Weighted fractional Fourier transform parameter estimation method based on maximum entropy criterion

Abstract

The invention relates to the technical field of signal processing, in particular to a weighted fractional Fourier transform parameter estimation method based on a maximum entropy criterion. The method comprises the following steps: solving a WFRFT of an input signal twiddle factor in a [0, 2] interval, and generating a local signal; solving an entropy value of a local signal of an input signal twiddle factor in a [0, 2] interval, and calculating a maximum value of the entropy value; constructing a sparse optimization function according to the input sparse coefficient, the entropy function of the local signal and the maximum value of the entropy function, solving the maximum peak value of the normalized entropy value in a twiddle factor [0, 2] range, and calculating the position of the maximum peak value of the twiddle factor; and estimating a twiddle factor according to the maximum position of the peak value. According to the method, effective estimation under unknown signal parameters is realized, a transformation order estimation problem is converted into a sparse optimization problem, a sparse reconstruction algorithm is used for solving, and relatively high parameter estimation accuracy can also be realized in a low signal-to-noise ratio environment.

Description

Technical Field

[0001] The invention relates to the technical field of signal processing, and in particular to a weighted fractional-order Fourier transform parameter estimation method based on a maximum entropy criterion. Background Art

[0002] In order to solve the multipath effect and Doppler effect of signals in non-ideal channels, a new system hybrid carrier (HC) system signal has emerged. The hybrid carrier system represented by weighted fractional Fourier transform includes both single-carrier and multi-carrier systems when transmitting information, which can realize the transmission of signals in the time and frequency domains, and integrate single-carrier system signals and multi-carrier system signals in terms of characteristics. The weighted fractional Fourier transform signal has good anti-detection performance and has achieved certain results in information concealment fields such as communication systems, information encryption, and image compression.

[0003] The emergence of weighted fractional Fourier transform (WFRFT) signals not only improves the confidentiality and anti-interference of wireless communications, but also puts higher requirements on the signal key parameter estimation technology; effective key parameter estimation of weighted fractional Fourier transform signals can provide a strong parameter basis for statistical analysis. In view of the parameter estimation problem of weighted fractional Fourier transform signals, reference [1] (Yang Zhenxin. Research on WFRFT signal parameter identification method [Dissertation]. Harbin: Harbin Institute of Technology, 2020) proposed a hybrid carrier system parameter identification method. Based on the analysis of typical hybrid carrier system characteristics and anti-scanning characteristics, this method estimates the rotation factor of the hybrid carrier system signal through high-order cumulants under known modulation modes; estimates the rotation factor of the hybrid carrier system signal through scanning method and decision method under unknown modulation modes, and uses neural network to optimize the proposed parameter estimation method, but this method has the disadvantage of large computational complexity. Reference [2] (Zhang Xiaoyu. Method for estimating rotation factors of weighted fractional Fourier transform signals based on cyclic correlation [J]. Acta Armamentarii, 2022, 43(7): 1646-1654) proposes a method for estimating rotation factors of mixed carrier signals based on cyclic correlation. This method uses cyclic correlation to replace traditional linear correlation to achieve effective estimation of signal parameters, and uses the fast FFT algorithm in circular correlation to reduce the complexity of multiplication operations, thereby shortening the time for parameter estimation. However, the accuracy of parameter estimation needs to be improved. Reference [3] (Li Zhiyang. Research on carrier keying technology based on WFRFT [Dissertation]. Harbin: Harbin Institute of Technology, 2021) proposes a method for estimating parameters of mixed carrier signals based on maximum constellation likelihood. This method can achieve rotation factor estimation through the maximum likelihood theory of the constellation diagram without prior knowledge, but its recognition rate needs to be improved at low signal-to-noise ratio. Summary of the invention

[0004] The technical problem to be solved by the present invention is: to overcome the shortcomings of the prior art and provide a weighted fractional-order Fourier transform parameter estimation method based on the maximum entropy criterion, by analyzing the generation mechanism of a mixed carrier signal, using the maximum entropy principle to analyze the peak value and its position of the maximum entropy value under different rotation factors, and proposing a parameter estimation method for a weighted fractional-order Fourier transform signal under unknown signal parameters; in order to improve the estimation accuracy under low signal-to-noise ratio, a sparse optimization function is constructed to realize the optimal order estimation of the parameters of the weighted fractional-order Fourier transform signal.

[0005] The technical solution adopted by the present invention is as follows: A weighted fractional Fourier transform parameter estimation method based on maximum entropy criterion comprises the following steps: S1. Calculate the WFRFT of the input signal with a rotation factor in the interval [0,2] to generate a local signal; S2. Calculate the entropy value of the local signal when the rotation factor of the input signal is in the interval [0,2], and calculate the maximum entropy value; S3. According to the input sparse coefficient, the entropy function of the local signal and the maximum value of the entropy function, a sparse optimization function is constructed, and the maximum peak value of the normalized entropy value under the rotation factor [0,2] range is calculated, and the position of the maximum peak value of the rotation factor is calculated; S4. Estimate the rotation factor based on the maximum peak position.

[0006] This technical solution realizes the effective estimation of signal rotation factors by introducing the entropy principle and sparse optimization method. Specifically, the fractional Fourier transform not only represents the conversion of the signal between the time domain and the frequency domain, but also represents the order of the signal between the two domains; the weighted fractional Fourier transform The weighting coefficient is introduced to enhance the applicability of the transformation. In signal processing, entropy is used to measure the information distribution characteristics of a signal under a certain transformation. The maximum entropy criterion is used to find a solution that maximizes the information entropy under given constraints. The rotation factor of the input signal varies in the interval [0,2]. Generate a local signal to convert the signal to different fractional Fourier domains so as to observe the change of its entropy value; calculate the entropy value of the local signal corresponding to each rotation factor, and find the maximum entropy value to find the rotation factor that makes the signal information most evenly distributed; construct a sparse optimization function based on the input sparse coefficient, the entropy function of the local signal and the maximum entropy function to convert the transformation order estimation problem into a sparse optimization problem so as to solve it using a sparse reconstruction algorithm; solve the sparse optimization function to find the maximum peak value and position of the normalized entropy value to determine the optimal rotation factor; estimating the rotation factor based on the maximum peak position is the ultimate goal of parameter estimation; the technical solution does not rely on the prior knowledge of the signal and realizes effective parameter estimation of the weighted fractional Fourier transform signal in the case of unknown signal parameters; by converting the transformation order estimation problem into a sparse optimization problem and solving it using a sparse reconstruction algorithm, a high parameter estimation accuracy can be obtained even under low signal-to-noise ratio; the technical solution performs effective parameter estimation on the rotation factor of the confidential signal of the new hybrid carrier system.

[0007] In addition, the weighted fractional Fourier transform parameter estimation method based on the maximum entropy criterion proposed in the present invention also has the following additional technical features: According to an embodiment of the present invention, in step S1, generating a local signal includes the following specific steps: S11, receiving the weighted fractional Fourier transform received signal to be estimated in the receiving channel The expression is as follows: (1) in: is the Gaussian white noise in the channel, is the weighted fractional Fourier transform received signal to be estimated and satisfies the following formula: (2) in: is the original baseband signal, is the fast Fourier transform, is the rotation factor of the transformation order, For the i The weighting factors of the components are: i =0, 1, 2, 3; The expression is as follows: (3) S12. Since the weighted fractional Fourier transform signal has the periodicity of the rotation factor, let is the rotation factor of the local signal of the receiver, which satisfies: (4) in: , is a value in the interval [0,2], To be valued, Take the values ​​one by one and get a set of periodic rotation factors for No. k Second value; S13, receiving signal After another cycle, the rotation factor After the weighted fractional Fourier transform of , The expression is as follows: (5).

[0008] In this technical solution, a local signal is generated by receiving a signal, utilizing the periodicity of a rotation factor and weighted fractional Fourier transform processing, and the entropy value of the local signal under different rotation factors is calculated, and the rotation factor corresponding to the maximum entropy value is found to estimate the signal parameters. Specifically, the received signal includes Gaussian white noise in the channel and the weighted fractional-order Fourier transform received signal to be estimated, indicating that the signal is interfered by noise during transmission, and the signal itself undergoes a weighted fractional-order Fourier transform; the weighted fractional-order Fourier transform signal has the periodicity of the rotation factor, indicating that when the rotation factor takes a value within a certain range, the transformation result will show a periodic change; there is a certain relationship between the rotation factor of the local signal of the receiver and the rotation factor to be estimated, and by adjusting the rotation factor of the local signal, the transformation result of the received signal under different rotation factors is simulated; the received signal is again subjected to a weighted fractional-order Fourier transform of a periodic rotation factor to obtain a local signal, which is used to convert the received signal to different fractional-order Fourier domains, so as to observe and analyze the performance of the signal under different rotation factors; by successively increasing the rotation factor, a set of periodic rotation factors is obtained, and then a series of weighted fractional-order Fourier transform local signals are obtained.

[0009] According to an embodiment of the present invention, in step S2, solving the entropy value of the local signal includes the following specific steps: S21. Since the entropy value can effectively measure the energy focusing of the signal, the entropy value expression of the discrete signal is as follows: (6) in: is the normalized received signal of sampling length M, is the entropy value of the discrete signal; S22, weighted fractional Fourier transform local signal Discretization can obtain the discretized local signal , then the WFRFT signal entropy value of the local signal for one rotation factor period is The expression is as follows: (7) The discretized local signal entropy value and the maximum position are used to estimate the rotation factor of the weighted fractional Fourier transform.

[0010] This technical solution generates a local signal by receiving a signal, utilizing the periodicity of a rotation factor, and performing a weighted fractional Fourier transform. The entropy value of the local signal under different rotation factors is calculated, and the rotation factor corresponding to the maximum entropy value is found to estimate the signal parameters.

[0011] According to one embodiment of the present invention, the step S22 The signal, whose rotation factor is related to the rotation angle of the time-frequency axis, has a certain periodicity; According to formula (4) and formula (6), The entropy values ​​of different signal rotation factors are also periodic, and the transformation period is also 4, showing a symmetrical distribution in the interval [0,2] and the interval [2,4]. That is, the entropy value results in half of the symmetrical interval are used to estimate the rotation factors of WFRFT.

[0012] This technical solution recognizes the correlation between the signal rotation factor and the rotation angle of the time-frequency axis, and uses the calculation results of formula (4) and formula (6) to reveal that the entropy value of the signal under different rotation factors also shows periodicity, and this period is exactly 4. The periodicity of the entropy value shows a symmetrical distribution in the interval [0,2] and the interval [2,4]. It is based on the existence of this symmetry that only using the entropy value results in half of the symmetrical interval is sufficient to The rotation factors can be accurately estimated.

[0013] According to an embodiment of the present invention, in step S2, the calculation of the maximum entropy value of the local signal includes the following specific steps: S23, when the energy is most concentrated, that is, different rotation factors When the signal entropy value is the largest, the rotation factor at this time is the parameter value to be estimated; For unknown rotation factors Signal, solve the optimization problem of parameter estimation by maximum entropy criterion: (8).

[0014] The technical solution first identifies the state of the signal when the energy is most concentrated, that is, when the signal entropy values ​​of different rotation factors reach the maximum, and determines the rotation factor at this time as the parameter value to be estimated. That is, there is a certain correlation between the energy focusing degree of the signal and its entropy value, and when the energy focusing is most concentrated, the entropy value of the signal will reach an extreme point; then, for signals with unknown rotation factors, the maximum entropy criterion is used to solve the optimization problem of parameter estimation values. The maximum entropy criterion is used as an optimization goal to find the rotation factor that maximizes the signal entropy value, thereby realizing the estimation of signal parameters; in order to solve this problem more accurately, a sparse optimization function is constructed; first, the entropy value exponential function expression under different rotation factors is defined, which establishes a mathematical relationship between the entropy value and the rotation factor, and introduces a sparse coefficient to regulate sparsity; the introduction of the sparse coefficient is to better retain the important features of the signal when processing signals with sparse characteristics, while removing redundant information; based on this entropy value exponential function expression, a sparse optimization function is constructed, the purpose of which is to find the solution that makes the maximum entropy value under different rotation factors reach the optimal solution under the constraint of a given sparse coefficient; by solving this sparse optimization function, a more accurate rotation factor estimate is obtained, thereby improving the accuracy of signal processing.

[0015] According to an embodiment of the present invention, in step S3, constructing a sparse optimization function includes the following specific steps: S31. The entropy exponential function expression under different rotation factors is as follows: (9) in: is the entropy exponential function, is a sparse coefficient, and is an integer greater than or equal to 1; S32. The expression of the constructed sparse optimization function is as follows: (10) in: is the maximum entropy value under different rotation factors, Optimizes the function for sparseness.

[0016] This technical solution first defines the entropy exponential function under different rotation factors, and establishes a mathematical model that links the rotation factor and the entropy value; the entropy exponential function not only reflects the change of the entropy value of the signal under different rotation factors, but also regulates the sparsity of the signal by introducing a sparse coefficient; based on the entropy exponential function, a sparse optimization function is constructed, the purpose of which is to find the optimal solution for the maximum entropy value under different rotation factors under the constraint of a given sparse coefficient. That is, by finding a rotation factor, the entropy value of the signal reaches the maximum after being transformed by this rotation factor, while meeting the sparsity requirement. The sparse optimization function not only takes the maximization of the entropy value of the signal as the optimization goal, but also considers the sparsity of the signal, so that better results can be achieved in signal processing; it is particularly suitable for processing signals with sparse characteristics, such as images, audio, etc., and can remove redundant information while retaining the important features of the signal, thereby improving the efficiency of signal processing.

[0017] According to one embodiment of the present invention, in step S32, the sparse optimization function By using formula (9) and formula (10), the entropy value change curve is transformed into a normalized single-peak curve, and the steepness of the curve under different sparsity coefficients is characterized differently; As the sparsity coefficient gradually increases, the single-peak curve becomes thinner, and the sparse characteristics of the constructed sparse optimization function become more obvious, which is more conducive to the maximum value estimation under low signal-to-noise ratio conditions, that is, it is conducive to the estimation of the rotation factor parameters of the weighted fractional Fourier transform signal.

[0018] This technical solution transforms the complex and changeable entropy value change curve into a normalized single-peak curve through a sparse optimization function, which not only simplifies the complexity of the problem, but also presents the change of entropy value in an intuitive and easy-to-analyze form; the steepness of the single-peak curve shows obvious differences under different sparsity coefficients; as the sparsity coefficient gradually increases, the single-peak curve becomes thinner, the curve changes more dramatically near the peak, and becomes smoother away from the peak. The more obvious the sparse characteristics are, the more the function tends to select solutions with significant features in the solution process, and ignore those redundant or minor solutions. Under low signal-to-noise ratio conditions, accurate estimation of the rotation factor parameters of the weighted fractional Fourier transform signal is achieved.

[0019] According to one embodiment of the present invention, in step S32, when the rotation factor parameter estimate is located near the optimal rotation factor, the sparse optimization function value at this time is 1; the sparse optimization function values ​​for other rotation factors are close to 0.

[0020] In this technical solution, under the optimal rotation factor, the signal has the maximum amount of information or the minimum redundancy, so the sparse optimization function value reaches the maximum; under other rotation factors, due to the difference between the signal and the optimal rotation factor, the amount of information is reduced or the redundancy is increased, so the sparse optimization function value approaches 0.

[0021] According to one embodiment of the present invention, in step S3, calculating the position of the maximum peak of the rotation factor includes the following specific steps: S33. Use a sparse function optimization method based on the maximum entropy criterion to estimate the rotation factor to improve the accuracy of parameter estimation in a low signal-to-noise ratio environment; use the periodicity of entropy values ​​in [0,2] and [2,4] to calculate the normalized entropy value of the local signal in the interval [0,2] to obtain the maximum entropy value under each rotation factor, and then use the sparse coefficient and entropy function to calculate the position of the maximum peak of the normalized entropy value of the rotation factor in the interval [0,2]; (11) in: is the weighted fractional Fourier transform received signal to be estimated, is the rotation factor of the local signal of the receiver, p is the sparse coefficient, To seek peace, To take the modulus, To take the logarithm, To obtain the maximum value, is the location of the maximum value, The location of the maximum value obtained.

[0022] This technical solution uses the maximum entropy criterion and sparse function optimization to improve the accuracy of parameter estimation in a low signal-to-noise ratio environment, uses the periodicity of entropy values ​​to reduce the amount of calculation, and infers the entropy value in the entire rotation factor space; controls the behavior of sparse function optimization by adjusting the sparse coefficient, thereby more accurately estimating the rotation factor; calculates the normalized entropy value under each rotation factor, and finds the maximum value therein to determine the position of the optimal rotation factor.

[0023] According to an embodiment of the present invention, in step S4, estimating the rotation factor according to the maximum position of the peak value includes the following specific steps: S41. The position of the maximum value is used to realize effective parameter estimation of the unknown rotation factor value. The expression of the rotation factor to be estimated and the position of the maximum value is as follows: (12) in: is the step size of the local rotation factor, is the final estimated weighted fractional Fourier transform parameter value; S42. By using the maximum entropy value position and step size of the sparse optimization function, the estimated value of the rotation factor parameter of the weighted fractional Fourier transform is finally obtained.

[0024] This technical solution realizes effective parameter estimation of unknown rotation factor values ​​through the rotation factor position when the sparse optimization function reaches the maximum entropy value; the final estimated weighted fractional Fourier transform parameter value represents the optimal rotation factor of the signal in the fractional Fourier transform domain; this position is fine-tuned using the step amount to obtain a more accurate rotation factor estimate, and the fine-tuning process is based on the properties of the sparse optimization function, that is, the function has a faster rate of change near the maximum entropy value position, so the optimal solution is quickly found by adjusting the step amount.

[0025] Compared with the prior art, the present invention has the following beneficial effects: (1) The maximum entropy principle is used to analyze the relationship between the entropy value and the transformation order, and effective estimation is achieved under unknown signal parameters without presetting the signal parameters, thereby improving the applicability of parameter estimation.

[0026] (2) By converting the transform order estimation problem into a sparse optimization problem and solving it using a sparse reconstruction algorithm, a high parameter estimation accuracy can be achieved even in a low signal-to-noise ratio environment. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 It is a flow diagram of the method of the present invention.

[0028] Figure 2 It is a graph showing the change in WFRFT signal entropy value under one period rotation factor of the local signal.

[0029] Figure 3 This is a comparison of the accuracy of the rotation factor estimation based on the maximum entropy method.

[0030] Figure 4 This is the normalized entropy value result graph after different sparse optimization coefficients.

[0031] Figure 5 This is a comparison chart of the estimated accuracy after different sparse optimization coefficients.

[0032] Figure 6 This is a comparison chart of the estimation accuracy of the maximum entropy sparse optimization method under different sampling times.

[0033] Figure 7 It is a curve diagram of the estimation accuracy of different rotation factors to be estimated.

[0034] Figure 8 This is a comparison chart of the estimation accuracy of different parameter estimation methods. DETAILED DESCRIPTION

[0035] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0036] Example 1 like Figure 1 As shown, this embodiment provides a weighted fractional Fourier transform parameter estimation method based on the maximum entropy criterion, comprising the following steps: S1. Calculate the WFRFT of the input signal with a rotation factor in the interval [0,2] to generate a local signal; S2. Calculate the entropy value of the local signal when the rotation factor of the input signal is in the interval [0,2], and calculate the maximum entropy value; S3. According to the input sparse coefficient, the entropy function of the local signal and the maximum value of the entropy function, a sparse optimization function is constructed, and the maximum peak value of the normalized entropy value under the rotation factor [0,2] range is calculated, and the position of the maximum peak value of the rotation factor is calculated; S4. Estimate the rotation factor based on the maximum peak position.

[0037] This technical solution realizes the effective estimation of signal rotation factors by introducing the entropy principle and sparse optimization method. Specifically, the fractional Fourier transform not only represents the conversion of the signal between the time domain and the frequency domain, but also represents the order of the signal between the two domains; the weighted fractional Fourier transform The weighting coefficient is introduced to enhance the applicability of the transformation. In signal processing, entropy is used to measure the information distribution characteristics of a signal under a certain transformation. The maximum entropy criterion is used to find a solution that maximizes the information entropy under given constraints. The rotation factor of the input signal varies in the interval [0,2]. Generate a local signal to convert the signal to different fractional Fourier domains so as to observe the change of its entropy value; calculate the entropy value of the local signal corresponding to each rotation factor, and find the maximum entropy value to find the rotation factor that makes the signal information most evenly distributed; construct a sparse optimization function based on the input sparse coefficient, the entropy function of the local signal and the maximum entropy function to convert the transformation order estimation problem into a sparse optimization problem so as to solve it using a sparse reconstruction algorithm; solve the sparse optimization function to find the maximum peak value and position of the normalized entropy value to determine the optimal rotation factor; estimating the rotation factor based on the maximum peak position is the ultimate goal of parameter estimation; the technical solution does not rely on the prior knowledge of the signal and realizes effective parameter estimation of the weighted fractional Fourier transform signal in the case of unknown signal parameters; by converting the transformation order estimation problem into a sparse optimization problem and solving it using a sparse reconstruction algorithm, a high parameter estimation accuracy can be obtained even under low signal-to-noise ratio; the technical solution performs effective parameter estimation on the rotation factor of the confidential signal of the new hybrid carrier system.

[0038] In addition, the weighted fractional Fourier transform parameter estimation method based on the maximum entropy criterion proposed in the present invention also has the following additional technical features: According to an embodiment of the present invention, in step S1, generating a local signal includes the following specific steps: S11, receiving the weighted fractional Fourier transform received signal to be estimated in the receiving channel The expression is as follows: (1) in: is the Gaussian white noise in the channel, is the weighted fractional Fourier transform received signal to be estimated and satisfies the following formula: (2) in: is the original baseband signal, is the fast Fourier transform, is the rotation factor of the transformation order, For the i The weighting factors of the components are: i =0, 1, 2, 3; The expression is as follows: (3) S12. Since the weighted fractional Fourier transform signal has the periodicity of the rotation factor, let is the rotation factor of the local signal of the receiver, which satisfies: (4) in: , is a value in the interval [0,2], To be valued, Take the values ​​one by one and get a set of periodic rotation factors for No. k Second value; S13, receiving signal After another cycle, the rotation factor After the weighted fractional Fourier transform of , The expression is as follows: (5).

[0039] In this technical solution, a local signal is generated by receiving a signal, utilizing the periodicity of a rotation factor and weighted fractional Fourier transform processing, and the entropy value of the local signal under different rotation factors is calculated, and the rotation factor corresponding to the maximum entropy value is found to estimate the signal parameters. Specifically, the received signal includes Gaussian white noise in the channel and the weighted fractional-order Fourier transform received signal to be estimated, indicating that the signal is interfered by noise during transmission, and the signal itself undergoes a weighted fractional-order Fourier transform; the weighted fractional-order Fourier transform signal has the periodicity of the rotation factor, indicating that when the rotation factor takes a value within a certain range, the transformation result will show a periodic change; there is a certain relationship between the rotation factor of the local signal of the receiver and the rotation factor to be estimated, and by adjusting the rotation factor of the local signal, the transformation result of the received signal under different rotation factors is simulated; the received signal is again subjected to a weighted fractional-order Fourier transform of a periodic rotation factor to obtain a local signal, which is used to convert the received signal to different fractional-order Fourier domains, so as to observe and analyze the performance of the signal under different rotation factors; by successively increasing the rotation factor, a set of periodic rotation factors is obtained, and then a series of weighted fractional-order Fourier transform local signals are obtained.

[0040] According to an embodiment of the present invention, in step S2, solving the entropy value of the local signal includes the following specific steps: S21. Since the entropy value can effectively measure the energy focusing of the signal, the entropy value expression of the discrete signal is as follows: (6) in: is the normalized received signal of sampling length M, is the entropy value of the discrete signal; S22, weighted fractional Fourier transform local signal Discretization can obtain the discretized local signal , then the WFRFT signal entropy value of the local signal for one rotation factor period is The expression is as follows: (7) The discretized local signal entropy value and the maximum position are used to estimate the rotation factor of the weighted fractional Fourier transform.

[0041] For example, the modulation mode is QPSK, the carrier amplitude is 1 V; the carrier frequency is 51.25 MHz; the sampling frequency is 1.025 GHz; the information code period is: 200 ns; the initial rotation factor is 0.22; under the condition that the channel is a Gaussian white noise channel, the WFRFT signal entropy value changes under one period rotation factor as follows: Figure 2 shown.

[0042] This technical solution generates a local signal by receiving a signal, utilizing the periodicity of a rotation factor, and performing a weighted fractional Fourier transform. The entropy value of the local signal under different rotation factors is calculated, and the rotation factor corresponding to the maximum entropy value is found to estimate the signal parameters.

[0043] According to one embodiment of the present invention, the step S22 The signal, whose rotation factor is related to the rotation angle of the time-frequency axis, has a certain periodicity; According to formula (4) and formula (6), The entropy values ​​of different signal rotation factors are also periodic, and the transformation period is also 4, showing a symmetrical distribution in the interval [0,2] and the interval [2,4]. That is, the entropy value results in half of the symmetrical interval are used to estimate the rotation factors of WFRFT.

[0044] This technical solution recognizes the correlation between the signal rotation factor and the rotation angle of the time-frequency axis, and uses the calculation results of formula (4) and formula (6) to reveal that the entropy value of the signal under different rotation factors also shows periodicity, and this period is exactly 4. The periodicity of the entropy value shows a symmetrical distribution in the interval [0,2] and the interval [2,4]. It is based on the existence of this symmetry that only using the entropy value results in half of the symmetrical interval is sufficient to The rotation factors can be accurately estimated.

[0045] According to an embodiment of the present invention, in step S2, the calculation of the maximum entropy value of the local signal includes the following specific steps: S23, when the energy is most concentrated, that is, different rotation factors When the signal entropy value is the largest, the rotation factor at this time is the parameter value to be estimated; For unknown rotation factors Signal, solve the optimization problem of parameter estimation by maximum entropy criterion: (8).

[0046] Under different signal-to-noise ratios, the signal processing length is 512 sampling points, the number of simulations is 500, and the comparison results of the rotation factor estimation accuracy of the maximum entropy method are as follows: Figure 3 shown.

[0047] The technical solution first identifies the state of the signal when the energy is most concentrated, that is, when the signal entropy values ​​of different rotation factors reach the maximum, and determines the rotation factor at this time as the parameter value to be estimated. That is, there is a certain correlation between the energy focusing degree of the signal and its entropy value, and when the energy focusing is most concentrated, the entropy value of the signal will reach an extreme point; then, for signals with unknown rotation factors, the maximum entropy criterion is used to solve the optimization problem of parameter estimation values. The maximum entropy criterion is used as an optimization goal to find the rotation factor that maximizes the signal entropy value, thereby realizing the estimation of signal parameters; in order to solve this problem more accurately, a sparse optimization function is constructed; first, the entropy value exponential function expression under different rotation factors is defined, which establishes a mathematical relationship between the entropy value and the rotation factor, and introduces a sparse coefficient to regulate sparsity; the introduction of the sparse coefficient is to better retain the important features of the signal when processing signals with sparse characteristics, while removing redundant information; based on this entropy value exponential function expression, a sparse optimization function is constructed, the purpose of which is to find the solution that makes the maximum entropy value under different rotation factors reach the optimal solution under the constraint of a given sparse coefficient; by solving this sparse optimization function, a more accurate rotation factor estimate is obtained, thereby improving the accuracy of signal processing.

[0048] According to an embodiment of the present invention, in step S3, constructing a sparse optimization function includes the following specific steps: S31. The entropy exponential function expression under different rotation factors is as follows: (9) in: is the entropy exponential function, is a sparse coefficient, and is an integer greater than or equal to 1; S32. The expression of the constructed sparse optimization function is as follows: (10) in: is the maximum entropy value under different rotation factors, is the sparse optimization function. Under different signal-to-noise ratios, the signal processing length is 512 sampling points, and the normalized entropy value results after different sparse optimization coefficients are as follows Figure 4 shown.

[0049] This technical solution first defines the entropy exponential function under different rotation factors, and establishes a mathematical model that links the rotation factor and the entropy value; the entropy exponential function not only reflects the change of the entropy value of the signal under different rotation factors, but also regulates the sparsity of the signal by introducing a sparse coefficient; based on the entropy exponential function, a sparse optimization function is constructed, the purpose of which is to find the optimal solution for the maximum entropy value under different rotation factors under the constraint of a given sparse coefficient. That is, by finding a rotation factor, the entropy value of the signal reaches the maximum after being transformed by this rotation factor, while meeting the sparsity requirement. The sparse optimization function not only takes the maximization of the entropy value of the signal as the optimization goal, but also considers the sparsity of the signal, so that better results can be achieved in signal processing; it is particularly suitable for processing signals with sparse characteristics, such as images, audio, etc., and can remove redundant information while retaining the important features of the signal, thereby improving the efficiency of signal processing.

[0050] According to one embodiment of the present invention, in step S32, the sparse optimization function By using formula (9) and formula (10), the entropy value change curve is transformed into a normalized single-peak curve, and the steepness of the curve under different sparsity coefficients is characterized differently; As the sparsity coefficient gradually increases, the single-peak curve becomes thinner, and the sparse characteristics of the constructed sparse optimization function become more obvious, which is more conducive to the maximum value estimation under low signal-to-noise ratio conditions, that is, it is conducive to the estimation of the rotation factor parameters of the weighted fractional Fourier transform signal.

[0051] This technical solution transforms the complex and changeable entropy value change curve into a normalized single-peak curve through a sparse optimization function, which not only simplifies the complexity of the problem, but also presents the change of entropy value in an intuitive and easy-to-analyze form; the steepness of the single-peak curve shows obvious differences under different sparsity coefficients; as the sparsity coefficient gradually increases, the single-peak curve becomes thinner, the curve changes more dramatically near the peak, and becomes smoother away from the peak. The more obvious the sparse characteristics are, the more the function tends to select solutions with significant features in the solution process, and ignore those redundant or minor solutions. Under low signal-to-noise ratio conditions, accurate estimation of the rotation factor parameters of the weighted fractional Fourier transform signal is achieved.

[0052] According to an embodiment of the present invention, in step S32, when the rotation factor parameter estimation is located near the optimal rotation factor, the sparse optimization function value at this time is 1; the sparse optimization function value for other rotation factors is close to 0. Under different signal-to-noise ratios, the signal processing length is 512 sampling points, the number of simulations is 500 times, and the parameter estimation accuracy comparison results of different sparse optimization coefficients are as follows Figure 5 shown.

[0053] In this technical solution, under the optimal rotation factor, the signal has the maximum amount of information or the minimum redundancy, so the sparse optimization function value reaches the maximum; under other rotation factors, due to the difference between the signal and the optimal rotation factor, the amount of information is reduced or the redundancy is increased, so the sparse optimization function value approaches 0.

[0054] According to one embodiment of the present invention, in step S3, calculating the position of the maximum peak of the rotation factor includes the following specific steps: S33. Use a sparse function optimization method based on the maximum entropy criterion to estimate the rotation factor to improve the accuracy of parameter estimation in a low signal-to-noise ratio environment; use the periodicity of entropy values ​​in [0,2] and [2,4] to calculate the normalized entropy value of the local signal in the interval [0,2] to obtain the maximum entropy value under each rotation factor, and then use the sparse coefficient and entropy function to calculate the position of the maximum peak of the normalized entropy value of the rotation factor in the interval [0,2]; (11) in: is the weighted fractional Fourier transform received signal to be estimated, is the rotation factor of the local signal of the receiver, p is the sparse coefficient, To seek peace, To take the modulus, To take the logarithm, To obtain the maximum value, is the location of the maximum value, The location of the maximum value obtained.

[0055] This technical solution uses the maximum entropy criterion and sparse function optimization to improve the accuracy of parameter estimation in a low signal-to-noise ratio environment, uses the periodicity of entropy values ​​to reduce the amount of calculation, and infers the entropy value in the entire rotation factor space; controls the behavior of sparse function optimization by adjusting the sparse coefficient, thereby more accurately estimating the rotation factor; calculates the normalized entropy value under each rotation factor, and finds the maximum value therein to determine the position of the optimal rotation factor.

[0056] According to an embodiment of the present invention, in step S4, estimating the rotation factor according to the maximum position of the peak value includes the following specific steps: S41. The position of the maximum value is used to realize effective parameter estimation of the unknown rotation factor value. The expression of the rotation factor to be estimated and the position of the maximum value is as follows: (12) in: is the step size of the local rotation factor, is the final estimated weighted fractional Fourier transform parameter value; S42. By using the maximum entropy value position and step size of the sparse optimization function, the estimated value of the rotation factor parameter of the weighted fractional Fourier transform is finally obtained.

[0057] The comparison results of the estimation accuracy of the maximum entropy sparse optimization method under different sampling times when the signal processing length is 512, 1024 and 2560 sampling points are as follows Figure 6 As shown in the figure, the signal processing length is 512 sampling points, the sparse coefficient is 10, and the estimated rotation factors are 0.1, 0.2 and 0.3 respectively. The curve of the accuracy of the rotation factor estimation under the condition of -15~5 dB is shown in the figure. Figure 7 As shown in the figure; when the signal processing length is 512 sampling points, the sparse coefficient is 10, and the estimated rotation factor is 0.22, the maximum entropy criterion method, the high-order cumulant method, and the cyclic correlation method have different rotation factor estimation accuracy rates under the conditions of -15~10 dB. Figure 8 shown.

[0058] This technical solution realizes effective parameter estimation of unknown rotation factor values ​​through the rotation factor position when the sparse optimization function reaches the maximum entropy value; the final estimated weighted fractional Fourier transform parameter value represents the optimal rotation factor of the signal in the fractional Fourier transform domain; this position is fine-tuned using the step amount to obtain a more accurate rotation factor estimate, and the fine-tuning process is based on the properties of the sparse optimization function, that is, the function has a faster rate of change near the maximum entropy value position, so the optimal solution is quickly found by adjusting the step amount.

[0059] Although the present invention is described in detail by referring to the accompanying drawings and in combination with the preferred embodiments, the present invention is not limited thereto. Without departing from the spirit and essence of the present invention, a person of ordinary skill in the art may make various equivalent modifications or substitutions to the embodiments of the present invention, and these modifications or substitutions shall be within the scope of the present invention. Any person of ordinary skill in the art may easily think of changes or substitutions within the technical scope disclosed by the present invention, and these shall be within the scope of protection of the present invention. Therefore, the scope of protection of the present invention shall be subject to the scope of protection of the claims.

Claims

1. A weighted fractional Fourier transform parameter estimation method based on maximum entropy criterion, characterized in that: The steps include: S1. Calculate the WFRFT of the input signal with a rotation factor in the interval [0,2] to generate a local signal; S2. Calculate the entropy value of the local signal when the rotation factor of the input signal is in the interval [0,2], and calculate the maximum entropy value; S3. According to the input sparse coefficient, the entropy function of the local signal and the maximum value of the entropy function, a sparse optimization function is constructed, and the maximum peak value of the normalized entropy value under the rotation factor [0,2] range is calculated, and the position of the maximum peak value of the rotation factor is calculated; S4. Estimate the rotation factor based on the maximum peak position.

2. The weighted fractional Fourier transform parameter estimation method based on the maximum entropy criterion as claimed in claim 1, characterized in that: In step S1, generating a local signal includes the following specific steps: S11, receiving the weighted fractional Fourier transform received signal to be estimated in the receiving channel The expression is as follows: （1） in: is the Gaussian white noise in the channel, is the weighted fractional Fourier transform received signal to be estimated and satisfies the following formula: （2） in: is the original baseband signal, is the fast Fourier transform, is the rotation factor of the transformation order, For the i The weighting factors of the components are: i =0, 1, 2, 3; The expression is as follows: （3） S12. Since the weighted fractional Fourier transform signal has the periodicity of the rotation factor, let is the rotation factor of the local signal of the receiver, which satisfies: （4） in: , is a value in the interval [0,2], To be valued, Take the values ​​one by one and get a set of periodic rotation factors for No. k Second value; S13, receiving signal After another cycle, the rotation factor After the weighted fractional Fourier transform of , The expression is as follows: （5）。 3. The weighted fractional Fourier transform parameter estimation method based on the maximum entropy criterion as claimed in claim 2, characterized in that: In step S2, solving the entropy value of the local signal includes the following specific steps: S21. Since the entropy value can effectively measure the energy focusing of the signal, the entropy value expression of the discrete signal is as follows: （6） in: is the normalized received signal of sampling length M, is the entropy value of the discrete signal; S22, weighted fractional Fourier transform local signal Discretization can obtain the discretized local signal , then the WFRFT signal entropy value of the local signal for one rotation factor period is The expression is as follows: （7） The discretized local signal entropy value and the maximum position are used to estimate the rotation factor of the weighted fractional Fourier transform.

4. The weighted fractional Fourier transform parameter estimation method based on maximum entropy criterion as claimed in claim 3, characterized in that: The step S22 The signal, whose rotation factor is related to the rotation angle of the time-frequency axis, has a certain periodicity; According to formula (4) and formula (6), The entropy values ​​of different signal rotation factors are also periodic, and the transformation period is also 4, showing a symmetrical distribution in the interval [0,2] and the interval [2,4]. That is, the entropy value results in half of the symmetrical interval are used to estimate the rotation factors of WFRFT.

5. The weighted fractional Fourier transform parameter estimation method based on maximum entropy criterion as claimed in claim 3, characterized in that: In step S2, the calculation of the maximum entropy value of the local signal includes the following specific steps: S23, when the energy is most concentrated, that is, different rotation factors When the signal entropy value is the largest, the rotation factor at this time is the parameter value to be estimated; For unknown rotation factors Signal, solve the optimization problem of parameter estimation by maximum entropy criterion: （8）。 6. The weighted fractional Fourier transform parameter estimation method based on maximum entropy criterion as claimed in claim 5, characterized in that: In step S3, constructing a sparse optimization function includes the following specific steps: S31. The entropy exponential function expression under different rotation factors is as follows: （9） in: is the entropy exponential function, is a sparse coefficient, and is an integer greater than or equal to 1; S32. The expression of the constructed sparse optimization function is as follows: （10） in: is the maximum entropy value under different rotation factors, Optimizes the function for sparseness.

7. The weighted fractional Fourier transform parameter estimation method based on maximum entropy criterion as claimed in claim 6, characterized in that: In step S32, the sparse optimization function By using formula (9) and formula (10), the entropy value change curve is transformed into a normalized single-peak curve, and the steepness of the curve under different sparsity coefficients is characterized differently; As the sparsity coefficient gradually increases, the single-peak curve becomes thinner, and the sparse characteristics of the constructed sparse optimization function become more obvious, which is more conducive to the maximum value estimation under low signal-to-noise ratio conditions, that is, it is conducive to the estimation of the rotation factor parameters of the weighted fractional Fourier transform signal.

8. The weighted fractional Fourier transform parameter estimation method based on maximum entropy criterion as claimed in claim 7, characterized in that: In step S32, when the rotation factor parameter estimate is located near the optimal rotation factor, the sparse optimization function value at this time is 1; the sparse optimization function values ​​for other rotation factors are close to 0.

9. The weighted fractional Fourier transform parameter estimation method based on maximum entropy criterion as claimed in claim 6, characterized in that: In step S3, calculating the position of the maximum peak of the rotation factor includes the following specific steps: S33. Use a sparse function optimization method based on the maximum entropy criterion to estimate the rotation factor to improve the accuracy of parameter estimation in a low signal-to-noise ratio environment; use the periodicity of entropy values ​​in [0,2] and [2,4] to calculate the normalized entropy value of the local signal in the interval [0,2] to obtain the maximum entropy value under each rotation factor, and then use the sparse coefficient and entropy function to calculate the position of the maximum peak of the normalized entropy value of the rotation factor in the interval [0,2]; （11） in: is the weighted fractional Fourier transform received signal to be estimated, is the rotation factor of the local signal of the receiver, p is the sparse coefficient, To seek peace, To take the modulus, To take the logarithm, To obtain the maximum value, is the location of the maximum value, The location of the maximum value obtained.

10. The weighted fractional Fourier transform parameter estimation method based on maximum entropy criterion as claimed in claim 9, characterized in that: In step S4, estimating the rotation factor according to the maximum position of the peak value includes the following specific steps: S41. The position of the maximum value is used to realize effective parameter estimation of the unknown rotation factor value. The expression of the rotation factor to be estimated and the position of the maximum value is as follows: （12） in: is the step size of the local rotation factor, is the final estimated weighted fractional Fourier transform parameter value; S42. By using the maximum entropy value position and step size of the sparse optimization function, the estimated value of the rotation factor parameter of the weighted fractional Fourier transform is finally obtained.

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