Spectral Estimation System

Abstract

To provide a spectrum estimation system that can estimate a frequency spectrum and an angular spectrum of received signals in a short time.SOLUTION: A spectrum estimation system estimates a frequency spectrum of received signals received by an array antenna and an angular spectrum of an arrival angle of the received signals, and the spectrum estimation system comprises: the array antenna that consists of a plurality of antennas receiving the received signals; a calculation unit that performs discrete Fourier transformation on the received signals received by the antennas and calculates a tensor consisting of a matrix based on the received signals on which the discrete Fourier transformation is performed; and an estimation unit that, on the basis of a dictionary based on compressed signals corresponding to sub-bands of the frequency and steering vectors for respective arrival angles and the tensor calculated by the calculation unit, estimates the frequency spectrum and the angular spectrum by using compressed sensing.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present invention relates to a spectral estimation system for the frequency spectrum and angular spectrum of a signal. [Background technology]

[0002] With the spread of 5G mobile communication systems, it is predicted that base stations using the millimeter wave band will be deployed densely by multiple operators. Accordingly, it will become important to know the center frequencies of signals transmitted from multiple base stations in order to prevent interference between operators and base stations and improve spectrum utilization efficiency. For this reason, technologies for knowing the center frequencies of signals transmitted from multiple base stations, such as those disclosed in Non-Patent Document 1 and Non-Patent Document 2, are attracting attention.

[0003] Non-Patent Document 1 discloses a frequency sweeping technique that uses a low-speed analog-to-digital converter (ADC) to sweep the center frequency. Non-Patent Document 2 discloses multi-coset sampling (MCS), which performs sampling at unequal intervals in order to reduce the conversion speed of the ADC required for estimation. [Prior art documents] [Non-patent literature]

[0004] [Non-Patent Document 1] H. Sun, A. Nallanathan, C. Wang, and Y. Chen, “Wideband Spectrum Sensing for Cognitive Radio Networks: A Survey,” IEEE Wirel. Commun., no. April, pp. 74-81, 2013. [Non-patent document 2] CP Yen, Y. Tsai, and X. Wang, “Wideband spectrum sensing based on sub-Nyquist sampling,” IEEE Trans. Signal Process., vol. 61, no.12, pp. 3028-3040, 2013. Summary of the Invention [Problem to be solved by the invention]

[0005] However, the frequency sweep method disclosed in Non-Patent Document 1 requires a long time for measurement because it uses a low-speed ADC, which poses a problem of enormous measurement costs.

[0006] Furthermore, when developing a CPS (cyber physical system) emulator, which requires an accurate model of the radio wave propagation channel including angular characteristics, information is required that includes not only the center frequency but also angular characteristics such as the angular spectrum of the arrival angle of the received signal.

[0007] However, Non-Patent Document 1 does not assume that information including angle characteristics will be swept, and therefore the technique disclosed in Non-Patent Document 1 has the problem that it is not possible to sweep information including angle characteristics.

[0008] Furthermore, the MCS algorithm described in Non-Patent Document 2 only supports the case of a single antenna and does not assume the use of multiple antennas. Therefore, the technology disclosed in Non-Patent Document 2 has the problem that it cannot estimate the angular spectrum.

[0009] The present invention has been devised to solve the above-mentioned problems, and has an object to provide a spectrum estimation system that can estimate the frequency spectrum and angular spectrum of a received signal in a short time. [Means for solving the problem]

[0010] The spectrum estimation system according to claim 1 is a spectrum estimation system for estimating a frequency spectrum of a received signal received by an array antenna and an angular spectrum of an angle of arrival of the received signal, the system comprising: an array antenna including a plurality of antennas for receiving the received signal; delay elements each having a different delay time, each connected to the antenna; Received by each antenna , delayed by the delay element a calculation unit that performs a discrete Fourier transform on the received signals and calculates a tensor consisting of a matrix based on each of the discrete Fourier transformed received signals; and an estimation unit that estimates the frequency spectrum and the angular spectrum using compressed sensing based on a dictionary based on compressed signals corresponding to each frequency subband and steering vectors for each angle of arrival, and the tensor calculated by the calculation unit.

[0011] The spectral estimation system according to claim 2 is the spectral estimation system according to claim 1, wherein the estimation unit uses the dictionary formed by a Hadamard product of a matrix whose components differ depending on the number of subbands and a matrix whose components differ depending on the angle of arrival.

[0012] The spectrum estimation system according to claim 3 is the spectrum estimation system according to claim 1 or claim 2, characterized in that the calculation unit calculates the tensor based on received signals received by two or more antennas selected from the antennas.

[0013] A spectrum estimation system according to claim 4 is the spectrum estimation system according to any one of claims 1 to 3, characterized in that the calculation unit calculates the three-dimensional tensor made up of a plurality of the matrices.

[0014] The spectrum estimation system according to claim 5 is the spectrum estimation system according to any one of claims 1 to 4, wherein the estimation unit calculates a difference value between a tensor calculated by the calculation unit and a pseudo-generated tensor, calculates a correlation between the calculated difference value and the dictionary, and repeats a plurality of times to estimate the frequency spectrum and angular spectrum of the received signal based on the subband of the frequency spectrum and the angle of arrival having the largest calculated correlation.

[0015] A spectrum estimation system according to claim 6 is a spectrum estimation system for estimating a frequency spectrum of a received signal received by an array antenna and an angular spectrum of an angle of arrival of the received signal, comprising: N receiving the received signal el an array antenna consisting of antennas; The delay time τ connected to each of the antennas p delay elements with different The signals are received by P antennas selected from the array antenna. , delayed by the delay element A calculation unit that calculates a three-dimensional tensor expressed by the following [Equation 1] based on each received signal; an estimation unit that estimates the angular spectrum and frequency spectrum of the received signal by finding a sparse solution of [Equation 3] below using the three-dimensional tensor calculated by the calculation unit and a two-dimensional tensor expressed by the following [Equation 2] A spectral estimation system comprising:

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[0016] According to the first to fifth inventions, the frequency spectrum and angular spectrum are estimated using compressed sensing based on a dictionary and a tensor. This makes it possible to reduce the ADC sampling frequency required for wideband frequency spectrum estimation, thereby reducing costs and the amount of calculations. Furthermore, it becomes possible to estimate the angular spectrum, allowing more spectral information to be grasped.

[0017] In particular, according to the second aspect of the present invention, the estimation unit uses a dictionary formed by the Hadamard product of a matrix whose components vary depending on the number of subbands and a matrix whose components vary depending on the angle of arrival, which makes it possible to reduce the ADC sampling frequency required for wider frequency spectrum estimation, thereby reducing costs and the amount of calculation.

[0018] In particular, according to the third aspect of the present invention, the calculation unit calculates a tensor based on signals received by two or more antennas randomly selected from among the antennas, thereby reducing ambiguity in searching for a spectral peak and enabling highly accurate spectrum estimation.

[0019] In particular, according to the fourth aspect of the present invention, the calculation unit calculates a three-dimensional tensor made up of a plurality of matrices, which makes it possible to estimate a spectrum based on a three-dimensional tensor made up of a plurality of snapshots, thereby enabling the spectrum to be estimated with high accuracy.

[0020] In particular, according to the fifth aspect of the present invention, the estimation unit repeats estimating the frequency spectrum and angular spectrum of the received signal multiple times in accordance with the calculated complex amplitude, thereby enabling highly accurate spectrum estimation using compressed sensing.

[0021] According to the sixth aspect of the present invention, the estimation unit estimates the angular spectrum and frequency spectrum of the received signal by using the two-dimensional tensor shown in [Equation 2]. This makes it possible to reduce the ADC sampling frequency required for wideband frequency spectrum estimation, thereby reducing costs and the amount of calculations. Furthermore, it becomes possible to estimate the angular spectrum, making it possible to grasp more spectral information. Furthermore, by finding a sparse solution to [Equation 3], it is possible to reduce costs and the amount of calculations. [Brief explanation of the drawings]

[0022] [Figure 1] FIG. 1 is a schematic diagram of the entire spectrum estimation system to which the present invention is applied. [Figure 2] FIG. 2 is a diagram showing the angles of arrival of received signals. [Figure 3] FIG. 3 is a diagram showing a schematic diagram of a three-dimensional tensor S(i,:,:). [Figure 4] FIG. 4 is a flowchart of the restoration algorithm. DETAILED DESCRIPTION OF THE INVENTION

[0023] A spectrum estimation system to which the present invention is applied will be described in detail below with reference to the drawings.

[0024] 1 is a schematic diagram of an overall spectrum estimation system 100 to which the present invention is applied. The spectrum estimation system 100 includes a base station 20 and a spectrum estimation device 1 that receives a signal transmitted from the base station 20.

[0025] The base station 20 serves as a wireless access point between communication devices and as an interface with public communication networks such as the Internet. That is, the base station 20 serves as a relay means that enables communication devices to transmit and receive data to and from public communication networks such as the Internet. The base station 20 transmits a signal to the spectrum estimating device 1.

[0026] The spectrum estimation device 1 receives a signal transmitted from a base station 20 and estimates the frequency spectrum and angular spectrum of the received signal. The spectrum estimation device 1 includes an array antenna 10 having a plurality of antenna elements, a plurality of delay elements 11 connected to each of the antenna elements of the array antenna 10, a plurality of ADCs 12 connected to each of the delay elements 11, a selection unit 13 connected to each of the ADCs 12, a processing unit 14 connected to the selection unit 13, an estimation unit 15 connected to the processing unit 14, and a storage unit 16 connected to the estimation unit 15.

[0027] The array antenna 10 is composed of a plurality of element antennas arranged to receive signals transmitted from the base station 20. The array antenna 10 is configured by N antenna elements arranged such that the spacing between the antenna elements is d. el The array antenna 10 outputs a received signal to a delay element 11.

[0028] The delay element 11 is composed of a delay circuit that delays the signal output from the array antenna 10. The delay element 11 has delay circuits that have different signal delay times connected to each antenna element included in the array antenna 10. The delay element 11 outputs the delayed signal to the ADC 12.

[0029] The ADC 12 samples the signal output from the delay element 11 and performs digital conversion. One ADC 12 is connected to each delay circuit of the delay element 11. For example, a converter with a conversion speed of 50 MHz is used as the ADC 12. The ADC 12 outputs the digitally converted signal to the selection unit 13.

[0030] The selection unit 13 selects a predetermined number of digital signals from among the plurality of digital signals respectively output from the ADC 12, and outputs the selected digital signals to the processing unit 14.

[0031] The processing unit 14 performs a discrete Fourier transform on each digital signal output from the selection unit 13, and calculates a tensor composed of a matrix based on each signal after the discrete Fourier transform. The processing unit 14 outputs the calculated tensor to the estimation unit 15.

[0032] The storage unit 16 stores various information including a dictionary of frequencies and arrival angles for estimating the frequency spectrum and the angle spectrum. The storage unit 16 outputs the pre-stored dictionary of frequencies and arrival angles to the estimation unit 15 as necessary.

[0033] The estimation unit 15 estimates the frequency spectrum and the angle spectrum of the received signal using compressed sensing based on the tensor output from the processing unit 14 and the dictionary output from the storage unit 16. Compressed Sensing (CS) is a method of reconstructing a signal from fewer observations than the originally required number of samplings. When the original signal is taken as a vector x ∈ R n when using an observation matrix Φ of size m × n (m < n), the observed (compressed) signal vector is y = Φ x ∈ R m becomes. At this time, if the dimension of the observed signal vector y is lower than the dimension of the original signal vector x, it becomes an ill-posed problem where the solution cannot be uniquely obtained, and the original signal vector x cannot be obtained from the observed signal vector y. However, when the original signal vector x can be converted into a signal vector having sparsity by using a dictionary matrix, the original signal can be restored by solving the signal vector so that it contains many zeros, spaces or sparseness, that is, by obtaining a sparse solution.

[0034] Next, the operation of estimating the frequency spectrum and the angle spectrum using the spectrum estimation system 100 will be described.

[0035] A frequency spectrum is a spectrum that indicates the intensity of each frequency of a signal. The frequency spectrum is divided into, for example, L subbands. In this case, if the spectral bandwidth is B, the bandwidth of one subband is B / L. The frequency spectrum may be a spectrum that indicates the intensity of frequencies included in the bandwidth of each subband.

[0036] The angular spectrum is a spectrum that indicates the intensity for each angle of the arrival angle of the signal received by the array antenna 10. As shown in Fig. 2, for example, the arrival angle is the angle φ of the direction γ perpendicular to the direction β in which the antennas 10a to 10c are arranged, with respect to the line α connecting the base station 20 and the array antenna 10. The arrival angle is also referred to as the AOA (Angle of Arrival).

[0037] First, in the spectrum estimation system 100, the array antenna 10 receives a signal transmitted from the base station 20. In this case, N el Each of the antenna elements in the array antenna 10 receives a signal. el The received signals are assigned numbers selected in order from {-1}. The numbers may be selected, for example, based on the order of the antennas, but are not limited to this and may be selected in any manner. The array antenna 10 outputs the received signals to the delay elements 11, respectively.

[0038] Next, the delay element 11 delays the output received signal. The delay time of each element provided in the delay element 11 is set to τ0 to τ0 according to the number of each connected antenna element. Nel-1 Also, τ p is shown by [Equation 5].

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[0039] Next, the ADC 12 converts the output received signal into a digital signal. The ADC 12 may perform sampling so that the sampling frequency is B / L, for example. The ADC 12 outputs the converted digital signal to the selection unit 13.

[0040] Next, the selector 13 selects a predetermined number of digital signals from the plurality of digital signals output from the ADC 12. For example, the selector 13 selects a predetermined number of digital signals from N el Alternatively, P antenna elements may be selected from the antenna elements, and a digital signal based on the received signal received by the selected antenna elements may be selected. In this case, the selector 13 may randomly select the P antenna elements using any random number. This reduces the ambiguity of the peak search in the restoration algorithm described below. The selector 13 outputs the selected signal to the processor 14.

[0041] Next, the processing unit 14 performs a discrete Fourier transform on the signals output from the selection unit 13, and calculates a tensor consisting of a matrix based on each of the discrete Fourier transformed signals. The processing unit 14 processes the signals, for example, and calculates a three-dimensional tensor as shown in [Equation 1]. Y where Y p [k] is y p [n] is the received signal obtained by the discrete Fourier transform, and y p [n] indicates the received signal received by the pth antenna, n indicates the sampling point of the sampled received signal, and k indicates the discrete Fourier transformed y p [n] denotes the sample points, and N b is Y p where k is the number of sampling points, L is the number of subbands in the frequency spectrum, and c p denotes an integer selected from the set {0···L-1}, and N s indicates the number of snapshots of the matrix. 3D tensor Yis the output signal y of ADC12 p [n] is subjected to a discrete Fourier transform and phase rotation, and then N b The samples of the frequency bins and P antenna elements are combined into one matrix, and the combined matrix is ​​then expressed as N s This allows processing without using correlation calculations, making it possible to estimate the complex amplitudes of the frequency spectrum and the angular spectrum.

[0042] In this case, the processing unit 14 first performs a discrete Fourier transform to obtain the received signal y p Then, the processing unit 14 calculates a matrix consisting of rows based on frequency bins with different sampling points based on received signals received by the same antenna, and columns based on frequency bins with the same sampling points based on received signals received by different antennas. s 3D tensor based on matrices Y In this case, N s The matrices may be snapshots of the same matrix, or may be a three-dimensional tensor consisting of multiple different matrices generated based on different received signals received by the array antenna 10 at different times. Y Also, N s is an integer equal to or greater than 1, and N s may be 1. In this case, the tensor expressed by [Equation 1] may be a two-dimensional tensor. The processing unit 14 outputs the calculated tensor to the estimation unit 15.

[0043] Next, the estimation unit 15 estimates the frequency spectrum and angular spectrum using compressed sensing based on the tensor output from the processing unit 14 and the dictionary of frequencies and angles of arrival output from the storage unit 16. For example, the estimation unit 15 estimates the frequency spectrum and angular spectrum of the received signal using the spectrum compressed by the processing unit 14 and the dictionary of frequencies and angles of arrival. For example, the estimation unit 15 estimates the frequency spectrum and angular spectrum of the received signal based on the three-dimensional tensor or third-order tensor shown in [Equation 1] output from the processing unit 14 and the dictionary shown in [Equation 2] output from the storage unit 16. In this case, the estimation unit 15 estimates the angular spectrum and frequency spectrum of the received signal by finding a sparse solution of [Equation 3] using the three-dimensional tensor shown in [Equation 1] and the dictionary of two-dimensional tensors shown in [Equation 2]. Here, f k denotes the frequency of the frequency bin of the kth sampling point, c denotes the speed of light, and φ g denotes the candidate value of the gth arrival angle, and C p is the set {0···N el −1}, d denotes the spacing between the array antennas, and l denotes the l-th subband. n denotes the Mode-n tensor-by-matrix product, S is expressed by the following [Equation 4]. g is from φ0 to φ Nφ-1 N up to φ candidate values ​​of the arrival angle may be prepared.

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[0044] Figure 3 shows a three-dimensional tensor SFIG. 1 is a diagram showing a schematic diagram of (i,:,:). S (1,:,:) is a 3-dimensional tensor S is cut along the vertical axis y, and the two-dimensional tensor of the first cut xz plane is shown. S (2,:,:) is a 3-dimensional tensor S is cut along the vertical axis y, and the two-dimensional tensor of the second cut xz plane is shown.

[0045] The two-dimensional tensor dictionary shown in [Equation 2] is a dictionary of frequencies and angles of arrival. The dictionary of frequencies and angles of arrival is based on compressed signals corresponding to each frequency subband and steering vectors for each angle of arrival. The steering vector is a vector indicating the phase relationship between the antenna elements of the array antenna 10. The dictionary of frequencies and angles of arrival may be, for example, a matrix formed by the Hadamard product of a matrix whose components vary depending on the number of subbands in the frequency spectrum and a matrix whose components vary depending on the angle of arrival. Using this dictionary makes it possible to reduce the ADC sampling frequency required for wideband frequency spectrum estimation, thereby reducing costs and the amount of calculation. Furthermore, it becomes possible to estimate the angular spectrum, allowing for the acquisition of more spectral information.

[0046] The estimation unit 15 may estimate a sparse solution to the equation shown in [Mathematical Expression 3], for example, using the compressed sensing iterative algorithm shown in Fig. 4. The estimation unit 15 calculates the difference between the tensor output from the processing unit 14 and the pseudo-generated tensor, calculates the correlation between the calculated difference and the above-mentioned dictionary, and repeats the process of estimating the angular spectrum and frequency spectrum of the received signal multiple times based on the subband of the frequency spectrum and the angle of arrival with the greatest calculated correlation. Detailed processing at each step in Fig. 4 will be described below.

[0047] First, in S10, the difference value between the three-dimensional tensor shown in [Equation 1] and the pseudo-generated three-dimensional tensor is calculated. R In this case, the three-dimensional tensor shown in [Equation 1] may be used as it is without calculating the difference value.

[0048] Next, in step S11, the difference value R and the correlation between the dictionary shown in [Equation 2] and [Equation 6] C Ask for.

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[0049] Next, in step S12, it is determined whether a stopping condition is met. If an error is included in the estimated spectrum, it will cause continuous errors when generating a pseudo-tensor, as will be described later. Therefore, when the difference value R is calculated in step S10, the detected signal cannot be completely removed, and a portion of the signal remains. This remaining signal may be detected as an erroneous signal in a later iteration. Therefore, in order to reduce the false alarm rate, a stopping condition is determined. If the stopping condition is met in step S12, the algorithm loop is interrupted.

[0050] The stopping condition is correlation C The value is determined by the frequency bin of the signal and the magnitude of the threshold γ given by [Equation 7].

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[0051] Also, it is determined whether the stopping condition satisfies the equation shown in [Equation 8].

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[0052] Next, in step S13, the subband and angle with the greatest correlation are searched for. In this case, the subband and angle with the greatest correlation are searched for using, for example, the formula shown in [Equation 9].

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[0053] Next, in step S14, the complex amplitude W of the spectrum is calculated. q In this case, for example, the complex amplitude W of the spectrum is estimated using the formula shown in [Mathematical formula 10]. q Estimate.

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[0054] Next, in step S15, it is determined whether the number of iterations exceeds the maximum number of iterations. If the number of iterations exceeds the maximum number of iterations, the algorithm loop is interrupted. If the number of iterations does not exceed the maximum number of iterations, the process proceeds to step S16, which will be described later.

[0055] Next, in step S16, a pseudo three-dimensional tensor is generated. Then, the process returns to step S10, where the differential value is calculated using the formula shown in [Mathematical formula 11]. R Calculate.

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[0056] This iterative algorithm is an algorithm that estimates a frequency spectrum and an angular spectrum that satisfy a condition by repeating steps S10 to S16 multiple times while increasing the value of k by 1. This makes it possible to estimate the spectrum with high accuracy using compressed sensing. Furthermore, the estimation unit 15 may estimate the frequency spectrum and the angular spectrum using any method other than compressed sensing using this iterative algorithm.

[0057] By performing the above-described operations, the operation of the spectrum estimation system 100 according to this embodiment is completed. This makes it possible to reduce the ADC sampling frequency required for wideband frequency spectrum estimation, thereby reducing costs and the amount of calculations. Furthermore, it becomes possible to estimate the angular spectrum, allowing more spectral information to be obtained.

[0058] Although an embodiment of the present invention has been described, this embodiment is presented as an example and is not intended to limit the scope of the invention. Such novel embodiments can be embodied in various other forms, and various omissions, substitutions, and modifications can be made without departing from the spirit of the invention. This embodiment and its modifications are included within the scope and spirit of the invention, and are also included in the inventions described in the claims and their equivalents. [Explanation of symbols]

[0059] 1. Spectral estimator 10 Array Antenna 11 Delay element 12 ADC 13 Selection section 14 Processing section 15 Estimation part 16 Memory section 20 base station 100 Spectral Estimation System

Claims

1. 1. A spectrum estimation system for estimating a frequency spectrum of a received signal received by an array antenna and an angular spectrum of an angle of arrival of the received signal, comprising: an array antenna including a plurality of antennas for receiving the received signals; delay elements each having a different delay time, each connected to the antenna; a calculation unit that performs a discrete Fourier transform on the received signals received by the antennas and delayed by the delay elements, and calculates a tensor consisting of a matrix based on the discrete Fourier transformed received signals; an estimation unit that estimates the frequency spectrum and the angular spectrum using compressed sensing based on a dictionary based on compressed signals corresponding to each frequency subband and steering vectors for each angle of arrival, and the tensor calculated by the calculation unit. A spectral estimation system comprising:

2. The estimation unit uses the dictionary formed by a Hadamard product of a matrix whose components differ depending on the number of subbands and a matrix whose components differ depending on the angle of arrival.

2. The spectral estimation system of claim 1 .

3. the calculation unit calculates the tensor based on each of received signals received by two or more antennas selected from the antennas.

3. The spectrum estimation system according to claim 1 or 2, wherein:

4. the calculation unit calculates the three-dimensional tensor made up of a plurality of the matrices.

4. The spectrum estimation system according to claim 1, wherein:

5. the estimation unit calculates a difference value between the tensor calculated by the calculation unit and a pseudo-generated tensor, calculates a correlation between the calculated difference value and the dictionary, and repeats a process of estimating the frequency spectrum and the angular spectrum of the received signal multiple times based on the subband of the frequency spectrum and the angle of arrival for which the calculated correlation is greatest.

5. The spectrum estimation system according to claim 1, wherein:

6. 1. A spectrum estimation system for estimating a frequency spectrum of a received signal received by an array antenna and an angular spectrum of an angle of arrival of the received signal, comprising: N receiving the received signal el an array antenna consisting of antennas; delay elements each having a different delay time τ p connected to the antenna; a calculation unit that calculates a three-dimensional tensor expressed by the following [Equation 1] based on each received signal received by P antennas selected from the array antenna and delayed by the delay element; an estimation unit that estimates the angular spectrum and frequency spectrum of the received signal by obtaining a sparse solution of the following [Equation 3] using the three-dimensional tensor calculated by the calculation unit and a two-dimensional tensor expressed by the following [Equation 2]. A spectral estimation system comprising: [Equation 1] Here, y p [n] denotes the received signal received by the p-th antenna, and Y p [k] is y p [n] represents the received signal obtained by the discrete Fourier transform, n represents the sampling point of the sampled received signal, and k represents the discrete Fourier transformed y p denotes the sample points of [n], and N b Is Y p [k] denotes the number of sampling points, L denotes the number of subbands in the frequency spectrum, and c p denotes an integer selected from the set {0...L-1}, and N s denotes the number of snapshots of the matrix, and τ p is given by the following [Equation 5]. [Equation 2] where f k denotes the frequency of the frequency bin of the kth sampling point, c denotes the speed of light, and φ g indicates the gth arrival angle candidate value, and C p is the set {0...N el −1}, and d represents the spacing between the array antennas. [Equation 3] Here, × n denotes the tensor-matrix product of the n-th mode, and S is given by the following [Equation 4]. [Equation 4] Here, ||| F denotes the Frobenius norm, S(i,:,:) denotes the matrix obtained by dividing an arbitrary three-dimensional tensor S along the first axis and extracting the i-th division, and S l,g [k] is the lth subband and the arrival angle φ g The spectrum at [Equation 5] Here, B denotes the spectral bandwidth.

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