Resolution improvement limit derivation method

Abstract

The invention discloses a resolution improvement limit derivation method. The method comprises the following steps: converting an echo signal into an echo signal matrix expression under a geometric diffraction model; echo vector probability distribution is obtained under the condition that preset noise is Gaussian complex noise, and an echo likelihood function is obtained according to the echo vector probability distribution; calculating a Fisher information matrix of the scattering center parameter; obtaining CRLB of the scattering center parameters according to the Fisher information matrix of the scattering center parameters; obtaining the minimum distance between the two scattering centers according to the pole phase difference of the two scattering centers under the pole model; obtaining the phase difference of two poles in the first sub-band and the phase difference of two poles in the double sub-bands according to the minimum distance between the two scattering centers; and obtaining a resolution improvement limit according to the phase difference minimum value of the two poles in the first sub-band and the phase difference minimum value of the two poles in the double sub-bands. The invention solves the problem that the existing resolution improvement limit is only suitable for a point scattering center and has a narrow application range.

Description

Technical Field

[0001] This invention belongs to the field of super-resolution microwave imaging technology, and in particular relates to a method for deriving the resolution improvement limit. Background Technology

[0002] Super-resolution microwave imaging technology can reconstruct a wideband signal with high-resolution microwave images from multiple subband signals with narrow bandwidth and low imaging resolution. However, existing research mainly focuses on fusion methods, with little research on the accuracy of scattering center parameter (SCP) estimation and the derivation of the resolution improvement limit. In 1999, Peleg derived the Cramer-Rao lower bound (CRLB) for constant amplitude polarization phase signals. In 2000, Smith derived the statistical resolution limit and the CRLB in the complex case. Both methods are applicable to point scattering center models. However, as the bandwidth and resolution of ISAR systems increase, the scattering centers received by radar include not only point scattering centers but also geometric diffraction theory (GTD) scattering centers. Traditional resolution limit derivation methods applicable to point scattering center models are no longer suitable. Therefore, research is needed on resolution improvement limit methods applicable to complex scattering centers. Summary of the Invention

[0003] The technical problem solved by this invention is to overcome the shortcomings of the prior art and provide a method for deriving the resolution improvement limit, which solves the problem that the existing resolution improvement limit is only applicable to point scattering centers and has a narrow range of applications, and lays the foundation for obtaining the resolution improvement limit for complex scattering centers.

[0004] The objective of this invention is achieved through the following technical solution: a method for deriving the resolution improvement limit, comprising: converting the echo signal into an echo signal matrix expression under a geometric diffraction model; obtaining the echo vector probability distribution under a preset noise condition of Gaussian complex noise, and obtaining the echo likelihood function based on the echo vector probability distribution; calculating the Fisher information matrix of the scattering center parameters; obtaining the CRLB of the scattering center parameters based on the Fisher information matrix of the scattering center parameters; obtaining the minimum distance between the two scattering centers based on the phase difference between the poles of the two scattering centers under a pole model; obtaining the phase difference between the two poles in the first sub-band and the phase difference between the two poles in the double sub-band based on the minimum distance between the two scattering centers; and obtaining the resolution improvement limit based on the minimum phase difference between the two poles in the first sub-band and the minimum phase difference between the two poles in the double sub-band.

[0005] In the above method for deriving the resolution enhancement limit, the echo signal is obtained through the following formula:

[0006]

[0007] Among them, Z(fq f is the value of the q-th frequency sampling point of the echo signal. q B is the sampling frequency. p f is the scattering amplitude of the p-th scattering center. c For the center frequency, a P Let β be the frequency dependence factor of the p-th scattering center, P be the number of scattering centers, β be the amplitude incoherence term between sub-band signals, η be the phase incoherence term between sub-band signals, and x be the frequency dependence factor of the p-th scattering center. p Let n be the position of the p-th scattering center, c be the speed of light, and n(f q ) represents complex Gaussian noise.

[0008] In the above derivation method for the resolution improvement limit, the expression for the echo signal matrix is:

[0009] Z = A(θ)B + n;

[0010] Where θ = (α, β, η, x) T ,

[0011] Where Z is the signal vector, A() is the steering matrix, θ is the set of scattering center parameters excluding B, B is the scattering amplitude vector, α is the frequency dependence factor vector, x is the position vector, n is the complex Gaussian noise vector, β is the amplitude incoherence term between sub-band signals, and η is the phase incoherence term between sub-band signals.

[0012] In the above derivation method for the resolution enhancement limit, the echo vector probability distribution is obtained by the following formula:

[0013]

[0014] Where Z is the signal vector, B is the scattering amplitude vector, A is the steering matrix, f(Z|A,B) is the echo vector probability distribution, N is the number of frequency sampling points, and R is the covariance matrix of the Gaussian noise vector.

[0015] In the above method for deriving the resolution enhancement limit, the echo likelihood function is obtained through the following formula:

[0016]

[0017] Where Z is the signal vector, B is the scattering amplitude vector, A is the steering matrix, f(Z|A,B) is the echo vector probability distribution, N is the number of frequency sampling points, and R is the covariance matrix of the Gaussian noise vector.

[0018] In the above method for deriving the resolution enhancement limit, the Fisher information matrix of the scattering center parameters is obtained by the following formula:

[0019]

[0020] in, Let be the Fisher information matrix of the scattering center parameters, E represent the expectation operation, and L be the echo likelihood function. Let be the second derivative of the echo likelihood function with respect to θ and B. For B and its conjugate Fisher matrix, Let G be the Fisher matrix of θ and B. θ Let θ be the Fisher matrix. The echo likelihood function is for B and its conjugate. The second derivative, Let be the second derivative of the echo likelihood function with respect to θ and B. Let be the second derivative of the echo likelihood function with respect to θ. It is the conjugate of B.

[0021] In the above method for deriving the resolution enhancement limit, the CRLB of the scattering center parameter is obtained by the following formula:

[0022]

[0023] Where B is the scattering amplitude vector, C α Let CRLB be the frequency dependence factor α, A be the turning matrix, and B be the directional matrix. H For the transpose of B, A is the transpose of the steering matrix corresponding to the frequency dependence factor α. [α] Let R be the steering matrix corresponding to the frequency-dependent factor, and let C be the covariance matrix of the Gaussian noise vector n. β CRLB is the amplitude incoherence term β between subband signals. A is the transpose of the steering matrix corresponding to the amplitude incoherence term β between subband signals. [β] C is the steering matrix corresponding to the amplitude incoherence term β between sub-band signals. η CRLB is the phase incoherence term η between subband signals. A is the transpose of the steering matrix corresponding to the phase incoherence term η between subband signals. [η] C is the steering matrix corresponding to the phase incoherence term η between sub-band signals. x CRLB is the scattering center position x. Let A be the transpose of the steering matrix corresponding to the scattering center position x. [x] Let x be the turning matrix corresponding to the scattering center position.

[0024] In the above derivation method for the resolution enhancement limit, the pole phase difference between the two scattering centers is obtained by the following formula:

[0025]

[0026] Wherein, Δ[phase(ρ 12 ] represents the pole phase difference between the two scattering centers, phase(·) indicates the phase taking operation, Δx is the position difference between the two scattering centers, α1 is the frequency dependence factor of the first scattering center, α2 is the frequency dependence factor of the second scattering center, Δf is the minimum frequency difference between the two frequency sampling points, and f c ρ is the center frequency. 12 The distance between the two scattering centers is denoted as .

[0027] In the above derivation method for the resolution enhancement limit, the phase difference between the two poles in the first sub-band is obtained by the following formula:

[0028]

[0029] The phase difference between the two poles in the twin subband is obtained by the following formula:

[0030]

[0031] Wherein, Δ[phase(ρ 12 )] zd1 Let f be the phase difference between the two poles in the first sub-band, α1 be the frequency dependence factor of the first scattering center, α2 be the frequency dependence factor of the second scattering center, and f be the frequency dependence factor of the second scattering center. c ρ is the center frequency. 12 Δx is the distance between the two scattering centers. zd1 Δf is the minimum distance difference between the two scattering centers in the first sub-band, c is the speed of light, and Δx is the minimum frequency difference. szd It represents the minimum distance difference between the two scattering centers in the twin sub-band.

[0032] In the above derivation method for the resolution improvement limit, the resolution improvement limit is obtained by the following formula:

[0033]

[0034] in, This represents the minimum phase difference between the two poles in the first sub-band. Δx represents the minimum phase difference between the two poles in the bipolar sub-band. zd1 Δx is the minimum distance difference between the two scattering centers in the first sub-band. szd Let f be the minimum distance difference between the two scattering centers in the twin sub-band, α1 be the frequency dependence factor of the first scattering center, α2 be the frequency dependence factor of the second scattering center, and f c ρ is the center frequency. 12 Let Δf be the distance between the two scattering centers, and Δf be the minimum frequency difference.

[0035] A resolution improvement limit derivation system includes: a first module for converting echo signals into echo signal matrix expressions under a geometric diffraction model; a second module for obtaining the echo vector probability distribution under a preset noise condition of Gaussian complex noise, and obtaining the echo likelihood function based on the echo vector probability distribution; a third module for calculating the Fisher information matrix of scattering center parameters; a fourth module for obtaining the CRLB of scattering center parameters based on the Fisher information matrix of scattering center parameters; a fifth module for obtaining the minimum distance between two scattering centers based on the phase difference between the poles of two scattering centers under a pole model; a sixth module for obtaining the phase difference between two poles in the first sub-band and the phase difference between two poles in the double sub-band based on the minimum distance between the two scattering centers; and a seventh module for obtaining the resolution improvement limit based on the minimum phase difference between the two poles in the first sub-band and the minimum phase difference between the two poles in the double sub-band.

[0036] Compared with the prior art, the present invention has the following advantages:

[0037] (1) This invention solves the problem that the existing resolution improvement limit is only applicable to point scattering centers and has a narrow range of applications, and lays the foundation for obtaining the resolution improvement limit corresponding to complex scattering centers.

[0038] (2) The resolution improvement derivation proposed in this invention takes into account the complexity of the scattering characteristics of the scattering center brought about by the increase in signal bandwidth, which is more consistent with the actual situation, has a wider range of applications, and is more practical. Attached Figure Description

[0039] Various other advantages and benefits will become apparent to those skilled in the art upon reading the following detailed description of preferred embodiments. The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Furthermore, the same reference numerals denote the same parts throughout the drawings. In the drawings:

[0040] Figure 1 This is a flowchart of the resolution improvement limit derivation method provided in the embodiments of the present invention;

[0041] Figure 2 This is a signal spectrum distribution diagram provided in an embodiment of the present invention;

[0042] Figure 3(a) is a graph showing the change of CRLB of the frequency dependence factor with signal-to-noise ratio when the fractional bandwidth is 0.1 according to the embodiment of the present invention;

[0043] Figure 3(b) is a graph showing the change of CRLB of the range position with signal-to-noise ratio when the fractional bandwidth is 0.1 according to the embodiment of the present invention;

[0044] Figure 3(c) is a graph showing the CRLB of the scattering amplitude as a function of the signal-to-noise ratio when the fractional bandwidth is 0.1 according to the embodiment of the present invention;

[0045] Figure 3(d) is a graph showing the CRLB of the amplitude incoherent term as a function of signal-to-noise ratio when the fractional bandwidth is 0.1 according to the embodiment of the present invention.

[0046] Figure 3(e) is a graph showing the CRLB of the phase incoherent term as a function of signal-to-noise ratio when the fractional bandwidth is 0.1 according to an embodiment of the present invention;

[0047] Figure 4(a) is a graph showing the change of CRLB of the frequency dependence factor with signal-to-noise ratio when the fractional bandwidth is 0.4 according to the embodiment of the present invention;

[0048] Figure 4(b) is a graph showing the change of CRLB of the range position with the signal-to-noise ratio when the fractional bandwidth is 0.4 according to the embodiment of the present invention;

[0049] Figure 4(c) is a graph showing the CRLB of the scattering amplitude as a function of the signal-to-noise ratio when the fractional bandwidth is 0.4 according to the embodiment of the present invention;

[0050] Figure 4(d) is a graph showing the CRLB of the amplitude incoherent term as a function of signal-to-noise ratio when the fractional bandwidth is 0.4 according to the embodiment of the present invention.

[0051] Figure 4(e) is a graph showing the change of CRLB of the phase incoherent term with signal-to-noise ratio when the fractional bandwidth is 0.4 according to the embodiment of the present invention. Detailed Implementation

[0052] Exemplary embodiments of the present disclosure will now be described in more detail with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided to enable a more thorough understanding of the present disclosure and to fully convey the scope of the disclosure to those skilled in the art. It should be noted that, unless otherwise specified, the embodiments and features described herein can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0053] Figure 1 This is a flowchart of the resolution enhancement limit derivation method provided in an embodiment of the present invention. Figure 1As shown, the method for deriving the resolution improvement limit includes: converting the echo signal into an echo signal matrix expression under a geometric diffraction model; obtaining the echo vector probability distribution under the pre-set noise condition of Gaussian complex noise, and obtaining the echo likelihood function based on the echo vector probability distribution; calculating the Fisher information matrix of the scattering center parameters; obtaining the CRLB of the scattering center parameters based on the Fisher information matrix of the scattering center parameters; obtaining the minimum distance between the two scattering centers based on the phase difference between the poles of the two scattering centers under the pole model; obtaining the phase difference between the two poles in the first sub-band and the phase difference between the two poles in the double sub-band based on the minimum distance between the two scattering centers; and obtaining the resolution improvement limit based on the minimum phase difference between the two poles in the first sub-band and the minimum phase difference between the two poles in the double sub-band.

[0054] The echo signal is obtained using the following formula:

[0055]

[0056] Among them, Z(f q f is the value of the q-th frequency sampling point of the echo signal. q B is the sampling frequency. p f is the scattering amplitude of the p-th scattering center. c For the center frequency, a P Let β be the frequency dependence factor of the p-th scattering center, P be the number of scattering centers, β be the amplitude incoherence term between sub-band signals, η be the phase incoherence term between sub-band signals, and x be the frequency dependence factor of the p-th scattering center. p Let n be the position of the p-th scattering center, c be the speed of light, and n(f q ) represents complex Gaussian noise.

[0057] The echo signal matrix expression is:

[0058] Z = A(θ)B + n;

[0059] Where θ = (α, β, η, x) T ,

[0060] Where Z is the signal vector, A() is the steering matrix, θ is the set of scattering center parameters excluding B, B is the scattering amplitude vector, α is the frequency dependence factor vector, x is the position vector, n is the complex Gaussian noise vector, β is the amplitude incoherence term between sub-band signals, and η is the phase incoherence term between sub-band signals.

[0061] The probability distribution of the echo vector is obtained by the following formula:

[0062]

[0063] Where Z is the signal vector, B is the scattering amplitude vector, A is the steering matrix, f(Z|A,B) is the echo vector probability distribution, N is the number of frequency sampling points, and R = E[nn] H ] is the covariance matrix of the Gaussian noise vector.

[0064] The echo likelihood function is obtained by the following formula:

[0065]

[0066] Where Z is the signal vector, B is the scattering amplitude vector, A is the steering matrix, L(Z|A,B) is the echo likelihood function, f(Z|A,B) is the echo vector probability distribution, N is the number of frequency sampling points, and R = E[nn H ] is the covariance matrix of the Gaussian noise vector.

[0067] The Fisher information matrix of the scattering center parameters is obtained by the following formula:

[0068]

[0069] in, Let be the Fisher information matrix of the scattering center parameters, E represent the expectation operation, and L be the echo likelihood function. Let be the second derivative of the echo likelihood function with respect to θ and B. For B and its conjugate Fisher matrix, Let G be the Fisher matrix of θ and B. θ Let θ be the Fisher matrix. The echo likelihood function is for B and its conjugate. The second derivative, Let be the second derivative of the echo likelihood function with respect to θ and B. Let be the second derivative of the echo likelihood function with respect to θ. It is the conjugate of B.

[0070] The CRLB of the scattering center parameter is obtained by the following formula:

[0071]

[0072]

[0073] Where B is the scattering amplitude vector, C α Let CRLB be the frequency dependence factor α, A be the turning matrix, and B be the directional matrix. H For the transpose of B, A is the transpose of the steering matrix corresponding to the frequency dependence factor α. [α]Let R be the steering matrix corresponding to the frequency-dependent factor, and let C be the covariance matrix of the Gaussian noise vector n. β CRLB is the amplitude incoherence term β between subband signals. A is the transpose of the steering matrix corresponding to the amplitude incoherence term β between subband signals. [β] C is the steering matrix corresponding to the amplitude incoherence term β between sub-band signals. η CRLB is the phase incoherence term η between subband signals. A is the transpose of the steering matrix corresponding to the phase incoherence term η between subband signals. [η] C is the steering matrix corresponding to the phase incoherence term η between sub-band signals. x CRLB is the scattering center position x. Let A be the transpose of the steering matrix corresponding to the scattering center position x. [x] Let x be the turning matrix corresponding to the scattering center position.

[0074] The phase difference between the poles of the two scattering centers is obtained by the following formula:

[0075]

[0076] Wherein, Δ[phase(ρ 12 ] represents the pole phase difference between the two scattering centers, phase(·) indicates the phase taking operation, Δx is the position difference between the two scattering centers, α1 is the frequency dependence factor of the first scattering center, α2 is the frequency dependence factor of the second scattering center, Δf is the minimum frequency difference between the two frequency sampling points, and f c ρ is the center frequency. 12 The distance between the two scattering centers is denoted as .

[0077] The phase difference between the two poles in the first sub-band is obtained by the following formula:

[0078]

[0079] The phase difference between the two poles in the twin subband is obtained by the following formula:

[0080]

[0081] Wherein, Δ[phase(ρ 12 )] zd1 Let f be the phase difference between the two poles in the first sub-band, α1 be the frequency dependence factor of the first scattering center, α2 be the frequency dependence factor of the second scattering center, and f be the frequency dependence factor of the second scattering center. c ρ is the center frequency. 12 Δx is the distance between the two scattering centers. zd1 Δf is the minimum distance difference between the two scattering centers in the first sub-band, c is the speed of light, and Δx is the minimum frequency difference.szd It represents the minimum distance difference between the two scattering centers in the twin sub-band.

[0082]

[0083] Where, Δx zd1 Δx is the minimum distance difference between the two scattering centers in the first sub-band. szd This represents the minimum distance difference between the two scattering centers in the twin sub-band. This represents the minimum phase difference between the two poles in the first sub-band. Zd1 represents the minimum phase difference between the two poles in the dual subband, and Zd1 represents the first subband, while Szd represents the dual subband.

[0084]

[0085] in, This represents the minimum phase difference between the two poles in the first sub-band. Δx represents the minimum phase difference between the two poles in the bipolar sub-band. zd1 Δx is the minimum distance difference between the two scattering centers in the first sub-band. szd Let f be the minimum distance difference between the two scattering centers in the twin sub-band, α1 be the frequency dependence factor of the first scattering center, α2 be the frequency dependence factor of the second scattering center, and f c ρ is the center frequency. 12 Let Δf be the distance between the two scattering centers, and Δf be the minimum frequency difference.

[0086] This embodiment proposes a method for deriving the resolution improvement limit. First, the scattering center parameter CRLB in a single narrowband signal and a reconstructed wideband signal is derived. Then, the resolution improvement limit is derived by comparing the CRLB parameters in the two signals.

[0087] The proposed method belongs to the derivation method of the resolution improvement limit. The overall process is as follows: Figure 1 Specifically, it includes the following steps.

[0088] Step 1: Representation of echo signal matrix.

[0089] The echo signal is converted into a matrix representation under the geometric diffraction (GTD) model.

[0090] Step 2: Calculate the echo likelihood function.

[0091] The probability distribution of the echo vector is obtained under the assumption that the noise is Gaussian complex noise, and the likelihood function is calculated based on this.

[0092] Step 3: Calculate the Fisher information matrix of the scattering center parameters.

[0093] Calculate the Fisher information matrix for parameters such as complex amplitude, position, and frequency dependence factor of the scattering center.

[0094] Step 4: Calculate the CRLB of the scattering center parameters.

[0095] Calculate the CRLB for parameters such as complex amplitude, position, and frequency dependence factor of the scattering center.

[0096] Step 5: Calculate the minimum distance between the two scattering centers.

[0097] The minimum distance between two scattering centers is estimated based on the pole phase difference between the two scattering centers under the pole model.

[0098] Step 6: Derive the resolution improvement limit.

[0099] The resolution improvement limit is derived by considering that the two sub-bands are located at the two ends of the reconstructed large bandwidth frequency band.

[0100] The specific steps of the derivation method for the resolution enhancement limit are as follows (see details). Figure 1 ):

[0101] Step 1: Representation of echo signal matrix.

[0102] For a single subband or a reconstructed wideband signal, the target echo can be expressed using the following method when considering the geometric diffraction (GTD) model:

[0103]

[0104] For simplicity of derivation, the above formula is transformed into a matrix expression:

[0105] Z=A(θ)B+n (2)

[0106] in, For signal vectors, Let θ be the steering matrix, where θ = (α, β, η, x). T Let A be the set of parameters for all scattering centers except B. A = [A1, A2, ..., A...] p ,…,A P ],and f is a frequency vector. According to... Figure 2 For subbands (small bandwidth), Bandwidth is f n = f0 + q·Δf, the steering matrix is ​​defined as A zd1 For large bandwidths, f = (f0, f1, ... f) q ,…f N-1 ) T The bandwidth is BW = f N-1 -f0, the steering matrix is ​​defined as Afull B = (B1, B2, ... B) P ) H α = (-1, -0.5, 0, 0.5, 1) H x = (x1, x2, ..., x P ) H , The scattering amplitude vector, For frequency-dependent factor vectors, For position vectors, Let β be the noise vector, and let β be the amplitude incoherence term between subband signals.

[0107] Step 2: Calculate the echo likelihood function.

[0108] Assuming the noise is Gaussian complex noise, E[nn] H If ] = R, then the probability distribution of Z satisfies:

[0109]

[0110] Taking the logarithm of the above formula yields the likelihood function:

[0111]

[0112] Step 3: Calculate the Fisher information matrix of the scattering center parameters.

[0113] All unknowns include θ and B (scattering amplitude vector). The CRLB of the known unknowns is the inverse of the Fisher information matrix, therefore the Fisher information matrices of θ and B should be determined first:

[0114]

[0115] Since B is a complex number, the CRLB derivation method for the complex field should be used to calculate B and its conjugate. Fisher matrix:

[0116]

[0117] Step 4: Calculate the CRLB of the scattering center parameters.

[0118] The CRLB of θ and B is the inverse of the Fisher information matrix:

[0119]

[0120] From the formula for finding the inverse of a reference matrix, we can obtain:

[0121]

[0122]

[0123] in,

[0124]

[0125]

[0126]

[0127] Further derivation yields:

[0128]

[0129]

[0130]

[0131]

[0132] Step 5: Calculate the minimum distance between the two scattering centers.

[0133] After estimating the scattering center parameters, incoherent terms are compensated, and a wide-bandwidth signal is reconstructed to achieve high-resolution ISAR imaging. Considering the impact of frequency dependence factors and noise on resolution improvement, the echo expression becomes:

[0134]

[0135] After discretization, the echo becomes:

[0136]

[0137] make The above equation then becomes a pole model:

[0138]

[0139] Convert the above expression into matrix form:

[0140] Z=A(ρ)B+n (20)

[0141] Where A=(ρ 1 ,…,ρ q ,…,ρ N ) T , B = (C1, C2, ... C P ) T .

[0142] Assuming the target has only two scattering centers, the pole difference between the two scattering centers is:

[0143]

[0144] In this equation, the subscript "12" on the left-hand side represents the difference between the first and second scattering poles; this pole difference naming convention will be used throughout the following text. The pole phase difference is:

[0145]

[0146] Where phase(·) represents the phase taking operation, Δx is the position difference between the two scattering centers, and the minimum value that Δx can take is the minimum distance between the two scattering centers.

[0147] Step 6: Derive the resolution improvement limit.

[0148] When the frequency dependence factors of the two scattering centers are the same, i.e., α1 = α2, the pole phase difference is proportional to the position difference between the two scattering centers:

[0149]

[0150] At this point, the pole phase difference of the first narrow-bandwidth signal is at its minimum. (Where the superscript zd1 represents the minimum phase error of the first narrow-bandwidth signal, and the subscript min indicates the minimum value operation; the same definition is used below.) Minimum phase difference between the poles of the two narrow-bandwidth signals. The ratio is the ratio of the minimum position difference between the two scattering centers in the first narrow-bandwidth signal and the two narrow-bandwidth signals. Since the minimum position difference between the two scattering centers is the resolution limit, dividing the minimum pole phase difference by the minimum pole phase difference yields the resolution improvement limit.

[0151]

[0152] The subscript 'max' indicates the operation of finding the maximum value.

[0153] When the frequency dependence factors of the two scattering centers are not the same, i.e., α1 ≠ α2, the phase difference of the poles corresponding to the first sub-band is: The phase difference between the poles corresponding to two narrow bandwidth signals is It can be seen that the limit of resolution improvement at this point is:

[0154]

[0155] The feasibility and efficiency of the calculation method in this invention will be verified through specific experiments below.

[0156] (1) Confirm the radar system and scattering center parameters, see Table 1;

[0157] Table 1 Main parameters of the experiment verifying the accuracy of scattering center parameter estimation

[0158] parameter Parameter value Center frequency 35GHz bandwidth 5GHz Large bandwidth sampling points N 1700 <![CDATA[Number of sampling points L for a single narrowband signal sb > 510 Fractional bandwidth 0.1，0.4 Signal-to-noise ratio In 5dB increments, from 0dB to 30dB

[0159] (2) Substituting the parameters into equations (13)-(16), we obtain the relationship between CRLB and signal-to-noise ratio for different parameters. See the graph below. Figure 2 ;

[0160] From Figures 3(a), 3(b), 3(c), 3(d), 3(e), 4(a), 4(b), 4(c), 4(d), and 4(e), it can be seen that: 1) the CRLB of the scattering center parameter decreases nonlinearly with the increase of the signal-to-noise ratio, and the CRLB of the scattering parameter changes in the same trend for a single narrow-bandwidth signal and multiple narrow-bandwidth signals; 2) the CRLB of the scattering parameter is lower with two narrow-bandwidth signals than with a single narrow-bandwidth signal, meaning that using multiple narrow-bandwidth signals can improve the scattering... 3) The ability of narrow-bandwidth signals to improve the accuracy of scattering parameter estimation is stronger at high signal-to-noise ratios (SNRs); 4) When the SNR is higher than 25dB, the CRLB decreases more slowly, meaning that to ensure high scattering parameter estimation accuracy, the SNR in the received signal should be higher than 25dB; 5) When the SNR is higher than 5dB, the larger the fractional bandwidth, the stronger the ability of multiple narrow-bandwidth signals to improve the accuracy of scattering parameter estimation; 6) Compared with other scattering parameters, the CRLB of scattering amplitude is unstable as the SNR decreases.

[0161] (3) When the frequency dependence factor α of the two scattering centers is the same, substituting the parameters into equation (25) yields a resolution improvement factor of 8.36 times; when the frequency dependence factors α of the two scattering centers are different, the resolution improvement factor is shown in the following equation. It can be seen from the equation that the resolution improvement limit is related to the type of scattering center. When the frequency dependence factors of the two scattering centers are 1 and -1 respectively, the resolution improvement factor is the highest.

[0162]

[0163] A resolution improvement limit derivation system includes: a first module for converting echo signals into echo signal matrix expressions under a geometric diffraction model; a second module for obtaining the echo vector probability distribution under a preset noise condition of Gaussian complex noise, and obtaining the echo likelihood function based on the echo vector probability distribution; a third module for calculating the Fisher information matrix of scattering center parameters; a fourth module for obtaining the CRLB of scattering center parameters based on the Fisher information matrix of scattering center parameters; a fifth module for obtaining the minimum distance between two scattering centers based on the phase difference between the poles of two scattering centers under a pole model; a sixth module for obtaining the phase difference between two poles in the first sub-band and the phase difference between two poles in the double sub-band based on the minimum distance between the two scattering centers; and a seventh module for obtaining the resolution improvement limit based on the minimum phase difference between the two poles in the first sub-band and the minimum phase difference between the two poles in the double sub-band.

[0164] This embodiment addresses the problem that existing resolution improvement limits are only applicable to point scattering centers and have a narrow range of applications, laying the foundation for deriving the resolution improvement limit corresponding to complex scattering centers. The resolution improvement derivation proposed in this embodiment takes into account the complexity of the scattering characteristics of the scattering center brought about by the increase in signal bandwidth, which is more consistent with the actual situation, has a wider range of application scenarios, and is more practical.

[0165] Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make possible changes and modifications to the technical solutions of the present invention by utilizing the methods and techniques disclosed above without departing from the spirit and scope of the present invention. Therefore, any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solutions of the present invention shall fall within the protection scope of the technical solutions of the present invention.

Claims

1. A method for deriving the resolution enhancement limit, characterized in that... include: Under the geometric diffraction model, the echo signal is converted into an echo signal matrix expression; The echo vector probability distribution is obtained under the preset noise condition of Gaussian complex noise, and the echo likelihood function is obtained based on the echo vector probability distribution; Calculate the Fisher information matrix of the scattering center parameters; The CRLB of the scattering center parameters is obtained from the Fisher information matrix of the scattering center parameters; The minimum distance between two scattering centers can be obtained based on the pole phase difference between the two scattering centers under the pole model; The phase difference between the two poles in the first sub-band and the phase difference between the two poles in the double sub-band are obtained based on the minimum distance between the two scattering centers. The resolution improvement limit is obtained by considering the minimum phase difference between the two poles in the first subband and the minimum phase difference between the two poles in both subbands.

2. The method for deriving the resolution improvement limit according to claim 1, characterized in that: The echo signal is obtained using the following formula: Among them, Z(f q f is the value of the q-th frequency sampling point of the echo signal. q B is the sampling frequency. p f is the scattering amplitude of the p-th scattering center. c For the center frequency, a P Let β be the frequency dependence factor of the p-th scattering center, P be the number of scattering centers, β be the amplitude incoherence term between sub-band signals, η be the phase incoherence term between sub-band signals, and x be the frequency dependence factor of the p-th scattering center. p Let n be the position of the p-th scattering center, c be the speed of light, and n(f q ) represents complex Gaussian noise.

3. The method for deriving the resolution improvement limit according to claim 1, characterized in that: The echo signal matrix expression is: Z = A(θ)B + n; Where, θ=(α,β,η,x) T , Where Z is the signal vector, A() is the steering matrix, θ is the set of scattering center parameters except B, B is the scattering amplitude vector, α is the frequency dependence factor vector, x is the position vector, n is the complex Gaussian noise vector, β is the amplitude incoherence term between sub-band signals, and η is the phase incoherence term between sub-band signals. The probability distribution of the echo vector is obtained by the following formula: Where Z is the signal vector, B is the scattering amplitude vector, A is the steering matrix, f(Z|A,B) is the echo vector probability distribution, N is the number of frequency sampling points, and R is the covariance matrix of the Gaussian noise vector.

4. The method for deriving the resolution improvement limit according to claim 1, characterized in that: The echo likelihood function is obtained by the following formula: Where Z is the signal vector, B is the scattering amplitude vector, A is the steering matrix, f(Z|A,B) is the echo vector probability distribution, N is the number of frequency sampling points, and R is the covariance matrix of the Gaussian noise vector.

5. The method for deriving the resolution improvement limit according to claim 1, characterized in that: The Fisher information matrix of the scattering center parameters is obtained by the following formula: in, Let be the Fisher information matrix of the scattering center parameters, E represent the expectation operation, and L be the echo likelihood function. Let be the second derivative of the echo likelihood function with respect to θ and B. For B and its conjugate Fisher matrix, Let G be the Fisher matrix of θ and B. θ Let θ be the Fisher matrix. The echo likelihood function is for B and its conjugate. The second derivative, Let be the second derivative of the echo likelihood function with respect to θ and B. Let be the second derivative of the echo likelihood function with respect to θ. It is the conjugate of B.

6. The method for deriving the resolution improvement limit according to claim 1, characterized in that: The CRLB of the scattering center parameter is obtained by the following formula: Where B is the scattering amplitude vector, C α Let CRLB be the frequency dependence factor α, A be the turning matrix, and B be the directional matrix. H For the transpose of B, A is the transpose of the steering matrix corresponding to the frequency dependence factor α. [α] Let R be the steering matrix corresponding to the frequency-dependent factor, and let C be the covariance matrix of the Gaussian noise vector n. β CRLB is the amplitude incoherence term β between subband signals. A is the transpose of the steering matrix corresponding to the amplitude incoherence term β between subband signals. [β] C is the steering matrix corresponding to the amplitude incoherence term β between sub-band signals. η CRLB is the phase incoherence term η between subband signals. A is the transpose of the steering matrix corresponding to the phase incoherence term η between subband signals. [η] C is the steering matrix corresponding to the phase incoherence term η between sub-band signals. x CRLB is the scattering center position x. Let A be the transpose of the steering matrix corresponding to the scattering center position x. [x] Let x be the turning matrix corresponding to the scattering center position.

7. The method for deriving the resolution improvement limit according to claim 1, characterized in that: The phase difference between the poles of the two scattering centers is obtained by the following formula: Wherein, Δ[phase(ρ 12 ] represents the pole phase difference between the two scattering centers, phase(·) indicates the phase taking operation, Δx is the position difference between the two scattering centers, α1 is the frequency dependence factor of the first scattering center, α2 is the frequency dependence factor of the second scattering center, Δf is the minimum frequency difference between the two frequency sampling points, and f c For the center frequency, ρ 12 The distance between the two scattering centers is denoted as .

8. The method for deriving the resolution improvement limit according to claim 1, characterized in that: The phase difference between the two poles in the first sub-band is obtained by the following formula: The phase difference between the two poles in the twin subband is obtained by the following formula: Wherein, Δ[phase(ρ 12 )] zd1 Let f be the phase difference between the two poles in the first sub-band, α1 be the frequency dependence factor of the first scattering center, α2 be the frequency dependence factor of the second scattering center, and f be the frequency dependence factor of the second scattering center. c For the center frequency, ρ 12 Δx is the distance between the two scattering centers. zd1 Δf is the minimum distance difference between the two scattering centers in the first sub-band, c is the speed of light, and Δx is the minimum frequency difference. szd It represents the minimum distance difference between the two scattering centers in the twin sub-band.

9. The method for deriving the resolution improvement limit according to claim 1, characterized in that: The limit for resolution improvement is obtained using the following formula: in, This represents the minimum phase difference between the two poles in the first sub-band. Δx represents the minimum phase difference between the two poles in the bipolar sub-band. zd1 Δx is the minimum distance difference between the two scattering centers in the first sub-band. szd Let f be the minimum distance difference between the two scattering centers in the bi-subband, α1 be the frequency dependence factor of the first scattering center, α2 be the frequency dependence factor of the second scattering center, and f c For the center frequency, ρ 12 Let Δf be the distance between the two scattering centers, and Δf be the minimum frequency difference.

10. A system for deriving the resolution enhancement limit, characterized in that... include: The first module is used to convert the echo signal into an echo signal matrix expression under the geometric diffraction model; The second module is used to obtain the echo vector probability distribution under the preset noise condition of Gaussian complex noise, and to obtain the echo likelihood function based on the echo vector probability distribution. The third module is used to calculate the Fisher information matrix of the scattering center parameters; The fourth module is used to obtain the CRLB of the scattering center parameters based on the Fisher information matrix of the scattering center parameters; The fifth module is used to obtain the minimum distance between two scattering centers based on the pole phase difference between the two scattering centers under the pole model; The sixth module is used to obtain the phase difference between the two poles in the first sub-band and the phase difference between the two poles in the double sub-band based on the minimum distance between the two scattering centers. The seventh module is used to determine the resolution improvement limit based on the minimum phase difference between the two poles in the first subband and the minimum phase difference between the two poles in both subbands.

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