MIMO radar mutual coupling error and DOA joint estimation method based on energy valley-compressed sensing

Abstract

The invention discloses an MIMO radar mutual coupling error and DOA joint estimation method based on energy valley-compressed sensing. According to the method, under a design framework of compressed sensing, aiming at the problem of poor target angle recognition accuracy caused by mutual coupling errors of an array model of an MIMO radar, an improved EVO-CS algorithm is designed by introducing different disturbance strategies into an EVO algorithm and introducing weighting and subspace filtering operations in a CS signal reconstruction process, and the target angle recognition accuracy is improved. According to the invention, the algorithm can obtain the global optimal solution to the greatest extent, the robustness of the algorithm under the severe conditions of low signal to noise ratio and the like is enhanced, the applicability of the compressed sensing algorithm in the MIMO radar is broadened, and the solving capability of the compressed sensing algorithm in target angle positioning in the MIMO radar scene is improved and enhanced. Examples show that a more accurate mutual coupling error coefficient and a DOA estimation result can be realized, the global optimization capability is stronger, the estimation accuracy is the highest, and the detection performance of an MIMO radar system for identifying a space target is effectively enhanced.

Description

Technical Field

[0001] The present invention relates to a technique for jointly estimating mutual coupling error coefficients and DOA by using an improved energy valley-compressed sensing method under a MIMO radar, and belongs to the field of radar array signal processing. Background Art

[0002] As a popular new radar mode, Multiple Input Multiple Output (MIMO) radar uses multiple transmit and receive antennas to process target echo signals and has been widely used in multiple fields such as autonomous driving, vital sign detection, and marine sonar. Compared with a single receiving-only array, MIMO radar expands the virtual array aperture by means of the co-array idea and achieves more accurate Direction of Arrival (DOA) estimation. However, the transceiver arrays of traditional MIMO radars usually use a uniform linear array structure. This dense array element layout pattern will cause serious mutual coupling effects and limited aperture length, affecting subsequent target estimation results. In addition, too many physical array elements will increase the hardware cost and processing complexity of the system, which is not conducive to real-time processing of engineering. Therefore, combining MIMO radar with sparse transceiver arrays has important research significance for improving the performance of radar systems.

[0003] In order to break through the limitations of the Nyquist sampling theorem, Compressed Sensing (CS) theory emerged as the times require. It utilizes the sparsity or compressibility characteristics of the signal itself and realizes signal reconstruction and parameter super-resolution estimation by means of few snapshots or even single snapshot signals. CS theory does not care about the sampling values of the signal itself but focuses on the information contained in the signal. Therefore, it can collect signals at a rate much lower than the Nyquist sampling law. In the application of MIMO radar, the target incoming wave signals received by the receiver are only limited to a few specific angles, and there are no signal incidences in most angular directions in the whole space. Therefore, the target signals are sparse in space. Therefore, it has important development significance to use CS technology to achieve accurate target angle estimation.

[0004] When a radar estimates the angle of a target echo signal, it usually assumes that the array steering matrix is not affected by anything. However, in practical applications, the mutual coupling effect between array elements is often inevitable. The radiation pattern distortion caused by the electromagnetic interaction between adjacent array elements directly affects the antenna radiation characteristics, reducing the angle estimation accuracy and algorithm stability. Therefore, how to design an optimization algorithm that can not only accurately compensate the mutual coupling error coefficients between array elements but also obtain accurate target angle estimation has become a key problem that urgently needs to be solved in the MIMO radar system.

[0005] Existing mutual coupling error coefficient and angle estimation methods, such as beamforming algorithms, subspace algorithms, deep learning algorithms, etc., are all subject to their respective limitations. Beamforming algorithms are simple but constrained by the Rayleigh limit, with low resolution and unable to resolve signal sources with close angles; subspace algorithms have high resolution but are sensitive to model assumptions; deep learning algorithms are highly adaptable but rely on data. In contrast, CS achieves high-precision angle estimation while reducing the hardware burden by exploiting signal sparsity, making it an ideal choice for resource-constrained scenarios. However, in the CS framework, the design of the dictionary matrix and the reconstruction algorithm has a crucial impact on the final signal recovery result. It not only needs to meet strict mathematical conditions but also takes into account computational efficiency, hardware implementation, and noise resistance. A reasonable design can achieve efficient and robust signal recovery using a small amount of sampled data. In addition, the Orthogonal Matching Pursuit (OMP) algorithm is still limited in terms of computational efficiency, noise robustness, and grid dependence in DOA estimation. Therefore, the present invention proposes a method for jointly estimating mutual coupling error and DOA of a MIMO radar based on Energy valley optimizer - CS (EVO - CS). This method first designs a new layout method for the sparse transceiver array, formulates a joint optimization problem for mutual coupling error coefficient and DOA estimation, and uses the improved EVO - CS method to obtain accurate mutual coupling error coefficients and signal angles under the cycle of rotation and iteration. Summary of the Invention

[0006] The object of the present invention is to provide a new method for jointly estimating mutual coupling error coefficient and DOA under a MIMO radar, which combines the EVO method and CS technology, and alternately estimates the mutual coupling error coefficient and signal angle during the cycle of rotation and iteration to improve the error compensation ability and DOA estimation ability of the improved algorithm.

[0007] Technical Solution: A method for jointly estimating mutual coupling error and DOA of a MIMO radar based on energy valley - compressive sensing, the implementation steps of this method include:

[0008] S1. Construct the element arrangement of the transmitting array and receiving array of the MIMO radar. The elements in the transmitting array are arranged according to the nested array model, and the receiving array uses a relatively sparse uniform array;

[0009] S2. According to the element positions of the transmitting array and receiving array, respectively model the steering vectors and steering matrices corresponding to the transmitting array and receiving array, so as to establish a mathematical model of the target echo data received by the MIMO radar;

[0010] S3. Construct the mutual coupling error matrix corresponding to the transmitting array with the mutual coupling error coefficient as the object according to the positional relationship of each element in the transmitting array;

[0011] S4. Determine the mathematical expression of the coupling coefficient according to the mutual coupling error matrix of the transmitting array;

[0012] S5. Construct the array received data of the MIMO radar under the influence of mutual coupling error;

[0013] S6. Construct the covariance matrix of the MIMO radar array received data;

[0014] S7. Use the vectorization operation of the matrix to obtain the virtual array received data corresponding to the virtual array formed by the MIMO radar;

[0015] S8. Design the dictionary matrix by using the sparsity of the target signal in the spatial angle domain;

[0016] S9. Construct the virtual array received data under the assumption of sparsity conditions;

[0017] S10. Under the CS framework, construct the joint optimization problem of the mutual coupling error coefficient and DOA estimation in the MIMO radar scenario, and the mathematical expression is:

[0018]

[0019] In the formula, represents the mutual coupling error coefficient, p = vec{R s} represents the vector after vectorizing R s The vector after vectorizing R s represents the signal autocorrelation matrix, F represents the coupling coefficient corresponding to the mutual coupling error of the transmitting array, ||·|| F represents the F norm, ||·||1 represents the l1 norm, represents the signal vector corresponding to the dictionary matrix, and respectively represent the i-th and j-th elements in the transmitting array, ε is a positive number, is the dictionary matrix.

[0020] Furthermore, the specific design of step S1 includes:

[0021] There are a total of N t elements in the transmitting array of the MIMO radar, and the element arrangement is in the form of a nested array. Among them, sub-array 1 has N t1 elements, the element spacing is d, and the position of each element can be described as Sub-array 2 has N t2 elements, the element spacing is (N t1 +1)d, and the distance between the two sub-arrays is d. The element positions of the transmitting array are represented as

[0022] The receiving array adopts a layout of a sparse uniform array, with M r elements, and the element spacing is (N t2 (N t1 +1))d. The receiving array

[0023] Furthermore, the specific process of step S2 includes:

[0024] According to the steering matrices of the transmitting and receiving matrices, establish a mathematical model of the received data of the MIMO radar after matched filtering, as follows:

[0025]

[0026] where a t (θ K ) represents the steering vector of the transmitting array, a r (θ K ) is the steering vector of the transmitting array, A = A t ⊙A r represents the steering matrix of the combined array, s(t) represents the signal vector, n(t) represents the noise vector, K represents the number of incident signals, represents the Kronecker product, and ⊙ represents the KR product.

[0027] Furthermore, the mathematical expression of the mutual coupling error matrix in the transmitting array in step S3 is as follows:

[0028]

[0029] where C1 = Toeplitz(c) represents the mutual coupling error matrix corresponding to the transmitting array, and its dimension is (M t +1)×(M t +1); Toeplitz(·) represents the operation of constructing a Toeplitz matrix with a vector as the row and column; represents the mutual coupling error coefficient, and satisfies 01 and 02 respectively represent all-zero matrices of dimensions (M t +1)×(N r -1) and (N r -1)×(M t +1); represents the identity matrix of dimension (N r -1)×(N r -1).

[0030] Furthermore, the coupling coefficient corresponding to the mutual coupling error of the transmitting array determined in step S4 is:

[0031]

[0032] where, ||·|| F denotes the Frobenius norm; diag(·) represents the operation of taking the diagonal elements of a matrix.

[0033] Furthermore, the array received data of the MIMO radar constructed in step S5 under the influence of mutual coupling error is:

[0034]

[0035] where, denotes the joint steering matrix affected by mutual coupling error; denotes the joint steering vector affected by mutual coupling error.

[0036] Furthermore, the covariance matrix of the array received data of the MIMO radar constructed in step S6 is:

[0037]

[0038] where, R s denotes the signal autocorrelation matrix; σ denotes the noise power; denotes an identity matrix of dimension M t N r ×M t N r ; E{·} represents the expectation operation; (·) H represents the conjugate transpose operation.

[0039] Furthermore, the virtual received data corresponding to the virtual array of the MIMO radar is constructed in step S7:

[0040]

[0041] where, denotes the joint virtual array steering matrix affected by mutual coupling error; denotes the joint virtual array steering vector affected by mutual coupling error; p = vec{R s} represents the vector after vectorizing R s ; represents the vector after vectorizing ; e i represents a column vector with 1 at the i-th position and 0 elsewhere; vec{·} represents the vectorization operation; (·) * represents the conjugate operation.

[0042] Furthermore, in step S8, using the sparsity of the target signal in the spatial angle domain, the dictionary matrix is designed as:

[0043]

[0044] Among them, represents the column vector of the dictionary matrix, and g = 1, 2, …, G; G represents the number of grid points for dividing the spatial domain angle, and G >> K.

[0045] Furthermore, under the assumption of the sparsity condition, step S9 constructs the ideal virtual array received data as:

[0046]

[0047] Among them, represents the signal vector corresponding to the dictionary matrix, and it only has non-zero values at the positions of the true signals, with zero values at the remaining positions, and has the characteristic of a sparsity of K.

[0048] The beneficial effects are: Compared with the prior art, the method of the present invention can accurately compensate for the mutual coupling error existing in the transmitting array, and at the same time achieve more accurate signal angle estimation, and has a stronger identification ability for spatial incident signals. Secondly, its substantial features and remarkable effects also include:

[0049] (1) The sparse transceiver array constructed by the present invention sparsely arranges the physical array element positions in the transmitting and receiving arrays of the MIMO radar. Its array element arrangement method can use fewer transceiver physical array elements to achieve a virtual array with a longer virtual aperture and the fewest redundant array elements, effectively saving the project construction cost and improving the angle resolution ability.

[0050] (2) By dividing the spatial angle domain, the present invention makes full use of the sparsity of signals in space, and under the framework of compressive sensing, by designing the dictionary matrix and reconstruction algorithm in the MIMO radar scenario, formulates the joint optimization problem of the mutual coupling error coefficient and the signal angle, breaks through the limitation of the Nyquist sampling rate, and greatly improves the signal angle resolution.

[0051] (3) Aiming at the defects that the EVO algorithm is prone to fall into local solutions and has weak search strength, the present invention embeds perturbation operations at different stages of the individual solution update to promote population diversity, effectively expands the degree of freedom of the solution space, and at the same time avoids the algorithm from falling into local convergence, enhancing the probability of obtaining the global optimal solution.

[0052] (3) For the RSOMP algorithm used by CS in the signal reconstruction process, a residual weighting operation and subspace prior information are introduced. By dynamically estimating the subspace, the performance of the OMP algorithm is improved to overcome the disadvantage of being strongly interfered by noise under low signal-to-noise ratio conditions. In the algorithm iteration process of the present invention, a weighted and subspace filtering operation is introduced. By dynamically estimating the signal subspace projection matrix, the noise components are suppressed, and the low-dimensional subspace where the signal is located is fully retained, greatly improving the optimization performance of the OMP algorithm. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 Layout of the transmitting and receiving arrays of the novel MIMO radar;

[0054] Figure 2 Flowchart of the combined estimation operation of the mutual coupling error coefficient and DOA of the improved EVO-CS;

[0055] Figure 3 Flowchart of the operation of the improved EVO algorithm;

[0056] Figure 4 Flowchart of the operation of the improved WSOMP algorithm. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0057] To describe in detail the technical solution disclosed by the present invention, the present invention will be further introduced in detail below in combination with specific embodiments and drawings.

[0058] What the present invention provides is a method for jointly estimating the mutual coupling error and DOA of a MIMO radar based on EVO-CS. Generally speaking in combination with the drawings, the physical element layout block diagram of the transmitting and receiving arrays of the novel MIMO radar is as Figure 1 shown, the overall operation flow of the improved EVO-CS method is as Figure 2 shown, the operation flow of the improved EVO method is as Figure 3 shown, and the operation flow of the improved weighted subspace OMP (Weight Subspace OMP, WSOMP) algorithm is as Figure 4 shown. Specifically, the implementation of this method includes the following steps:

[0059] Step 1: Design the element arrangement mode of the transmitting array and the receiving array of the MIMO radar. In the transmitting array of the MIMO radar, the elements are arranged according to the nested array model, while the receiving array adopts a sparse uniform array. This sparse transceiver array model can effectively expand the virtual array aperture of the MIMO radar, making the virtual array generate the least number of redundant elements, thereby improving the angle resolution ability of the array and greatly enhancing the angle measurement accuracy of the radar system.

[0060] Design the element layout mode of the transmitting array and the receiving array of the MIMO radar. There are a total of N elements in the transmitting array of the MIMO radart elements, and the element arrangement is in a nested array form, where sub-array 1 has N t1 elements, the element spacing is d, and the position of each element can be described as Sub-array 2 has N t2 elements, the element spacing is (N t1 +1)d, and the distance between the two sub-arrays is d, then the positions of the elements of the transmitting array can be expressed as The receiving array adopts a sparse uniform array layout, with M r elements, and the element spacing is (N t2 (N t1 +1))d, This sparse transceiver array model can generate the least number of redundant elements, effectively expand the virtual array aperture of the MIMO radar, thereby improving the angle resolution ability of the array and greatly enhancing the angle measurement accuracy of the radar system.

[0061] Step 2: Construct the array received data of the MIMO radar. According to the element positions of the transmitting array Model the steering vector of the transmitting array and the steering matrix A t =[a t (θ1) a t (θ2) … a t (θ K )]. According to the element positions of the receiving array Model the steering vector of the receiving array and the steering matrix A r =[a r (θ1) a r (θ2) … a r (θ K )]. Using the steering matrices of the transmitting and receiving arrays, establish the mathematical model of the MIMO radar received data after matched filtering, as shown below:

[0062]

[0063] where, A = A t ⊙ A r represents the steering matrix of the combined array; s(t) represents the signal vector; n(t) represents the noise vector; K represents the number of incident signals; represents the Kronecker product; ⊙ represents the KR product.

[0064] Step 3: Construct the mutual coupling error matrix in the transmitting array, and its expression is as follows:

[0065]

[0066] Among them, C1 = Toeplitz(c) represents the mutual coupling error matrix corresponding to the transmitting array, and its dimension is (M t +1) × (M t +1); Toeplitz(·) represents the operation of constructing a Toeplitz matrix with vectors as rows and columns; represents the mutual coupling error coefficient and satisfies 01 and 02 respectively represent all-zero matrices with dimensions of (M t +1) × (N r -1) and (N r -1) × (M t +1); represents the identity matrix with dimensions of (N r -1) × (N r -1).

[0067] Step 4: Determine the coupling coefficient corresponding to the mutual coupling error of the transmitting array:

[0068]

[0069] Among them, ||·|| F represents the Frobenius norm; diag(·) represents the operation of taking the diagonal elements of a matrix.

[0070] Step 5: Construct the array received data of the MIMO radar under the influence of mutual coupling error:

[0071]

[0072] Among them, represents the joint steering matrix affected by mutual coupling error; represents the joint steering vector affected by mutual coupling error.

[0073] Step 6. Construct the covariance matrix of the array received data of the MIMO radar:

[0074]

[0075] Among them, R s represents the signal autocorrelation matrix; σ represents the noise power; represents the identity matrix with dimensions of M t N r ×M t N r ; E{·} represents the expectation operation; (·) H represents the conjugate transpose operation.

[0076] Step 7. Construct the virtual received data corresponding to the virtual array of the MIMO radar:

[0077]

[0078] Among them, represents the joint virtual array steering matrix affected by mutual coupling error; represents the joint virtual array steering vector affected by mutual coupling error; p = vec{R s} represents the vector after vectorizing R s after; represents the vector after vectorizing after, e i represents a column vector with 1 at the i-th position and 0 at the rest; vec{·} represents the vectorization operation; (·) * represents the conjugate operation.

[0079] Step 8: Utilize the sparsity of the target signal in the spatial angle domain to design a dictionary matrix:

[0080]

[0081] Among them, represents the column vector of the dictionary matrix, and g = 1, 2,..., G; G represents the number of grid points for dividing the spatial domain angle, and G >> K.

[0082] Step 9: Under the assumption of sparsity conditions, construct the ideal virtual array received data:

[0083]

[0084] Among them, represents the signal vector corresponding to the dictionary matrix, and it only has values at the positions of the true signals and is 0 at the remaining positions, with the characteristic of a sparsity of K.

[0085] Step 10: Under the CS framework, construct the joint optimization problem of the mutual coupling error moment coefficient and DOA estimation in MIMO radar:

[0086]

[0087] Among them, B represents the maximum action interval of the mutual coupling error; and respectively represent the i-th and j-th array elements in the transmit array; ||·||1 represents the l1 norm; ε represents a very small positive number.

[0088] Step 11: Use the improved EVO-CS algorithm to solve the above optimization problem:

[0089]

[0090]

[0091] The operation steps of the improved EVO algorithm used in the above ① can be described as follows:

[0092]

[0093]

[0094] The operation steps of the improved WSOMP algorithm used in the above ② can be described as follows:

[0095]

[0096]

[0097] The specific implementation of the present invention has been described in detail above. It should be understood that individual details are not limited to the above specific implementation manners, and those skilled in the art can make various deformations or modifications within the scope of the claims without affecting the essence of the present invention.

Claims

1. A method for jointly estimating mutual coupling error and DOA of MIMO radar based on energy valley-compressive sensing, characterized in that The implementation steps of this method include: S1. Construct the element arrangement modes of the transmitting array and the receiving array of the MIMO radar. The elements in the transmitting array are arranged according to the nested array model, and the receiving array adopts a relatively sparse uniform array; S2. According to the element positions of the transmitting array and the receiving array, respectively model the steering vectors and steering matrices corresponding to the transmitting array and the receiving array, so as to establish a mathematical model of the target echo data received by the MIMO radar; S3. According to the positional relationship of each element in the transmitting array, taking the mutual coupling error coefficient as the object, construct the mutual coupling error matrix corresponding to the transmitting array; S4. According to the mutual coupling error matrix of the transmitting array, determine the mathematical expression of the coupling coefficient; S5. Construct the array received data of the MIMO radar under the influence of mutual coupling error; S6. Construct the covariance matrix of the array received data of the MIMO radar; S7. Using the vectorization operation of the matrix, obtain the virtual array received data corresponding to the virtual array formed by the MIMO radar; S8. Using the sparsity of the target signal in the spatial angle domain, design the dictionary matrix; S9. Under the assumption of sparsity conditions, construct the virtual array received data; S10. Under the CS framework, construct the joint optimization problem of the mutual coupling error coefficient and DOA estimation in the MIMO radar scenario, and the mathematical expression is: Wherein, represents the mutual coupling error coefficient, p = vec{R s} represents the vectorized R s after that, R s represents the signal autocorrelation matrix, F represents the coupling coefficient corresponding to the mutual coupling error of the transmitting array, ||·|| F represents the F norm, ||·||1 represents the l1 norm, represents the signal vector corresponding to the dictionary matrix, and respectively represent the i-th and j-th array elements in the transmitting array, ε is a positive number, is the dictionary matrix.

2. The method for jointly estimating mutual coupling error and DOA of MIMO radar according to claim 1, characterized in that The specific design of step S1 includes: There are a total of N elements in the transmit array of the MIMO radar t The elements are arranged in a nested array form, where sub-array 1 has N t1 elements, the element spacing is d, and the position of each element is described as Sub-array 2 has N t2 elements, the element spacing is (N t1 +1)d, and the distance between the two sub-arrays is d, The positions of the elements in the transmit array are represented as The receiving array adopts a layout of a sparse uniform array, with M r array elements, and the element spacing is (N t2 (N t1 +1))d. The receiving array is expressed as 3. The method for jointly estimating mutual coupling error and DOA of MIMO radar according to claim 1, characterized in that, The specific process of step S2 includes: According to the steering matrices of the transmitting and receiving matrices, establish the mathematical model of the received data of the MIMO radar after matched filtering as follows: where a t (θ K ) represents the steering vector of the transmitting array, a r (θ K ) is the steering vector of the transmitting array, A = A t ⊙A r represents the steering matrix of the combined array, s(t) represents the signal vector, n(t) represents the noise vector, K represents the number of incident signals, denotes the Kronecker product, and ⊙ denotes the KR product.

4. The method for jointly estimating mutual coupling error and DOA of MIMO radar according to claim 1, wherein The mathematical expression of the mutual coupling error matrix in the transmitting array in step S3 is as follows: where \(C1 = Toeplitz(c)\) represents the mutual coupling error matrix corresponding to the transmitting array, and its dimension is \((M t + 1)\times(M t + 1)\); \(Toeplitz(\cdot)\) represents the operation of constructing a Toeplitz matrix with vectors as rows and columns; represents the mutual coupling error coefficient and satisfies \(01\) and \(02\) represent all-zero matrices of dimensions \((M t + 1)\times(N r - 1)\) and \((N r - 1)\times(M t + 1)\) respectively; represents the identity matrix of dimension \((N r - 1)\times(N r - 1)\).

5. The method for jointly estimating mutual coupling error and DOA of MIMO radar according to claim 1, characterized in that The coupling coefficient corresponding to the mutual coupling error of the transmitting array determined in step S4 is: Among them, ||·|| F represents the Frobenius norm; diag(·) represents the operation of taking the diagonal elements of a matrix.

6. The method for jointly estimating mutual coupling errors and DOA of an MIMO radar according to claim 1, characterized in that The array received data of the MIMO radar constructed in step S5 under the influence of mutual coupling error is: Among them, represents the combined steering matrix affected by mutual coupling error; represents the combined steering vector affected by mutual coupling error.

7. The method for jointly estimating mutual coupling error and DOA of MIMO radar according to claim 1, characterized in that The covariance matrix of the array received data of the MIMO radar constructed in step S6 is: Among them, R s represents the signal autocorrelation matrix; σ represents the noise power; represents a matrix of dimension M t N r ×M t N r identity matrix; E{·} represents the expectation operation; (·) H represents the conjugate transpose operation.

8. The method for jointly estimating mutual coupling error and DOA of MIMO radar according to claim 1, characterized in that The virtual received data corresponding to the virtual array of the MIMO radar constructed in step S7: Among them, represents the joint virtual array steering matrix affected by mutual coupling error; represents the joint virtual array steering vector affected by mutual coupling error; p = vec{R s} represents the vector after vectorizing R s ; represents the vector after vectorizing ; e i represents the column vector with 1 at the i-th position and 0 elsewhere; vec{·} represents the vectorization operation; (·) * represents the conjugate operation.

9. The method for jointly estimating mutual coupling error and DOA of MIMO radar according to claim 1, characterized in that The dictionary matrix designed in step S8 using the sparsity of the target signal in the spatial angle domain is: Among them, represents the column vector of the dictionary matrix, where g = 1, 2, …, G; G represents the number of grid points divided by the spatial domain angle, and G >> K.

10. The method for jointly estimating mutual coupling error and DOA of MIMO radar according to claim 1, wherein, The ideal virtual array received data constructed in step S9 under the assumption of sparsity conditions is: Among them, represents the signal vector corresponding to the dictionary matrix, and it only has non-zero values at the positions of the true signals, with zero values at the remaining positions, and has the characteristic of sparsity K.

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