A frequency diversity MIMO radar parameter estimation method based on matrix beam principle

Abstract

The present invention discloses a frequency diversity MIMO radar parameter estimation method based on the matrix beam principle, the method comprising: firstly, the receiving signal model of the frequency diversity MIMO radar is derived and the coupling relationship between the angle and the distance in the transmit-receive steering vector is analyzed; secondly, the augmented matrix beam is constructed using the echo signal obtained after matched filtering at the receiving end; then, the augmented matrix beam is subjected to matrix decomposition and transformation to obtain the diagonal phase matrix containing the angle and distance information respectively. Finally, considering the coupling and matching problem of the angle and distance parameters of the target, the same characteristic vector is used to achieve matching and realize accurate positioning of the target. Simulation results show that compared with the prior art, the present invention exhibits superior estimation performance under low snapshot number and low signal-to-noise ratio.

Description

Technical Field

[0001] The invention belongs to the technical field of radar signal processing, and in particular relates to a frequency diversity MIMO radar parameter estimation method based on the matrix beam principle. Background Art

[0002] Frequency diversity array is a new array radar technology developed in the field of radar signal processing in recent years. Frequency diversity array uses a smaller frequency offset in different transmitting array elements to obtain a distance-dependent beam. Since its beam pattern is related to time, angle, distance and frequency offset, it has become the focus of many researchers. The combination of frequency diversity radar and MIMO (Multiple Input Multiple Output) radar, namely frequency diversity MIMO (FDA-MIMO) radar, shows significant advantages in parameter estimation and beam synthesis. Since the single-base frequency diversity MIMO radar has the advantages of flexibility and convenient joint control of transmission and reception, it can be applied to unmanned aerial vehicles, unmanned ships, automobiles and other fields. For the parameter estimation of single-base frequency diversity MIMO radar, since the transmitting array adopts frequency diversity technology, a distance-angle-dependent beam can be obtained, but the problem of distance and angle coupling also arises. In addition, the current parameter estimation problems of frequency diversity MIMO radar are mainly concentrated on subspace algorithms. For example, in the literature (Li B, Bai W, Zheng G. Successful ESPRIT algorithm for joint DOA-range-polarization estimation with polarization sensitive FDA-MIMO radar[J]. IEEE Access, 2018, 6: 36376-36382.), the transmitting array is divided into two sub-arrays, and the rotation invariant technology (ESPRIT) is used to realize the two-dimensional joint estimation of angle and distance.

[0003] However, since the algorithm is based on the covariance matrix of complex numbers, it does not fully utilize the mutual correlation information in the covariance matrix, and the complexity required by the algorithm is relatively high. In addition, under low signal-to-noise ratio and low snapshot conditions, the parameter estimation accuracy of this type of EPSPRIT algorithm is greatly reduced or even fails.

[0004] In addition, a real-valued ESPRIT algorithm is provided in the literature (Liu Feilong, Wang Xianpeng, Huang Mengxing, et al. A NovelUnitary ESPRIT Algorithm for Monostatic FDA-MIMO Radar [J]. Sensors, 2020, 20(3): 827-843) to realize the joint estimation of angle and distance of frequency diversity MIMO radar.

[0005] However, since this algorithm is also based on the decomposition of the covariance matrix of the received data, and the algorithm is prone to failure in low snapshot and low signal-to-noise ratio scenarios, the above method will be greatly limited in practical applications. Summary of the invention

[0006] In order to solve the above problems existing in the prior art, the present invention provides a frequency diversity MIMO radar parameter estimation method based on the matrix beam principle. The technical problem to be solved by the present invention is achieved by the following technical solutions:

[0007] A frequency diversity MIMO radar parameter estimation method based on the matrix beam principle, the frequency diversity MIMO radar parameter estimation method comprising:

[0008] Step 1, M transmitting array elements transmit orthogonal signals;

[0009] Step 2, the N receiving array elements perform matched filtering on the reflected echo signal of the target using the mutual orthogonality of the orthogonal signals to obtain a filtered receiving signal;

[0010] Step 3, obtaining an augmented matrix beam according to the filtered received signal;

[0011] Step 4: performing singular value decomposition on the augmented matrix bundle to decompose and obtain left singular vectors;

[0012] Step 5: Process the left singular vector using the least square method to obtain a phase matrix including an angle;

[0013] Step 6: Estimate the angle of the target according to the phase matrix to obtain an angle estimation result;

[0014] Step 7: Perform row permutation on the left singular vector, and use the same fuzzy vector and least square method to obtain a matching distance phase matrix;

[0015] Step 8: Estimate the distance of the target according to the distance phase matrix to obtain a distance estimation result.

[0016] In one embodiment of the present invention, the orthogonal signal is:

[0017]

[0018] Among them, s m (t) is the orthogonal signal of the mth transmitting element at the tth moment, E is the total power of the signals transmitted by the M transmitting elements, f m is the carrier frequency of the mth transmitting array element, φ m (t) is the baseband signal of the mth transmitting array element, and T is the pulse duration.

[0019] In one embodiment of the present invention, the baseband signals of the M transmitting array elements satisfy the following:

[0020]

[0021] Where τ is the delay, is the baseband signal of the kth transmitting array element, and Δf is the frequency increment between adjacent transmitting antennas.

[0022] In one embodiment of the present invention, for the orthogonal signal sent by the mth transmitting array element, the received signal of the nth receiving array element is:

[0023]

[0024] Where ξ is the complex scattering coefficient of the target, r is the distance, c is the speed of light, f0 is the emission frequency of the first transmitting element, and d t is the spacing between the transmitting array elements, d r is the spacing between receiving array elements, θ is the azimuth of the target, and Δf is the frequency increment between adjacent transmitting antennas;

[0025] All the received signals are arranged in vector form as follows:

[0026]

[0027] Among them, x mn is the received signal sent by the mth transmitting array element and received by the nth receiving array element, N is the noise vector, and and, is a NM×1 dimensional complex vector, is the Kronecker product,

[0028] a t (θ, r) is expressed as:

[0029]

[0030] a r (θ) is expressed as:

[0031]

[0032] Among them, a t (θ, r) is the launch guidance vector, a r (θ) is the receiving steering vector.

[0033] In one embodiment of the present invention, the augmented matrix bundle is:

[0034]

[0035]

[0036] Among them, R e is the augmented matrix beam, P is the number of equivalent transmitting array elements, R m For R e The mth element in R m is a Q×(N-Q+1) Hankel matrix, where Q is the number of equivalent receiving array elements.

[0037] In one embodiment of the present invention, R e The formula for singular value decomposition is expressed as:

[0038]

[0039] Among them, E S is the signal subspace, E N is the noise subspace, E S and E N is the left singular vector, Λ s and Λ N Corresponding to R e The first K larger and remaining singular values ​​of , where K is the target number.

[0040] In one embodiment of the present invention, the phase matrix is:

[0041]

[0042] Among them, E S1 For E S The matrix consisting of the first (Q-1)P rows, E S2 For E S The matrix consisting of (Q-1)P rows in the middle and back is In order to seek false rebellion.

[0043] In one embodiment of the present invention, the calculation formula for the angle estimation of the target is:

[0044]

[0045] Among them, Φ θis a diagonal matrix whose diagonal elements contain the target angle information, Ψ θ =T -1 Φ θ T, T stands for E S The coefficient matrix between and the steering vector A.

[0046] In one embodiment of the present invention, the distance phase matrix is:

[0047]

[0048]

[0049] Among them, U S1 For U S The matrix consisting of the first (P-1)Q rows, U S2 For U S The matrix consisting of (P-1)Q rows in the middle and back is To find the pseudo-inverse, T represents E S and the coefficient matrix between the steering vector A, Δf is the frequency increment between adjacent transmitting antennas.

[0050] Beneficial effects of the present invention:

[0051] The present invention provides a frequency diversity MIMO radar parameter estimation method based on the matrix beam principle, which can realize low-complexity and high-precision joint estimation of target parameters when the sampling data is small. By performing matrix decomposition on the constructed augmented matrix beam, the angle estimation of the target can be obtained. The obtained angle estimation is then used to solve the problem of angle and distance coupling in the guidance vector, and the same column fuzzy vector is used to achieve the correct pairing of angle and distance parameters. Since the augmented matrix beam essentially makes full use of the cross-correlation characteristics between data to achieve the purpose of expanding the number of snapshots, the method proposed by the present invention can show superior positioning performance compared to the prior art under low signal-to-noise ratio and small number of snapshots.

[0052] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is a flow chart of a frequency diversity MIMO radar parameter estimation method based on the matrix beam principle provided by an embodiment of the present invention;

[0054] Figure 2 It is a signal model diagram of a single-base frequency diversity MIMO radar system provided by an embodiment of the present invention;

[0055] Figure 3 is an estimation result diagram in a far-field three-target scenario provided by an embodiment of the present invention;

[0056] Figure 4 is a trend diagram of a target angle estimation root mean square error changing with a signal-to-noise ratio provided by an embodiment of the present invention;

[0057] Figure 5 is a trend diagram of a target distance estimation root mean square error changing with a signal-to-noise ratio provided by an embodiment of the present invention;

[0058] Figure 6 is a trend diagram of a target angle estimation root mean square error changing with the number of snapshots provided by an embodiment of the present invention;

[0059] Figure 7 This is a trend diagram of a target distance estimation root mean square error changing with the number of snapshots provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0060] The present invention is further described in detail below with reference to specific embodiments, but the embodiments of the present invention are not limited thereto.

[0061] Embodiment 1

[0062] See also Figure 1 , Figure 1 1 is a flow chart of a frequency diversity MIMO radar parameter estimation method based on the matrix beam principle provided by an embodiment of the present invention. An embodiment of the present invention provides a frequency diversity MIMO radar parameter estimation method based on the matrix beam principle, and the frequency diversity MIMO radar parameter estimation method includes steps 1 to 8, wherein:

[0063] Step 1: M transmitting array elements transmit orthogonal signals.

[0064] In the embodiment of the present invention, a single-base signal model of a frequency diversity MIMO radar is considered, such as Figure 2 As shown, the transmitting end uses M transmitting array elements, the array element arrangement adopts a uniform linear array structure, and the array element spacing is d t The receiving end uses N receiving array elements, and the array element arrangement also adopts a uniform linear array structure, with an array element spacing of d r The first array element at the transmitting end is set as the reference array element, and its transmitting frequency is f0. Then the carrier frequency of the mth transmitting array element of the frequency diversity MIMO radar can be expressed as:

[0065] f m =f0+mΔf,m=0,1,…,M-1

[0066] Where Δf is the frequency increment between adjacent transmitting antennas. Usually, this frequency increment is much smaller than the reference carrier frequency. Therefore, the following approximate expression is available:

[0067]

[0068] Where λ is the signal wavelength, c is the speed of light, θ is the azimuth of the target, and r is the distance. In order to eliminate the phase aliasing effect, the array element spacing is set to half of the maximum wavelength, that is, d t =d r =0.5λ max ,λ max is the maximum wavelength. At the tth moment, the transmission signal of the mth transmitting array element can be expressed as:

[0069]

[0070] Among them, s m (t) is the orthogonal signal of the mth transmitting element at the tth moment, E is the total power of the signals transmitted by the M transmitting elements, f m is the carrier frequency of the mth transmitting array element, φ m (t) is the baseband signal of the mth transmitting array element, and T is the pulse duration.

[0071] In the present invention, it is assumed that the transmitted waveforms are mutually orthogonal and normalized, which can be expressed as:

[0072]

[0073] Where τ is the delay, is the baseband signal of the kth transmitting array element, and Δf(mk) is the frequency increment between the kth transmitting array element and the mth transmitting array element.

[0074] Step 2: The N receiving array elements use the mutually orthogonal characteristics of the orthogonal signals to perform matched filtering on the reflected echo signals of the target to obtain filtered received data.

[0075] In this embodiment, the first transmitting array element and the receiving array element are respectively regarded as reference array elements. In the present embodiment, the target considered in the far field is assumed to be a point target. The output of each receiving array element is the superposition of all the transmitting signals at the array element. For the orthogonal signal emitted by the mth transmitting array element, the receiving signal of the nth receiving array element can be expressed as:

[0076]

[0077] Where ξ is the complex scattering coefficient of the target, d t and d r They represent the spacing between the transmitting array element and the receiving array element, θ is the azimuth of the target, and c is the speed of light.

[0078] The data after receiving array pulse compression is arranged in vector form and can be expressed as:

[0079]

[0080] Among them, x mn is the received signal sent by the mth transmitting array element and received by the nth receiving array element, N is the noise vector, and and, represents a NM×1-dimensional complex vector, represents the Kronecker product, a t The specific form of (θ, r) can be expressed as:

[0081]

[0082] Among them, a t (θ, r) is the launch steering vector.

[0083] a r The specific form of (θ) can be expressed as:

[0084]

[0085] Among them, a r (θ) is the receiving steering vector.

[0086] Therefore, it can be seen from the calculation formula of the transmission steering vector that for the transmission steering vector, the angle and distance parameters are naturally coupled, so a corresponding decoupling method must be used when implementing parameter estimation. In the present invention, since the angle estimation parameter can be obtained first, the decoupling can be achieved by substituting the obtained angle estimation into the corresponding received data.

[0087] Step 3: Obtain an augmented matrix beam according to the filtered received signal.

[0088] According to the vector form of the received data after filtering, it can be defined as follows:

[0089]

[0090] Among them, E[·] is the statistical expectation, x m,n is the ((n-1)M+m)th row of x. The constructed augmented matrix bundle can be expressed as:

[0091]

[0092]

[0093] Among them, R e is the augmented matrix beam, P is the number of new equivalent transmitting array elements after adopting the matrix beam principle, R m For R e The mth element in R mis a Q×(N-Q+1) Hankel matrix, where Q is the number of new equivalent receiving array elements after adopting the matrix beam principle. The selection of P and Q is to construct an augmented matrix, which is equivalent to increasing the number of snapshots. m Perform the following matrix decomposition:

[0094] R m =V L SX θ V R

[0095]

[0096]

[0097] Among them, S represents the signal correlation matrix, X θ =exp(jπsinθ).

[0098] Step 4: Perform singular value decomposition on the augmented matrix bundle to decompose and obtain left singular vectors.

[0099] R e Performing singular value decomposition, we can get:

[0100]

[0101] Among them, E S and E N denote the signal subspace and the noise subspace respectively, Λ s and Λ N Corresponding to R e The first K larger and remaining singular values ​​of , where K is the target number.

[0102] Step 5: Use the least squares method to process the left singular vectors and obtain the phase matrix containing the angle.

[0103] Due to R m and R e Therefore, E S The matrix E consisting of the first (Q-1)P rows of S1 and the matrix E formed by the following (Q-1)P rows S2 The following relationship exists:

[0104]

[0105] Among them, θ is the phase matrix containing the angle, and is Ψ θ =T -1 Φ θ T, T stands for E S The coefficient matrix between the steering vector A, Φ θThe phase of the diagonal elements represents a diagonal matrix containing the target angle information.

[0106] Step 6: Estimate the angle of the target according to the phase matrix to obtain an angle estimation result.

[0107] Specifically, the estimation of the target angle can be expressed as:

[0108]

[0109] in, is the estimated target angle.

[0110] Step 7: Perform row permutation on the left singular vectors and use the same fuzzy vector and least square method to obtain the matching distance phase matrix.

[0111] Specifically, E S Multiply the permutation matrix π on the left to reconstruct and get U S .U S The matrix U consisting of the first (P-1)Q rows of S1 and the matrix U consisting of the (P-1)Q rows after S2 The following relationship exists:

[0112]

[0113] Since the phase matrices of angle and distance have the same column ambiguity vector T, the distance and angle pairing of the target can be automatically achieved in the present invention. The estimated distance of the target is:

[0114]

[0115] Where T represents E S The coefficient matrix between and the steering vector A.

[0116] Step 8: Estimate the distance of the target according to the distance phase matrix to obtain a distance estimation result.

[0117] Specifically, the estimated angle parameters are substituted into the range phase matrix to achieve distance estimation.

[0118] The effect of the present invention is further described below in conjunction with simulation experiments.

[0119] To evaluate the performance of this method, consider Figure 1 The single-base frequency diversity MIMO radar parameter estimation method shown in FIG. 1 has system simulation parameters as shown in Table 1.

[0120] Table 1 Frequency diversity MIMO radar system simulation parameters

[0121]

[0122]

[0123] In addition, in order to highlight the estimation performance advantage of the present invention, the ESPRIT algorithm applicable to single-base frequency diversity MIMO radar and the real-valued EPSPRIT algorithm are selected as comparison algorithms in the simulation.

[0124] Figure 3 This is a target positioning diagram of the present invention when there are three point targets in the far field. Figure 3 In the simulation experiment, the signal-to-noise ratio SNR = 0dB, the number of Monte Carlo experiments is 200, and the noise is uniform complex Gaussian white noise. Figure 3 It can be seen from the simulation experiment results that, compared with the parameter information of the real targets, the distance and angle parameters of the three targets can be accurately estimated and correctly matched by the present invention.

[0125] Figure 4 and Figure 5 The following are the comparison diagrams of the root mean square error of angle and distance of the present invention and the comparison algorithm as the signal-to-noise ratio changes. In this part of the experiment, the number of snapshots L = 20, and the signal-to-noise ratio SNR increases from -10dB to 10dB in steps of 2dB. Figure 4 The results show that as the signal-to-noise ratio increases, the angle parameter estimation performance of the present invention is higher than that of the comparison algorithm, and at low signal-to-noise ratios, the parameter estimation performance of the present invention and the real-valued ESPRIT algorithm is higher, while the performance of the ESPRIT algorithm is severely reduced. Figure 5 It can be seen from the results that, under the same signal-to-noise ratio, the root mean square error curve of the distance parameter of the present invention is closer to the ideal Cramer-Rao bound than the root mean square error curve of the comparison algorithm, so the present invention has a more superior estimation performance.

[0126] Figure 6 and Figure 7 The following are comparison diagrams of the root mean square error of angle and distance of the present invention and the comparison algorithm as the number of snapshots changes. Figure 6 and Figure 7 In the simulation experiment, the signal-to-noise ratio SNR = 0 dB. In order to explore the estimation performance of the present invention under low snapshot number, the snapshot number L is increased from 1 to 41 in steps of 2. Figure 6 It can be seen that under low snapshot conditions, the parameter estimation of the comparison algorithm has failed, while the angle estimation of the present invention is still effective, which proves the effectiveness of the present invention in angle parameter estimation. Figure 7 It can be seen from the results that, under low snapshot conditions, the comparison algorithm can no longer estimate the distance parameter, while the present invention can still accurately obtain the estimated value of the distance parameter.

[0127] The present invention provides a frequency diversity MIMO radar parameter estimation method based on the matrix beam principle, which can realize low-complexity and high-precision joint estimation of target parameters when the sampling data is small. By performing matrix decomposition on the constructed augmented matrix beam, the angle estimation of the target can be obtained. The obtained angle estimation is then used to solve the problem of angle and distance coupling in the guidance vector, and the same column fuzzy vector is used to achieve the correct pairing of angle and distance parameters. Since the augmented matrix beam essentially makes full use of the cross-correlation characteristics between data to achieve the purpose of expanding the number of snapshots, the method proposed by the present invention can show superior positioning performance compared to the prior art under low signal-to-noise ratio and small number of snapshots.

[0128] A person of ordinary skill in the art can understand that all or part of the steps of implementing the above-mentioned method embodiment can be completed by hardware related to program instructions, and the aforementioned program can be stored in a computer-readable storage medium. When the program is executed, it executes the steps of the above-mentioned method embodiment; and the aforementioned storage medium includes: ROM, RAM, magnetic disk or optical disk, etc., various media that can store program codes.

[0129] In the description of this specification, the description with reference to the terms "one embodiment", "some embodiments", "example", "specific example", or "some examples" etc. means that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described may be combined in any one or more embodiments or examples in a suitable manner. In addition, those skilled in the art may combine and combine different embodiments or examples described in this specification.

[0130] Although the present application is described herein in conjunction with various embodiments, in the process of implementing the claimed application, those skilled in the art may understand and implement other variations of the disclosed embodiments by viewing the drawings, the disclosure, and the appended claims. In the claims, the word "comprising" does not exclude other components or steps, and "one" or "an" does not exclude multiple situations. A single processor or other unit may implement several functions listed in a claim. Certain measures are recorded in different dependent claims, but this does not mean that these measures cannot be combined to produce good results.

[0131] The above contents are further detailed descriptions of the present invention in combination with specific preferred embodiments, and it cannot be determined that the specific implementation of the present invention is limited to these descriptions. For ordinary technicians in the technical field to which the present invention belongs, several simple deductions or substitutions can be made without departing from the concept of the present invention, which should be regarded as falling within the protection scope of the present invention.

Claims

1. A frequency diversity MIMO radar parameter estimation method based on matrix beam principle, characterized in that: The frequency diversity MIMO radar parameter estimation method comprises: Step 1, M transmitting array elements transmit orthogonal signals; Step 2, the N receiving array elements perform matched filtering on the reflected echo signal of the target using the mutual orthogonality of the orthogonal signals to obtain a filtered receiving signal; Step 3, obtaining an augmented matrix beam according to the filtered received signal; Step 4: Perform singular value decomposition on the augmented matrix bundle to decompose and obtain left singular vectors, which are represented by E S ; Step 5: Process the left singular vector using the least square method to obtain a phase matrix containing angles; the phase matrix is: in, For E S The matrix consisting of the first (Q-1)P rows, E S2 For E S The matrix consists of (Q-1)P rows, where P is the number of equivalent transmitting array elements and Q is the number of equivalent receiving array elements. It is the pseudo-inverse operation of the matrix; Step 6: Estimate the angle of the target according to the phase matrix to obtain an angle estimation result; the calculation formula for the angle estimation of the target is: Among them, Φ θ is a diagonal matrix whose diagonal elements contain the target angle information, ψ θ =T -1 Φ θ T,ψ θ is the phase matrix containing the angle, T represents E S and the column fuzzy vector between the steering vector A; Step 7: Perform row permutation on the left singular vector, and use the same column fuzzy vector and least square method to obtain a matching distance phase matrix; the distance phase matrix is: Among them, U S1 For U S The matrix consisting of the first (P-1) Q rows, U S2 For U S The matrix composed of (P-1)Q rows in the middle and back is To obtain the pseudo-inverse, Δf is the frequency increment between adjacent transmitting antennas; Step 8: Estimate the distance of the target according to the distance phase matrix to obtain a distance estimation result.

2. The frequency diversity MIMO radar parameter estimation method according to claim 1, characterized in that: The orthogonal signals are: Among them, s m (t) is the orthogonal signal of the mth transmitting element at the tth moment, E is the total power of the signals transmitted by the M transmitting elements, f m is the carrier frequency of the mth transmitting array element, φ m (t) is the baseband signal of the mth transmitting array element, and T is the pulse duration.

3. The frequency diversity MIMO radar parameter estimation method according to claim 2, characterized in that: The baseband signals of the M transmitting array elements satisfy: Where τ is the delay, is the baseband signal of the kth transmitting array element, and Δf is the frequency increment between adjacent transmitting antennas.

4. The frequency diversity MIMO radar parameter estimation method according to claim 1, characterized in that: For the orthogonal signal sent by the mth transmitting array element, the received signal of the nth receiving array element is: Where ξ is the complex scattering coefficient of the target, r is the distance, c is the speed of light, f0 is the emission frequency of the first transmitting element, and d t is the spacing between the transmitting array elements, d r is the spacing between receiving array elements, θ is the azimuth of the target, and Δf is the frequency increment between adjacent transmitting antennas; All the received signals are arranged in vector form as follows: Among them, x mn is the received signal sent by the mth transmitting array element and received by the nth receiving array element, N is the noise vector, and and, is a NM×1 dimensional complex vector, is the Kronecker product, a t (θ, r) is expressed as: a r (θ) is expressed as: Among them, a t (θ, r) is the launch guidance vector, a r (θ) is the receiving steering vector.

5. The frequency diversity MIMO radar parameter estimation method according to claim 1, characterized in that: The augmented matrix bundle is: Among them, R e is the augmented matrix beam, P is the number of equivalent transmitting array elements, R m For R e The mth element in R m It is a Q×(N-Q+1) Hankel matrix, where Q is the number of equivalent receiving array elements.

6. The frequency diversity MIMO radar parameter estimation method according to claim 1, characterized in that: R e The formula for singular value decomposition is expressed as: Among them, E S is the signal subspace, E N is the noise subspace, E S and E N is the left singular vector, Λ s and Λ N Corresponding to R e The first K larger and remaining singular values ​​of , where K is the target number.

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