A self-correction method for amplitude and phase errors based on the covariance matrix of a uniform linear array signal

Abstract

This invention discloses a self-correction method for amplitude and phase errors based on the covariance matrix of a uniform linear array signal, relating to the field of array signal processing technology. The method includes: deriving a system of linear equations concerning amplitude and phase errors from the received signal of the sensor array, wherein the received signal considers internal receiver noise; calculating the statistical characteristics of the noise error based on the estimated covariance matrix of the received signal; constructing boundary conditions based on prior phase information; constructing a weighted least squares estimation problem with equality constraints concerning amplitude and phase errors based on the system of linear equations, the statistical characteristics of the noise error, and the boundary conditions; solving the weighted least squares estimation problem to obtain an estimate of the amplitude and phase errors; and using the estimate of the amplitude and phase errors to correct the received signal. This invention considers a more generalized signal model containing internal noise and constructs a high-precision amplitude and phase error correction algorithm with lower computational cost, which is more consistent with actual conditions compared to existing algorithms.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing technology, specifically relating to a self-correction method for amplitude and phase errors based on the covariance matrix of a uniform linear array signal. Background Technology

[0002] Direction-of-arrival (DOA) estimation algorithms can accurately determine the direction of arrival of target signals to a sensor array. This technology has significant application value in the field of array signal processing and is one of the main tasks in many fields such as radar, communication, and sonar. Existing high-precision estimation algorithms can achieve very high estimation accuracy and can be applied under harsh conditions of low snapshot number and low signal-to-noise ratio.

[0003] However, these high-resolution estimation algorithms require the assumption that the array steering vector is accurate and error-free. In reality, due to various factors, the mathematical model does not match the actual situation, leading to a significant decrease in the performance of traditional high-resolution estimation algorithms. Array error has become an unavoidable factor limiting the performance of various high-precision algorithms.

[0004] Self-correction of amplitude and phase errors based on the covariance matrix of the received signal from a uniform linear array is an effective method to reduce computational complexity. Existing methods, based on the Tolliterz property of the covariance matrix of an error-free uniform linear array, construct a system of linear equations related to the amplitude and phase error parameters. They also consider the measurement deviation between the calculated covariance matrix and the actual signal covariance matrix due to limited samples, thereby improving estimation accuracy. Ultimately, the original problem is transformed into a weighted least squares estimation problem, yielding a closed-form solution. However, in order to simply linearly represent the amplitude and phase error parameters, these methods typically only consider the influence of external receiver noise, making the noise signal also affected by the amplitude and phase errors. Furthermore, they simply assume that external noise follows a complex Gaussian distribution, but in reality, external noise often has certain spectral characteristics and is not simply complex Gaussian white noise. Moreover, in high-frequency radar systems, the influence of internal receiver noise is crucial. Summary of the Invention

[0005] To address the aforementioned problems in the existing technology, this invention provides a self-correction method for amplitude and phase errors based on the covariance matrix of a uniform linear array signal. The technical problem to be solved by this invention is achieved through the following technical solution:

[0006] This invention provides a self-correction method for amplitude and phase errors based on the covariance matrix of a uniform linear array signal, comprising:

[0007] A set of linear equations concerning amplitude and phase error is obtained based on the received signal from the sensor array, wherein the received signal is a signal that takes into account the internal noise of the receiver;

[0008] The statistical characteristics of the noise error are calculated based on the estimated covariance matrix of the received signal.

[0009] Construct boundary conditions based on phase prior information;

[0010] Based on the linear equations, the statistical characteristics of the noise error, and the boundary conditions, a weighted least squares estimation problem with respect to amplitude and phase errors containing equality constraints is constructed.

[0011] Solve the weighted least squares estimation problem to obtain an estimate of the amplitude and phase error, and use the estimate of the amplitude and phase error to correct the received signal.

[0012] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0013] The amplitude and phase error self-correction method based on the covariance matrix of a uniform linear array signal of this invention, within the framework of amplitude and phase error correction based on the covariance matrix of the received signal of a linear array, considers a more generalized signal model containing internal noise. By introducing additional parameters, the covariance matrix of the received data is linearly represented by amplitude error and phase error, constructing a set of equations containing noisy errors. Furthermore, to address the shortcomings of amplitude and phase error correction for linear arrays, prior phase information is introduced as boundary conditions. Through a constrained weighted least squares algorithm, a high-precision amplitude and phase error correction algorithm with lower computational cost is constructed, which is more in line with the actual situation compared to existing algorithms.

[0014] The above description is merely an overview of the technical solution of the present invention. In order to better understand the technical means of the present invention and to implement it in accordance with the contents of the specification, and to make the above and other objects, features and advantages of the present invention more apparent and understandable, preferred embodiments are described in detail below with reference to the accompanying drawings. Attached Figure Description

[0015] Figure 1 This is a flowchart of a self-correction method for amplitude and phase errors based on the covariance matrix of a uniform linear array signal provided in an embodiment of the present invention;

[0016] Figure 2 This is a performance graph of amplitude estimation provided in an embodiment of the present invention;

[0017] Figure 3 This is a phase estimation performance diagram provided by an embodiment of the present invention;

[0018] Figure 4 This is a diagram showing the DOA estimation results using the MUSIC algorithm before and after amplitude and phase error correction, provided by an embodiment of the present invention. Detailed Implementation

[0019] To further illustrate the technical means and effects adopted by the present invention to achieve the intended purpose, the following describes in detail, with reference to the accompanying drawings and specific embodiments, a self-correction method for amplitude and phase errors based on the covariance matrix of a uniform linear array signal proposed according to the present invention.

[0020] The foregoing and other technical contents, features, and effects of the present invention will be clearly presented in the following detailed description of specific embodiments in conjunction with the accompanying drawings. Through the description of the specific embodiments, a more in-depth and concrete understanding can be gained of the technical means and effects adopted by the present invention to achieve its intended purpose. However, the accompanying drawings are for reference and illustration only and are not intended to limit the technical solutions of the present invention.

[0021] This invention provides a self-correction method for amplitude and phase errors based on the covariance matrix of a uniform linear array signal. Please refer to [link to relevant documentation]. Figure 1 , Figure 1 This is a flowchart of a self-correction method for amplitude and phase errors based on the covariance matrix of a uniform linear array signal provided in an embodiment of the present invention, as shown below. Figure 1 As shown, the amplitude and phase error self-correction method based on the covariance matrix of a uniform linear array signal in this embodiment may include the following steps:

[0022] Step 1: Obtain a set of linear equations for amplitude and phase error based on the received signal from the sensor array, where the received signal is a signal that takes into account the internal noise of the receiver.

[0023] Consider a uniform linear array containing M omnidirectional sensors with different amplitude and phase errors, with an element spacing of d. There are K far-field sources in space, each with an angle θ = [θ1, θ2, ..., θ]. K A narrowband signal with wavelength λ is transmitted into the array. The k-th source signal is denoted as s. k (t), k=1,2,…,K. Then, the received signal of the array at time t, i.e., the array observation signal, can be expressed as:

[0024] x(t)=ΓA(θ)s(t)+n(t)=ΨΦA(θ)s(t)+n(t)

[0025] In the formula, 1≤t≤T, T is the number of snapshots, s(t) represents the signal source in space, and s(t)=[s1(t),s2(t),…,s K (t)] T K represents the number of signal sources in space, s(t) follows a zero-mean, and the covariance matrix is... The circularly symmetric complex Gaussian distribution, Let n(t) represent the variance of the k-th signal, and n(t) be the receiver's internal noise, where n(t) ~ CN(0,Q). n ) is a matrix with a mean of zero and a covariance matrix of Q.n Circularly symmetric complex Gaussian white noise, Let I be the noise variance, and let Γ be the noise variance. The noise and signal are independent of each other. It is worth noting that the internal noise of the receiver is not affected by the amplitude and phase errors. Let I be the identity matrix, Γ be the diagonal matrix of the array amplitude and phase errors, Γ = ΨΦ, where Ψ represents the diagonal matrix of the amplitude error and Φ represents the diagonal matrix of the phase error. Let A(θ) be the array manifold matrix.

[0026] Among them, Ψ=diag([ψ1,…,ψ M ]), where ψ m ,m=1,2,…,M represents the amplitude error of the m-th array element; Φ=diag([φ1,…,φ M ]), where φ m Let m = 1, 2, ..., M represent the phase error of the m-th element. Without loss of generality, let the first element be the reference element, i.e., ψ1 = 1, φ1 = 0. A(θ) = [a(θ1), a(θ2), ..., a(θ...] K )] is the array manifold matrix, a(θ) k ) is the array steering vector, represented as:

[0027]

[0028] In this embodiment, a set of linear equations concerning amplitude and phase error is obtained based on the received signals from the sensor array, including:

[0029] Step i: Calculate the estimated covariance matrix of the received signal based on the received signal from the sensor array; the estimated covariance matrix of the received signal is calculated according to the following formula:

[0030]

[0031] In the formula, Let represent the estimated covariance matrix of the received signal, T be the number of snapshots, t be the time, x(t) be the received signal of the sensor array at time t, and H be the matrix conjugate.

[0032] Step ii: Take the logarithm of the estimated covariance matrix, define the unknown parameter vector, and obtain the linear equation system based on the unknown parameter vector and the real and imaginary parts of the estimated covariance matrix after taking the logarithm.

[0033] First, the process of obtaining a system of linear equations without "measurement error" based on the true covariance matrix of the received signal from the linear array is explained.

[0034] Assume the signal s has zero mean and the covariance matrix is ​​R. s The complex Gaussian distribution, i.e.:

[0035] s(t)~CN(0,R s);

[0036] Therefore, the array-received signal also follows a complex Gaussian distribution with a mean of 0 and a covariance matrix of R, i.e.:

[0037] x(t)~CN(0,R);

[0038] In the formula,

[0039] Therefore, the element in the i-th row and j-th column of the covariance matrix is:

[0040]

[0041] Taking its logarithm yields the linear equations for the amplitude and phase errors:

[0042]

[0043] In the formula, K represents the number of signal sources in space. For noise variance, Let C represent the variance of the k-th signal. And since C is the Topplitz matrix in a uniform linear array, we define c... i Let i = 1, 2, ..., M be the M constituent elements of matrix C. Taking their real and imaginary parts, we can obtain a system of linear equations concerning the logarithm of the amplitude and the phase, respectively:

[0044] Get the real part:

[0045]

[0046] Take the imaginary part:

[0047]

[0048] Define an unknown parameter vector:

[0049]

[0050] in:

[0051] φ=[φ1,…,φ M ] T ;

[0052]

[0053] Δ=[Δ1,…,Δ M ] T ;

[0054] In the formula, Let ψ be the logarithm of the amplitude error parameter.m m = 1, 2, ..., M, representing the amplitude error of the m-th array element, and φ is the phase error parameter. m Let m = 1, 2, ..., M, where m represents the phase error of the m-th element, M represents the number of elements, and ρ, l, and Δ are redundant parameters. Indicates taking the real part, c represents taking the imaginary part. i Let i = 1, 2, ..., M be the M constituent elements of matrix C, and C be the Topplitz matrix.

[0055] in, And φ, that is, the first 2M parameters are amplitude and phase error parameters, which are also the target parameters, while ρ, ι, Δ, that is, the last 3M parameters are redundant parameters.

[0056] In practice, considering the solvability of the equation, since the redundant parameters Δ and ρ1 only appear in the main diagonal elements of the covariance matrix, let Δ′=Δ+ρ1. At the same time, since C is a Hermitian matrix, ι1=0, thereby reducing the number of redundant parameters. Therefore, the final number of parameters that need to be solved is 5M-2.

[0057] Based on this, a system of linear equations without measurement error can be obtained:

[0058]

[0059] In the formula, The coefficient matrix, This represents the vector of all redundant parameters.

[0060] Since the true covariance matrix is ​​often difficult to obtain directly, an estimated covariance matrix can be obtained by estimating the received data, which can then be used to replace the true covariance matrix.

[0061] Considering the measurement noise ε introduced when calculating the covariance matrix due to the finite sample size, the estimated covariance matrix can be expressed as:

[0062]

[0063] Similarly, take the logarithm of the estimated covariance matrix and then take the real and imaginary parts:

[0064]

[0065] Get the real part:

[0066]

[0067] Take the imaginary part:

[0068]

[0069] In the formula, The noise error of the equation can be obtained by taking its real and imaginary parts. And let the noise error vector ξ = [ε T ,∈ T ] T Where, ε=[…,ε ij ,…] T ,∈=[…,∈ ij ,…] T .

[0070] Therefore, the linear equation system to be solved, which contains noise error, can be expressed as: y = HΘ + ξ, where ξ is the noise error vector.

[0071] Step 2: Calculate the statistical characteristics of the noise error based on the estimated covariance matrix of the received signal.

[0072] In this embodiment, the statistical characteristics of the noise error include the mean and covariance matrix of the noise error vector. The mean of the noise error vector is calculated using the following formula:

[0073]

[0074] The covariance matrix of the noise error vector is calculated according to the following formula:

[0075]

[0076] In the formula, Expressing expectations, Indicates taking the real part, Indicates taking the imaginary part, ε ij Indicates to The real part of the noise error, ε, after taking the logarithm of the element in the i-th row and j-th column. kl Indicates to The real part of the noise error, ∈ [the element in row k, column l], is obtained by taking the logarithm of the element in row l. ij Indicates to The imaginary part of the noise error after taking the logarithm of the element in the i-th row and j-th column is ∈ kl Indicates to The imaginary part of the noise error after taking the logarithm of the element in the k-th row and l-th column. This represents the estimation of the covariance matrix, where ε represents the result of ε. ij The vector formed by ∈ represents the vector composed of ∈ ij The vectors formed, i, j, k, l represent respectively Row and column indexes in * represents conjugate, and T represents transpose.

[0077] Step 3: Construct boundary conditions based on phase prior information.

[0078] In practical signal processing scenarios, prior information about system errors can often be obtained. For phase errors, which generally fluctuate within a certain range, it can be assumed that the phase information follows a zero-mean uniform distribution, i.e., φ ~ U[-φ]. e ,φ e ], where φ e The probability density function of the maximum magnitude of the phase fluctuation is:

[0079]

[0080] Since the system of linear equations to be solved is an underdetermined system of equations, it does not have a unique unbiased solution. Therefore, considering that the phase error is relatively small, we can statistically approximate the mean of all phase errors as follows:

[0081]

[0082] Using this as a boundary condition, we solve the system of linear equations, and take the first array element as the reference array element, i.e.

[0083] Step 4: Based on the linear equations, the statistical characteristics of the noise error, and the boundary conditions, construct a weighted least squares estimation problem for the amplitude and phase errors with equality constraints.

[0084] In this embodiment, the weighted least squares estimation problem with respect to amplitude and phase errors, which includes equality constraints, can be expressed as:

[0085]

[0086]

[0087] In the formula, st represents the constraint condition.

[0088] Transform the constraints into matrix form:

[0089]

[0090] In the formula,

[0091] Step 5: Solve the weighted least squares estimation problem to obtain the estimated value of amplitude and phase error, and use the estimated value of amplitude and phase error to correct the received signal.

[0092] In this embodiment, the Lagrange multiplier method is used to solve the weighted least squares estimation problem to obtain a constrained weighted least squares solution. The first 2M values ​​of the weighted least squares solution are used as estimates of the amplitude and phase errors, and the received signal is corrected based on the estimates of the amplitude and phase errors.

[0093] Specifically, the Lagrange function is:

[0094]

[0095] Differentiating the parameters Θ and the Lagrange multipliers λ and setting them to zero, we obtain the final system of linear equations:

[0096]

[0097] Solving this system of linear equations yields a constrained weighted least squares solution:

[0098]

[0099] Finally, we can obtain weighted least squares estimates of the amplitude and phase errors:

[0100]

[0101] The received data is corrected using the estimated amplitude and phase error results. The corrected received signal is represented as follows: Based on the corrected received signal, the corrected data can be processed by other high-resolution direction-of-arrival (DOA) estimation algorithms to obtain more accurate DOA estimation results.

[0102] The amplitude and phase error self-correction method based on the covariance matrix of a uniform linear array signal in this invention, within the framework of amplitude and phase error correction based on the covariance matrix of the received signal from a linear array, considers a more generalized signal model containing internal noise. By introducing additional parameters, the covariance matrix of the received data is linearly represented by amplitude error and phase error, constructing a set of equations containing noise error. Furthermore, to address the shortcomings of amplitude and phase error correction for linear arrays, prior phase information is introduced as boundary conditions. Through a constrained weighted least squares algorithm, a high-precision amplitude and phase error correction algorithm with lower computational cost is constructed, which is more consistent with actual conditions compared to existing algorithms.

[0103] Furthermore, the amplitude and phase error self-correction method based on the covariance matrix of a uniform linear array signal provided in this embodiment of the invention is further illustrated through simulation experiments.

[0104] Specifically, a uniform linear array of 8 elements is used, with an element spacing of half a wavelength, i.e., d = λ / 2, and carrier frequencies of f = 3 × 10⁻⁶ are distributed in space. 8 A signal source with a Hz frequency and a carrier wavelength λ = c / f = 1 m.

[0105] Please see Figure 2 and Figure 3 , Figure 2This is a performance graph of amplitude estimation provided in an embodiment of the present invention; Figure 3 This is a phase estimation performance diagram provided in an embodiment of the present invention.

[0106] In this example, three signal sources with incident angles of [25°, 30°, 70°] are placed in space. The array's amplitude error is [1, 1.15, 1.3, 1.2, 1.25, 0.8, 0.8, 1.3], and its phase error is [0, 15°, 9°, -13°, 20°, -5°, -15°, -11°]. The performance metric is measured by the estimated mean square error (RMSE), defined as follows:

[0107] Amplitude parameters:

[0108]

[0109] Phase parameters:

[0110]

[0111] In the formula, L represents the number of Monte Carlo experiments, and CCRLB (Constraints Cramer-Rao Lower Bound) represents the Cramer-Rao lower bound with equality constraints, serving as a reference standard for performance estimation. Figure 2 and 3 As shown, the method of the present invention does not require iterative calculations, and its estimation performance for amplitude and phase can approach that of CCRLB.

[0112] Please see Figure 4 , Figure 4 This is an embodiment of the invention providing the DOA estimation results using the MUSIC algorithm before and after amplitude and phase error correction. In this example, two signal sources with DOAs of [30°, 70°] are placed in space, with a signal-to-noise ratio of 10dB. Figure 4 It can be seen that before the array is corrected, the estimation result of the MUSIC algorithm has a large error and the spectral peaks are relatively inconspicuous. After correction by the method of this invention, the spectral peaks are more obvious and the estimation performance is improved.

[0113] It should be noted that, in this document, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations are intended to cover non-exclusive inclusion, such that an article or device comprising a list of elements includes not only those elements but also other elements not expressly listed. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the article or device comprising said element. Terms such as "connected" or "linked" are not limited to physical or mechanical connections but can include electrical connections, whether direct or indirect. The orientations or positional relationships indicated by terms such as "upper," "lower," "left," and "right" are based on the orientations or positional relationships shown in the accompanying drawings and are used only for the convenience of describing the invention and for simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be construed as limiting the invention.

[0114] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features or characteristics described may be combined in any suitable manner in one or more embodiments or examples. In addition, those skilled in the art can combine and integrate the different embodiments or examples described in this specification.

[0115] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions or substitutions can be made without departing from the concept of the present invention, and all such modifications and substitutions should be considered within the scope of protection of the present invention.

Claims

1. A self-correction method for amplitude and phase errors based on the covariance matrix of a uniform linear array signal, characterized in that, include: A set of linear equations concerning amplitude and phase error is obtained based on the received signal from the sensor array, wherein the received signal is a signal that takes into account the internal noise of the receiver; The statistical characteristics of the noise error are calculated based on the estimated covariance matrix of the received signal. Construct boundary conditions based on phase prior information; Based on the linear equations, the statistical characteristics of the noise error, and the boundary conditions, a weighted least squares estimation problem with respect to amplitude and phase errors containing equality constraints is constructed. Solve the weighted least squares estimation problem to obtain an estimate of the amplitude and phase error, and use the estimate of the amplitude and phase error to correct the received signal; Based on the received signals from the sensor array, a set of linear equations concerning the amplitude and phase errors is obtained, including: The estimated covariance matrix of the received signal is calculated based on the received signal from the sensor array. Take the logarithm of the estimated covariance matrix, define the unknown parameter vector, and obtain the linear equation system based on the unknown parameter vector and the real and imaginary parts of the estimated covariance matrix after taking the logarithm. The unknown parameter vector is represented as follows: ； in: ; ； ； ； ； In the formula, This is the logarithm of the amplitude error parameter. , , indicating the first The amplitude error of each array element For phase error parameters, , , indicating the first Phase error of each array element Indicates the number of array elements. , , All of these are redundant parameters. Indicates taking the real part, This indicates taking the imaginary part. , For matrix of Each component element Let the Topulitz matrix be defined. , Indicates the number of signal sources in space. For noise variance, Indicates the first The variance of each signal; The statistical characteristics of the noise error include the mean and covariance matrix of the noise error vector, wherein the mean of the noise error vector is calculated according to the following formula: ； The covariance matrix of the noise error vector is calculated according to the following formula: ； In the formula, , , , Expressing expectations, Indicates taking the real part, This indicates taking the imaginary part. Indicates to No. Line number The real part of the noise error after taking the logarithm of the elements in the column. Indicates to No. OK The real part of the noise error after taking the logarithm of the elements in the column. Indicates to No. Line number The imaginary part of the noise error after taking the logarithm of the elements of the column. Indicates to No. OK The imaginary part of the noise error after taking the logarithm of the elements of the column. This represents the estimated covariance matrix of the received signal. Indicates by The vector formed Indicates by The vector formed They represent Row and column indexes in Indicates conjugate. Indicates transpose. This refers to the number of snapshots; The boundary conditions are expressed as follows: ； ； In the formula, Indicates the number of array elements. Indicates the first Phase error of each array element This represents the logarithm of the magnitude error of the first array element. This represents the phase error of the first array element, which is the reference array element.

2. The amplitude and phase error self-correction method based on the covariance matrix of a uniform linear array signal according to claim 1, characterized in that, exist The received signal from the time sensor array is represented as follows: ； In the formula, , Represents a signal source in space. , Indicates the number of signal sources in space. It follows a mean of 0, and its covariance matrix is... The circularly symmetric complex Gaussian distribution, Indicates the first The variance of a signal, For receiver internal noise, To ensure that the mean is zero, the covariance matrix is: Circularly symmetric complex Gaussian white noise, , For noise variance, It is the identity matrix. This is a diagonal matrix representing the amplitude and phase errors of the array. , The diagonal matrix representing the amplitude error. A diagonal matrix representing the phase error. It is an array manifold matrix.

3. The amplitude and phase error self-correction method based on the covariance matrix of a uniform linear array signal according to claim 1, characterized in that, The estimated covariance matrix of the received signal is calculated according to the following formula: ； In the formula, This represents the estimated covariance matrix of the received signal. For the number of snapshots, Indicates time, Indicates in The received signal of the time sensor array, This represents matrix conjugation.

4. The amplitude and phase error self-correction method based on the covariance matrix of a uniform linear array signal according to claim 1, characterized in that, The system of linear equations is expressed as follows: ； In the formula, The coefficient matrix, , Represents the vector of all redundant parameters. This is the noise error vector.

5. The amplitude and phase error self-correction method based on the covariance matrix of a uniform linear array signal according to claim 1, characterized in that, The weighted least squares estimation problem with respect to amplitude and phase errors, which contains equality constraints, is expressed as: ； ； ； In the formula, This indicates a constraint condition.

6. The amplitude and phase error self-correction method based on the covariance matrix of a uniform linear array signal according to claim 5, characterized in that, Solving the weighted least squares estimation problem to obtain an estimate of the amplitude and phase error, and then using the estimate of the amplitude and phase error to correct the received signal, includes: The weighted least squares estimation problem is solved using the Lagrange multiplier method to obtain a constrained weighted least squares solution. The first... One value is used as an estimate of the amplitude and phase error, and the received signal is corrected based on the estimate of the amplitude and phase error.

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