CN107843875A - Bayes's compressed sensing Radar Data Fusion method based on singular value decomposition noise reduction - Google Patents
Bayes's compressed sensing Radar Data Fusion method based on singular value decomposition noise reduction Download PDFInfo
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- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
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Abstract
The invention discloses a kind of Bayes's compressed sensing Radar Data Fusion method based on singular value decomposition noise reduction, step are as follows：The echo data under radar target different frequent points different orientations is obtained using actual measurement or emulation mode, the echo data obtained to emulation, adds different degrees of noise, use to be tested；Using singular value decomposition noise-reduction method, noise reduction process is carried out to obtained radar echo signal；Data fusion is carried out to multiple sparse band radar echo datas after noise reduction process using Bayes's compression sensing method, reconstructs the echo data of Whole frequency band；Using the Whole frequency band radar return data reconstructed through data fusion, inverse synthetic aperture radar imaging is carried out.The present invention carries out noise reduction process before multiband radar data fusion is carried out to input radar return, so as to significantly reduce the relative error of Bayes's compression sensing method reconstruct Whole frequency band radar signal, improves resolution ratio and the degree of accuracy of radar imagery.
Description
Technical Field
The invention belongs to the technical field of high-resolution imaging of radar targets, and particularly relates to a Bayesian compressed sensing multiband radar data fusion high-resolution imaging method based on singular value decomposition and noise reduction.
Background
Because the resolution of a single radar imaging system is constrained by the signal bandwidth and the coherent accumulation angle, a breakthrough is sought from the algorithm, and the realization of the target high-resolution imaging is particularly important. In recent years, a multi-radar data fusion technology is taken as an emerging radar imaging technology, so that the technology has a very high significance in military affairs and has a wide application prospect. Different from a single radar, multiple radars can form different observation networks to observe the target in all directions. The multi-radar data fusion technology integrates radar echo data with different visual angles and different frequency bands, so that the technology of obtaining high-precision target model parameters breaks through the constraint of single radar resolution, and further high-resolution imaging of a target is realized. For the radar multiband data fusion problem in the optical area, the radar data fusion problem can be converted into a signal sparse representation problem by using a geometric diffraction theory model, and when the method is applied to radar imaging processing, target scattering center parameters can be accurately estimated, so that the final imaging quality is greatly improved, and subsequent analysis and processing are facilitated.
However, it should be noted that the traditional bayesian compressed sensing multiband radar data fusion technology only considers noise influence during modeling according to sparse prior information of echoes, so that the algorithm has a certain anti-manufacturing performance during signal reconstruction calculation. However, when the influence of noise is too large, the noise resisting capability of the sparse bayesian algorithm is not enough to completely eliminate the noise, so that the error of the reconstructed full-band radar signal is larger than that of the original full-band signal, and further a fuzzy or false scattering center appears in radar imaging.
Disclosure of Invention
The invention aims to provide a Bayesian compressed sensing radar data fusion method based on singular value decomposition noise reduction, which has very small reconstruction errors for multiband radar data fusion under the condition of small signal-to-noise ratio, thereby providing an important way for radar target high-resolution imaging.
The technical solution for realizing the purpose of the invention is as follows: a Bayesian compressed sensing radar data fusion method based on singular value decomposition noise reduction comprises the following steps:
firstly, acquiring echo data of a radar target under different frequency points and different azimuth angles by using an actual measurement or simulation method, and adding noises with different degrees into the echo data obtained by simulation to be tested and used;
secondly, noise reduction processing is carried out on the obtained radar echo signal by using a singular value decomposition noise reduction method;
thirdly, performing data fusion on the echo data of the sparse band radar subjected to noise reduction processing by using a Bayesian compressed sensing method, and reconstructing echo data of a full band;
and fourthly, performing inverse synthetic aperture radar imaging by using the full-band radar echo data reconstructed by data fusion in the third step.
Compared with the prior art, the invention has the following remarkable advantages: (1) the noise resistance is enhanced: after large noise is added into an echo, the Bayesian compressed sensing multiband radar data fusion method based on singular value decomposition noise reduction has high noise resistance; (2) the echo reconstruction precision is improved, and high-resolution imaging is realized: performing anti-noise processing on each azimuth echo to reduce the final echo reconstruction error so as to realize high-resolution radar imaging of the target; (3) the noise reduction processing is carried out on the input radar echo before the multi-band radar data fusion is carried out, so that the relative error of a Bayes compressed sensing method for reconstructing a full-band radar signal is obviously reduced, and the resolution and the accuracy of radar imaging are improved. The present invention is described in further detail below with reference to the attached drawing figures.
Drawings
Fig. 1 is a model diagram of three radar targets, wherein (a) is a warhead model diagram, (B) is a reentry aircraft model diagram, and (c) is a B2 aircraft model diagram.
Fig. 2 is a comparison graph of signal reconstruction of the warhead structure under different snr environments by the bayesian learning method of the present invention and the conventional bayesian learning method, wherein (a) is a graph of the fused full band data by the conventional bayesian learning method when the snr is 0dB, (b) is a graph of the fused full band data by the method of the present invention when the snr is 0dB, (c) is a graph of the fused full band data by the conventional bayesian learning method when the snr is-3 dB, and (d) is a graph of the fused full band data by the method of the present invention when the snr is-3 dB.
FIG. 3 is a comparison graph of signal reconstruction for a reentry vehicle structure under different SNR environments, where (a) is a graph of fused full-band data using a conventional Bayesian learning method when the SNR is 0dB, (b) is a graph of fused full-band data using the method of the present invention when the SNR is 0dB, (c) is a graph of fused full-band data using a conventional Bayesian learning method when the SNR is-3 dB, and (d) is a graph of fused full-band data using the method of the present invention when the SNR is-3 dB.
Detailed Description
The invention relates to a Bayesian compressed sensing multiband radar data fusion method based on singular value decomposition noise reduction. The method comprises the steps of firstly, realizing noise reduction of radar echo data through a singular value decomposition noise reduction technology, then realizing sparse estimation of a target strong scattering center by utilizing a Bayesian compressed sensing technology, and further obtaining a multiband data fusion result. The method can improve the data fusion anti-noise performance and can realize sparse band radar high-resolution imaging.
The invention discloses a Bayesian compressed sensing radar data fusion method based on singular value decomposition noise reduction, which comprises the following steps of:
firstly, acquiring echo data of a radar target under different frequency points and different azimuth angles by using an actual measurement or simulation method, and adding noises with different degrees into the echo data obtained by simulation to be tested and used;
the echo data of the radar target under different frequency points and different azimuth angles are echo data acquired based on a short pulse radar system, so that a target geometric model can be established, full-band radar echo data under a plurality of azimuth angle angles are acquired by utilizing a rapid simulation method in computational electromagnetism, the selection rule of the plurality of azimuth angle angles is that the azimuth angle is continuously changed from 0 degree to 360 degrees, and the angle intervals of adjacent azimuth angles are equal and are all 0.25-1 degree; simulating radar echo data of a full frequency band at each angle, wherein the frequency interval between adjacent frequency points in the full frequency band is 20 MHz-50 MHz; and adding noise into the obtained radar echo data according to the empirical value of the signal-to-noise ratio in the actual radar echo.
Secondly, performing noise reduction processing on the obtained radar echo signal by using a singular value decomposition noise reduction method, and specifically comprising the following steps:
(2.1) the obtained radar echo data at a certain angle is expressed as an L-dimensional vector Y ═ Y1,y2,...,yLIn which y isiRepresenting the size of an echo electric field corresponding to the ith frequency point;
(2.2) constructing a m × n dimensional Hankel matrix as:
wherein H is a Hankel matrix, m is an embedding dimension, m is less than or equal to n, and m + n-1 is L;
(2.3) carrying out singular value decomposition on the Hankel matrix constructed in the step (2.2):
H＝UΣVT
the system comprises a matrix, a sigma main diagonal element, a sigma matrix, a matrix and a matrix, wherein U is an m multiplied by m dimension orthogonal matrix, V is an n multiplied by n dimension orthogonal matrix, sigma is an n multiplied by n dimension matrix, and the sigma main diagonal element is a matrix singular value and is arranged from large to small;
(2.4) in the diagonal matrix sigma obtained in the step (2.3), the diagonal element is sigma1，σ2，σ3，…，σp，…，...σnIf it satisfies σ1/σp≥10-3If the first p singular values reflect useful signals, the rest singular values reflect noise, and the noise in the signals can be removed by carrying out zero setting operation on the singular values of the reaction noise; and then obtaining a reconstruction matrix by utilizing a singular value decomposition inverse process so as to obtain a signal subjected to noise reduction.
Thirdly, performing data fusion on the multiple noise-reduced sparse band radar echo data by using a Bayesian compressed sensing method, and reconstructing echo data of a full band, wherein the method specifically comprises the following steps:
(3.1) for the full-band signal after noise reduction, taking low-frequency sub-band data and high-frequency sub-band data according to a frequency band loss rate of 60%, and carrying out multi-band radar data fusion;
(3.2) one-dimensional GTD modeling of the target:
wherein f isq＝f0+ Δ f is the Q-th frequency point, Q0, 10Is the initial frequency, delta f represents the frequency modulation interval, K is the number of strong scattering centers, and c is the speed of light; model parameter sigmakIs the scattering coefficient of the kth strong scattering center, rkα, the distance of the kth scattering center relative to a zero-phase reference point in the radar target coordinate system, i.e. the radial distance of the target relative to the radarkIs of scattering center type, is an integral multiple of 1/2, and is [ -1, -1/2,0,1/2,1]；
The one-dimensional GTD model of equation (1) is rewritten as:
wherein,when D is greater than or equal to 300, using the setDispersingFrom this, a dictionary matrix Ψ of size qxm is constructed, M ═ 5D,
therein, ΨaDenotes the frequency-dependent factor, ΨaThe matrix elements of
The full-band broadband radar data Y is approximately expressed as
Y≈Ψσ (5)
Wherein Y ═ Y (f)0),Y(f1),…,Y(fQ-1)]Tσ is an unknown coefficient vector, and σ contains K non-zero elements, corresponding to { σk′|k＝1,…K}；
(3.3) estimating a sparse coefficient by using a Bayes compressed sensing method, realizing full-band data fusion, and obtaining a reconstructed radar echo signal; the sparse problem is expressed by y ═ Φ x + epsilon, and the variance is sigma assuming that the noise obeys the mean value of 02The probability density of the vector y is given by the fact that each element is independent from each other:
based on Bayes compressed sensing theory, supposing that each element in the coefficient vector x obeys 0-mean Gaussian distribution, introducing a hyperparameter for each weight coefficient in the vector x, wherein phi is a dictionary matrix, N is the dimension of the coefficient vector x, and the prior distribution of the coefficient vector x is as follows:
wherein α ═ a1,a2,...,aM)T，aiAs a hyperparameter corresponds to xiTo control each xiThe probability density of x is then:
p(x|y；a,σ2)＝N(x|μ,Σ)， (8)
the mean μ and covariance Σ are as follows:
μ＝σ-2ΣΦTy (9)
Σ＝(A+σ-2ΦTΦ)-1(10)
wherein A ═ diag (a)1,a2,…,aM) I.e. (a)1,a2,…,aM) Forming a diagonal matrix;
from equations (6) to (7), an edge likelihood function p, which is an edge probability density function of y, is obtained as:
wherein, C ═ σ2Ι+ΦA-1ΦT(ii) a By maximum likelihood estimation, the hyper-parameters a and sigma are obtained-2And obtaining an estimation formula of the hyperparameter by using a partial derivative method:
wherein, γi＝1-ai∑ii，∑iiIs the ith diagonal element of the covariance matrix;
the equations (9), (10), (12) and (13) form an iterative solution process, when the iterative convergence criterion is satisfied, x is approximately expressed by mu, so that y is approximately equal to phi mu; therefore, firstly, the data of a plurality of sparse frequency bands are used for estimating the sparse scattering center, and then the full-frequency band sparse dictionary is used for obtaining the reconstructed full-frequency band echo signal.
And fourthly, performing inverse synthetic aperture radar imaging by using the full-band radar echo data reconstructed by data fusion in the third step, and comparing the imaging with the original full-band echo signal.
In order to verify the effectiveness of the method, simulation experiments are compared by using the traditional Bayesian compressed sensing multiband data fusion technology and the Bayesian compressed sensing multiband data fusion technology based on singular value decomposition and noise resistance provided by the invention in combination with different targets. Three radar targets, a mesostructure warhead, an aircraft, and B2, were first modeled separately using commercially available software ANSYS, as shown in fig. 1. And then, calculating a single-station scattered field of the target in a full frequency band by using a physical optical method, wherein the sweep intervals are respectively 25MHz, 50MHz and 20MHz, the pitch angle is generally 5 degrees, the azimuth angle is changed from 0 degree to 360 degrees, and the interval is 1 degree. In the following experiment, the echo at a certain specific angle is used as reference data, the band loss rate is 60%, the low-frequency sub-band and the high-frequency sub-band are used for data fusion to obtain a reconstructed signal, and relative reconstruction errors obtained by two different methods are compared.
Fig. 2 shows the comparison of the signal reconstruction of the warhead structure under different signal-to-noise ratios by the bayesian learning method. Fig. 2(a) shows the signal reconstruction of the warhead structure by the conventional method when the signal-to-noise ratio is 0dB, fig. 2(b) shows the signal reconstruction of the warhead structure by the method of the present invention when the signal-to-noise ratio is 0dB, fig. 2(c) shows the signal reconstruction of the warhead structure by the conventional method when the signal-to-noise ratio is-3 dB, and fig. 2(d) shows the signal reconstruction of the warhead structure by the method of the present invention when the signal-to-noise ratio is-3 dB. FIG. 3 shows the comparison of the signal reconstruction of the reentry vehicle structure under different SNR environments according to the present invention and the traditional Bayesian learning method. FIG. 3(a) is a signal reconstruction of a reentrant aircraft structure by a conventional method with a signal-to-noise ratio of 0dB, FIG. 3(b) is a signal reconstruction of a reentrant aircraft structure by a method according to the invention with a signal-to-noise ratio of 0dB, FIG. 3(c) is a signal reconstruction of a reentrant aircraft structure by a conventional method with a signal-to-noise ratio of-3 dB, and FIG. 3(d) is a signal reconstruction of a reentrant aircraft structure by a method according to the invention with a signal-to-noise ratio of-3 dB.
TABLE 1B2 reconstruction error comparison of targets
Table 1 shows the comparison of the signal reconstruction errors of the B2 aircraft structure under different signal-to-noise ratios according to the present invention and the traditional bayesian learning method. Through observation, the relative reconstruction errors of the method are much smaller than those of the traditional Bayes compressed sensing method when the frequency band loss rate of the three radar target models is 60% (the low-frequency sub-band is 12GHz-13.2GHz, the high-frequency sub-band is 16.8GHz-18GHz) and the signal-to-noise ratio is 0dB or-3 dB, which proves that the radar imaging resolution and the anti-noise capability can be obviously improved.
Claims (4)
1. A Bayesian compressed sensing radar data fusion method based on singular value decomposition noise reduction is characterized by comprising the following steps:
firstly, acquiring echo data of a radar target under different frequency points and different azimuth angles by using an actual measurement or simulation method, and adding noises with different degrees into the echo data obtained by simulation to be tested and used;
secondly, noise reduction processing is carried out on the obtained radar echo signal by using a singular value decomposition noise reduction method;
thirdly, performing data fusion on the echo data of the sparse band radar subjected to noise reduction processing by using a Bayesian compressed sensing method, and reconstructing echo data of a full band;
and fourthly, performing inverse synthetic aperture radar imaging by using the full-band radar echo data reconstructed by data fusion in the third step.
2. The Bayesian compressed sensing radar data fusion method based on singular value decomposition noise reduction according to claim 1, characterized in that: in the first step, the echo data of the radar target under different frequency points and different azimuth angles are echo data acquired based on a short pulse radar system, the azimuth angles are continuously changed from 0 degree to 360 degrees according to a rule of selecting a plurality of azimuth angles, and the equal angle intervals of adjacent azimuth angles are 0.25-1 degree; simulating radar echo data of a full frequency band at each angle, wherein the frequency interval between adjacent frequency points in the full frequency band is 20 MHz-50 MHz; and adding noise into the obtained radar echo data according to the empirical value of the signal-to-noise ratio in the actual radar echo.
3. The Bayesian compressed sensing radar data fusion method based on singular value decomposition noise reduction according to claim 1, characterized in that: in the second step, the method for decomposing and denoising by using singular values comprises the following specific steps:
(2.1) the obtained radar echo data at a certain angle is expressed as an L-dimensional vector Y ═ Y1,y2,...,yLIn which y isiRepresenting the size of an echo electric field corresponding to the ith frequency point;
(2.2) constructing a m × n dimensional Hankel matrix as:
<mrow> <msub> <mi>H</mi> <mrow> <mi>m</mi> <mo>&times;</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>y</mi> <mn>1</mn> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mn>2</mn> </msub> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <msub> <mi>y</mi> <mi>n</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mn>3</mn> </msub> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <msub> <mi>y</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mi>m</mi> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <msub> <mi>y</mi> <mi>L</mi> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow>
wherein H is a Hankel matrix, m is an embedding dimension, m is less than or equal to n, and m + n-1 is L;
(2.3) carrying out singular value decomposition on the Hankel matrix constructed in the step (2.2):
H＝U∑VT
the system comprises a matrix, a sigma main diagonal element, a sigma matrix, a matrix and a matrix, wherein U is an m multiplied by m dimension orthogonal matrix, V is an n multiplied by n dimension orthogonal matrix, sigma is an n multiplied by n dimension matrix, and the sigma main diagonal element is a matrix singular value and is arranged from large to small;
(2.4) in the diagonal matrix sigma obtained in the step (2.3), the diagonal element is sigma1，σ2，σ3，…，σp，…，...σnIf it satisfies σ1/σp≥10-3If the first p singular values reflect useful signals, the rest singular values reflect noise, and the noise in the signals can be removed by carrying out zero setting operation on the singular values of the reaction noise; and then obtaining a reconstruction matrix by utilizing a singular value decomposition inverse process so as to obtain a signal subjected to noise reduction.
4. The Bayesian compressed sensing radar data fusion method based on singular value decomposition noise reduction according to claim 1, characterized in that: thirdly, performing data fusion on the multiple noise-reduced sparse band radar echo data by using a Bayesian compressed sensing method to reconstruct full-band echo data, and specifically comprising the following steps:
(3.1) for the full-band signal after noise reduction, taking low-frequency sub-band data and high-frequency sub-band data according to a frequency band loss rate of 60%, and carrying out multi-band radar data fusion;
(3.2) one-dimensional GTD modeling of the target:
<mrow> <mi>Y</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>q</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <msub> <mi>&sigma;</mi> <mi>k</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>j</mi> <mfrac> <msub> <mi>f</mi> <mi>q</mi> </msub> <msub> <mi>f</mi> <mn>0</mn> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>&alpha;</mi> <mi>k</mi> </msub> </msup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>4</mn> <mi>&pi;</mi> </mrow> <mi>c</mi> </mfrac> <msub> <mi>f</mi> <mi>q</mi> </msub> <msub> <mi>r</mi> <mi>k</mi> </msub> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
wherein f isq＝f0+ Δ f is the Q-th frequency point, Q0, 10Is the initial frequency, delta f represents the frequency modulation interval, K is the number of strong scattering centers, and c is the speed of light; model parameter sigmakIs the scattering coefficient of the kth strong scattering center, rkα, the distance of the kth scattering center relative to a zero-phase reference point in the radar target coordinate system, i.e. the radial distance of the target relative to the radarkIs of scattering center type, is an integral multiple of 1/2, and is [ -1, -1/2,0,1/2,1]；
The one-dimensional GTD model of equation (1) is rewritten as:
<mrow> <mi>Y</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>q</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <msubsup> <mi>&sigma;</mi> <mi>k</mi> <mo>&prime;</mo> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>j</mi> <mfrac> <msub> <mi>f</mi> <mi>q</mi> </msub> <msub> <mi>f</mi> <mn>0</mn> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>&alpha;</mi> <mi>k</mi> </msub> </msup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>4</mn> <mi>&pi;</mi> <mfrac> <mrow> <msub> <mi>&Delta;fr</mi> <mi>k</mi> </msub> </mrow> <mi>c</mi> </mfrac> <mi>q</mi> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
wherein,when D is greater than or equal to 300, using the setDispersingFrom this, a dictionary matrix Ψ of size qxm is constructed, M ═ 5D,
<mrow> <mi>&Psi;</mi> <mo>=</mo> <mo>&lsqb;</mo> <msup> <mi>&Psi;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>&Psi;</mi> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo>,</mo> <msup> <mi>&Psi;</mi> <mn>0</mn> </msup> <mo>,</mo> <msup> <mi>&Psi;</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> <mo>,</mo> <msup> <mi>&Psi;</mi> <mn>1</mn> </msup> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
therein, ΨaDenotes the frequency-dependent factor, ΨaThe matrix elements of
<mrow> <msubsup> <mi>&psi;</mi> <mrow> <mi>q</mi> <mi>d</mi> </mrow> <mi>a</mi> </msubsup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>j</mi> <mfrac> <msub> <mi>f</mi> <mi>q</mi> </msub> <msub> <mi>f</mi> <mn>0</mn> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>&alpha;</mi> </msup> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <mi>e</mi> <mi>&pi;</mi> <mfrac> <mi>d</mi> <mi>D</mi> </mfrac> <mi>q</mi> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
The full-band broadband radar data Y is approximately expressed as
Y≈Ψσ (5)
Wherein Y ═ Y (f)0),Y(f1),…,Y(fQ-1)]Tσ is an unknown coefficient vector, and has K non-zero elementsPlain, corresponding to { σ'k|k＝1,…K}；
(3.3) estimating a sparse coefficient by using a Bayes compressed sensing method, realizing full-band data fusion, and obtaining a reconstructed radar echo signal; the sparse problem is expressed by y ═ Φ x + epsilon, and the variance is sigma assuming that the noise obeys the mean value of 02The probability density of the vector y is given by the fact that each element is independent from each other:
<mrow> <mi>p</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>|</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mn>2</mn> <msup> <mi>&pi;&sigma;</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>|</mo> <mo>|</mo> <mi>y</mi> <mo>-</mo> <mi>&Phi;</mi> <mi>x</mi> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
based on Bayes compressed sensing theory, supposing that each element in the coefficient vector x obeys 0-mean Gaussian distribution, introducing a hyperparameter for each weight coefficient in the vector x, wherein phi is a dictionary matrix, N is the dimension of the coefficient vector x, and the prior distribution of the coefficient vector x is as follows:
<mrow> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>|</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <mi>N</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>|</mo> <mn>0</mn> <mo>,</mo> <msubsup> <mi>a</mi> <mi>i</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
wherein α ═ a1,a2,...,aM)T，aiAs a hyperparameter corresponds to xiTo control each xiThe probability density of x is then:
p(x|y；a,σ2)＝N(x|μ,Σ)， (8)
the mean μ and covariance Σ are as follows:
μ＝σ-2ΣΦTy (9)
Σ＝(A+σ-2ΦTΦ)-1(10)
wherein A ═ diag (a)1,a2,…,aM) I.e. (a)1,a2,…,aM) Forming a diagonal matrix;
from equations (6) to (7), an edge likelihood function p, which is an edge probability density function of y, is obtained as:
<mrow> <mi>p</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>|</mo> <mi>a</mi> <mo>,</mo> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>&pi;</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mi>N</mi> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>|</mo> <mi>C</mi> <msup> <mo>|</mo> <mrow> <mo>-</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>y</mi> <mi>T</mi> </msup> <msup> <mi>C</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>y</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
wherein, C ═ σ2Ι+ΦA-1ΦT(ii) a By maximum likelihood estimation, the hyper-parameters a and sigma are obtained-2And obtaining an estimation formula of the hyperparameter by using a partial derivative method:
<mrow> <msubsup> <mi>a</mi> <mi>i</mi> <mrow> <mi>n</mi> <mi>e</mi> <mi>w</mi> </mrow> </msubsup> <mo>=</mo> <mfrac> <msub> <mi>&gamma;</mi> <mi>i</mi> </msub> <msubsup> <mi>&mu;</mi> <mi>i</mi> <mn>2</mn> </msubsup> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msup> <mrow> <mo>(</mo> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mi>e</mi> <mi>w</mi> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mo>|</mo> <mo>|</mo> <mi>y</mi> <mo>-</mo> <mi>&Phi;</mi> <mi>&mu;</mi> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <msub> <mi>&gamma;</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
wherein, γi＝1-ai∑ii，∑iiIs the ith diagonal element of the covariance matrix;
the equations (9), (10), (12) and (13) form an iterative solution process, when the iterative convergence criterion is satisfied, x is approximately expressed by mu, so that y is approximately equal to phi mu; therefore, firstly, the data of a plurality of sparse frequency bands are used for estimating the sparse scattering center, and then the full-frequency band sparse dictionary is used for obtaining the reconstructed full-frequency band echo signal.
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