Signal positioning method and system of airborne antenna array

Abstract

The invention relates to a wireless communication technology, and discloses a signal positioning method and system of an airborne antenna array, so as to solve many problems caused by rank deficiency of a covariance matrix in a traditional DOA (Direction of Arrival) method. The method comprises the following steps: processing signals received by an antenna array by using a norm analysis method to obtain the number of target signals; reconstructing an output signal of the antenna array by using a Toeplitz matrix to obtain an equivalent covariance matrix, wherein noise items in the equivalent covariance matrix are distributed on a main diagonal line; performing eigenvalue decomposition based on the reconstructed equivalent covariance matrix to realize separation of a signal subspace and a noise subspace; and constructing a spatial spectrum function according to a separation result, and selecting angles respectively corresponding to series value points which take a maximum value point as the first and have the same number as the target signals from large to small as the direction of arrival of each target signal.

Description

Technical Field

[0001] The present invention relates to the field of wireless communication technology, and in particular to a signal positioning method and system for an airborne antenna array. Background Art

[0002] Drones are usually used to track and locate unknown target signals. The common method is to carry an antenna array, collect signals through the antenna, and then determine the direction of the target signal through algorithm processing, which is usually called signal angle of arrival (DOA, Direction Of Arrival) estimation.

[0003] However, due to the existence of multipath effects, signals are usually not independent, but have strong correlation or coherence. This will cause the covariance matrix of the signal to be rank-deficient, resulting in a significant decrease in the performance of the subspace-based DOA estimation algorithm. In addition, the existing DOA estimation algorithm requires the number of target signals to be known in advance, which also limits the performance of the algorithm to a certain extent. Summary of the invention

[0004] The present invention aims to disclose a signal positioning method and system for an airborne antenna array to solve many problems of the traditional DOA method Y caused by the lack of covariance matrix rank.

[0005] To achieve the above object, the present invention discloses a signal positioning method of an airborne antenna array, which is characterized by comprising:

[0006] Step S1, using a normal mode analysis method to process the signal received by the antenna array to obtain the number of target signals;

[0007] Step S2, reconstructing the output signal of the antenna array using the Toeplitz matrix to obtain an equivalent covariance matrix, wherein the noise terms in the equivalent covariance matrix are distributed on the main diagonal;

[0008] Step S3, perform eigenvalue decomposition based on the reconstructed equivalent covariance matrix to achieve separation of the signal subspace and the noise subspace; then construct a spatial spectrum function based on the separation result, and then select, in descending order, the angles corresponding to the series of value points headed by the maximum value point and the number of which is consistent with the number of target signals as the angular arrival directions corresponding to each target signal.

[0009] Preferably, the step S1 specifically includes:

[0010] Step S11: Determine that the data model received by the antenna array is:

[0011] X = AS + N;

[0012] WX = S;

[0013] W = pinv(A);

[0014] Where X represents the output signal of each antenna, A represents the response matrix, N is the noise matrix, W is the weight vector, S is the signal matrix, and pinv represents the generalized inverse matrix of the matrix;

[0015] Step S12: Divide the output array X using the singular value decomposition method so that: SVD(X)=UΓV * ;make is the subspace constructed by the non-zero singular vectors in V, then A set of orthogonal bases that form the output array X is obtained as follows:

[0016]

[0017] Where z is the number of signals, including target signals and non-target signals; Related to the subspace of non-zero singular value vectors, w i is the weight vector corresponding to the i-th signal; since v n Expressed as The nth column of , we get the following expression:

[0018]

[0019] If defined but:

[0020]

[0021] Let l j =vec(p j ) T , L T =[l1,l2…l n ] T , The following expression is obtained:

[0022]

[0023] Then use the Householder transformation to transform it, assuming the following matrix:

[0024]

[0025] After that, multiply Q by L on the left to obtain the following expression:

[0026]

[0027] calculate get:

[0028]

[0029] Then, L is decomposed by SVD to obtain a set of orthogonal bases of the null space, and the singular value with a value of zero is taken as the solution; in the presence of noise, there is a corresponding number of series of small singular values ​​whose distance from 0 is within the set threshold range, and the number of these small singular values ​​obtained is the number of target signals.

[0030] Preferably, the antenna array is composed of 2M+1 array elements, and the spacing between the array elements is d=λ2, and the step S2 includes:

[0031] Step S21: Assume that k narrowband signals are incident on the antenna array from a distance, with an angle of θ k , k = 1, 2, ..., K; and k < M + 1; under noise-free conditions, the output of the matrix is ​​expressed as:

[0032]

[0033] When Gaussian white noise is present, the array output is expressed as:

[0034]

[0035] Where s(t) is a K×1 dimensional signal matrix, A=[a(θ1),L,a(θ K )] is a (2M+1)×K-dimensional matrix (2M+1)×1-dimensional matrix;

[0036] The kth signal direction response is expressed as:

[0037] Step S22: Divide the output matrix y(t) into two parts, As(t) and n(t), and construct a (M+1)×(M+1) Vitoplitz matrix R X (t), expressed as:

[0038]

[0039] Among them, x M (t) The Mth antenna output of the antenna array; Assume S(t) = diag{s1(t),…,s K (t)}, which is simplified to:

[0040]

[0041] in, is the steering vector matrix composed of the elements from row 0 to row -M of A, and the steering vector in column k

[0042] Step S23: Establish R X(t) and the signal x obtained by the i-th array element i The relationship between (t) is:

[0043]

[0044] in,

[0045] Step S24: construct the same form of Toeplitz matrix for the array output matrix and the noise signal, denoted as y(t), R N (t), and the signal R obtained by the i-th array element of the output matrix y(t) Yi The relationship between (t) is:

[0046]

[0047] Among them, σ 2 is the Gaussian white noise variance; is a (M+1)×(M+1)-dimensional matrix. Except for the i-th diagonal element which is 1, the rest of the matrix elements are 0. When i=0, we get is the unit matrix, and the signal subspace can be directly obtained by eigenvalue decomposition; when i is non-zero, the noise mainly comes from

[0048] Step S25, solving R Yi (t) The sum of the squares of the matrix, i∈[-M,M], obtains the preliminary equivalent covariance matrix As follows:

[0049]

[0050] The expression of the preliminary equivalent covariance matrix after sorting is:

[0051]

[0052] Among them, R S =E[s(t)s H (t)], definition get:

[0053]

[0054] Step S26: Use FBSS (Forward and Backward Spatial Smoothing, bidirectional spatial smoothing algorithm) technology to smooth the preliminary equivalent covariance matrix to obtain the final equivalent covariance matrix R, and the calculation formula is:

[0055]

[0056] Wherein, J is a (M+1)×(M+1) matrix whose anti-diagonal is 1 and the remaining matrix elements are 0.

[0057] Preferably, step S3 comprises:

[0058] Step S31, perform eigenvalue decomposition on R to obtain the following formula:

[0059]

[0060] Among them, U S =[u1,…,u K ] is the signal subspace, U n =[u K ,…,u M+1 ] is the noise subspace, and then the decomposed eigenvalues ​​are arranged from large to small, Λ S is a matrix composed of larger eigenvalues, the number of which is consistent with the number of target signals determined in step S1, n is the matrix composed of the remaining smaller eigenvalues; we get:

[0061]

[0062] Where a(θ) is the steering vector;

[0063] Step S32: construct a one-dimensional spatial spectrum function:

[0064]

[0065] Among them, P MUSIC (θ) The final azimuth of the signal; in descending order, the angles corresponding to the series of value points headed by the maximum value point and the number of which is the same as the number of target signals are selected as the direction of the arrival angle corresponding to each target signal;

[0066] Step S33: If In the constructed two-dimensional case, the signal subspace and the steering vector have the following relationship:

[0067]

[0068] Among them, θ i , Represents the two-dimensional direction angle corresponding to the i-th signal; the constructed two-dimensional spatial spectrum function is expressed as:

[0069]

[0070] To achieve the above objectives, the present invention also discloses a signal positioning system for an airborne antenna array, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the above method when executing the computer program.

[0071] The present invention has the following beneficial effects:

[0072] The output signal of the antenna array is reconstructed using the Toeplitz matrix to obtain an equivalent covariance matrix, in which the noise terms are distributed on the main diagonal. This solves the rank-deficient problem of the covariance matrix caused by the presence of correlated and coherent signals. There is no need for pre-denoising, which also avoids the impact of denoising and improves efficiency. At the same time, the accurate signal source arrival angle can be obtained based on the equivalent covariance matrix.

[0073] The present invention will be further described in detail below with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] The drawings constituting a part of this application are used to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0075] Figure 1 It is a schematic flow chart of a signal positioning method of an airborne antenna array disclosed in an embodiment of the present invention.

[0076] Figure 2 It is a schematic diagram of obtaining the number of target signals by processing the signals received by the antenna array based on the normal mode analysis method disclosed in an embodiment of the present invention.

[0077] Figure 3 It is a schematic diagram of the arrangement of the uniform linear array disclosed in the embodiment of the present invention.

[0078] Figure 4 It is a schematic diagram of the arrangement of the L-shaped array disclosed in the embodiment of the present invention.

[0079] Figure 5 It is a schematic diagram of the arrangement of the circular array disclosed in the embodiment of the present invention.

[0080] Figure 6 It is a result diagram of the uniform linear array based on the one-dimensional spatial spectrum disclosed in the embodiment of the present invention.

[0081] Figure 7 It is a result diagram based on the two-dimensional spatial spectrum of the L-type array disclosed in the embodiment of the present invention.

[0082] Figure 8 It is a result diagram based on the two-dimensional spatial spectrum of the circular array disclosed in the embodiment of the present invention. DETAILED DESCRIPTION

[0083] The embodiments of the present invention are described in detail below with reference to the accompanying drawings, but the present invention can be implemented in many different ways as defined and covered by the claims.

[0084] Example 1

[0085] For ease of description, the meanings of some parameters and symbols in this embodiment are first stated as follows:

[0086] E[x]: mathematical expectation of random variable x; vec(x): vectorization in column direction of x; mat(x): matrixization in column direction of x; X -1 : The inverse matrix of matrix X; X T : Transpose of matrix X; X H : conjugate transpose of matrix X; rank(X): rank of matrix X; Kronecker product; SVD(X): singular value decomposition of X matrix; EVD(X): eigenvalue decomposition of X matrix. In addition, the transpose of a single signal replaces the "H" symbol with a superscript "*", such as described below: etc.; furthermore, some characters have a "~" symbol on top. Unless otherwise stated, the meaning they represent is similar to but different from the same symbol without the "~". The two can be understood as related relationships such as reconstruction, mapping or transformation; or, some characters are case-sensitive. Usually, uppercase characters represent the entire matrix, while lowercase characters represent a single element, a single row or a single column in the entire matrix; similarly, unless otherwise stated, the difference between the same character with and without a subscript is that the character without a subscript is usually used to represent the entire data set, while the character with a subscript is limited to specifying the specific element mapped by the subscript; the aforementioned expressions are commonly used by those skilled in the art and will not be elaborated on later.

[0087] This embodiment discloses a signal positioning method of an airborne antenna array, such as Figure 1 As shown, the following steps are included:

[0088] Step S1: using a normal mode analysis method to process the signal received by the antenna array to obtain the number of target signals.

[0089] Optionally, this step specifically includes:

[0090] Step S11: Determine that the data model received by the antenna array is:

[0091] X = AS + N;

[0092] WX = S;

[0093] W = pinv(A);

[0094] Among them, X represents the output signal of each antenna, A represents the response matrix, N is the noise matrix, W is the weight vector, S is the signal matrix, and pinv represents the generalized inverse matrix of the matrix.

[0095] Step S12: Divide the output array X using the singular value decomposition method so that: SVD(X)=UΓV * ;make is the subspace constructed by the non-zero singular vectors in V, then A set of orthogonal bases that form the output array X is obtained as follows:

[0096]

[0097] Where z is the number of signals, including target signals and non-target signals; Related to the subspace of non-zero singular value vectors, w i is the weight vector corresponding to the i-th signal; since v n Expressed as The nth column of , we get the following expression:

[0098]

[0099] If defined but:

[0100]

[0101] Let l j =vec(p j ) T , L T =[l1,l2…l n ] T , The following expression is obtained:

[0102]

[0103] Then use the Householder transformation to transform it, assuming the following matrix:

[0104]

[0105] After that, multiply Q by L on the left to obtain the following expression:

[0106]

[0107] calculate get:

[0108]

[0109] Then, L is decomposed by SVD to obtain a set of orthogonal bases of the null space, and the singular value with a value of zero is taken as the solution; in the presence of noise, there is a corresponding number of series of small singular values ​​whose distance from 0 is within the set threshold range, and the number of these small singular values ​​obtained is the number of target signals.

[0110] like Figure 2 As shown, after receiving the signal, the antenna array converts the signal into a normal mode signal, and according to the characteristics of the normal mode signal, the number of target signals is obtained by using the normal mode analysis method. The analysis result is shown in Figure 6 As shown in the figure, it can be seen that there are four data points that are significantly lower than other data points. The number of these smaller data points is the number of signals.

[0111] Step S2: Reconstruct the output signal of the antenna array using the Toeplitz matrix to obtain an equivalent covariance matrix, wherein the noise terms in the equivalent covariance matrix are distributed on the main diagonal.

[0112] Usually, if Figure 3 As shown, the antenna array consists of 2M+1 array elements, and the spacing between the array elements is d=λ2. This step can be specifically divided into the following processes:

[0113] Step S21: Assume that k narrowband signals are incident on the antenna array from a distance, with an angle of θ k , k = 1, 2, ..., K; and k < M + 1; under noise-free conditions, the output of the matrix is ​​expressed as:

[0114]

[0115] When Gaussian white noise is present, the array output is expressed as:

[0116]

[0117] Where s(t) is a K×1 dimensional signal matrix, A=[a(θ1),L,a(θ K )] is a (2M+1)×K dimensional matrix, n(t) is a (2M+1)×1 dimensional matrix; the k-th signal direction response is expressed as:

[0118] Step S22: Divide the output matrix y(t) into two parts, As(t) and n(t), and construct a (M+1)×(M+1) Vitoplitz matrix R X (t), expressed as:

[0119]

[0120] Among them, x M(t) The Mth antenna output of the antenna array; Assume S(t) = diag{s1(t),…,s K (t)}, diag represents the diagonal, which can be simplified to:

[0121]

[0122] in, is the steering vector matrix composed of the elements from row 0 to row -M of A, and the steering vector in column k

[0123] Step S23: Establish R X (t) and the signal x obtained by the i-th array element i The relationship between (t) is:

[0124]

[0125] in,

[0126] Step S24: construct the same form of Toeplitz matrix for the array output matrix and the noise signal, denoted as y(t), R N (t), and the signal R obtained by the i-th array element of the output matrix y(t) Yi The relationship between (t) is:

[0127]

[0128] Among them, σ 2 is the Gaussian white noise variance; is a (M+1)×(M+1)-dimensional matrix. Except for the i-th diagonal element which is 1, the rest of the matrix elements are 0. When i=0, we get is the unit matrix, and the signal subspace can be directly obtained by eigenvalue decomposition; when i is non-zero, the noise mainly comes from

[0129] Step S25, solving R Yi (t) The sum of the squares of the matrix, i∈[-M,M], obtains the preliminary equivalent covariance matrix As follows:

[0130]

[0131] The expression of the preliminary equivalent covariance matrix after sorting is:

[0132]

[0133] Among them, R S =E[s(t)s H (t)], definition get:

[0134] Step S26: Use FBSS technology to smooth the preliminary equivalent covariance matrix to obtain the final equivalent covariance matrix R, and the calculation formula is:

[0135]

[0136] Wherein, J is a (M+1)×(M+1) matrix whose anti-diagonal is 1 and the remaining matrix elements are 0.

[0137] Step S3, perform eigenvalue decomposition based on the reconstructed equivalent covariance matrix to achieve separation of the signal subspace and the noise subspace; then construct a spatial spectrum function based on the separation result, and then select, in descending order, the angles corresponding to the series of value points headed by the maximum value point and the number of which is consistent with the number of target signals as the angular arrival directions corresponding to each target signal.

[0138] This step may specifically include:

[0139] Step S31, perform eigenvalue decomposition on R to obtain the following formula:

[0140]

[0141] Among them, U S =[u1,…,u K ] is the signal subspace, U n =[u K ,…,u M+1 ] is the noise subspace, and then the decomposed eigenvalues ​​are arranged from large to small, Λ S is a matrix composed of larger eigenvalues, the number of which is consistent with the number of target signals determined in step S1, n is the matrix composed of the remaining smaller eigenvalues; we get:

[0142]

[0143] Among them, a(θ) is the steering vector.

[0144] Step S32: construct a one-dimensional spatial spectrum function:

[0145]

[0146] Among them, P MUSIC (θ) The final azimuth of the signal; in descending order, the angles corresponding to the series of value points headed by the maximum value point and whose number is the same as the number of target signals are selected as the angle of arrival direction corresponding to each target signal.

[0147] Step S33: If In the constructed two-dimensional case, the signal subspace and the steering vector have the following relationship:

[0148]

[0149] Among them, θ i , Represents the two-dimensional direction angle corresponding to the i-th signal; the constructed two-dimensional spatial spectrum function is expressed as:

[0150]

[0151] In this embodiment, the antenna array can be divided into a one-dimensional antenna array model and a two-dimensional antenna array model according to the dimension. The one-dimensional antenna array model is mainly a uniform array (such as Figure 3 ), the two-antenna array model is mainly a circular array (such as Figure 4 ) and L-type arrays (such as Figure 5 ). The estimated results of the corresponding spatial spectrum function are distributed as follows: Figures 6 to 8 .

[0152] In summary, this embodiment first obtains the number of target signals by the method of normal mode analysis, and obtains a DOA estimation technology with high resolution and low complexity by combining the advantages of FBSS technology, signal subspace and Toeplitz matrix reconstruction algorithm. This technology has the characteristics of good robustness and strong environmental adaptability, and this technology can process single or multiple coherent signals at the same time, which is not available in other technologies. It also has the advantages of good estimation accuracy, high resolution, accurate signal number estimation, etc., has better noise resistance and robustness under low signal-to-noise ratio or low snapshot number conditions, and has low complexity and is easy to implement in engineering.

[0153] Example 2

[0154] Corresponding to the above embodiment, this embodiment discloses a signal positioning system for an airborne antenna array, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the above method is implemented when the processor executes the computer program.

[0155] In summary, the methods and systems disclosed in the above embodiments of the present invention respectively have at least the following technical effects:

[0156] The output signal of the antenna array is reconstructed using the Toeplitz matrix to obtain an equivalent covariance matrix, in which the noise terms are distributed on the main diagonal. This solves the rank-deficient problem of the covariance matrix caused by the presence of correlated and coherent signals. There is no need for pre-denoising, which also avoids the impact of denoising and improves efficiency. At the same time, the accurate signal source arrival angle can be obtained based on the equivalent covariance matrix.

[0157] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A signal positioning method for an airborne antenna array, characterized in that: include: Step S1, using a normal mode analysis method to process the signal received by the antenna array to obtain the number of target signals; Step S2, reconstructing the output signal of the antenna array using the Toeplitz matrix to obtain an equivalent covariance matrix, wherein the noise terms in the equivalent covariance matrix are distributed on the main diagonal; Step S3, perform eigenvalue decomposition based on the reconstructed equivalent covariance matrix to achieve separation of the signal subspace and the noise subspace; then construct a spatial spectrum function based on the separation result, and then select, in descending order, the angles corresponding to the series of value points headed by the maximum value point and the number of which is consistent with the number of target signals as the angular arrival directions corresponding to each target signal.

2. The method according to claim 1, characterized in that The step S1 specifically includes: Step S11: Determine that the data model received by the antenna array is: X = AS + N; WX = S; W = pinv(A); Where X represents the output signal of each antenna, A represents the response matrix, N is the noise matrix, W is the weight vector, S is the signal matrix, and pinv represents the generalized inverse matrix of the matrix. Step S12: Divide the output array X using singular value decomposition so that: SVD(X)=UΓV * ;make is the subspace constructed by the non-zero singular vectors in V, then A set of orthogonal bases that form the output array X are obtained as follows: Where z is the number of signals, including target signals and non-target signals; Related to the subspace of non-zero singular value vectors, w i is the weight vector corresponding to the i-th signal; since v n Expressed as The nth column of , we get the following expression: If defined but: Let l j =vec(p j ) T , L T =[l1,l2…l n ] T , The following expression is obtained: Then use the Householder transformation to transform it, assuming the following matrix: After that, multiply Q by L on the left to obtain the following expression: calculate get: Then, L is decomposed by SVD to obtain a set of orthogonal bases of the null space, and the singular value with a value of zero is taken as the solution; in the presence of noise, there is a corresponding number of series of small singular values ​​whose distance from 0 is within the set threshold range, and the number of these small singular values ​​obtained is the number of target signals.

3. The method according to claim 2, characterized in that The antenna array is composed of 2M+1 array elements, and the spacing between the array elements is d=λ2. The step S2 includes: Step S21: Assume that k narrowband signals are incident on the antenna array from a distance, with an angle of θ k , k = 1, 2, ..., K; and k < M + 1; under noise-free conditions, the output of the matrix is ​​expressed as: When Gaussian white noise is present, the array output is expressed as: Where s(t) is a K×1 dimensional signal matrix, A=[a(θ1),L,a(θ K )] is a (2M+1)×K dimensional matrix, and n(t) is a (2M+1)×1 dimensional matrix; The kth signal direction response is expressed as: Step S22: Divide the output matrix y(t) into two parts, As(t) and n(t), and construct a (M+1)×(M+1) Vitoplitz matrix R X (t), expressed as: Among them, x M (t) The Mth antenna output of the antenna array; Assume S(t) = diag{s1(t),…,s K (t)}, which is simplified to: in, is the steering vector matrix composed of the elements from row 0 to row -M of A, and the steering vector in column k Step S23: Establish R X (t) and the signal x obtained by the i-th array element i The relationship between (t) is: in, Step S24: construct the same form of Toeplitz matrix for the array output matrix and the noise signal, denoted as y(t), R N (t), and the signal R obtained by the i-th array element of the output matrix y(t) Yi The relationship between (t) is: Among them, σ 2 is the Gaussian white noise variance; is a (M+1)×(M+1)-dimensional matrix. Except for the i-th diagonal element which is 1, the rest of the matrix elements are 0. When i=0, we get is the unit matrix, and the signal subspace can be directly obtained by eigenvalue decomposition; when i is non-zero, the noise mainly comes from Step S25, solving R Yi (t) The sum of the squares of the matrix, i∈[-M,M], obtains the preliminary equivalent covariance matrix As follows: The expression of the preliminary equivalent covariance matrix after sorting is: Among them, R S =E[s(t)s H (t)], definition get: Step S26: Use FBSS technology to smooth the preliminary equivalent covariance matrix to obtain the final equivalent covariance matrix R, and the calculation formula is: Wherein, J is a (M+1)×(M+1) matrix whose anti-diagonal is 1 and the remaining matrix elements are 0.

4. The method according to claim 3, characterized in that The step S3 comprises: Step S31, perform eigenvalue decomposition on R to obtain the following formula: Among them, U S =[u1,…,u K ] is the signal subspace, U n =[u K ,…,u M+1 ] is the noise subspace, and then the decomposed eigenvalues ​​are arranged from large to small, Λ S is a matrix composed of larger eigenvalues, the number of which is consistent with the number of target signals determined in step S1, n is the matrix composed of the remaining smaller eigenvalues; we get: Where a(θ) is the steering vector; Step S32: construct a one-dimensional spatial spectrum function: Among them, P MUSIC (θ) The final azimuth of the signal; in descending order, the angles corresponding to the series of value points headed by the maximum value point and the number of which is the same as the number of target signals are selected as the direction of the arrival angle corresponding to each target signal; Step S33: If In the constructed two-dimensional case, the signal subspace and the steering vector have the following relationship: Among them, θ i , Represents the two-dimensional direction angle corresponding to the i-th signal; the constructed two-dimensional spatial spectrum function is expressed as:

5. A signal positioning system for an airborne antenna array, comprising a memory, a processor, and a computer program stored in the memory and capable of running on the processor, characterized in that: When the processor executes the computer program, the method according to any one of claims 1 to 4 is implemented.

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