Frequency control array radar RF stealth and target detection method based on frequency offset design

Abstract

The application discloses a frequency control array radar radio frequency stealth and target detection method based on frequency offset design, which comprises the following steps: constructing a transmitting end array and a receiving end array of a frequency control array radar whose frequency offset is greater than the bandwidth of a baseband waveform; presetting a candidate frequency offset number and calculating array element transmitting signals of the transmitting end array; obtaining target echo signals according to the receiving end array structure and the array element transmitting signals; designing a cost function and constructing a radio frequency stealth and target detection integrated beam pattern design problem about a frequency offset selection matrix and a receiving weight vector according to the obtained target echo signals; designing an optimization algorithm according to the constructed beam pattern design problem to obtain an optimal frequency offset selection matrix and a receiving weight vector, and completing beam forming in the distance and angle dimensions to enhance the energy of a target signal. Through the method, the integration of the frequency control array radar radio frequency stealth and target detection is realized.

Description

Technical Field

[0001] This invention relates to the field of radar signal processing technology, and in particular to a method for radio frequency stealth and target detection of frequency-controlled array radar based on frequency offset design. Background Technology

[0002] By introducing a frequency offset (or simply fundamental frequency offset), each transmitting element of a frequency diversity array (FDA, also known as a frequency-controlled array radar) is excited by a carrier wave of a different frequency. The frequency difference between the elements results in a phase difference in the echo signal, thus generating a time-varying transmitted beam. Frequency-controlled array radar has potential advantages in target detection, joint range and angle estimation, and main lobe interference suppression.

[0003] Currently, most radar radio frequency (RF) stealth technologies employ low-sidelobe antenna designs, power control, or complex waveform designs to reduce the probability of interception by enemy passive detection systems, often neglecting the radar's own target detection capabilities. In actual battlefield environments, to ensure radar operational effectiveness, it is crucial to maximize the radar's combat power while simultaneously using RF stealth technology to make the radar's emitted signals difficult for enemy passive detection systems to intercept. Current research on frequency-controlled array radars focuses on two aspects: firstly, active detection, studying key technologies such as target detection and tracking, range ambiguity clutter suppression, and main lobe interference suppression; and secondly, RF stealth, researching the RF stealth and positioning deception performance of frequency-controlled array radars. However, currently, there is no integrated technology for RF stealth and target detection in frequency-controlled array radars. Summary of the Invention

[0004] This invention proposes a frequency-controlled array radar radio frequency stealth and target detection method based on frequency offset design, realizing the integration of radio frequency stealth and target detection of frequency-controlled array radar.

[0005] The frequency-offset-based frequency-controlled array radar radio frequency stealth and target detection method includes the following basic steps:

[0006] Step S1: Construct the transmitter array and receiver array of a frequency-controlled array radar with a frequency offset greater than the baseband waveform bandwidth;

[0007] Step S2: Preset the candidate frequency offset and calculate the transmitted signal of the array elements of the transmitter array;

[0008] Step S3: Obtain the target echo signal based on the receiver array structure and the transmitted signal of the array element;

[0009] Step S4: Based on the obtained target echo signal, design the cost function and construct a beam pattern design problem integrating radio frequency stealth and target detection with respect to the frequency offset selection matrix and the receiver weight vector;

[0010] Step S5: Design an optimization algorithm based on the constructed beam pattern design problem to obtain the optimal frequency offset selection matrix and receiver weight vector, and complete beamforming in the distance and angle dimensions to enhance the energy of the target signal.

[0011] Furthermore, step S1 specifically includes the following sub-steps:

[0012] Step S11: Preset the number of array elements at the transmitting and receiving ends of the frequency-controlled array radar to M and N respectively, and set the element spacing between the transmitting and receiving array elements according to the center frequency of the transmitted signal:

[0013]

[0014] Where, d t d represents the spacing between the transmitting elements. r For the spacing between receiving array elements, wavelength f c The center carrier frequency of the transmitted signal, where c is the speed of light;

[0015] Step S12: Configure M mixers for each array element at the receiver and add a matched filter to each mixer.

[0016] Furthermore, step S2 specifically includes the following sub-steps:

[0017] Step S21: Calculate the carrier signal of the m-th array element based on the preset candidate frequency offset U:

[0018]

[0019] in,(·) T This indicates a transpose operation, where wm is a U×1 dimensional frequency offset selection vector assigned to the m-th transmit element, with only one element being 1 and the rest being 0; c(Δf,t)=[e j2πΔft ,e j2π2Δft ,...,e j2πUΔft ] T Here, Δf is the frequency offset vector;

[0020] Step S22: Calculate the transmitted signal of each element in the transmitter array based on the number of transmitter array elements and the baseband waveform. The calculation formula is as follows:

[0021]

[0022] in, and These represent the baseband waveform and the transmitted signal of the m-th array element, respectively.

[0023] Furthermore, step S3 specifically includes the following sub-steps:

[0024] Step S31: Based on the signal transmitted by the m-th array element, calculate the echo signal y reflected from the target (r, θ) to the n-th receiving array element. m,n (t), the formula is:

[0025]

[0026] Where ξ is the complex reflection coefficient of the target;

[0027] Step S32: Based on the set frequency control array receiving structure, obtain the output r of the m′th channel of the nth receiving array element. m′,n The formula is:

[0028]

[0029] Where m' = 1, 2, ..., M, Let be the complex reflection coefficient of the fused matrix, (·) c The asterisks and '*' represent conjugation and convolution operations, respectively.

[0030] Step S33: Combine the outputs of all N receiving array elements to obtain the echo vector r after receiver processing, as shown in the formula:

[0031]

[0032] In the formula, diag{a T (θ)} diagonal matrix, Represents Kronecker matrix multiplication, r n The output of the nth receiving element is expressed as:

[0033]

[0034] a T (θ) is the guiding vector of the transmitting array, and its expression is:

[0035]

[0036] a R (θ) is the receiving array steering vector, and its expression is:

[0037]

[0038] W is an M×U dimensional frequency offset selection matrix, and its expression is:

[0039] W = [w1, w2, ..., w M ] T .

[0040] Furthermore, step S4 includes the following sub-steps:

[0041] Step S41: Based on the output data obtained in step S3, obtain the equivalent beam pattern of the frequency control array receiver, represented as:

[0042] P(r,θ)=|z H h(r,θ;W)| 2 =z H H(r,θ;W)z;

[0043] in,(·) H For the conjugate transpose operation, z is the beam filtering vector at the receiver, and H(r,θ;W) is specifically:

[0044] H(r,θ;W)=h(r,θ;W)h H (r,θ;W)

[0045]

[0046] Step S42: Based on the output data obtained in step S3, obtain the cross-correlation sidelobes of the beam, expressed as:

[0047]

[0048] in, Specifically, it is expressed as follows:

[0049]

[0050] Step S43: Ensure the frequency offset Δf is greater than the baseband waveform bandwidth to guarantee the radar's radio frequency stealth. Optimize the frequency offset selection matrix W and the beam filtering vector z to ensure optimal target detection, making the beam pattern P(r,θ) approximate the desired radiation pattern d(r,θ) in the least-squares sense and minimizing the cross-correlation sidelobes.

[0051] Step S44: Based on step S43, select the cost function J(η,z,W), whose expression is:

[0052]

[0053] Where η is the scale factor. A(r k ,θ k ;W) and The expression is:

[0054]

[0055]

[0056]

[0057] in, and Weighting coefficients respectively;

[0058] Step S45: Based on step S43 and the cost function from step S44, construct the beam pattern design problem integrating radio frequency stealth and target detection, whose expression is:

[0059]

[0060]

[0061] in, express The i-th element, w m (u) represents w m The u-th element.

[0062] Furthermore, step S5 specifically includes the following sub-steps:

[0063] Step S51: Obtain the suboptimal solution to the problem through iterative loops, namely the frequency offset selection matrix and the receiver weight vector;

[0064] Step S52: Fix variable w m and The problem concerning the other M elements and η is transformed into an unconstrained problem, yielding an analytical solution, expressed as:

[0065]

[0066] Step S53: For variables The cost function for the other M elements is equivalent to:

[0067]

[0068] Among them, f i g i , and Specifically:

[0069]

[0070]

[0071]

[0072]

[0073] Among them, b k,i and d k,i Specifically:

[0074]

[0075]

[0076] Among them, b p,q,i and d p,q,i The specific form is the same as b k,i and d k,i ,(·) c For conjugate operations;

[0077] Step S54: For variables The other M elements are updated, and the optimal solution can be efficiently obtained using numerical methods. The specific expression is as follows:

[0078]

[0079] Step S55: With the variable fixed When, the expression for the problem concerning W is:

[0080]

[0081]

[0082] Step S56: Solve each row of matrix W one by one. For the m-th row w of matrix W... m Set its uth m,opt If one element is 1 and the other elements are 0, then u m,opt The expression is:

[0083]

[0084] in, Specifically:

[0085]

[0086] Finally, the optimal frequency offset selection matrix W and the receiving weight vector z are calculated.

[0087] The beneficial effects of this invention are as follows: This invention proposes a method for radio frequency stealth and target detection of frequency-controlled array radar based on frequency offset design, realizing the integration of radio frequency stealth and target detection of frequency-controlled array radar.

[0088] (1) Select a frequency offset greater than the baseband waveform bandwidth so that the radar emits an omnidirectional beam, reducing the probability of interception and achieving radio frequency stealth.

[0089] (2) Configure the frequency offset selection matrix so that the frequency offset used by the frequency control array radar is an integer multiple of the basic frequency offset, which facilitates hardware implementation.

[0090] (3) Jointly optimize the frequency offset selection matrix and the receiving weight vector to focus the energy of the echo signal at the desired distance and angle position, thereby achieving energy enhancement of the target signal. Attached Figure Description

[0091] Figure 1 This is a structural diagram of the transmitter and receiver array provided in an embodiment of the present invention;

[0092] Figure 2 This is a flowchart of a frequency-controlled array radar radio frequency stealth and target detection method based on frequency offset design, provided in an embodiment of the present invention.

[0093] Figure 3 A graph showing the relationship between the cost function and the number of algorithm iterations provided in an embodiment of the present invention;

[0094] Figure 4 This is a scatter plot of frequency offset for a frequency-controlled array radar provided in an embodiment of the present invention.

[0095] Figure 5 The equivalent beam pattern of the frequency-controlled array radar receiver provided in the embodiments of the present invention. Detailed Implementation

[0096] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments of the present invention will now be described with reference to the accompanying drawings.

[0097] This invention proposes a method for radio frequency stealth and target detection of frequency-controlled array radar based on frequency offset design, achieving the integration of radio frequency stealth and target detection in frequency-controlled array radar. Figure 1 As shown, the basic steps include:

[0098] Step S1: Construct the transmitter array and receiver array of a frequency-controlled array radar with a frequency offset greater than the baseband waveform bandwidth;

[0099] Step S2: Preset the candidate frequency offset and calculate the transmitted signal of the array elements of the transmitter array;

[0100] Step S3: Obtain the target echo signal based on the receiver array structure and the transmitted signal of the array element;

[0101] Step S4: Based on the obtained target echo signal, design the cost function and construct a beam pattern design problem integrating radio frequency stealth and target detection with respect to the frequency offset selection matrix and the receiver weight vector;

[0102] Step S5: Design an optimization algorithm based on the constructed beam pattern design problem to obtain the optimal frequency offset selection matrix and receiver weight vector, and complete beamforming in the distance and angle dimensions to enhance the energy of the target signal.

[0103] In this embodiment, step S1 involves constructing a transmitter array and a receiver array for a frequency-controlled array radar with a frequency offset greater than the baseband waveform bandwidth; specifically:

[0104] S11. The number of array elements at the transmitting and receiving ends of the frequency-controlled array radar is preset to M and N, respectively, and the element spacing between the transmitting and receiving array elements is set according to the center frequency of the transmitted signal:

[0105]

[0106] Where, d t d represents the spacing between the transmitting elements. r For the spacing between receiving array elements, wavelength f c The center carrier frequency of the transmitted signal, where c is the speed of light.

[0107] S12. Configure M mixers for each array element at the receiving end, and add a matched filter to each mixer.

[0108] In practice, the transmission and structure of a frequency-controlled array radar are obtained by constructing a transmitter array and a receiver array, such as... Figure 1 As shown.

[0109] Step S2: Preset candidate frequency offset numbers and calculate the transmitted signals of the array elements in step S1; In this embodiment, step S2 specifically includes the following sub-steps:

[0110] S21. Calculate the carrier signal of the m-th array element based on the preset candidate frequency offset U:

[0111]

[0112] in,(·) T This indicates a transpose operation. wm is a U×1-dimensional frequency offset selection vector assigned to the m-th transmit element, with only one element being 1 and the rest being 0.

[0113]

[0114] Let f be the frequency offset vector, and Δf be the frequency offset.

[0115] S22. Calculate the transmitted signal of the transmitting array element based on the number of transmitting array elements and the baseband waveform:

[0116]

[0117] in, and These represent the baseband waveform and the transmitted signal of the m-th array element, respectively.

[0118] In practice, the signals transmitted by each array element at the transmitting end are s m(t) is a pulse train signal.

[0119] Step S3: Obtain the target echo signal based on the receiver array structure in step S1 and the array element transmission signal in step S2; in this embodiment, step S3 specifically involves:

[0120] S31. Calculate the echo signal y reflected from the target (r, θ) to the nth receiving element based on the signal emitted by the m-th array element. m,n (t):

[0121]

[0122] Where ξ is the target complex reflection coefficient.

[0123] S32. Based on the set frequency control array receiving structure, obtain the output r of the m′th channel of the nth receiving array element. m′,n :

[0124]

[0125] Where m' = 1, 2, ..., M, Let be the complex reflection coefficient of the fused matrix, (·) c The asterisks and the asterisks represent conjugation and convolution operations, respectively.

[0126] S33. Combining the outputs of all N receiver array elements yields the echo vector r after receiver processing:

[0127]

[0128] In the formula, diag{a T (θ)} diagonal matrix, This represents Kronecker matrix multiplication. n The output of the nth receiving element is expressed as:

[0129] r n =[r 1,n ,r 2,n ,...,r M,n ] T

[0130] a T (θ) is the guiding vector of the transmitting array, and its expression is:

[0131]

[0132] a R (θ) is the receiving array steering vector, and its expression is:

[0133]

[0134] W is an M×U dimensional frequency offset selection matrix, and its expression is:

[0135] W = [w1, w2, ..., w M ] T

[0136] In practice, the echo signal received by the receiving array is calculated for the far-field midpoint target (r,θ).

[0137] In this embodiment, step S4 involves designing a cost function and constructing a beammap design problem integrating radio frequency stealth and target detection based on the signal obtained in step S3. Step S4 specifically includes the following steps:

[0138] S41. Based on the output data obtained in step S3, the equivalent beam pattern of the frequency control array receiver can be obtained, which can be represented as:

[0139] P(r,θ)=|z H h(r,θ;W)| 2 =z H H(r,θ;W)z

[0140] in,(·) H For the conjugate transpose operation, z is the beam filtering vector at the receiver, and H(r,θ;W) is specifically:

[0141] H(r,θ;W)=h(r,θ;W)h H (r,θ;W)

[0142]

[0143] Step S42: Based on the output data obtained in step S3, obtain the cross-correlation sidelobes of the beam, expressed as:

[0144]

[0145] in, and, Specifically:

[0146]

[0147] Step S43: A frequency offset Δf greater than the baseband waveform bandwidth ensures the radar's radio frequency stealth. Optimal target detection can be achieved by optimizing the frequency offset selection matrix W and the beam filtering vector z. The scheme is to make the beam pattern P(r,θ) approximate the desired radiation pattern d(r,θ) in a least-squares sense while minimizing the cross-correlation sidelobes.

[0148] Step S44: Based on step S43, select the cost function J(η,z,W), whose expression is:

[0149]

[0150] Where η is the scale factor. A(r k ,θ k ;W) and The expression is:

[0151]

[0152]

[0153]

[0154] in, and Weighting coefficients are applied separately.

[0155] Step S45: Based on step S43 and the cost function from step S44, construct the beam pattern design problem integrating radio frequency stealth and target detection, expressed as:

[0156]

[0157]

[0158] in, express The i-th element, w m (u) represents w m The u-th element.

[0159] In practice, the beam pattern design problem shown in step S45 is constructed according to the above steps.

[0160] Step S5: Based on the problem constructed in step S4, design an optimization algorithm to obtain the optimal frequency offset selection matrix and receiver weight vector, and perform beamforming in the range and angle dimensions to enhance the energy of the target signal. In this embodiment, step S5 specifically involves:

[0161] The cost function in steps S51 and S45 is a fourth-order polynomial with non-convex constraints, making it an NP-hard problem. A suboptimal solution, namely the frequency offset selection matrix and the receiver weight vector, is obtained through iterative iteration.

[0162] Step S52, with the variable w fixed m and When there are the other M elements, the problem concerning η is transformed into an unconstrained problem, and an analytical solution can be obtained, expressed as:

[0163]

[0164] Step S53, for variables The cost function for the other M elements is equivalent to the following expression:

[0165]

[0166] Among them, f i g i , and Specifically

[0167]

[0168]

[0169]

[0170]

[0171] Among them, b k,i and d k,i Specifically

[0172]

[0173]

[0174] b p,q,i and d p,q,i The specific form is the same as b k,i and d k,i ,(·) c This is a conjugate operation.

[0175] Step S54: According to step S53, the variable The updates to the other M elements can be obtained by solving the following problem, expressed as:

[0176]

[0177] The optimal solution can be obtained efficiently using numerical methods, such as the fminbnd function in MATLAB.

[0178] Step S55, with fixed variables When, the expression for the problem concerning W is:

[0179]

[0180]

[0181] Step S55: Based on the problem in step S54, solve it by addressing each row of matrix W one by one. For the m-th row w of matrix W... m Set its uth m,opt One element is 1, and all other elements are 0. m,opt The expression is:

[0182]

[0183] in, Specifically

[0184]

[0185] In practice, the algorithm is executed according to the above steps, and then the optimal frequency offset selection matrix W and the receiving weight vector z are calculated.

[0186] In this embodiment of the invention, the following simulation experiments provide further illustration, with configurations as follows: Figure 1 The diagram shows the transmit and receive array structure of a frequency-controlled radar. The preset transmit array elements M = 20, the receive array elements N = 8, the candidate frequency offset U = 15, and the basic frequency offset Δf = 10MHz. The preset baseband waveform bandwidth is 10MHz. Therefore, the radar range resolution r... reso =15m. Preset grid size: 0.5° angle, 0.15m distance. Preset weighting coefficients. and Preset beam main lobe

[0187]

[0188] The flowchart of the frequency-controlled array radar radio frequency stealth and target detection method based on frequency offset design in this invention is shown in the following embodiment. Figure 2 The simulation results for the number of algorithm iterations and the cost function are given in [the document / reference]. Figure 3 The results show that the designed algorithm has monotonic convergence and a fast convergence speed. Figure 4 This demonstrates the frequency offset design of an integrated solution for frequency-controlled array radar. The beam pattern output at the receiver is shown in... Figure 5 As shown in the figure, focusing the echo energy in the desired area significantly enhances the target's detection performance. Using the scheme designed in this invention, the frequency-controlled array radar transmitter achieves radio frequency stealth through its omnidirectional radiation pattern, and at the radar receiver, the echo energy is concentrated in the desired area, thereby enhancing target detection.

[0189] The foregoing description and illustrations have shown and described the basic principles, main features, and advantages of this invention. Those skilled in the art should understand that this invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to this invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.

Claims

1. A method for radio frequency stealth and target detection of frequency-controlled array radar based on frequency offset design, characterized in that, The basic steps include the following: Step S1: Construct the transmitter array and receiver array of a frequency-controlled array radar with a frequency offset greater than the baseband waveform bandwidth; Step S2: Preset the candidate frequency offset and calculate the transmitted signal of the array elements of the transmitter array; Step S3: Obtain the target echo signal based on the receiver array structure and the transmitted signal of the array element, specifically including the following sub-steps: Step S31: According to the first The signal emitted by each array element is calculated to reach the target. The echo signal reflected to the nth receiving element The formula is: ； in, The complex reflection coefficient of the target. The center carrier frequency of the transmitted signal, At the speed of light, To be assigned to the Each launch element A frequency offset selection vector of dimension 1, which has only one element. The remaining elements are ; It is the frequency offset vector. For frequency offset; Step S32: Based on the set frequency control array receiving structure, obtain the first... The first receiving element of the array Output of each channel The formula is: ； in, , For the complex reflection coefficient of the fusion, and These represent conjugation and convolution operations, respectively. This indicates the transpose operation. This represents the number of mixers. Step S33: Combine all The output of each receiving array element yields the echo vector processed by the receiver. The formula is: ； In the formula, It is a diagonal matrix. This represents Kronecker matrix multiplication. For the first The output of each receiving array element is expressed as follows: ； The vector for guiding the transmission array is expressed as follows: ； The receiving array steering vector is expressed as follows: ； for The frequency offset selection matrix of dimension 1 is expressed as follows: ； Step S4: Based on the obtained target echo signal, design the cost function and construct a beammap design problem integrating RF stealth and target detection with respect to the frequency offset selection matrix and receiver weight vector, including the following sub-steps: Step S41: Based on the output data obtained in step S3, obtain the equivalent beam pattern of the frequency control array receiver, represented as: ； in, This is the conjugate transpose operation. The beam filtering vector at the receiving end. Specifically: ； Step S42: Based on the output data obtained in step S3, obtain the cross-correlation sidelobes of the beam, expressed as: ； in, , , Specifically, it is expressed as follows: ； Step S43: Ensure frequency offset A bandwidth greater than the baseband waveform ensures the radar's radio frequency stealth by optimizing the frequency offset selection matrix. and beam filter vector Vectors ensure optimal target detection, resulting in a beam pattern. The pattern approximates the desired orientation in the least squares sense. And minimize cross-correlation sidelobes ; Step S44: Select the cost function according to step S43. Its expression is: ； in, As a scale factor, , and The expression is: ； in, and Weighting coefficients respectively; Step S45: Based on step S43 and the cost function from step S44, construct the beam pattern design problem integrating radio frequency stealth and target detection, whose expression is: ； in, express The One element, express The One element, The preset candidate frequency offset; Step S5: Design an optimization algorithm based on the constructed beam pattern design problem to obtain the optimal frequency offset selection matrix and receiver weight vector, and complete beamforming in the distance and angle dimensions to enhance the energy of the target signal.

2. The frequency-controlled array radar radio frequency stealth and target detection method based on frequency offset design according to claim 1, characterized in that, Step S1 specifically includes the following sub-steps: Step S11: Preset the number of array elements at the transmitter and receiver of the frequency-controlled array radar as follows: and The spacing between the transmitting and receiving array elements is set according to the center frequency of the transmitted signal: ； in, The spacing between the transmitting elements, For the spacing between receiving array elements, wavelength , The center carrier frequency of the transmitted signal, The speed of light; Step S12: Configure each array element at the receiver. Each mixer has a matching filter added to it.

3. The frequency-controlled array radar radio frequency stealth and target detection method based on frequency offset design according to claim 2, characterized in that, Step S2 specifically includes the following sub-steps: Step S21: Based on the preset candidate frequency offset Calculate the first Carrier signals of each array element: ； in, This indicates the transpose operation. To be assigned to the Each launch element A frequency offset selection vector of dimension 1, which has only one element. The remaining elements are ; It is the frequency offset vector. For frequency offset; Step S22: Calculate the transmitted signal of each element in the transmitter array based on the number of transmitter array elements and the baseband waveform. The calculation formula is as follows: ； in, and They represent the first The baseband waveform and transmitted signal of each array element.

4. The method for radio frequency stealth and target detection of frequency-controlled array radar based on frequency offset design according to claim 1, characterized in that, Step S5 specifically includes the following sub-steps: Step S51: Obtain the suboptimal solution to the problem through iterative loops, namely the frequency offset selection matrix and the receiving weight vector; Step S52: Fix variables and Other One element, about The problem is transformed into an unconstrained problem, and an analytical solution is obtained, expressed as: ； Step S53: For variables Other With elements, the cost function is equivalent to: ； in, , , and Specifically: ； in, and Specifically: ； in, and The specific form is the same as and , This is a conjugate operation; Step S54: For variables other The elements are updated, and the optimal solution can be efficiently obtained using numerical methods. The specific expression is: ； Step S55: With the variable fixed At that time, regarding The expression for the problem is: ； Step S56: Solve the matrix one by one For each row of the matrix The OK , set its first The elements are Other elements are ,but The expression is: ； in, Specifically: ； Finally, the optimal frequency offset selection matrix is ​​calculated. and receiving weight vector .

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