a multi-target receiving signal with 4 targets, the input signal-to-noise ratio is -3dB, and the actual delay is d 1 =112,d 2 =120,d 3 =130,d 4 = 135 (unit is the number of sampling points). After J = 100 independent Monte Carlo experiments, Figure 3 As shown, the sparse reconstruction delay estimation method based on the exponential filter proposed in the present invention has a stronger peak resolution for multiple targets, a more accurate estimation of the true delay, and fewer interference pseudo peaks, compared with the sparse reconstruction delay estimation method based on the matched filter (i.e., the cross-correlation function). Therefore, it will have a higher multi-target delay estimation accuracy.

[0138] Simulation Example 3

[0139] For the received signals with different input signal-to-noise ratios in simulation example 2, after J=100 independent Monte Carlo experiments, the root mean square error of the delay estimation method based on the exponential filter and the sparse reconstruction delay estimation method based on the matched filter proposed in the present invention is calculated, as shown in FIG. Figure 4 Assume that the real delay vector is a row vector with dimension M = 200 If and only if m = d i ,i=1,2,3,4, other cases The delay estimation vector given by the jth Monte Carlo experiment given by the algorithm is The delay estimation error calculation formula is: l represents the vector 2 Norm. Figure 4 As shown, under the condition of a certain input signal-to-noise ratio, the sparse reconstruction delay estimation method based on exponential filter proposed in the present invention has higher multi-objective delay estimation accuracy than the sparse reconstruction delay estimation method based on matched filter.

Claims

1. A multi-objective time delay sparse reconstruction estimation method based on exponential filter, It is characterized in that The following steps are involved: S1. Use the radiation source reference signal to construct an exponential filter H with the optimal order p p ; S2, input the received multi-target reception signal into the exponential filter H p , get the exponential order cross-correlation function of the received signal; S3, performing wavelet soft threshold denoising on the exponential order cross-correlation function; S4. Construct a multi-objective delay parameter sparse reconstruction optimization model using the denoised exponential order cross-correlation function; S5. Use l 1 The norm sparse reconstruction algorithm solves the multi-objective delay parameter sparse reconstruction optimization model, obtains the multi-objective delay estimation value, and completes the sparse reconstruction estimation of the multi-objective delay.

2. According to the multi-objective time delay sparse reconstruction estimation method based on exponential filter in claim 1, It is characterized in that The expression of the exponential order cross-correlation function is as follows: Among them, r p [m] represents the mth sample value of the p-order exponential cross-correlation function of the reference signal r(t), h p [m] and h p [k] represents the m-th sampling value and the k-th sampling value of the unit impulse response of the p-order exponential filter, M represents the total number of sampling values, r represents the discrete sampling signal of the reference signal r(t), * represents the discrete convolution operation, and r[m] represents the m-th sampling value of the reference signal r(t).

3. The multi-objective time delay sparse reconstruction estimation method based on exponential filter according to claim 2, It is characterized in that The expression of the multi-objective delay parameter sparse reconstruction optimization model is as follows: in, represents the multi-objective delay parameter sparse reconstruction optimization model, y p represents the linear observation sparse model of the received signal on the exponential correlation domain, || || 1 ,|| || 2 Respectively represent the vector l 1 and l 2 norm, λ represents a hyperparameter, represents the observation matrix, It represents the N-dimensional target amplitude vector after zero-filling expansion, and its nth component is n=1,2,...,N, where N represents The total number of components, i.e., the dimension, express The Nth component of p Represents the noise vector, whose mth component is the mth sampling value of the noise Indicates w p The Mth sample value of represents the p-th exponential cross-correlation function of the reference signal r(t). sampling values, T represents the vector transpose operation.

4. The multi-objective time delay sparse reconstruction estimation method based on exponential filter according to claim 3, It is characterized in that The utilization 1 The norm sparse reconstruction algorithm solves the multi-objective delay parameter sparse reconstruction optimization model, which is specifically: A1. Use l 1 The norm sparse reconstruction algorithm transforms the multi-objective delay parameter sparse reconstruction optimization model into the following convex quadratic optimization problem with linear inequality constraints: So that: Among them, u n Indicates the nth component a of the N-dimensional target magnitude vector a to be sought n The constraint boundary of A2. From the initial value t = 1 / λ and u = [1,...,1] T ∈R N Start the iteration with Iterative sequence approximation solution Get the multi-objective delay estimation value, where R N represents N-dimensional Euclidean space, and u represents the constraint boundary vector.

5. According to claim 4, the multi-objective time delay sparse reconstruction estimation method based on exponential filter, It is characterized in that Said Iterative sequence approximation solution Specifically: B1. Use the conjugate gradient iteration algorithm to solve the following linear equations to give the iteration direction vectors Δa, Δu of each step of the target amplitude vector a and the constraint boundary vector u: Among them, t represents the time, the initial value is given in the first iteration, represents the transpose of the observation matrix, D 1 , D 2 , g 1 and g 2 represents the intermediate variable, Δa, Δu represent the iteration vector u is the iteration direction of each step, diag[·] represents the diagonal matrix, Represents the iteration vector The nth component, n=1,2,...,N, where N represents The total number of components, i.e., the dimension; B2. Calculate s = β ρ The value of is used to obtain the step size of each iteration, where α and β represent preset constants, ρ represents the smallest positive integer that satisfies the following inequality, and s represents the calculated intermediate variable: in, Among them, φ t (·) represents a multivariate function, Δa, Δu represent vectors respectively u is the iteration direction of each step, represents the nth component of the iteration direction vector Δa, Δu n represents the nth component of the iteration direction vector Δu, n=1,2,...,N, N is and the total number of components of u, i.e., the dimension, β ρ represents β to the power of ρ; B3, according to s and Iterate once to update and u: in, represents the column vector of Δa and Δu, Δa and Δu represent vectors respectively uThe iteration direction of each step; B4. If the iteration error ξ is less than the preset threshold ε rel , then output the approximate solution Get the multi-target delay estimation value, if the iteration error ξ is greater than or equal to the preset threshold ε rel , then update t according to the following formula and return to step B1 to continue calculating iterations: Among them, μ and s min Both represent preset hyperparameters, N represents the total number of components, η and v both represent intermediate calculation variables, and v T represents the transpose of v, Representation vector The mth component of p ) (m) Represents vector y p The mth component of , where M represents the total number of components.

Images