BPSK-FDA-MIMO radar target high-precision angle and distance measurement method

Abstract

The invention provides a BPSK (Binary Phase Shift Keying)-FDA-MIMO (Frequency Division Multiple Input Multiple Output) radar target high-precision angle and distance measurement method, which comprises the following steps of: 1, designing FDA-MIMO radar transmitting signal waveforms in combination with a BPSK coding technology, so that transmitting signals among different array elements are mutually orthogonal; step 2, orthogonal signal separation is carried out on the received signals by using a matched filtering method, and the same array element is separated to receive signals of different array elements; 3, receiving data of the same array element by using different array elements as a data combination, constructing a corresponding transmitting-receiving vector, and performing traversal search on the data combination by using the vector to obtain a target angle dimension estimation result; and 4, constructing a corresponding transmitting-receiving vector by using all data in one pulse, and performing traversal search on a data combination by using the vector to obtain a target distance dimension estimation result. The angle and distance estimated by the method provided by the invention are high in measurement precision.

Description

Technical Field

[0001] This application relates to the technical field of radar signal processing, and particularly relates to a method for high-precision angle and range measurement of targets by a BPSK-FDA-MIMO radar. Background Art

[0002] A Frequency Diverse Array (FDA), also known as a frequency diversity radar, is different from a phased array in that the carrier frequencies of the transmitted signals of the array elements are increasing, and this frequency increment is fixed and much smaller than the transmitted carrier frequency. Due to the existence of this frequency increment, range information is introduced into the array antenna. Different from the phased linear array pattern being only controlled by angle, the FDA transmitted pattern is jointly controlled by angle and range, presenting a periodic distortion in space, so that the radiated spatial beam has a range-angle dependence characteristic.

[0003] Since the FDA transmitted beam has coupling in the two dimensions of angle-range, it is impossible to accurately locate the target. In order to decouple it, a dual-pulse method has been proposed for target positioning, that is, the first pulse transmits a single-carrier frequency signal. At this time, the array can be equivalent to a phased array, so as to measure the angle of the target. The second pulse is transmitted in the form of an FDA, and the range of the target is measured according to the echo signal. However, this method requires transmitting two pulses, which has limitations in time, and due to the small number of degrees of freedom of the FDA radar and limited received data, this method needs to be improved in terms of measurement accuracy. Summary of the Invention

[0004] This application provides a method for high-precision angle and range measurement of targets by a BPSK-FDA-MIMO radar, which can be used to solve the technical problem of low precision of existing methods. The method provided by this application uses each array element to transmit mutually orthogonal BPSK codes, that is, the transmitted signals are orthogonal signals, so that each array element in the receiving array can separate the signals transmitted by different array elements, and realize the separation of the target echo signals. Furthermore, combined with the FDA-MIMO radar system, the separated received echo signals are processed in different dimensions. Using the echo data of the same array element received by different array elements, the target angle information is estimated with high precision, and then using the echo data of different array elements received by different array elements, the target range information is estimated with high precision, completing the decoupling of the target range-angle, and finally realizing high-precision angle and range measurement of the target.

[0005] This application provides a method for high-precision angle and range measurement of targets by a BPSK-FDA-MIMO radar, and the method includes:

[0006] Step 1: Combining the BPSK coding technology, designing the transmitted signal waveform of the FDA-MIMO radar to make the transmitted signals between different array elements orthogonal;

[0007] Step 2: Use the matched filtering method to separate the orthogonal signals of the received signals, and separate the signals received by different array elements from the same array element;

[0008] Step 3: Estimate the angle of the target: Use the data received by different array elements from the same array element as the data combination γ, construct the corresponding transmit-receive vector, and use the transmit-receive vector to perform a traversal search on the data combination γ to obtain the estimated result of the target angle dimension

[0009] Step 4: Estimate the target distance: Use all the data within one pulse, that is, the data received by different array elements from different array elements as the data combination λ, construct the corresponding transmit-receive vector, and use the transmit-receive vector to perform a traversal search on the data combination λ to obtain the estimated result of the target distance dimension

[0010] The present invention has the following advantages:

[0011] (1) By combining BPSK coding and MIMO technology, the present invention designs the transmit signals between array elements to be orthogonal signals, which is conducive to separating the received signals through de-carrier frequency processing and matched filtering processing, that is, separating the signals received by different array elements from the same array element.

[0012] (2) Although the target angle information estimation and distance information estimation of the present invention are carried out separately, only the data of one pulse is required, and multiple pulses are not needed.

[0013] (3) Under the condition of very low signal-to-noise ratio, the present invention can achieve high accuracy of the angle and distance estimation results and high measurement accuracy. Description of the Drawings

[0014] Figure 1 is the flowchart of angle and distance measurement of the BPSK-based FDA-MIMO radar provided by the embodiment of the present application.

[0015] Figure 2 is the basic structure of the FDA radar provided by the embodiment of the present application.

[0016] Figure 3 is the BPSK signal coding waveform and transmit signal waveform diagram provided by the embodiment of the present application.

[0017] Figure 4 is the autocorrelation diagram of the single-array element transmit signal and the cross-correlation diagram of the transmit signals between array elements provided by the embodiment of the present application.

[0018] Figure 5 is the sampling waveform diagram of the echo signal of the first receiving array element within 1 PRI provided by the embodiment of the present application.

[0019] Figure 6 It is the waveform diagram of the echo signal after the de - carrier processing provided by the embodiment of the present application.

[0020] Figure 7 It is the time - domain diagram after the matched filtering processing provided by the embodiment of the present application.

[0021] Figure 8 It is the estimation result diagram of the target angle and distance provided by the embodiment of the present application;

[0022] Among them, Figure 8 (a) is the target angle estimation result diagram; Figure 8 (b) is the target distance estimation result diagram; Figure 9 (a) is the target angle estimation result diagram;

[0023] Figure 9 It is the angle and distance estimation result diagram at a fixed distance or a fixed angle provided by the embodiment of the present application.

[0024] Figure 10 It is the stability degree diagram of the method at different angles and distances provided by the embodiment of the present application.

[0025] Figure 11 It is the estimation performance diagram of the target angle and distance under different signal - to - noise ratios provided by the embodiment of the present application. Detailed implementation manners

[0026] To make the objectives, technical solutions and advantages of the present application clearer, the following will further describe the embodiments of the present application in detail with reference to the accompanying drawings.

[0027] The following first introduces the embodiments of the present application with reference to the accompanying drawings.

[0028] The present application provides a high - precision angle and distance measurement method for BPSK - FDA - MIMO radar targets. The method includes:

[0029] Step 1: Combining the BPSK coding technology, design the transmitting signal waveform of the FDA - MIMO radar so that the transmitting signals between different array elements are orthogonal to each other;

[0030] Step 2: Use the matched filtering method to separate the orthogonal signals of the received signals, and separate the signals received by the same array element from different array elements;

[0031] Step 3: Estimate the angle of the target: Use the data received by different array elements from the same array element as the data combination γ, construct the corresponding transmit - receive vector, and perform a traversal search on the data combination γ using the transmit - receive vector to obtain the target angle - dimension estimation result

[0032] Step 4: Estimate the target distance: Use all the data within a pulse, that is, the data received by different array elements as the data combination λ, construct the corresponding transmit-receive vector, and perform a traversal search on the data combination λ using the transmit-receive vector to obtain the target distance dimension estimation result. Step 1: Combine the BPSK coding technology to design the transmit signal waveform of the FDA-MIMO radar so that the transmit signals between different array elements are orthogonal to each other.

[0033] When the number of transmit array elements is M, the transmit signal s m (t) of the m-th element of the FDA radar is:

[0034] s m (t) = u(t)d m (t)exp(j2πf m t), m = 1, 2,...M (1)

[0035] Where, T is the transmit pulse width, d m (t) is the symbol sequence after the m-th element maps "0" and "1" to "-1" and "1"; f m is the carrier frequency of the transmit signal of the m-th element:

[0036] f m = f0 + (m - 1)Δf, m = 1, 2,…,M (2)

[0037] Where, f0 is the carrier frequency of the reference element of the FDA radar, that is, the first element, and Δf is the small frequency step introduced between the elements of the transmit signal, and numerically Δf is much smaller than the carrier frequency;

[0038] If each element requires l symbols as the transmit symbols, then M elements require Ml symbols. Therefore, design an m-sequence with a length of Ml symbols, and divide this m-sequence into M equal parts, each part with l symbols, as the transmit symbols d m (t) of the m elements, m = 1, 2,...M; Utilize the autocorrelation function characteristics and cross-correlation function characteristics of the m-sequence, and the BPSK codings d m (t) of different elements, m = 1, 2,...M satisfy orthogonal sequences and are used as the transmit signal symbols of the FDA-MIMO radar.

[0039] Step 2: Use the matched filtering method to separate the orthogonal signals of the received signals and separate the signals received by the same element from different elements.

[0040] Step 21, the FDA radar transceiver shares one array, i.e., self-transmitting and self-receiving. Then, the signal transmitted by the m-th array element reaches the far-field target, and the echo reflected by the target signal is received by the n-th receiving array element. The time delay is expressed as:

[0041]

[0042] where c represents the speed of light, r represents the distance between the target and the first array element, θ represents the azimuth angle of the target, and d represents the array element spacing;

[0043] Then, the signal received by the n-th receiving array element from the m-th transmitting array element is expressed as s m,n (t):

[0044] s m,n (t) = u(t - τ m,n )d m (t - τ m,n )exp(j2πf m (t - τ m,n )) (4)

[0045] where m, n = 1, 2, …, M, u(t - τ m,n ) represents the time delay of u(t) by τ m,n , and d m (t - τ m,n ) represents the time delay of d m (t) by τ m,n ;

[0046] Step 22, orthogonal signal separation is performed by the matched filtering method, i.e., separating the signals received by the same array element from different array elements. The specific operation is as follows:

[0047] Step 221, the received signal s m,n (t) of each receiving array element is de-carrier processed; the signal expression after processing is:

[0048] y m,n (t) = u(t - τ m,n )d m (t - τ m,n )exp(j2π(t - τ m,n ), m = 1, 2, …, M, n = 1, 2, …, M (5)

[0049] Step 222, perform matched filtering processing on each pulse. Take the matched filter as the anti-fold conjugate of the BPSK coding d m (t) of different transmitting array elements, m = 1, 2,...M; design M different matched filters for each receiving array element to separate M different transmitted signals. Then, the m-th matched filter h m(t) is expressed as:

[0050] h m (t) = (u(t)d m (t)) * , m = 1, 2,... M (6)

[0051] Among them, () * represents the conjugate operation;

[0052] After performing matched filtering on the received signal, the m-th signal separated by the n-th receiving array element after passing through the filter is expressed as:

[0053]

[0054] Among them, represents the convolution operator, and h m,n (t) is the m-th matched filter of the n-th receiving array element.

[0055] Step 3: Estimate the angle of the target: Use the data received by different array elements from the same array element as the data combination γ, construct the corresponding transmit-receive vector, and perform a traversal search on the data combination γ using the transmit-receive vector to obtain the target angle dimension estimation result

[0056] Considering the distance and azimuth of the far-field target, that is, the distance between the target and the first array element is r, the azimuth angle is θ, and the array element spacing is d, the transmit steering vector of the FDA-MIMO radar is expressed as:

[0057]

[0058] Using Step 2, separate the signals of different array elements received by the same array element. Therefore, the receive steering vector of the FDA-MIMO radar is expressed as an M×M matrix. The first M represents the transmit array elements, and the second M represents the receive array elements. The receive steering vector is expressed as:

[0059]

[0060] Among them, represents the receive vector of the n-th array element receiving the signal transmitted by the m-th array element; then the received signal s m,n (t) is re-expressed as:

[0061] s m,n (t) = α s u m,n (θ, r) + n(t) (10)

[0062] Among them, α s represents the amplitude of the target signal, n(t) represents white noise, and u m,n(θ,r) is the element in the m-th row and n-th column of the transmit-receive steering vector u(θ,r) of the radar. The expression of u(θ,r) is:

[0063] u(θ,r) = w T H a(θ,r)b(θ,r) (11)

[0064] where w T = a(θ,r) is equivalent to the transmit beamforming weight, and (·) H represents the conjugate transpose operation;

[0065] To estimate the angle information of the target, the signals received by different array elements from the transmission of the same array element are extracted for processing. Among them, the transmission signal of the first array element received by all array elements is represented by the peak point after matched filtering of the corresponding data, that is, x in formula (7) is extracted 1,1 ,x 1,2 ,…,x 1,M , let:

[0066] y1(θ0,r0) = [x 1,1 ,x 1,2 ,…,x 1,M (12)

[0067] Then b in formula (9) is extracted 1,1 ,b 1,2 ,…,b 1,M Combined into γ1, let:

[0068] γ1(θ,r) = [b 1,1 ,b 1,2 ,…,b 1,M (13)

[0069] At this time, the receive-transmit vector u1(θ,r) is expressed as:

[0070] u1(θ,r) = (w T H a(θ,r))γ1(θ,r) (14)

[0071] At this time, u1(θ,r) does not contain range information, and the unknown quantity is only the angle. Therefore, the angle information is extracted from it. Let w1 = u1(θ), and spectral search is performed using formula (15) to obtain the angle information of the target

[0072]

[0073] Select the transmission signals of the m-th array element received by each receive array element for combination, and search for the angle of the target. There are M groups of data that can be used, which greatly improves the angle estimation accuracy.

[0074] Step 4: Estimate the target distance: Use all the data within a pulse, that is, the data received by different array elements as the data combination λ, construct the corresponding transmit-receive vector, and perform a traversal search on the data combination λ using the transmit-receive vector to obtain the target distance dimension estimation result

[0075] After completing the angle estimation using Step 3, only the distance parameter in the FDA-MIMO radar receive vector is unknown. Therefore, rearrange b(θ,r) to obtain an M 2 ×1 one-dimensional matrix and set it equal to λ:

[0076] λ = [b 1,1 , b 2,1 , …, b M,1 , b 1,2 , b 2,2 , …, b M,2 , …, b 1,M , b 2,M , …, b M,M T (16)

[0077] At this time, use all the data within a pulse, that is, the data of M receiving array elements. The peak points after each array element's received data passes through M matched filters total M×M peak points. As shown in Equation (7), rearrange them and denote as y2(θ0,r0):

[0078] y2(θ0,r0) = [x 1,1 , x 2,1 , …, x M,1 , x 1,2 , x 2,2 , …, x M,2 , …, x 1,M , x 2,M , …, x M,M T (17)

[0079] At this time, the receive-transmit vector u2(θ,r) is expressed as:

[0080] u2(θ,r) = (w T H a(θ,r))λ(θ,r) (18)

[0081] Since the angle parameter in u2 is the same as that in u1, so let Use the spectral search method of Equation (19) to perform a traversal search to obtain the estimated value of the target distance

[0082] ​​

[0083] At this time, both the angle parameter and the distance parameter of the target are estimated.

[0084] Embodiment

[0085] The effects of the present invention are further illustrated by the following simulation experiments.

[0086] In the index design of the simulation experiment, the target distance is set at 4005 m and the angle is 17.62°. According to the design method of the specific implementation, the FDA-MIMO radar array parameters and the transmitted waveform parameters are designed as shown in Table 1:

[0087] Table 1

[0088]

[0089] Design the radar transmitted waveform according to Table 1, and the results are as Figure 3 shown. Among them, the left figure is the signal coding waveform, and the right figure is the transmitted signal waveform. In order to verify that the signals transmitted by each array element are orthogonal signals, they are tested, and the results are as Figure 4 shown. It can be seen that the autocorrelation result of the signal transmitted by a single array element is 1, while the cross-correlation result of the signals transmitted between array elements is relatively low, indicating that the designed transmitted signal has good autocorrelation performance and cross-correlation performance, and is suitable for the orthogonal signal of the FDA-MIMO radar.

[0090] According to Step 2, use the matched filter to separate the received signals of the array. Among them, the sampling waveform of the echo signal of the first receiving array element within 1 PRI is as Figure 5 shown; perform the de-carrier frequency processing on it, and the results are as Figure 6 shown; finally perform the matched filtering on it to separate the signals received by different array elements of the first array element, and the separated results are as Figure 7 shown.

[0091] When the SNR is 0 dB, Figure 8 this method is used to perform positioning estimation on a target with an angle of 17.62° and a distance of 4005 m, where Figure 8 (a) is a schematic diagram of the estimated result of the target angle information, Figure 8 (b) is a schematic diagram of the estimated result of the target distance information. It can be seen that the maximum values of the two figures correspond to the target angle and distance, indicating that this method can correctly and accurately measure the target angle and distance.

[0092] Although Figure 8The results prove that the method has good performance in measuring the target angle and distance. However, the amount of data is relatively small. Therefore, experiments were conducted by changing the target distance at a fixed target angle and changing the target angle at a fixed target distance. To ensure the correctness and authenticity of the data, 1000 Monte Carlo experiments were performed for each simulation condition. The results are as Figure 9 shown. Among them, Figure 9 (a) shows the angle estimation results at different distances, Figure 9 (b) shows the distance estimation results at different angles. It can be seen that the estimation results of this method are close to the true values. In terms of the mean values of the estimated angle and distance, this method has good performance.

[0093] To verify the stability of this method, multiple measurements were taken for targets at different positions, and the standard deviation (SD) of the estimation results was used as a parameter to judge the stability. The results are as Figure 10 shown. Among them, Figure 10 (a) shows the results of obtaining the SD values by performing 1000 Monte Carlo experiments when the target is at different angles, Figure 10 (b) shows the results of obtaining the SD values by performing 1000 Monte Carlo experiments when the target is at different distances. It can be seen that this method has high stability in the angle dimension estimation and the distance dimension estimation, and the results are also satisfactory.

[0094] Estimation performance of the azimuth angle and distance of the target relative to SNR: Figure 11 Shows the angle estimation performance of this method at different signal-to-noise ratios. Among them, Figure 11 (a) shows the azimuth angle estimation results of the target at different SNRs, Figure 11 (b) shows the distance estimation results of the target at different SNRs. The results show that this method can adapt to multi-signal-to-noise ratio situations, and the angle and distance measurement results are relatively accurate and stable.

[0095] The embodiments of the present application described above do not constitute a limitation to the protection scope of the present application.

Claims

1. A high-precision angle and distance measurement method for BPSK-FDA-MIMO radar targets, characterized in that The method includes: Step 1: Combining with the BPSK coding technology, design the transmit signal waveform of the FDA-MIMO radar to make the transmit signals between different array elements orthogonal to each other. Step 2: Use the matched filtering method to separate the orthogonal signals of the received signals, separating the signals received by the same array element from different array elements. Step 3: Estimate the angle of the target: Use different array elements to receive the data of the same array element as the data combination γ, construct the corresponding transmit-receive vector, and perform a traversal search on the data combination γ using the transmit-receive vector to obtain the estimated result of the target angle dimension Step 4: Estimate the target distance: Use all the data within one pulse, that is, the data received by different array elements as the data combination λ, construct the corresponding transmit-receive vector, and perform a traversal search on the data combination λ using the transmit-receive vector to obtain the target distance dimension estimation result 2. The method according to claim 1, wherein Step 1: Combining with the BPSK coding technology, design the transmit signal waveform of the FDA-MIMO radar to make the transmit signals between different array elements orthogonal to each other, including: When the number of elements in the transmitting array is M, the transmitted signal s m (t) of the m-th element of the FDA radar is s m (t) = u(t)d m (t) exp(j2πf m t), m = 1, 2,... M (1) Among them, T is the transmission pulse width, d m (t) is the symbol sequence after the m-th array element maps "0" and "1" to "-1" and "1"; f m is the carrier frequency of the signal transmitted by the m-th array element: f m = f0 + (m - 1)Δf, m = 1, 2, …, M (2) Where f0 is the carrier frequency of the reference array element of the FDA radar, that is, the first array element, and Δf is the small frequency step introduced between the array elements of the transmit signal, and numerically Δf is much smaller than the carrier frequency. If each array element requires l code elements as transmitted code elements, then M array elements require Ml code elements. Therefore, an m-sequence with a length of Ml code elements is designed, and the m-sequence is equally divided into M parts, each part having l code elements, which serve as the transmitted code elements d m (t), where m = 1, 2,... M; by utilizing the autocorrelation function characteristics and cross-correlation function characteristics of the m-sequence, the BPSK coding d m (t), where m = 1, 2,... M satisfies the orthogonal sequence and is used as the transmitted signal code element of the FDA-MIMO radar.

3. The method according to claim 2, wherein Step 2: Use the matched filtering method to separate the orthogonal signals of the received signals, separating the signals received by the same array element from different array elements, including: Step 21: The FDA radar uses the same array for transmitting and receiving, that is, self-transmitting and self-receiving. Then the signal transmitted by the m-th array element reaches the far-field target signal, and then the target signal is reflected back to the n-th receiving array element for reception. The time delay is expressed as: Where c represents the speed of light, r represents the distance between the target and the first array element, θ represents the azimuth angle of the target, and d represents the array element spacing. Then the signal received by the nth receiving array element from the mth transmitting array element is expressed as s m,n (t): s m,n (t) = u(t - τ m,n )d m (t - τ m,n ) exp(j2πf m (t - τ m,n )) (4) where m, n = 1, 2, …, M, u(t - τ m,n ) represents the time delay of τ m,n for u(t), and d m (t - τ m,n ) represents the time delay of τ m for d m,n (t); Step 22: Perform orthogonal signal separation through the matched filtering method, that is, separate the signals received by the same array element from different array elements. Step 221, perform de - frequency - offset processing on the received signal s m,n (t) of each receiving element; the signal expression after processing is obtained as: y m,n h(t) = u(t - τ m,n )d m (t - τ m,n )exp(j2π(t - τ m,n )), m = 1, 2, …, M, n = 1, 2, …, M (5) Step 222, perform matched filtering on each pulse, and take the matched filter as the anti-fold conjugate of the BPSK coding d m (t) of different transmitting array elements, where m = 1, 2,... M; design M different matched filters for each receiving array element to separate M different transmitted signals, then the m-th matched filter h m (t) is expressed as: h m (t) = (u(t)d m (t)) * , m = 1, 2,... M (6) where, () * represents the conjugate operation; After performing matched filtering processing on the received signals, the m-th signal separated by the n-th receiving array element after passing through the filter is expressed as: Among them, represents the convolution operator, and h m,n (t) is the m-th matched filter of the n-th receiving array element.

4. The method according to claim 3, wherein Step 3: Estimate the angle of the target, including: The distance between the target and the first array element is r, the azimuth angle is θ, the array element spacing is d, and the transmit steering vector of the FDA-MIMO radar is expressed as: Separate the signals of different array elements received by the same array element. Therefore, the receive steering vector of the FDA-MIMO radar is expressed as an M×M matrix. The first M represents the transmit array elements, and the second M represents the receive array elements. The receive steering vector is expressed as: Among them, represents the receiving vector for the nth array element to receive the signal transmitted by the mth array element; then the received signal s m,n (t) is re-expressed as: s m,n (t) = α s u m,n (θ, r) + n(t) (10) Among them, α s represents the amplitude of the target signal, n(t) represents white noise, and u m,n (θ,r) is the element in the m-th row and n-th column of the transmit-receive steering vector u(θ,r) of the radar. The expression of u(θ,r) is: u(θ,r) = w T H a(θ,r)b(θ,r) (11) where, w T = a(θ,r) is equivalent to the transmit beamforming weight, and (·) H represents the conjugate transpose operation; To estimate the angle information of the target, signals transmitted by the same array element received by different array elements are extracted for processing; among them, the transmitted signals of the first array element received by all array elements are represented by the peak points after matched filtering of the corresponding data, that is, x in formula (7) is extracted 1,1 , x 1,2 , …, x 1,M , let: y1(θ0,r0) = [x 1,1 , x 1,2 , …, x 1,M (12) Then extract b in formula (9) 1,1 , b 1,2 , …, b 1,M Combine them into γ1, and let: γ1(θ,r) = [b 1,1 ,b 1,2 ,…,b 1,M (13) At this time, the receive-transmit vector u1(θ,r) is expressed as: u1(θ,r) = (w T H a(θ,r))γ1(θ,r) (14) At this time, u1(θ,r) does not contain distance information and the unknown quantity is only the angle. Therefore, the angle information is extracted from it. Let w1 = u1(θ), and use Equation (15) for spectral search to obtain the angle information of the target. Select the transmit signals of the m-th array element received by each receiving array element for combination to search for the angle of the target.

5. The method according to claim 4, characterized in that Step 4: Use all the data within a pulse, that is, the data received by different array elements as the data combination λ, construct the corresponding transmit-receive vector, and perform a traversal search on the data combination λ using the transmit-receive vector to obtain the target range dimension estimation result Rearrange \(b(\theta,r)\) to obtain \(M\). 2 A one-dimensional matrix of \(×1\), let it be equal to \(\lambda\): λ = [b 1,1 , b 2,1 , …, b M,1 , b 1,2 , b 2,2 , …, b M,2 , …, b 1,M , b 2,M , …, b M,M T (16)​ At this time, use all the data within one pulse, that is, the data of M receiving array elements. The peak points of each array element's received data after passing through M matched filters are a total of M×M peak points, as shown in Equation (7). Rearrange them and denote them as y2(θ0,r0): y2(θ0,r0) = [x 1,1 , x 2,1 , …, x M,1 , x 1,2 , x 2,2 , …, x M,2 , …, x 1,M , x 2,M , …, x M,M T (17)​ At this time, the receive-transmit vector u2(θ,r) is expressed as: u2(θ,r) = (w T H a(θ,r))λ(θ,r)(18) Since the angular parameter in u2 is the same as that in u1, let Perform a traversal search using the spectral search method of Equation (19) to obtain an estimated value of the target distance

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