CN104679976A

CN104679976A - Contractive linear and contractive generalized linear complex-valued least squares algorithm for signal processing

Info

Publication number: CN104679976A
Application number: CN201410606028.XA
Authority: CN
Inventors: 黄磊; 石运梅; 王永华; 尤琳
Original assignee: Harbin Institute of Technology Shenzhen
Current assignee: Harbin Institute of Technology Shenzhen
Priority date: 2014-10-31
Filing date: 2014-10-31
Publication date: 2015-06-03
Anticipated expiration: 2034-10-31
Also published as: CN104679976B

Abstract

In order to solve the traditional fixed update step size in practical applications, as well as the slow convergence speed and large mean square error when the non-circularity of the signal is not considered, the present invention proposes a contracted linear and contracted generalized linear complex least squares algorithm. With adaptive beamforming, the variable step size of the weight update is used to minimize the instantaneous average error of the posterior error when the noise is not considered, and the contracted generalized linear complex least squares algorithm also considers the expected signal non-circularity. These two methods increase the convergence speed and greatly reduce the steady-state mean square error.

Description

Shrinking Linear and Shrinking Generalized Linear Complex Least Squares Algorithms for Signal Processing

技术领域 technical field

本发明涉及阵列信号处理技术领域，尤其涉及一种收缩线性和收缩广义线性复最小二乘算法。 The invention relates to the technical field of array signal processing, in particular to a contracted linear and contracted generalized linear complex least squares algorithm.

背景技术 Background technique

阵列信号处理是信号处理领域中的一个重要分支，经过几十年的发展已日趋成熟并且在雷达、生物医疗、勘探及天文等多个军事和国民经济领域都有着广泛的应用。其工作原理是将多个传感器组成传感器阵列，并利用这一阵列对空间信号进行接收和处理，目的是抑制干扰和噪声，提取信号的有用信息。与一般的信号处理方式不同，阵列信号处理是通过布置在空间的传感器组接收信号，并且利用信号的空域特性来滤波及提取信息。因此，阵列信号处理也常被成为空域信号处理。此外，阵列信号处理有着灵活的波束控制、很强的抗干扰能力与极高的空间超分辨能力等优点，因而受到了众多学者的关注，其应用范围也不断地增大。 Array signal processing is an important branch in the field of signal processing. After decades of development, it has become increasingly mature and has been widely used in many military and national economic fields such as radar, biomedicine, exploration and astronomy. Its working principle is to form a sensor array with multiple sensors, and use this array to receive and process space signals, with the purpose of suppressing interference and noise, and extracting useful information of signals. Different from general signal processing methods, array signal processing is to receive signals through sensor groups arranged in space, and use the spatial characteristics of signals to filter and extract information. Therefore, array signal processing is often referred to as spatial domain signal processing. In addition, array signal processing has the advantages of flexible beam steering, strong anti-interference ability, and extremely high spatial super-resolution ability, so it has attracted the attention of many scholars, and its application range has also continued to increase.

在阵列信号处理领域，最重要的两个研究方向是自适应滤波和空间谱估计，其中自适应滤波技术先于空间谱估计产生，而且其应用在工程系统中已十分广泛。然而，对于空间谱估计虽然在近30年中得到了快速的发展，相关研究内容十分广泛，但其工程应用系统却不多见。这里，自适应滤波技术是阵列信号处理领域中的一个重要概念。 In the field of array signal processing, the two most important research directions are adaptive filtering and spatial spectrum estimation. Among them, adaptive filtering technology is produced before spatial spectrum estimation, and its application in engineering systems has been very extensive. However, although spatial spectrum estimation has developed rapidly in the past 30 years, and the related research content is very extensive, its engineering application system is rare. Here, adaptive filtering technique is an important concept in the field of array signal processing.

自适应滤波可以应用到模型化、均衡、控制、回声消除器和自适应波束形成中。复值的最小二乘算法是一种自适应估计和预测技术，可以实现性能收敛到最优维纳解。自适应波束形成器权矢量可以基于不同的设计准则来计算，常用的准则有最小均方误差、最小方差和恒模准则，本发明采用的是最小均方误差的准则。 Adaptive filtering can be applied in modeling, equalization, steering, echo cancellers, and adaptive beamforming. The complex-valued least-squares algorithm is an adaptive estimation and prediction technique that achieves performance convergence to an optimal Wiener solution. The weight vector of the adaptive beamformer can be calculated based on different design criteria. Commonly used criteria include the minimum mean square error, minimum variance and constant modulus criteria. The present invention adopts the minimum mean square error criterion.

经典的自适应阵列利用的是圆信号，通常可以找到一个线性时不变的复值滤波器w，滤波器的输出在确定性约束的条件下最优化二阶准则。但是，在实际应用中，非圆信号已经广泛地应用到许多现代通信系统中。由于经典的自适应波束形成器对圆信号来说是最优的，但对非圆信号来说是次优的。因此，广义复值最小二乘法利用扩展信号可以得到更低的波束器输出与期望信号之间的均方误差。而且复值最小二乘法对于二阶非圆信号也是次优的，因此如何保证得到非圆信号的最优值，以及提高收敛速度和输出信干噪比，降低均方误差为问题的重点。 The classic adaptive array uses the circular signal, usually a linear time-invariant complex-valued filter w can be found, and the output of the filter Optimizing a second-order criterion subject to deterministic constraints. However, in practical applications, non-circular signals have been widely used in many modern communication systems. Since the classical adaptive beamformer is optimal for circular signals, it is suboptimal for non-circular signals. Therefore, the generalized complex-valued least squares method utilizes the extended signal A lower mean square error between the beamer output and the desired signal can be obtained. Moreover, the complex-valued least squares method is also suboptimal for second-order non-circular signals, so how to ensure the optimal value of non-circular signals, improve the convergence speed and output SINR, and reduce the mean square error is the focus of the problem.

发明内容 Contents of the invention

为了解决在实际应用中传统的固定更新步长以及未考虑信号的非圆性时，收敛速度慢和均方误差大等问题，本发明提出了一种收缩线性和收缩广义线性的复值最小二乘算法，该方法在考虑非圆性的条件下，使得不考虑噪声时的后验误差的瞬时平均误差最小化，得到近似最优的可变步长，从而提高收敛速度，降低均方误差。 In order to solve the problems of slow convergence speed and large mean square error when the traditional fixed update step size and the non-circularity of the signal are not considered in practical applications, the present invention proposes a contracted linear and contracted generalized linear complex-valued least squares Multiplication algorithm, under the condition of considering non-circularity, this method minimizes the instantaneous average error of the posterior error when no noise is considered, and obtains an approximately optimal variable step size, thereby increasing the convergence speed and reducing the mean square error.

本发明通过如下技术方案实现： The present invention realizes through following technical scheme:

本发明的收缩线性复最小二乘法和收缩广义线性复最小二乘法，与以前方法不同的是既利用了非圆性又利用了收缩算法来得到可变的更新步长值，则从下面两方面来提高性能： The shrinking linear complex least squares method and the shrinking generalized linear complex least squares method of the present invention are different from the previous methods in that both non-circularity and contraction algorithm are utilized to obtain variable update step length values, then from the following two aspects to improve performance:

1.利用非圆性来提高收敛速度和降低均方误差。具体实施步骤如下： 1. Use non-circularity to improve convergence speed and reduce mean square error. The specific implementation steps are as follows:

首先，当期望信号是BPSK、QPSK以及PAM这样的信号时，则此时期望信号可分解为则此时接收到的信号与它的共轭是相关的，即共轭里面包含了期望信号的有用信息，因此，C_x＝E[x(k)x^T(k)]≠0_M×M。则此时的扩展信号为 First of all, when the desired signal is a signal such as BPSK, QPSK and PAM, then the desired signal can be decomposed into Then the signal received at this time is related to its conjugate, that is, the conjugate contains useful information of the desired signal, therefore, C _x =E[x(k)x ^T (k)]≠0 _M×M . Then the extended signal at this time is

则再由均方误差准则来得出阵列输出与期望信号之间误差的代价函数可求得此时的权值。此时，扩展的阵列权值为则此时利用了非圆信息，提高收敛速度，降低了均方误差。 Then the cost function of the error between the array output and the expected signal is obtained by the mean square error criterion, and the weight value at this time can be obtained. At this point, the expanded array weight is At this time, the non-circular information is used to improve the convergence speed and reduce the mean square error.

2.再利用收缩的方法。具体实施步骤如下： 2. Reuse the method of contraction. The specific implementation steps are as follows:

在收缩线性最小二乘法的求变化的步长的过程中，则由于步长为通常情况下在实际应用中，通常E[|x(k)||²]是已知的，E[|E(k)||²]是通过估计出的，其中λ是遗忘因子且0＜＜λ≤1。然而，e_f(k)是未知的，很难通过求解E[e_f(k)|²]来解上述步长。可以用收缩去噪方法，通过先验误差e(k)恢复出无噪声时的先验误差e_f(k)。使f[e_f(k)]＝0.5|e_f(k)-e(k)|²+α|e_f(k)|最小，则可恢复出e_f(k)。此时，则可得到可变步长的值，从而得到权值的更新过程。从而可以提高收敛速度，降低均方误差。 In the process of shrinking the step size of the linear least squares method, since the step size is Usually in practical applications, usually E[|x(k)|| ² ] is known, and E[|E(k)|| ² ] is passed Estimated, where λ is the forgetting factor and 0<<λ≤1. However, e _f (k) is unknown, and it is difficult to solve the above step size by solving E[ _ef (k)| ² ]. The shrinkage denoising method can be used to restore the prior error e _f (k) without noise through the prior error e(k). Make f[e _f (k)]=0.5|e _f (k)-e(k)| ² +α|e _f (k)| the minimum, then e _f (k) can be recovered. At this time, the value of the variable step can be obtained, thereby obtaining the update process of the weight. Therefore, the convergence speed can be improved and the mean square error can be reduced.

附图说明 Description of drawings

图1是本发明的收缩线性和收缩广义线性复最小二乘算法A估计方法流程图； Fig. 1 is contraction linear and contraction generalized linear complex least squares algorithm A estimation method flowchart of the present invention;

图2(a)和图2(b)是在Q不同时，本发明的算法与复最小二乘法、广义线性复最小二乘法的输出信干躁比随迭代次数的变化曲线图； Fig. 2 (a) and Fig. 2 (b) are when Q is different, the output signal-to-noise ratio of the algorithm of the present invention and complex least squares method, generalized linear complex least squares ratio changes curve figure with number of iterations;

图3(a)和图3(b)是在Q固定、步长不同时，本发明的算法与复最小二乘法、广义线性复最小二乘法的输出信干躁比随迭代次数的变化曲线图； Fig. 3 (a) and Fig. 3 (b) are when Q is fixed, step size is different, the output signal-to-noise ratio of the algorithm of the present invention and complex least squares method, generalized linear complex least squares method changes the curve figure with number of iterations ;

图4是本发明的算法与VSS、CNLMS、WL-CNLMS和WL-VSS算法的输出信干躁比随迭代次数的变化曲线图； Fig. 4 is the change curve graph of the output signal-to-noise ratio of the algorithm of the present invention and VSS, CNLMS, WL-CNLMS and WL-VSS algorithm with the number of iterations;

图5(a)和图5(b)是在Q不同时，本发明的算法与复最小二乘法、广义线性复最小二乘法的均方误差随迭代次数的变化曲线图； Fig. 5 (a) and Fig. 5 (b) are when Q is different, the change curve graph of the mean square error of algorithm of the present invention and complex least squares method, generalized linear complex least squares method with number of iterations;

图6(a)和图6(b)是步长不同时，本发明的算法与复最小二乘法、广义线性复最小二乘法的均方误差随迭代次数的变化曲线图； Fig. 6 (a) and Fig. 6 (b) are when step length is different, the mean square error of algorithm of the present invention and complex least squares method, generalized linear complex least squares method is with the variation curve figure of number of iterations;

图7是本发明的算法与VSS、CNLMS、WL-CNLMS和WL-VSS算法的均方误差随迭代次数的变化曲线图。 Fig. 7 is a graph showing the variation of the mean square error of the algorithm of the present invention and the VSS, CNLMS, WL-CNLMS and WL-VSS algorithms with the number of iterations.

具体实施方式 Detailed ways

下面结合附图说明及具体实施方式对本发明进一步说明。 The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

考虑一M阵元的均匀线阵，接收一个远场窄带信号s₀(k)，对应的波达角为θ_d。这个信号是零均值，二阶非圆的。则阵列输出数据可以表示为： Consider a uniform linear array with M array elements to receive a far-field narrowband signal s ₀ (k), and the corresponding angle of arrival is θ _d . This signal is zero-mean, second-order non-circular. Then the array output data can be expressed as:

x(k)＝a(θ_d)s₀(k)+n(k) x(k)=a(θ _d )s ₀ (k)+n(k)

其中， $a (θ_{d}) = {[1, e^{j 2 πΔ \sin (θ_{d}) / λ}, . . ., e^{j (M - 1) 2 πΔ \sin (θ_{d}) / λ}]}^{T}$ 为期望信号的导向矢量，Δ代表相邻阵元间的阵列间隔，λ代表波长，n(k)＝[n₁(k)，…，n_M(k)]^T为加性噪声矢量，它由背景噪声和干扰组成，可以表达为： in, $a (θ_{d}) = {[1, e^{j 2 πΔ \sin (θ_{d}) / λ}, . . ., e^{j (m - 1) 2 πΔ \sin (θ_{d}) / λ}]}^{T}$ is the steering vector of the desired signal, Δ represents the array spacing between adjacent array elements, λ represents the wavelength, n(k)=[n ₁ (k),...,n _M (k)] ^T is the additive noise vector, it Composed of background noise and disturbances, it can be expressed as:

$n no ((k k)) = = {Σ Σ}_{i i = = 11}^{P P} a a (({θ θ}_{i i})) {s the s}_{i i} ((k k)) + + η η ((k k))$

其中，P个统计不相关的非圆干扰，它们的复包络为s_i(k)，i＝1，2，...，P，以及对应的导向矢量为a(θ_i)，i＝１，２，...，P，η(k)是与期望信号和干扰均不相关的背景噪声。 Among them, P statistically irrelevant non-circular interferences, their complex envelopes are s _i (k), i=1, 2,..., P, and the corresponding steering vectors are a(θ _i ), i= 1, 2, ..., P, η(k) is the background noise that is uncorrelated with the desired signal and interference.

当权值向量为w＝[w₁，...，w_M]^T时，则最优权矢量可通过使波束形成器的输出与理想信号s_d(k)的均方误差最小得到 When the weight vector is w=[w ₁ ,...,w _M ] ^T , the optimal weight vector can be obtained by minimizing the mean square error between the output of the beamformer and the ideal signal s _d (k)

$\begin{matrix} {w w}_{opt opt} = = arg arg \underset{w w}{min min} E E. [[{| | e e ((k k)) | |}^{22}]] \\ = = arg arg \underset{w w}{min min} E E. [[{| | {s the s}_{d d} ((k k)) - - y the y ((k k)) | |}^{22}]] \\ = = arg arg \underset{w w}{min min} E E. [[{| | {s the s}_{00} ((k k)) - - y the y ((k k)) | |}^{22}]] \end{matrix} - - - - - - ((11))$

其中，s_d(k)＝s₀(k)，通过一些运算可解得最优权值为： Among them, s _d (k) = s ₀ (k), the optimal weight can be solved by some operations:

${w w}_{opt opt} = = {R R}_{x x}^{- - 11} {P P}_{x x}$

其中， $R_{x} \overset{Δ}{=} E [x (k) x^{H} (k)], P_{x} \overset{Δ}{=} E [x (k) s_{0}^{*} (k)],$ 使瞬时方差的功率J[w(k)]最小 in, $R_{x} \overset{Δ}{=} E. [x (k) x^{h} (k)], P_{x} \overset{Δ}{=} E. [x (k) {the s}_{0}^{*} (k)],$ Minimize the power J[w(k)] of the instantaneous variance

$J J [[ω ω ((k k))]] \overset{Δ Δ}{= =} {| | {s the s}_{00} ((k k)) - - y the y ((k k)) | |}^{22} = = {| | {s the s}_{00} ((k k)) - - ω ω {((k k))}^{H h} x x ((k k)) | |}^{22} - - - - - - ((22))$

从而得到权值的更新过程 So as to get the update process of the weight

w(k+1)＝w(k)+μe^*(k)x(k)。 (3) w(k+1)=w(k)+μe ^* (k)x(k). (3)

当期望信号是非圆信号时，例如BPSK、QPSK、PAM等，则此时期望信号矢量可表示为通常，C_x＝E[x(k)x^T(k)]≠0_M×M，为了利用非圆性，扩充向量可表示为 When the desired signal is a non-circular signal, such as BPSK, QPSK, PAM, etc., the desired signal vector can be expressed as Usually, C _x ＝E[x(k)x ^T (k)]≠0 _M×M , in order to take advantage of the non-circularity, the extended vector can be expressed as

其中，分别为扩展的导向矢量和噪声矢量。与复最小二乘法类似的广义复最小二乘法的代价函数为 in, are the expanded steering vector and noise vector, respectively. The cost function of the generalized complex least squares method similar to the complex least squares method is

$\begin{matrix} J J ((\overset{~ ~}{w w} ((k k)))) \overset{Δ Δ}{= =} {| | \overset{~ ~}{e e} ((k k)) | |}^{22} \\ = = {| | {s the s}_{00} ((k k)) - - \overset{~ ~}{y the y} ((k k)) | |}^{22} \\ = = {| | {s the s}_{00} ((k k)) - - \overset{~ ~}{w w} {((k k))}^{H h} \overset{~ ~}{x x} ((k k)) | |}^{22} \end{matrix} - - - - - - ((55))$

其中，为扩展的瞬时误差，代表扩展的波束形成器的输出。 in, is the extended instantaneous error, represents the output of the expanded beamformer.

此时，广义权矢量的更新过程为 at this time, The update process of the generalized weight vector is

$\begin{matrix} {w w}_{11} ((k k + + 11)) = = {w w}_{11} ((k k)) + + μ μ {\overset{~ ~}{e e}}^{* *} ((k k)) x x ((k k)),, \\ {w w}_{22} ((k k + + 11)) = = {w w}_{22} ((k k)) + + μ μ {\overset{~ ~}{e e}}^{* *} ((k k)) x x {((k k))}^{* *} . . \end{matrix} - - - - - - ((66))$

使最小化，得到最优的广义权矢量 make Minimize to get the optimal generalized weight vector

${\overset{~ ~}{w w}}_{opt opt} = = {\overset{~ ~}{R R}}_{x x}^{- - 11} {\overset{~ ~}{P P}}_{x x}$

其中， ${\tilde{R}}_{x} = [\begin{matrix} R_{x} & C_{x} \\ C_{x}^{*} & R_{x}^{*} \end{matrix}],$ ${\tilde{P}}_{x} \overset{Δ}{=} E [\tilde{x} (k) s_{0}^{*} (k)] .$ in, ${\tilde{R}}_{x} = [\begin{matrix} R_{x} & C_{x} \\ C_{x}^{*} & R_{x}^{*} \end{matrix}],$ ${\tilde{P}}_{x} \overset{Δ}{=} E. [\tilde{x} (k) {the s}_{0}^{*} (k)] .$

将式(3)中的μ换成可变步长μ_k，则此时权矢量的更新过程为 Replace μ in formula (3) with variable step size μ _k , then the update process of the weight vector at this time is

$\begin{matrix} w w ((k k + + 11)) = = w w ((k k)) + + {μ μ}_{k k} {| | s the s}_{00} ((k k)) - - y the y {| |}^{* *} x x ((k k)) \\ = = [[{I I}_{M m} - - {μ μ}_{k k} x x ((k k)) {x x}^{H h} ((k k))]] w w ((k k)) + + {μ μ}_{k k} {s the s}_{00}^{* *} ((k k)) x x ((k k)) . . \end{matrix} - - - - - - ((77))$

假设序列对{x(k)，s₀(k)}是广义平稳的，因此最优权矢量w_opt(k)是时不变的，即ω_opt(k)＝w_opt。设权矢量的误差向量v(k)＝w(k)-w_opt，则可以得到v(k)的更新过程为 Assume that the sequence pair {x(k), s ₀ (k)} is broadly stationary, so the optimal weight vector w _opt (k) is time-invariant, ie ω _opt (k)=w _opt . Assuming the error vector v(k) of the weight vector=w(k)-w _opt , the updating process of v(k) can be obtained as

$v v ((k k + + 11)) = = [[{I I}_{M m} - - {μ μ}_{k k} x x ((k k)) {x x}^{H h} ((k k))]] v v ((k k)) + + {μ μ}_{k k} {&Element; &Element;}_{opt opt}^{* *} ((k k)) x x ((k k)) - - - - - - ((88))$

其中，在k时刻时，波束形成器的输出与期望信号s₀(k)之间的误差为 in, At time k, the error between the output of the beamformer and the desired signal s ₀ (k) is

$\begin{matrix} e e ((k k)) = = {s the s}_{00} ((k k)) - - {w w}^{H h} ((k k)) x x ((k k)) \\ = = {&Element; &Element;}_{opt opt} ((k k)) + + {w w}_{opt opt}^{H h} x x ((k k)) - - {w w}^{H h} ((k k)) x x ((k k)) \\ = = {&Element; &Element;}_{opt opt} ((k k)) + + {e e}_{f f} ((k k)) \end{matrix} - - - - - - ((99))$

其中， $e_{f} (k) = w_{opt}^{H} x (k) - w^{H} (k) x (k) = - v^{H} (k) x (k)$ 是无噪声时的先验误差。另外，后验误差可表示为ε(k)＝∈_opt(k)+ε_f(k)，其中 in, $e_{f} (k) = w_{opt}^{h} x (k) - w^{h} (k) x (k) = - v^{h} (k) x (k)$ is the prior error when there is no noise. In addition, the posterior error can be expressed as ε(k)= _∈opt (k)+ _εf (k), where

$\begin{matrix} {ϵ ϵ}_{f f} ((k k)) = = {w w}_{opt opt}^{H h} x x ((k k)) - - {w w}^{H h} ((k k + + 11)) x x ((k k)) \\ = = - - {v v}^{H h} ((k k + + 11)) x x ((k k)) \end{matrix} - - - - - - ((1010))$

为无噪声时的后验误差，将式(8)共轭转置后两边同时右乘x(k)，再代入上式，得到 is the posterior error when there is no noise, after the conjugate transposition of formula (8), both sides are multiplied by x(k) at the same time, and then substituted into the above formula, we get

ε_f(k)＝(1-μ_k||x(k)||²)e_f(k)-μ_k∈_opt(k)||x(k)||². (11) ε _f (k)＝(1-μ _k ||x(k)|| ² )e _f (k)-μ _k ∈ _opt (k)||x(k)|| ² . (11)

瞬时的无噪后验误差的能量可表达为 The energy of the instantaneous noise-free posterior error can be expressed as

$\begin{matrix} {| | {ϵ ϵ}_{f f} ((k k)) | |}^{22} = = {| | {e e}_{f f} ((k k)) | |}^{22} - - {μ μ}_{k k} [[22 {| | | | x x ((k k)) | | | |}^{22} {| | {e e}_{f f} ((k k)) | |}^{22} \\ + + {&Element; &Element;}_{opt opt}^{* *} ((k k)) {e e}_{f f} ((k k)) {| | | | x x ((k k)) | | | |}^{22} + + {&Element; &Element;}_{opt opt} ((k k)) {e e}_{f f}^{* *} ((k k)) {| | | | x x ((k k)) | | | |}^{22}]] \\ + + {μ μ}_{k k}^{22} [[{| | | | x x ((k k)) | | | |}^{44} {| | {e e}_{f f} ((k k)) | |}^{22} + + {| | | | x x ((k k)) | | | |}^{44} {e e}_{f f} ((k k)) {&Element; &Element;}_{opt opt}^{* *} ((k k)) \\ + + {| | | | x x ((k k)) | | | |}^{44} {e e}_{f f}^{* *} ((k k)) {&Element; &Element;}_{opt opt} ((k k)) + + {| | | | x x ((k k)) | | | |}^{44} {| | {&Element; &Element;}_{opt opt} ((k k)) | |}^{22}]] . . \end{matrix} - - - - - - ((1212))$

将式(12)两边同时对μ_k求导后等于0，则得到 After deriving both sides of formula (12) with respect to μ _k , it is equal to 0, then we get

$\begin{matrix} 22 {| | {e e}_{f f} ((k k)) | |}^{22} + + {&Element; &Element;}_{opt opt}^{* *} ((k k)) {e e}_{f f} ((k k)) + + {&Element; &Element;}_{opt opt} ((k k)) {e e}_{f f}^{* *} ((k k)) \\ = = 22 {μ μ}_{k k} {| | | | x x ((k k)) | | | |}^{22} [[{| | {e e}_{f f} ((k k)) + + {&Element; &Element;}_{opt opt} ((k k)) | |}^{22}]] . . \end{matrix} - - - - - - ((1313))$

将式(9)代入式(13)，得到 Substituting formula (9) into formula (13), we get

$\begin{matrix} 22 {| | {e e}_{f f} ((k k)) | |}^{22} + + {&Element; &Element;}_{opt opt}^{* *} ((k k)) {e e}_{f f} ((k k)) + + {&Element; &Element;}_{opt opt} ((k k)) {e e}_{f f}^{* *} ((k k)) \\ = = 22 {μ μ}_{k k} {| | | | x x ((k k)) | | | |}^{22} {| | e e ((k k)) | |}^{22} . . \end{matrix} - - - - - - ((1414))$

对共轭再右乘x(k)后两边同时求期望，则 right After the conjugate is multiplied to the right by x(k), both sides are expected at the same time, then

$\begin{matrix} E E. [[{&Element; &Element;}_{opt opt}^{* *} ((k k)) x x ((k k))]] = = E E. [[{s the s}_{00}^{* *} ((k k)) x x ((k k))]] - - E E. [[x x ((k k)) {x x}^{H h} ((k k))]] {w w}_{opt opt} \\ = = {P P}_{x x} - - {R R}_{x x} {R R}_{x x}^{- - 11} {P P}_{x x} \\ = = 00 . . \end{matrix} - - - - - - ((1515))$

因此，输入信号x(k)与是统计垂直的。当非常小以及在稳态时变化很慢时，e^*(k)与x(k)是不相关的，可得到 Therefore, the input signal x(k) and is statistically vertical. when When it is very small and changes slowly in steady state, e ^* (k) and x(k) are irrelevant, and it can be obtained

E[||x(k)||²|e(k)|²]＝E[||x(k)||²]E[|e(k)|²]. (16) E[||x(k)|| ² |e(k)| ² ]＝E[||x(k)|| ² ]E[|e(k)| ² ]. (16)

观察到在j＜k时，输入序列为独立时，v(k)仅与{x(j)，s₀(j)}有关，而独立于当前的输入信号x(k)，可得到 Observe that when j<k, when the input sequence is independent, v(k) is only related to {x(j), s ₀ (j)}, but independent of the current input signal x(k), we can get

$\begin{matrix} E E. [[{&Element; &Element;}_{opt opt}^{* *} ((k k)) {e e}_{f f} ((k k))]] = = - - E E. [[{&Element; &Element;}_{opt opt}^{* *} ((k k)) {v v}^{H h} ((k k)) x x ((k k))]] \\ = = 00 . . \end{matrix} - - - - - - ((1717))$

对式(14)的两边求期望并结合式(15)到(17)的结果，得到 Finding the expectation on both sides of formula (14) and combining the results of formulas (15) to (17), we get

$E E. [[{μ μ}_{k k}]] = = \frac{E E. [[{| | {e e}_{f f} ((k k)) | |}^{22}]]}{E E. [[{| | | | x x ((k k)) | | | |}^{22}]] E E. [[{| | e e ((k k)) | |}^{22}]]} - - - - - - ((1818))$

其中，E[μ_k||x(k)||²|e(k)|²]＝E[μ_k]E[||x(k)||²|e(k)|²]。(19) Wherein, E[μ _k ||x(k)|| ² |e(k)| ² ]=E[μ _k ]E[||x(k)|| ² |e(k)| ² ]. (19)

当μ_k是常数时，上式肯定成立。实际上，在稳态时，μ_k与x(k)、e(k)相比变化比较慢。因此，可以认为μ_k与x(k)、e(k)是近似不相关的，即式(19)是近似成立的。 When μ _k is a constant, the above formula must be true. In fact, in steady state, μ _k changes relatively slowly compared with x(k) and e(k). Therefore, it can be considered that μ _k is approximately uncorrelated with x(k) and e(k), that is, formula (19) is approximately true.

由于E[|e_f(k)|²]是由权矢量和最优权矢量之间的误差引起的多余均方误差。 Since E[| _ef (k)| ² ] is the redundant mean square error caused by the error between the weight vector and the optimal weight vector.

${μ μ}_{k k} = = \frac{E E. [[{| | {e e}_{f f} ((k k)) | |}^{22}]]}{E E. [[{| | | | x x ((k k)) | | | |}^{22}]] E E. [[{| | e e ((k k)) | |}^{22}]]} - - - - - - ((2020))$

在式(3)中，用μ_k代替μ，可以得到复最小二乘法的权值更新过程。 In formula (3), replace μ by μ _k , and the weight update process of the complex least squares method can be obtained.

在实际应用中，通常E[||x(k)||²]是已知的，E[|e(k)|²]是通过下式估计出的 In practical applications, usually E[||x(k)|| ² ] is known, and E[|e(k)| ² ] is estimated by the following formula

${σ σ}_{e e}^{22} ((k k)) = = λ λ {σ σ}_{e e}^{22} ((k k - - 11)) + + ((11 - - λ λ)) {| | e e ((k k)) | |}^{22} - - - - - - ((21 twenty one))$

其中λ是遗忘因子且0＜＜λ≤1。然而，e_f(k)是未知的，很难通过求解E[|e_f(k)|²]来解(20)。可以用收缩去噪方法，通过先验误差e(k)恢复出无噪声时的先验误差e_f(k)。 Where λ is the forgetting factor and 0<<λ≤1. However, e _f (k) is unknown, and it is difficult to solve (20) by solving E[|e _f (k)| ² ]. The shrinkage denoising method can be used to restore the prior error ef(k) without noise through the prior error _e (k).

f[e_f(k)＝0.5|e_f(k)-e(k)|²+α|e_f(k)| (22)使上式关于e_f(k)最小化，可以得到 f[e _f (k)＝0.5|e _f (k)-e(k)| ² +α|e _f (k)| (22) To minimize the above formula with respect to e _f (k), we can get

${\overset{^^}{e e}}_{f f} ((k k)) = = sign sign [[e e ((k k))]] max max ((| | e e ((k k)) | | - - α α,, 00)) - - - - - - ((23 twenty three))$

由此可知α的选择非常重要。假设背景噪声为高斯白噪声，协方差为干扰偏离期望信号的主瓣可以抑制干扰部分的大部分能量则有 It can be seen that the choice of α is very important. Assuming that the background noise is Gaussian white noise, the covariance is If the interference deviates from the main lobe of the desired signal, most of the energy of the interference part can be suppressed, then there is

$\begin{matrix} {w w}_{opt opt}^{H h} a a (({θ θ}_{d d})) \approx \approx 11 \\ {w w}_{opt opt}^{H h} a a (({θ θ}_{i i})) \approx \approx 00,, i i = = 1,2 1,2,, . . . . . .,, P P . . \end{matrix} - - - - - - ((24 twenty four))$

通过(9)和(24)可得 Through (9) and (24) can get

$\begin{matrix} E E. [[{| | {&Element; &Element;}_{opt opt} ((k k)) | |}^{22}]] = = E E. [[{| | {e e}_{f f} ((k k)) - - e e ((k k)) | |}^{22}]] \\ = = E E. [[{| | {s the s}_{00} ((k k)) - - {w w}_{opt opt}^{H h} x x ((k k)) | |}^{22}]] \\ \approx \approx E E. [[{| | {s the s}_{00} ((k k)) - - {s the s}_{00} - - {w w}_{opt opt}^{H h} η η ((k k)) | |}^{22}]] \\ = = {σ σ}_{η η}^{22} {| | | | {w w}_{opt opt} | | | |}^{22} . . \end{matrix} - - - - - - ((2525))$

在均匀线性阵列中，为了保证在期望信号的波达角的波束图为1同时最大化||w_opt||²，通常设由此得到 In a uniform linear array, in order to ensure that the beam pattern of the angle of arrival of the desired signal is 1 while maximizing ||w _opt || ² , it is usually set From this we get

$E E. [[{| | {e e}_{f f} ((k k)) - - e e ((k k)) | |}^{22}]] &GreaterEqual; &Greater Equal; \frac{{σ σ}_{n no}^{22}}{M m} - - - - - - ((2626))$

综上所述，我们则可把其中，Q是一个参数，用来补偿上述的近似。类似于E[|e(k)|²]，可以得到 In summary, we can put the Among them, Q is a parameter used to compensate the above approximation. Similar to E[|e(k)| ² ], we can get

${σ σ}_{ef ef}^{22} ((k k)) = = λ λ {σ σ}_{ef ef}^{22} ((k k - - 11)) + + ((11 + + λ λ)) {| | {\overset{^^}{e e}}_{f f} ((k k)) | |}^{22} - - - - - - ((2727))$

将上式中的结果和(21)中的结果代入(20)，可得到此时的权值更新步长为 Substituting the result in the above formula and the result in (21) into (20), the weight update step at this time can be obtained as

${μ μ}_{k k} = = \frac{{σ σ}_{ef ef}^{22} ((k k))}{E E. [[{| | | | x x ((k k)) | | | |}^{22}]] {σ σ}_{e e}^{22} ((k k))} - - - - - - ((2828))$

将式(28)中的μ_k代替(3)中的μ，即可得到复最小二乘法的权矢量的更新过程。 Substituting μ _k in formula (28) for μ in (3), the update process of the weight vector of the complex least squares method can be obtained.

类似地，将(6)中的两式子的两端分别减去w₁、w₂的最优权值w_1，opt、w_2，opt，可以得到权矢量误差的更新过程如下所示： Similarly, by subtracting the optimal weights w ₁ _{, opt} and w ₂ _{, opt of w 1 and w 2} from both ends of the two equations in (6), the update process of the weight vector error can be obtained as follows:

${v v}_{11} ((k k + + 11)) = = {v v}_{11} ((k k)) - - {μ μ}_{k k} {\overset{~ ~}{e e}}^{* *} ((k k)) x x ((k k))$

${v v}_{22} ((k k + + 11)) = = {v v}_{22} ((k k)) - - {μ μ}_{k k} {\overset{~ ~}{e e}}^{* *} ((k k)) {x x}^{* *} ((k k)) - - - - - - ((2929))$

此时用可变步长μ_k来替代(6)中的μ。根据(5)和(28)，得到权误差矢量的向量矩阵形式 In this case, use variable step size μ _k to replace μ in (6). According to (5) and (28), the vector matrix form of the weight error vector is obtained

$\begin{matrix} [\begin{matrix} {v v}_{11} ((k k + + 11)) \\ {v v}_{22} ((k k + + 11)) \end{matrix}] = = [\begin{matrix} {I I}_{M m} - - {μ μ}_{K K} x x ((k k)) {x x}^{H h} ((k k)) & - - {μ μ}_{k k} x x ((k k)) {x x}^{T T} ((k k)) \\ - - {μ μ}_{k k} {x x}^{* *} ((k k)) {x x}^{H h} ((k k)) & {I I}_{M m} - - {μ μ}_{k k} {x x}^{* *} ((k k)) {x x}^{T T} ((k k)) \end{matrix}] \\ \times \times [\begin{matrix} {v v}_{11} ((k k)) \\ {v v}_{22} ((k k)) \end{matrix}] - - {μ μ}_{k k} [\begin{matrix} x x ((k k)) \\ {x x}^{* *} ((k k)) \end{matrix}] {\overset{~ ~}{&Element; &Element;}}_{opt opt}^{* *} ((k k)) \end{matrix} - - - - - - ((3030))$

其中， ${\tilde{&Element;}}_{opt} (k) \overset{Δ}{=} s_{0} (k) - w_{1, opt}^{H} (k) x (k) - w_{2, opt}^{H} x^{*} (k)$ 是扩展的权矢量的波形输出与期望信号之间的误差。将(30)两边同时进行共轭转置再右乘 $\tilde{x} (k) = {[x {(k)}^{T}, x^{H} (k)]}^{T}$ 得 in, ${\tilde{&Element;}}_{opt} (k) \overset{Δ}{=} {the s}_{0} (k) - w_{1, opt}^{h} (k) x (k) - w_{2, opt}^{h} x^{*} (k)$ is the error between the waveform output of the expanded weight vector and the desired signal. Perform conjugate transposition on both sides of (30) at the same time and then multiply $\tilde{x} (k) = {[x {(k)}^{T}, x^{h} (k)]}^{T}$ have to

${\overset{~ ~}{ϵ ϵ}}_{f f} ((k k)) = = ((11 - - 22 {μ μ}_{k k} {| | | | x x ((k k)) | | | |}^{22})) {\overset{~ ~}{e e}}_{f f} ((k k)) - - 22 {μ μ}_{k k} {\overset{~ ~}{&Element; &Element;}}_{opt opt} {| | | | x x ((k k)) | | | |}^{22} - - - - - - ((3131))$

由于扩展无噪的后验误差和先验误差分别为 Due to the extended noiseless posterior error and prior error are respectively

$\begin{matrix} {\overset{~ ~}{ϵ ϵ}}_{f f} ((k k)) = = {[[{w w}_{11,, opt opt} - - {w w}_{11} ((k k + + 11))]]}^{H h} x x ((k k)) \\ + + {[[{w w}_{22,, opt opt} - - {w w}_{22} ((k k + + 11))]]}^{H h} {x x}^{* *} ((k k)) \\ = = - - {v v}_{11}^{H h} ((k k + + 11)) x x ((k k)) - - {v v}_{22}^{H h} ((k k + + 11)) {x x}^{* *} ((k k)) \end{matrix} - - - - - - ((3232))$

$\begin{matrix} {\overset{~ ~}{e e}}_{f f} ((k k)) = = {[[{w w}_{11,, opt opt} - - {w w}_{11} ((k k))]]}^{H h} x x ((k k)) \\ + + {[[{w w}_{22,, opt opt} - - {w w}_{22} ((k k))]]}^{H h} {x x}^{* *} ((k k)) \\ = = - - {v v}_{11}^{H h} ((k k)) x x ((k k)) - - {v v}_{22}^{H h} ((k k)) {x x}^{* *} ((k k)) \end{matrix} - - - - - - ((3333))$

与收缩的线性复最小二乘法类似，瞬时扩展的无噪后验误差的平方为 Similar to the contracted linear complex least squares method, the square of the instantaneously expanded noise-free posterior error is

$\begin{matrix} {| | {\overset{~ ~}{ϵ ϵ}}_{f f} ((k k)) | |}^{22} = = {| | {\overset{~ ~}{e e}}_{f f} ((k k)) | |}^{22} - - {μ μ}_{k k} [[44 {| | | | x x ((k k)) | | | |}^{22} {| | {\overset{~ ~}{e e}}_{f f} ((k k)) | |}^{22} \\ + + 22 {\overset{~ ~}{&Element; &Element;}}_{opt opt}^{* *} ((k k)) {\overset{~ ~}{e e}}_{f f} ((k k)) {| | | | x x ((k k)) | | | |}^{22} + + 22 {\overset{~ ~}{&Element; &Element;}}_{opt opt} ((k k)) {\overset{~ ~}{e e}}_{f f}^{* *} ((k k)) {| | | | x x ((k k)) | | | |}^{22}]] \\ + + {μ μ}_{k k}^{22} [[44 {| | | | x x ((k k)) | | | |}^{44} {| | {\overset{~ ~}{e e}}_{f f} ((k k)) | |}^{22} + + 44 {| | | | x x ((k k)) | | | |}^{44} {\overset{~ ~}{e e}}_{f f} ((k k)) {\overset{~ ~}{&Element; &Element;}}_{opt opt}^{* *} ((k k)) \\ + + 44 {| | | | x x ((k k)) | | | |}^{44} {\overset{~ ~}{e e}}_{f f}^{* *} ((k k)) {\overset{~ ~}{&Element; &Element;}}_{opt opt} ((k k)) + + 44 {| | | | x x ((k k)) | | | |}^{44} {| | {\overset{~ ~}{&Element; &Element;}}_{opt opt} ((k k)) | |}^{22}]] . . \end{matrix} - - - - - - ((3434))$

把上式作为代价函数，对其求关于μ_k的导数，并将其等于0，可以得到 Take the above formula as the cost function, find its derivative with respect to μ _k , and set it equal to 0, you can get

$\begin{matrix} 22 {| | {\overset{~ ~}{e e}}_{f f} ((k k)) | |}^{22} + + {\overset{~ ~}{&Element; &Element;}}_{opt opt}^{* *} ((k k)) {\overset{~ ~}{e e}}_{f f} ((k k)) + + {\overset{~ ~}{&Element; &Element;}}_{opt opt} ((k k)) {\overset{~ ~}{e e}}_{f f}^{* *} ((k k)) \\ = = 44 {μ μ}_{k k} {| | | | x x ((k k)) | | | |}^{22} [[{| | {\overset{~ ~}{e e}}_{f f} ((k k)) + + {\overset{~ ~}{&Element; &Element;}}_{opt opt} ((k k)) | |}^{22}]] \end{matrix} - - - - - - ((3535))$

在k时刻的瞬时误差为 The instantaneous error at time k is

$\begin{matrix} \overset{~ ~}{e e} ((k k)) \overset{Δ Δ}{= =} {s the s}_{00} ((k k)) - - {\overset{~ ~}{w w}}^{H h} ((k k)) \overset{~ ~}{x x} ((k k)) \\ = = {\overset{~ ~}{e e}}_{f f} ((k k)) + + {\overset{~ ~}{&Element; &Element;}}_{opt opt} ((k k)) \end{matrix} - - - - - - ((3636))$

将(36)代入(35)中得到 Substitute (36) into (35) to get

$\begin{matrix} 22 {| | {\overset{~ ~}{e e}}_{f f} ((k k)) | |}^{22} + + {\overset{~ ~}{&Element; &Element;}}_{opt opt}^{* *} ((k k)) {\overset{~ ~}{e e}}_{f f} ((k k)) + + {\overset{~ ~}{&Element; &Element;}}_{opt opt} ((k k)) {\overset{~ ~}{e e}}_{f f}^{* *} ((k k)) \\ = = 44 {μ μ}_{k k} {| | | | x x ((k k)) | | | |}^{22} {| | \overset{~ ~}{e e} ((k k)) | |}^{22} \end{matrix} - - - - - - ((3737))$

通过 ${\tilde{&Element;}}_{opt} (k) \overset{Δ}{=} s_{0} (k) - w_{1, opt}^{H} (k) x (k) - w_{2, opt}^{H} x^{*} (k),$ 则对其两边同时求共轭转置再右乘再两边同时求期望则有 pass ${\tilde{&Element;}}_{opt} (k) \overset{Δ}{=} {the s}_{0} (k) - w_{1, opt}^{h} (k) x (k) - w_{2, opt}^{h} x^{*} (k),$ Then calculate the conjugate transpose of both sides and multiply it right If we ask for expectations on both sides at the same time, we have

$\begin{matrix} E E. [[{\overset{~ ~}{&Element; &Element;}}_{opt opt}^{* *} ((k k)) \overset{~ ~}{x x} ((k k))]] = = E E. [[{s the s}_{00}^{* *} ((k k)) \overset{~ ~}{x x} ((k k))]] - - E E. [[\overset{~ ~}{x x} ((k k)) {\overset{~ ~}{x x}}^{H h} ((k k))]] {\overset{~ ~}{w w}}_{opt opt} \\ = = {\overset{~ ~}{P P}}_{x x} - - {\overset{~ ~}{R R}}_{x x} {\overset{~ ~}{R R}}_{x x}^{- - 11} {\overset{~ ~}{P P}}_{x x} \\ = = 00 \end{matrix} - - - - - - ((3838))$

假设也与不相关，由上式可知与扩展的输入信号垂直以及则可的先验误差与也是不相关的，即 Suppose also with irrelevant, it can be seen from the above formula with extended input signal vertical and a priori error and is also irrelevant, i.e.

$E E. [[{| | | | x x ((k k)) | | | |}^{22} {| | \overset{~ ~}{e e} ((k k)) | |}^{22}]] = = E E. [[{| | | | x x ((k k)) | | | |}^{22}]] E E. [[{| | \overset{~ ~}{e e} ((k k)) | |}^{22}]] - - - - - - ((3939))$

由(29)知v₁(k)，v₂(k)分别与x(k)，x^*(k)是不相关的。设由(33)和(38)的结果知 It is known from (29) that v ₁ (k), v ₂ (k) are irrelevant to x(k), x ^* (k) respectively. set up From the results of (33) and (38) we know

$\begin{matrix} E E. [[{\overset{~ ~}{&Element; &Element;}}_{opt opt}^{* *} ((k k)) {\overset{~ ~}{e e}}_{f f} ((k k))]] = = E E. {{[[- - {v v}_{11}^{H h} ((k k)) x x ((k k)) - - {v v}_{22}^{H h} ((k k)) {x x}^{* *} ((k k))]] {\overset{~ ~}{&Element; &Element;}}_{opt opt}^{* *} ((k k))}} \\ = = E E. [[- - {\overset{~ ~}{v v}}^{H h} ((k k)) \overset{~ ~}{x x} ((k k)) {\overset{~ ~}{&Element; &Element;}}_{opt opt}^{* *} ((k k))]] \\ = = 00 \end{matrix} - - - - - - ((4040))$

对(37)两边同时求期望，再利用(38)和(39)的结果得到 Find the expectation on both sides of (37) at the same time, and then use the results of (38) and (39) to get

$E E. [[{μ μ}_{k k}]] = = \frac{E E. [[{| | {\overset{~ ~}{e e}}_{f f} ((k k)) | |}^{22}]]}{22 E E. [[{| | | | x x ((k k)) | | | |}^{22}]] E E. [[{| | \overset{~ ~}{e e} ((k k)) | |}^{22}]]} - - - - - - ((4141))$

上式子是基于这样的假设 The above formula is based on the assumption that

$E E. [[{{μ μ}_{k k} | | | | x x ((k k)) | | | |}^{22} {| | \overset{~ ~}{e e} ((k k)) | |}^{22}]] = = E E. {[[{μ μ}_{k k}]] E E. [[| | | | x x ((k k)) | | | |}^{22} {| | \overset{~ ~}{e e} ((k k)) | |}^{22}]] - - - - - - ((4242))$

把E[μ_k]作为μ_k的估计代入(30)中，就得到广义线性复最小二乘法。 Substituting E[μ _k ] as the estimate of μ _k into (30), the generalized linear complex least squares method is obtained.

式(41)中的估计可通过下式得 In formula (41) can be estimated by the following formula

${\overset{~ ~}{σ σ}}_{e e}^{22} ((k k)) = = λ λ {\overset{~ ~}{σ σ}}_{e e}^{22} ((k k - - 11)) + + ((11 - - λ λ)) {| | \overset{~ ~}{e e} ((k k)) | |}^{22} - - - - - - ((4343))$

扩展的无噪先验误差平方的均值 Extended noise-free prior error mean of squares

${\overset{~ ~}{σ σ}}_{ef ef}^{22} ((k k)) = = λ λ {\overset{~ ~}{σ σ}}_{ef ef}^{22} ((k k - - 11)) + + ((11 - - λ λ)) {| | {\overset{~ ~}{e e}}_{f f} ((k k)) | |}^{22} - - - - - - ((4444))$

来代替(41)中的扩展无噪先验误差可通过恢复过来 to replace in (41) Extended noise-free prior error accessible recovered

${\overset{\overset{^^}{~ ~}}{e e}}_{f f} ((k k)) = = sign sign [[\overset{~ ~}{e e} ((k k))]] max max ((| | \overset{~ ~}{e e} ((k k)) | | - - α α,, 00)) - - - - - - ((4545))$

然后，需要如何选择门限α，由于干扰和背景噪声都与期望信号是不相关的，则可表示为 Then, how to choose the threshold α, since the interference and background noise are irrelevant to the desired signal, then can be expressed as

$\begin{matrix} {\overset{~ ~}{P P}}_{x x} = = E E. [[\overset{~ ~}{x x} ((k k)) {s the s}_{00}^{* *} ((k k))]] \\ = = E E. [[\overset{~ ~}{a a} (({θ θ}_{d d})) {σ σ}_{s the s}^{22} + + n no ((k k)) {s the s}_{00}^{* *} ((k k))]] \\ = = {σ σ}_{s the s}^{22} \overset{~ ~}{a a} (({θ θ}_{d d})) \end{matrix} - - - - - - ((4646))$

其中，是期望信号的功率。则此时的最优权矢量为 in, is the power of the desired signal. Then the optimal weight vector at this time is

${\overset{~ ~}{w w}}_{opt opt} = = {σ σ}_{s the s}^{22} {\overset{~ ~}{R R}}_{x x}^{- - 11} \overset{~ ~}{a a} (({θ θ}_{d d})) - - - - - - ((4747))$

这个结果类似于广义最优的最小方差无失真响应 This result is similar to the generalized optimal minimum variance distortion-free response

${\overset{~ ~}{w w}}_{MVDR MVDR} = = κ κ {\overset{~ ~}{R R}}_{x x}^{- - 11} \overset{~ ~}{a a} (({θ θ}_{d d})) - - - - - - ((4848))$

其中，与仅常数部分不同。当干扰的波达角与期望信号的波达角相差很大时，则有 in, and Only the constant part is different. When the angle of arrival of the interference is very different from the angle of arrival of the desired signal, then there is

$\begin{matrix} {\overset{~ ~}{w w}}_{opt opt}^{H h} \overset{~ ~}{a a} (({θ θ}_{d d})) \approx \approx 11 \\ {\overset{~ ~}{w w}}_{opt opt}^{H h} \overset{~ ~}{a a} (({θ θ}_{i i})) \approx \approx 00,, i i = = 1,2 1,2,, . . . . . .,, P P \end{matrix} - - - - - - ((4949))$

其中，其中是第i个干扰的初始相位。的近似结果是 in, in is the initial phase of the i-th disturbance. The approximate result of

$\begin{matrix} E E. [[{| | {\overset{~ ~}{&Element; &Element;}}_{opt opt} ((k k)) | |}^{22}]] = = E E. [[{| | {s the s}_{00} ((k k)) - - {\overset{~ ~}{w w}}_{opt opt}^{H h} \overset{~ ~}{x x} ((k k)) | |}^{22}]] \\ \approx \approx E E. [[{| | {s the s}_{00} ((k k)) - - {s the s}_{00} ((k k)) - - {\overset{~ ~}{w w}}_{opt opt}^{H h} \overset{~ ~}{η η} ((k k)) | |}^{22}]] \\ = = {σ σ}_{η η}^{22} {| | | | {\overset{~ ~}{w w}}_{opt opt} | | | |}^{22} \end{matrix} - - - - - - ((5050))$

其中，由于将这代入(50)中则得到 in, because Substituting this into (50) gives

$E E. [[{| | {\overset{~ ~}{&Element; &Element;}}_{opt opt} ((k k)) | |}^{22}]] &GreaterEqual; &Greater Equal; \frac{{σ σ}_{η η}^{22}}{22 M m} - - - - - - ((5151))$

由此，可得到将这些代入(3-41)中，则有 From this, it can be obtained Substituting these into (3-41), we have

${μ μ}_{k k} = = \frac{{\overset{~ ~}{σ σ}}_{ef ef}^{22} ((k k))}{22 E E. [[{| | | | x x ((k k)) | | | |}^{22} {\overset{~ ~}{σ σ}}_{e e}^{22} ((k k))} - - - - - - ((5252))$

将(52)中的结果代入(6)中则可得到收缩的广义复最小二乘法的权值更新过程。 Substituting the result in (52) into (6), the weight update process of the contracted generalized complex least squares method can be obtained.

如附图1所示，本发明的收缩线性最小二乘算法包括如下步骤： As shown in accompanying drawing 1, shrinkage linear least square algorithm of the present invention comprises the following steps:

1.计算无噪声时的后验误差ε_f(k)的能量，再对将(3-12)两边同时对μ_k求导后等于0再将(3-9)代入(3-13)中并且结合(3-15)(3-16)则得到 $μ_{k} = \frac{E [{| e_{f} (k) |}^{2}]}{E [{| | x (k) | |}^{2}] E [{| e (k) |}^{2}]};$ 1. Calculate the energy of the posterior error ε _f (k) when there is no noise, and then take the derivative of both sides of (3-12) for μ _k and equal to 0, then substitute (3-9) into (3-13) And combine (3-15)(3-16) to get $μ_{k} = \frac{E. [{| e_{f} (k) |}^{2}]}{E. [{| | x (k) | |}^{2}] E. [{| e (k) |}^{2}]};$

2.由于通常E[||x(k)||²]是已知的，E[|e(k)|²]是通过(3-21)估计得到，而可以用收缩去噪方法，即使(3-22)最小得到 2. Since E[||x(k)|| ² ] is usually known, E[|e(k)| ² ] is estimated by (3-21), and shrinkage denoising method can be used, even (3-22) The minimum is obtained

3.通过(3-27)可得到E[|e_f(k)|²]的估计量； 3. The estimator of E[|e _f (k)| ² ] can be obtained through (3-27);

4.将估计量的结果代入(1)中μ_k式子中可得到 4. Substituting the result of the estimator into the μ _k formula in (1) can be obtained

5.得到权值更新过程w(k+1)＝w(k)+μ_ke^*(k)x(k)。 5. Obtain the weight update process w(k+1)=w(k)+μ _k e ^* (k)x(k).

本发明的收缩广义线性最小二乘算法包括如下步骤： Shrinkage generalized linear least squares algorithm of the present invention comprises the following steps:

1.计算瞬时扩展无噪声时的后验误差的平方，再对将(3-34)两边同时对μ_k求导后等于0，再将(3-36)代入(3-35)中并且结合(3-38)(3-39)则得到 $E [μ_{k}] = \frac{E [{| {\tilde{e}}_{f} (k) |}^{2}]}{2 E [{| | x (k) | |}^{2}] E [{| \tilde{e} (k) |}^{2}]};$ 1. Calculate the posterior error when the instantaneous expansion is noise-free The square of (3-34) is equal to 0 after taking the derivative of μ _k on both sides of (3-34), then substituting (3-36) into (3-35) and combining (3-38)(3-39) to get $E. [μ_{k}] = \frac{E. [{| {\tilde{e}}_{f} (k) |}^{2}]}{2 E. [{| | x (k) | |}^{2}] E. [{| \tilde{e} (k) |}^{2}]};$

2.由于通常E[||x(k)||²]是已知的，是通过(3-43)估计得到，而可以用收缩去噪方法，即使(3-45)最小得到 ${\hat{\tilde{e}}}_{f} (k) = sign [\tilde{e} (k)] \max (| \tilde{e} (k) | - α, 0);$ 2. Since E[||x(k)|| ² ] is usually known, It is estimated by (3-43), but shrinkage denoising method can be used, even if (3-45) is the smallest ${\hat{\tilde{e}}}_{f} (k) = sign [\tilde{e} (k)] \max (| \tilde{e} (k) | - α, 0);$

3.通过(3-24)可得到的估计量； 3. Can be obtained through (3-24) the estimate of

5.得到权值更新过程： 5. Get the weight update process:

${w w}_{11} ((k k + + 11)) = = {w w}_{11} ((k k)) + + μ μ {\overset{~ ~}{e e}}^{* *} ((k k)) x x ((k k)),, {w w}_{22} ((k k + + 11)) = = {w w}_{22} ((k k)) + + μ μ {\overset{~ ~}{e e}}^{* *} ((k k)) x x ((k k)) . .$

考虑一均匀线阵，阵元数为M＝4，阵列间距为四个等功率的BPSK信号，它们的非圆系数是1，初始相位均为0°，期望信号入射到阵列时的波达角为θ_d＝-45°，信噪比为10dB，其它三个干扰信号的波达角分别为θ₁＝8°、θ₂＝-13°、θ₃＝30°，干噪比(INR)固定为10dB，所有的仿真结果均由500次蒙特卡洛实验获得。在收缩的线性最小二乘法的初始值为以及w（0）＝0_M×1。而且，收缩的广义线性复最小二乘法的初始值为w₁(0)＝0_M×1以及w₂(0)＝0_M×1。遗忘因子λ是固定的，λ＝0.95。 Consider a uniform linear array, the number of array elements is M=4, and the array spacing is Four equal-power BPSK signals, their non-circular coefficients are 1, the initial phase is 0°, the angle of arrival of the desired signal when it is incident on the array is θ _d =-45°, the signal-to-noise ratio is 10dB, and the other three The angles of arrival of the interference signals are θ ₁ =8°, θ ₂ =-13°, θ ₃ =30°, and the interference-to-noise ratio (INR) is fixed at 10dB. All simulation results are obtained by 500 Monte Carlo experiments . The initial value of the contracted linear least squares method is And w(0)=0 _M×1 . Moreover, the initial value of the contracted generalized linear complex least squares method is w ₁ (0)=0 _M×1 and w ₂ (0)=0 _M×1 . The forgetting factor λ is fixed, λ=0.95.

实验1输出信干躁比随迭代次数的变化。 Experiment 1 output signal to noise ratio changes with the number of iterations.

在这个仿真中将比较本发明提出的算法与复最小二乘法、广义线性复最小二乘法随迭代次数的变化情况，当复最小二乘法与广义线性复最小二乘法的步长分别为0.001，0.0005，可以观察到在这两种情况下，考虑广义(非圆性)时的收敛速度要快于不考虑非圆性的情况，并且考虑非圆性时的输出信干噪比也较高，从附图2可以看出，Q对线性收缩的复最小二乘法的稳态性质影响较小。而且，收缩的广义线性复最小二乘法在Q＝2时比Ｑ＝1时的性能有所提高。附图3是在Q固定时，附图3(a)中复最小二乘法与广义线性复最小二乘法的步长分别为0.008，0.004，而附图3(b)中复最小二乘法与广义线性复最小二乘法的步长分别为0.0005，0.00025时得到的。从附图3(a)中可以看出，当步长比附图2(b)中的步长大8倍时，复最小二乘法与广义线性复最小二乘法收敛速度要大于附图2(b)。由于较大的步长可以提高收敛速度。然而，在稳态时，广义线性复最小二乘法与复最小二乘法此时的输出信干躁比较小。如果我们把步长固定为附图2(b)的μ的一半，从附图3(b)中，广义线性复最小二乘法与复最小二乘法在稳态时达到与本发明所提出的算法同样的输出信干噪比。然而，这两种方法达到稳态时分别需要200，400次迭代次数，而附图2(b)中的则仅分别需要100，200次即可达到稳态。这是由于步长较小，降低收敛速度。从附图3中可以看出本发明的算法大约分别需要60，100次迭代达到稳态。而且本发明的算法输出的信干噪比分别近似达到最优广义线性信干噪比，最优信干噪比。因此本发明的算法相对于广义线性复最小二乘法和复最小二乘法而言，有较快的收敛速度和较高的输出信干噪比。 In this simulation, the algorithm proposed by the present invention will be compared with the variation of the complex least squares method and the generalized linear complex least squares method with the number of iterations. , it can be observed that in both cases, the convergence speed when considering the generalized (non-circularity) is faster than that without considering the non-circularity, and the output SINR is also higher when considering the non-circularity, from It can be seen from Figure 2 that Q has little influence on the steady-state properties of the complex least squares method of linear shrinkage. Moreover, the contracted generalized linear complex least squares method has improved performance when Q=2 compared to Q=1. Accompanying drawing 3 is when Q is fixed, the step size of complex least squares method and generalized linear complex least squares method is respectively 0.008,0.004 in accompanying drawing 3 (a), and complex least squares method and generalized linear method in accompanying drawing 3 (b) The step sizes of the linear complex least squares method are 0.0005 and 0.00025 respectively. As can be seen from accompanying drawing 3(a), when the step size is 8 times larger than that in accompanying drawing 2(b), the convergence speed of complex least squares method and generalized linear complex least squares method is greater than that of accompanying drawing 2(b ). Convergence speed can be improved due to larger step size. However, in the steady state, the output signals of the generalized linear complex least squares method and the complex least squares method are relatively small. If we fix the step size as half of μ of accompanying drawing 2 (b), from accompanying drawing 3 (b), generalized linear complex least squares method and complex least squares method reach the algorithm proposed by the present invention in steady state The same output SINR. However, these two methods require 200 and 400 iterations respectively to reach the steady state, while the one in Figure 2(b) only needs 100 and 200 iterations respectively to reach the steady state. This is due to the smaller step size, which slows down the convergence rate. It can be seen from Fig. 3 that the algorithm of the present invention needs about 60 and 100 iterations respectively to reach a steady state. Moreover, the SINR output by the algorithm of the present invention approximately reaches the optimal generalized linear SINR and the optimal SINR respectively. Therefore, compared with the generalized linear complex least squares method and the complex least squares method, the algorithm of the present invention has faster convergence speed and higher output SINR.

附图4为比较本发明所提出的收缩线性复最小二乘法与标准复最小二乘法(CNLMS)、变步长法(VSS)，以及本发明所提出的比较收缩广义线性复最小二乘法与广义线性标准复最小二乘法(WL-CNLMS)、广义线性变步长法(WL-VSS)的输出信干噪比。设定复标准最小二乘法与广义线性复标准最小二乘法的步长分别为0.2，0.1。变步长的参数为μ_min＝e^-6、μ_max＝3e^-3、和广义变步长参数为μ_min＝5e^-7、μ_max＝1.5e^-3、和从附图4中可以看出，与复标准最小二乘法和广义复最小二乘法相比，本发明的算法收敛速度更快。而且，与变步长法和广义变步长法相比，本发明的算法具有更高的输出信干噪比。还可以看出，考虑非圆性时的这几种方法要优于不考虑非圆性时的算法。 Accompanying drawing 4 is for comparing contraction linear complex least squares method proposed by the present invention and standard complex least squares method (CNLMS), variable step size method (VSS), and the comparison contraction generalized linear complex least square method proposed by the present invention and generalized Output signal-to-interference-noise ratio of linear standard complex least squares method (WL-CNLMS) and generalized linear variable step size method (WL-VSS). Set the step size of the complex standard least squares method and the generalized linear complex standard least squares method to be 0.2 and 0.1, respectively. The parameters of variable step size are μ _min ＝e ^-6 , μ _max ＝3e ^-3 , and The generalized variable step size parameters are μ _min ＝5e ^-7 , μ _max ＝1.5e ^-3 , and It can be seen from accompanying drawing 4 that, compared with the complex standard least squares method and the generalized complex least squares method, the algorithm of the present invention has a faster convergence speed. Moreover, compared with the variable step size method and the generalized variable step size method, the algorithm of the present invention has a higher output SINR. It can also be seen that these methods when considering non-circularity are better than the algorithm when non-circularity is not considered.

实验2输出均方误差与迭代次数的关系。 Experiment 2 outputs the relationship between the mean square error and the number of iterations.

在实验中，每个变量的值得设置与上个实验是一样的。从附图5可以看出本发明的算法相对于广义线性复最小二乘法和复最小二乘法而言，收敛速度较快。本发明的收缩线性复最小二乘法和复最小二乘法在稳态时近似有相等的均方误差，同样，本发明的收缩广义线性复最小二乘法和广义复最小二乘法在稳态时近似有相等的均方误差。而且，由于考虑了非圆性，基于广义线性的方法有更好的性质。因为扩展的阵列孔径使收缩广义复最小二乘法对α的选择更敏感，则从附图5(a)中可以看出，当Q＝1时，本发明的收缩的广义线性复最小二乘法在稳态时要稍次于广义复最小二乘法。从附图5(b)中可以看出，当Q＝2时，本发明的收缩广义线性复最小二乘法在稳态时要稍优于广义复最小二乘法。而且不像基于线性的算法，Q值得大小对收缩线性复最小二乘法和复最小二乘法影响很小。附图6是比较本发明的算法与具有不同步长值的复最小二乘法和广义复最小二乘法。与附图5(b)相比，附图6(a)中的复最小二乘法和广义线性复最小二乘法具有更快的收敛速度，因为它们的步长值分别为0.008，0.004，是附图5(b)中的8倍。从附图6(a)中可以看出，广义复最小二乘法和复最小二乘法分别收敛到-4dB和0dB。然而，在附图5(b)中，μ＝0.001时，则它们分别收敛到-8dB和-4dB。附图6(b)为显示本发明的算法与步长分别为μ＝0.00025，μ＝0.0005的广义线性复最小二乘法与复最小二乘法的均方误差的比较。尽管复最小二乘法与广义线性复最小二乘法可以与本发明算法达到相同的稳态值，但是与附图5(b)相比，它们需要更大的迭代次数。前者分别需要200，400次迭代，后者分别需要150，250次迭代。因此本发明的的算法无论在收敛速度还是在均方误差方面都要优于复最小二乘法和广义线性复最小二乘法。 In the experiment, the value setting of each variable is the same as the previous experiment. As can be seen from accompanying drawing 5, algorithm of the present invention is relative to generalized linear complex least squares method and complex least squares method, and convergence rate is faster. Contraction linear complex least squares method of the present invention and complex least squares method approximately have equal mean square error in steady state, similarly, contraction generalized linear complex least squares method of the present invention and generalized complex least squares method approximately have in steady state equal mean square error. Moreover, methods based on generalized linearity have better properties due to the consideration of non-circularity. Because the extended array aperture makes the shrinkage generalized complex least squares method more sensitive to the selection of α, then as can be seen from accompanying drawing 5 (a), when Q=1, the generalized linear complex least squares method of the present invention shrinks in It is slightly inferior to the generalized complex least squares method in steady state. It can be seen from accompanying drawing 5 (b) that when Q=2, the shrinkage generalized linear complex least squares method of the present invention is slightly better than the generalized complex least squares method in steady state. And unlike linear-based algorithms, the size of the Q value has little effect on contracted linear complex least squares and complex least squares. Figure 6 is a comparison of the algorithm of the present invention with the complex least squares method and the generalized complex least squares method with different step length values. Compared with Figure 5(b), the complex least squares method and generalized linear complex least squares method in Figure 6(a) have faster convergence speeds, because their step values are 0.008 and 0.004 respectively, which are attached 8 times in Fig. 5(b). It can be seen from Figure 6(a) that the generalized complex least squares method and the complex least squares method converge to -4dB and 0dB respectively. However, in Fig. 5(b), when μ=0.001, they converge to -8dB and -4dB respectively. Accompanying drawing 6 (b) shows the algorithm of the present invention and the comparison of the mean square error of the generalized linear complex least squares method and the complex least squares method that the step size is respectively μ=0.00025 and μ=0.0005. Although the complex least squares method and the generalized linear complex least squares method can achieve the same steady-state value as the algorithm of the present invention, they require a larger number of iterations compared with the accompanying drawing 5(b). The former requires 200, 400 iterations, respectively, and the latter requires 150, 250 iterations, respectively. Therefore, the algorithm of the present invention is superior to the complex least squares method and the generalized linear complex least squares method in terms of convergence speed and mean square error.

在附图7中，为比较本发明的算法与复标准最小二乘法、广义线性复标准最小二乘法、变步长法和广义线性变步长法的均方误差。复标准最小二乘法、广义线性复标准最小二乘法、变步长法和广义线性变步长法的参数设置与实验1中是相同的。从附图7中可以看出，本发明的的收缩线性复最小二乘法和收缩广义线性复最小二乘法分别需要60，50次迭代达到稳态。然而，复标准最小二乘法和义线性复标准最小二乘法分别需要100，150次迭代达到稳态。因此考虑非圆性时的收敛速度要快于不考虑非圆性时。尽管，变步长法和广义线性变步长法与本文算法有近似的收敛速度，但是，它们在稳态时，有很大的失调。由此可知，本发明算法有较高的收敛速度和较小的均方误差。 In accompanying drawing 7, in order to compare the mean square error of algorithm of the present invention and complex standard least squares method, generalized linear complex standard least squares method, variable step method and generalized linear variable step method. The parameter settings of complex standard least squares method, generalized linear complex standard least squares method, variable step method and generalized linear variable step method are the same as those in Experiment 1. It can be seen from FIG. 7 that the contracted linear complex least squares method and the contracted generalized linear complex least squares method of the present invention require 60 and 50 iterations respectively to reach a steady state. However, the complex standard least squares method and the linear complex standard least squares method require 100 and 150 iterations, respectively, to reach a steady state. Therefore, the convergence speed is faster when non-circularity is considered than when non-circularity is not considered. Although the variable step size method and the generalized linear variable step size method have similar convergence speeds to the algorithm in this paper, they have a large misalignment in the steady state. It can be seen from this that the algorithm of the present invention has a higher convergence speed and a smaller mean square error.

以上内容是结合具体的优选实施方式对本发明所作的进一步详细说明，不能认定本发明的具体实施只局限于这些说明。对于本发明所属技术领域的普通技术人员来说，在不脱离本发明构思的前提下，还可以做出若干简单推演或替换，都应当视为属于本发明的保护范围。 The above content is a further detailed description of the present invention in conjunction with specific preferred embodiments, and it cannot be assumed that the specific implementation of the present invention is limited to these descriptions. For those of ordinary skill in the technical field of the present invention, without departing from the concept of the present invention, some simple deduction or replacement can be made, which should be regarded as belonging to the protection scope of the present invention.

Claims

1. A method of systolic linear complex least squares for signal processing, the method being applied in beamforming, characterized by: the method comprises the following steps:

1) considering a uniform linear array of M array elements, let x (k) denote the sample data matrix received by it, and obtain x (k) ═ a (θ)_d)s₀(k) + n (k), wherein

For the steering vector of the desired signal, Δ represents the array spacing between adjacent array elements and λ represents the wavelength

2) Computing the output of the array y w (k)^Hx (k) derived from the minimum mean square error criterion

Make J [ w (k)]At minimum, we get w (k +1) = w (k) + mue^*(k)x(k)；

3) With variable step size mu_kInstead of the above-mentioned μ value, to obtain

Let the weight error vector be v (k) ═ w (k) — w_optThen get the updating process of the weight error vector

<math> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mo>[</mo> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>-</mo> <msub> <mi>μ</mi> <mi>k</mi> </msub> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>]</mo> <mi>v</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>μ</mi> <mi>k</mi> </msub> <msubsup> <mo>&Element;</mo> <mi>opt</mi> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

Wherein,

<math> <mrow> <msub> <mo>&Element;</mo> <mi>opt</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>s</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>w</mi> <mi>opt</mi> <mi>H</mi> </msubsup> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

4) at time k, the a priori error between the output of the beamformer and the desired signal is

<math> <mrow> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>s</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>w</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mo>&Element;</mo> <mi>opt</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>w</mi> <mi>opt</mi> <mi>H</mi> </msubsup> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>w</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mo>&Element;</mo> <mi>opt</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>e</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

Wherein the prior error in the absence of noise is

e_{f} (k) = w_{opt}^{H} x (k) - w^{H} (k) x (k) = - v^{H} (k) x (k);

5) Similarly, the a posteriori error is (k) ∈_opt(k)+_f(k) Wherein the posterior error in the absence of noise is

6) Can be obtained by calculation_f(k) And e_f(k) The relationship between them is:

_f(k)＝(1-μ_k||x(k)||²)e_f(k)-μ_k∈_opt(k)||x(k)||²，

to both sides of its square, simultaneously to mu_kTaking the derivative and making the derivative equal to 0 then

<math> <mrow> <mn>2</mn> <msup> <mrow> <mo>|</mo> <msub> <mi>e</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mo>&Element;</mo> <mi>opt</mi> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mo>&Element;</mo> <mi>opt</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mi>e</mi> <mi>f</mi> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <msub> <mi>μ</mi> <mi>k</mi> </msub> <msup> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mo>[</mo> <mo>|</mo> <msub> <mi>e</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mo>&Element;</mo> <mi>opt</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>]</mo> <mo>;</mo> </mrow> </math>

7) The step (6) is simplified according to the formulaWhen mu is_kIf it is a constant, the above equation is certainly true; in fact, in steady state, μ_kChanges are relatively slow compared to x (k), e (k), and therefore, considering them as approximately uncorrelated, one can obtain

8) In practical application, generally E [ | | x (k) | purple²]Is known, E [ | E (k) ] non-combustible²]Is obtained byEstimated, however, e_f(k) Is unknown and difficult to solve by solving for E [ E ]_f(k)|²]Solving the step length;

9) recovering e (k) from e (k) by shrinkage denoising method_f(k) Utilize immediately

f[e_f(k)]＝0.5|e_f(k)-e(k)|²+α|e_f(k)|，

By minimizing this formula, it is possible to obtainThereby substituting into

To obtain E [ E ]_f(k)|²]An estimate of (a);

10) substituting the above estimators to obtain a step value by contracting linear least square methodThereby replacing the mu in the step 2 to obtain the weight updating process of the shrinkage linear complex least square method.

2. A method of shrinking generalized linear complex least squares for signal processing, the method being applied to beamforming, characterized by: the method comprises the following steps:

2) When C is present_x＝E[x(k)x^T(k)]≠0_M×MThen, the non-circularity of the signal needs to be considered at this time, and the expanded data at this time is:

computing the output of an array

3) From a minimum mean square error quasi-then

Is at a minimum, andthen the weight value updating process is

4) V. the₁(k)＝w₁-w_opt，v₂(k)＝w₂-w_optThen get the update procedure of the weight vector error

The matrix form can be written as:

<math> <mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>v</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>-</mo> <msub> <mi>μ</mi> <mi>k</mi> </msub> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mo>-</mo> <msub> <mi>μ</mi> <mi>k</mi> </msub> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <msub> <mi>μ</mi> <mi>x</mi> </msub> <msup> <mi>x</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> <mtd> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>-</mo> <msub> <mi>μ</mi> <mi>k</mi> </msub> <msup> <mi>x</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>×</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>v</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <msub> <mi>μ</mi> <mi>k</mi> </msub> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <msubsup> <mover> <mo>&Element;</mo> <mo>~</mo> </mover> <mi>opt</mi> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

wherein,

<math> <mrow> <msub> <mover> <mo>&Element;</mo> <mo>~</mo> </mover> <mi>opt</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mo>=</mo> <mi>Δ</mi> </mover> <msub> <mi>s</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>w</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>opt</mi> </mrow> <mi>H</mi> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>w</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>opt</mi> </mrow> <mi>H</mi> </msubsup> <msup> <mi>x</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

5) the relation between the extended noiseless posterior error and the prior error is obtained by the formula

<math> <mrow> <msub> <mover> <mi>ϵ</mi> <mo>~</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mrow> <mn>2</mn> <mi>μ</mi> </mrow> <mi>k</mi> </msub> <msup> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>)</mo> </mrow> <msub> <mover> <mi>e</mi> <mo>~</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>μ</mi> <mi>k</mi> </msub> <msub> <mover> <mo>&Element;</mo> <mo>~</mo> </mover> <mi>opt</mi> </msub> <msup> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>,</mo> </mrow> </math>

The noise-free posterior error and the prior error are respectively as follows:

6) the instantaneous posterior error is squared and then related to mu_kIs 0, is obtained

<math> <mrow> <mn>2</mn> <msup> <mrow> <mo>|</mo> <msub> <mover> <mi>e</mi> <mo>~</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mover> <mo>&Element;</mo> <mo>~</mo> </mover> <mi>opt</mi> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msub> <mover> <mi>e</mi> <mo>~</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mover> <mo>&Element;</mo> <mo>~</mo> </mover> <mi>opt</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mover> <mi>e</mi> <mo>~</mo> </mover> <mi>f</mi> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <msub> <mi>μ</mi> <mi>k</mi> </msub> <msup> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mo>[</mo> <mo>|</mo> <msub> <mover> <mi>e</mi> <mo>~</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mover> <mo>&Element;</mo> <mo>~</mo> </mover> <mi>opt</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>]</mo> <mo>;</mo> </mrow> </math>

7) According to the above formula and simplifying to obtainWhen mu is_kIf it is a constant, the above equation is certainly true; in fact, in steady state, μ_kWith x (k),Relatively slowly compared to the change, and therefore, considering them as approximately uncorrelated, one can obtain

8) Wherein,can pass through

The results were obtained, however,is unknown and difficult to solve bySolving the step length; by a shrinkage denoising method to obtainIt is substituted into the following formula to obtainEstimated value of (a):

9) substituting the above estimators to obtain a contraction generalized linear complex least square method to obtain a step valueThereby substituting the update process for obtaining the weight.