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: Shenzhen Graduate School Harbin Institute of Technology
Current assignee: Shenzhen Graduate School Harbin Institute of Technology
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

The invention provides a contractive linear and contractive generalized linear complex-valued least squares algorithm for signal processing and aims to solve the problems such as slow convergence rate and a big mean square error since step size is fixedly updated and signal noncircularity is unconsidered during actual application. The contractive linear and contractive generalized linear complex-valued least squares algorithm is applicable to adaptive beamforming and minimizes instantaneous average errors of posterior errors, without consideration of noise, by the aid of the variable step sizes during weight updating; besides, the contractive generalized linear complex-values least squares algorithm further considers noncircularity of desired signals, thus convergence rate is increased, and steady-state mean square error is greatly decreased.

Description

Shrinking linear and shrinking generalized linear complex least square algorithm for signal processing

Technical Field

The invention relates to the technical field of array signal processing, in particular to a contraction linearity and contraction generalized linear complex least square algorithm.

Background

Array signal processing is an important branch in the field of signal processing, has become mature over several decades and has been widely applied in a plurality of military and national economic fields such as radar, biomedical, exploration and astronomy. The working principle is that a plurality of sensors form a sensor array, and the array is used for receiving and processing space signals, so that interference and noise are suppressed, and useful information of the signals is extracted. Unlike a general signal processing method, the array signal processing is to receive signals through a sensor group arranged in a space, and filter and extract information by using spatial characteristics of the signals. Therefore, the array signal processing is also often referred to as spatial domain signal processing. In addition, the array signal processing has the advantages of flexible beam control, strong anti-interference capability, extremely high spatial super-resolution capability and the like, so that the array signal processing is concerned by a plurality of scholars, and the application range of the array signal processing is continuously enlarged.

In the field of array signal processing, the two most important research directions are adaptive filtering and spatial spectrum estimation, wherein the adaptive filtering technique is generated before the spatial spectrum estimation, and the application thereof in engineering systems is very extensive. However, although the spatial spectrum estimation has been rapidly developed in the last 30 years, the related research content is very extensive, but the engineering application systems are not common. Here, the adaptive filtering technique is an important concept in the field of array signal processing.

Adaptive filtering may be applied in modeling, equalization, control, echo cancellers and adaptive beamforming. The least square algorithm of complex value is a self-adaptive estimation and prediction technology, and can realize the performance convergence to the optimal wiener solution. The adaptive beamformer weight vector may be calculated based on different design criteria, commonly used criteria are minimum mean square error, minimum variance, and constant modulus criteria, which is used by the present invention.

Classical adaptive arrays use circular signals, usually a linear time-invariant complex filter w is found, the output of which isUnder the condition of deterministic constraintAnd optimizing a second-order criterion. However, in practice, non-circular signals have been widely used in many modern communication systems. Since the classical adaptive beamformer is optimal for round signals but suboptimal for non-round signals. Thus, the generalized complex least squares method utilizes the spread signalA lower mean square error between the output of the beamformer and the desired signal can be obtained. And the complex least square method is suboptimal for the second-order non-circular signal, so how to ensure to obtain the optimal value of the non-circular signal, improve the convergence rate and the output signal-to-interference-and-noise ratio, and reduce the mean square error is the key point of the problem.

Disclosure of Invention

In order to solve the problems of low convergence speed, large mean square error and the like when the traditional fixed updating step length and the non-circularity of a signal are not considered in practical application, the invention provides a complex least square algorithm of contraction linearity and contraction generalized linearity.

The invention is realized by the following technical scheme:

the shrinkage linear complex least square method and the shrinkage generalized linear complex least square method of the invention are different from the prior method in that the non-circularity and the shrinkage algorithm are utilized to obtain the variable updating step value, and the performance is improved from the following two aspects:

1. non-circularity is used to improve convergence speed and reduce mean square error. The specific implementation steps are as follows:

first, when the desired signal is a signal such as BPSK, QPSK, and PAM, the desired signal can be decomposed intoThe received signal is now correlated with its conjugate, i.e. the conjugate contains the useful information of the desired signal, and therefore C_x＝E[x(k)x^T(k)]≠0_M×M. The spread signal at this time is

Then the cost function of the error between the array output and the expected signal is obtained by the mean square error criterion to obtain the weight value at this time. At this time, the expanded array weight isThen the non-circular information is utilized, the convergence rate is increased, and the mean square error is reduced.

2. The method of shrinkage is reused. The specific implementation steps are as follows:

in the process of solving the changing step length by the contraction linear least square method, the step length isIn general, in practical application, E [ | x (k) | ceiling fume is usually²]Is known, E [ | E (k) | non-combustible²]Is obtained byEstimated, where λ is a forgetting factor and 0 < λ ≦ 1. However, e_f(k) Is unknown and difficult to solve by solving for E [ E ]_f(k)|²]To solve the step size. The prior error e in the noise-free state can be recovered by a shrinkage denoising method through the prior error e (k)_f(k) In that respect Let f [ e ]_f(k)]＝0.5|e_f(k)-e(k)|²+α|e_f(k) If l is minimum, e can be recovered_f(k) In that respect At this time, the value of the variable step length can be obtained, so as to obtain the updating process of the weight value. Therefore, the convergence rate can be improved, and the mean square error can be reduced.

Drawings

FIG. 1 shows a schematic view of aThe invention relates to a method for estimating shrinkage linear and shrinkage generalized linear complex least square algorithm ADrawing (A)；

FIG. 2(a) AndFIG. 2(b) When Q is different, the variation curve of the output signal-to-interference-noise ratio of the algorithm, the complex least square method and the generalized linear complex least square method along with the iteration timesDrawing (A)；

FIG. 3(a) AndFIG. 3(b) When Q is fixed and the step length is different, the variation curve of the output signal-to-interference-noise ratio of the algorithm, the complex least square method and the generalized linear complex least square method along with the iteration timesDrawing (A)；

FIG. 4The variation curve of the output signal-to-interference-noise ratio of the algorithm of the invention and the VSS, CNLMS, WL-CNLMS and WL-VSS algorithms along with the iteration timesDrawing (A)；

FIG. 5(a) AndFIG. 5(b) When Q is different, the variation curve of mean square error of the algorithm, the complex least square method and the generalized linear complex least square method along with the iteration timesDrawing (A)；

FIG. 6(a) AndFIG. 6(b) When the step length is different, the variation curve of the mean square error of the algorithm, the complex least square method and the generalized linear complex least square method along with the iteration timesDrawing (A)；

FIG. 7The variation curve of mean square error of the algorithm of the invention and VSS, CNLMS, WL-CNLMS and WL-VSS algorithm along with the iteration numberDrawing (A)。

Detailed Description

The invention is further described with reference to the following description and embodiments in conjunction with the accompanying drawings.

Consider an M matrixA uniform linear array of elements receiving a far-field narrow-band signal s₀(k) Corresponding to an angle of arrival of theta_d. This signal is zero-mean, second-order non-circular. The array output data can be expressed as:

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, λ represents the wavelength, and n (k) is [ n ]₁(k)，…，n_M(k)]^TIs an additive noise vector, which consists of background noise and interference, and can be expressed as:

wherein, P non-circular interferences which are not related statistically have complex envelopes of s_i(k) 1, 2, and P, and the corresponding steering vector is a (θ)_i) 1, 2, P, η (k) is background noise that is uncorrelated with both the desired signal and the interference.

When the weight vector is w ═ w₁，...，w_M]^TThen the optimal weight vector can be formed by combining the beamformer output with the ideal signal s_d(k) Minimum mean square error of

\begin{matrix} w_{opt} = \arg \min_{w} E [{| e (k) |}^{2}] \\ = \arg \min_{w} E [{| s_{d} (k) - y (k) |}^{2}] \\ = \arg \min_{w} E [{| s_{0} (k) - y (k) |}^{2}] \end{matrix} - - - (1)

Wherein s is_d(k)＝s₀(k) The optimal weight can be solved through some operations as follows:

w_{opt} = R_{x}^{- 1} P_{x}

wherein,

power of instantaneous variance J [ w (k)]Minimum size

Update procedure to obtain weight

w(k+1)＝w(k)+μe^*(k)x(k)。 (3)

When the desired signal is a non-circular signal, such as BPSK, QPSK, PAM, etc., then the desired signal vector may be represented at this time asIn general, C_x＝E[x(k)x^T(k)]≠0_M×MTo exploit non-circularity, vectors are extendedCan be expressed as

Wherein,respectively, an extended steering vector and a noise vector. The cost function of the generalized complex least squares method similar to the complex least squares method is

Wherein,in order to extend the instantaneous error of the signal,representing the output of the extended beamformer.

At this time, the process of the present invention,the generalized weight vector is updated by

Make itMinimizing to obtain optimal generalized weight vector

{\tilde{w}}_{opt} = {\tilde{R}}_{x}^{- 1} {\tilde{P}}_{x}

Wherein,

{\tilde{R}}_{x} = [\begin{matrix} R_{x} & C_{x} \\ C_{x}^{*} & R_{x}^{*} \end{matrix}],

changing mu in formula (3) into variable step size mu_kThen the update process of the weight vector is

Suppose a sequence pair { x (k), s₀(k) Is generalized stationary, so the optimal weight vector w_opt(k) Is time-invariant, i.e. ω_opt(k)＝w_opt. Weighted vector error vector v (k) w (k) -w_optThen the update procedure of v (k) can be obtained as

<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> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow></math>

Wherein,at time k, the output of the beamformer is coupled to the desired signal s₀(k) An error of

<math><mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <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> </mtd> </mtr> <mtr> <mtd> <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> </mtd> </mtr> <mtr> <mtd> <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> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow></math>

Wherein,

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

is the a priori error in the absence of noise. In addition, the a posteriori error can be expressed as (k) ═ e_opt(k)+_f(k) Wherein

For the posterior error without noise, the two sides of the conjugate transpose of the formula (8) are simultaneously multiplied by x (k) and then substituted into the above formula to obtain

_f(k)＝(1-μ_k||x(k)||²)e_f(k)-μ_k∈_opt(k)||x(k)||². (11)

The energy of the instantaneous noise-free A posteriori error can be expressed as

<math><mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msup> <mrow> <mo>|</mo> <msub> <mi>ϵ</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <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> <msub> <mi>μ</mi> <mi>k</mi> </msub> <mo>[</mo> <mn>2</mn> <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> <msub> <mi>e</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mtd> </mtr> <mtr> <mtd> <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> <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> <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> <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> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msubsup> <mi>μ</mi> <mi>k</mi> <mn>2</mn> </msubsup> <mo>[</mo> <msup> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>4</mn> </msup> <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> <msup> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>4</mn> </msup> <msub> <mi>e</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mo>&Element;</mo> <mi>opt</mi> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msup> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>4</mn> </msup> <msubsup> <mi>e</mi> <mi>f</mi> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msub> <mo>&Element;</mo> <mi>opt</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>4</mn> </msup> <msup> <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> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow></math>

Simultaneously aligning two sides of formula (12) to mu_kAfter derivation equals 0, then

<math><mfenced open='' close='' separators=' '> <mtable> <mtr> <mtd> <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> </mtd> </mtr> <mtr> <mtd> <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> <mo>[</mo> <msup> <mrow> <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> </mtd> </mtr> </mtable> <mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow> </mfenced></math>

By substituting formula (9) for formula (13) to give

<math><mfenced open='' close='' separators=' '> <mtable> <mtr> <mtd> <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> </mtd> </mtr> <mtr> <mtd> <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> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>.</mo> </mtd> </mtr> </mtable> <mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow> </mfenced></math>

To pairConjugation is then multiplied by x (k) to obtain two simultaneous expectations, then

<math><mfenced open='' close='' separators=' '> <mtable> <mtr> <mtd> <mi>E</mi> <mo>[</mo> <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> <mo>=</mo> <mi>E</mi> <mo>[</mo> <msubsup> <mi>s</mi> <mn>0</mn> <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> <mo>-</mo> <mi>E</mi> <mo>[</mo> <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> <msub> <mi>w</mi> <mi>opt</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <msub> <mi>P</mi> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>R</mi> <mi>x</mi> </msub> <msubsup> <mi>R</mi> <mi>x</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>P</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mtd> </mtr> </mtable> <mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow> </mfenced></math>

Thus, the input signals x (k) andare statistically vertical. When in useVery small and very slowly changing at steady state, e^*(k) Is independent of x (k), can be obtained

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

Observe that when the input sequence is independent for j < k, v (k) is only associated with { x (j), s%₀(j) Relative, and independently of the current input signal x (k), is obtained

<math><mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <mi>E</mi> <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> <mo>=</mo> <mo>-</mo> <mi>E</mi> <mo>[</mo> <msubsup> <mo>&Element;</mo> <mi>opt</mi> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msup> <mi>v</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> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow></math>

The results of anticipating both sides of the formula (14) and combining the formulae (15) to (17) were obtained

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

When mu is_kIf it is constant, the above equation is certainly satisfied. In fact, in steady state, μ_kThe change is relatively slow compared to x (k), e (k). Therefore, it can be said that μ_kAre approximately uncorrelated with x (k), e (k), i.e., equation (19) holds approximately true.

Since E [ | E_f(k)|²]Is the unwanted mean square error caused by the error between the weight vector and the optimal weight vector.

In formula (3), the molar ratio is determined by_kInstead of μ, a weight update procedure of the complex least squares method can be obtained.

In practical application, generally E [ | | x (k) | purple²]Is known, E [ | E (k) ] non-combustible²]Is estimated by the following formula

Wherein λ is forgetting factor0 < lambda < 1. However, e_f(k) Is unknown and is difficult to solve by solving for E [ | E [ ]_f(k)|²]To solve (20). The prior error e (k) in the noise-free state can be recovered by a shrinkage de-noising method through the prior error e (k)_f(k)。

f[e_f(k)＝0.5|e_f(k)-e(k)|²+α|e_f(k) [ 22 ] relating the above formula to e_f(k) Minimize, can obtain

From this, it is understood that the selection of α is very important. The background noise is assumed to be white Gaussian noise and the covariance isThe interference deviates from the main lobe of the desired signal and most of the energy of the interference part can be suppressed

Can be obtained by (9) and (24)

<math><mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <mi>E</mi> <mo>[</mo> <msup> <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> <mi>E</mi> <mo>[</mo> <msup> <mrow> <mo>|</mo> <msub> <mi>e</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>]</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mi>E</mi> <mo>[</mo> <msup> <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> <mn>2</mn> </msup> <mo>]</mo> </mtd> </mtr> <mtr> <mtd> <mo>≈</mo> <mi>E</mi> <mo>[</mo> <msup> <mrow> <mo>|</mo> <msub> <mi>s</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>s</mi> <mn>0</mn> </msub> <mo>-</mo> <msubsup> <mi>w</mi> <mi>opt</mi> <mi>H</mi> </msubsup> <mi>η</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>]</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <msubsup> <mi>σ</mi> <mi>η</mi> <mn>2</mn> </msubsup> <msup> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>w</mi> <mi>opt</mi> </msub> <mo>|</mo> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow></math>

In a uniform linear array, to ensure that the beam pattern at the angle of arrival of the desired signal is 1 while maximizing | | w_opt||²Is usually provided withThereby obtaining

<math><mrow> <mi>E</mi> <mo>[</mo> <msup> <mrow> <mo>|</mo> <msub> <mi>e</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>]</mo> <mo>&GreaterEqual;</mo> <mfrac> <msubsup> <mi>σ</mi> <mi>n</mi> <mn>2</mn> </msubsup> <mi>M</mi> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow></math>

In summary, we can get the followingWhere Q is a parameter used to compensate for the approximation. Like E | E (k) does not dust²]Can obtain

Substituting the result in the above equation and the result in (21) into (20) can obtain the weight update step size at this time as

Mu in formula (28)_kAnd (4) replacing the mu in the step (3), obtaining the updating process of the weight vector of the complex least square method.

Similarly, w is subtracted from both ends of the two equations in (6), respectively₁、w₂Is the optimal weight w_1，opt、w_2，optThe update procedure for the weight vector error can be obtained as follows:

at this time with variable step size mu_kInstead of μ in (6). According to (5) and (28), a vector matrix form of the weight error vector is obtained

<math><mrow> <mfenced open='' close=' '> <mtable> <mtr> <mtd> <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> <mi></mi> <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>k</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> <mi></mi> <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> </mtd> </mtr> <mtr> <mtd> <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> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> </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> </mrow></math>

is the error between the waveform output of the spread weight vector and the desired signal. Conjugate transpose is carried out on two sides of (30) at the same time, and then right multiplication is carried out

\tilde{x} (k) = {[x {(k)}^{T}, x^{H} (k)]}^{T}

To obtain

<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> <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> <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> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> </mrow></math>

The posterior error and the prior error due to the extended noise are respectively

\begin{matrix} {\tilde{e}}_{f} (k) = {[w_{1, opt} - w_{1} (k)]}^{H} x (k) \\ + {[w_{2, opt} - w_{2} (k)]}^{H} x^{*} (k) \\ = - v_{1}^{H} (k) x (k) - v_{2}^{H} (k) x^{*} (k) \end{matrix} - - - (33)

Similar to the linear complex least squares method of shrinkage, the square of the instantaneous extended noiseless posterior error is

<math><mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msup> <mrow> <mo>|</mo> <msub> <mover> <mi>ϵ</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> <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> <msub> <mi>μ</mi> <mi>k</mi> </msub> <mrow> <mo>[</mo> <mn>4</mn> <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> <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> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mn>2</mn> <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> <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> <mn>2</mn> <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> <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> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msubsup> <mi>μ</mi> <mi>k</mi> <mn>2</mn> </msubsup> <mo>[</mo> <mn>4</mn> <msup> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>4</mn> </msup> <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> <mn>4</mn> <msup> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>4</mn> </msup> <msub> <mover> <mi>e</mi> <mo>~</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mover> <mo>&Element;</mo> <mo>~</mo> </mover> <mi>opt</mi> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mn>4</mn> <msup> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>4</mn> </msup> <msubsup> <mover> <mi>e</mi> <mo>~</mo> </mover> <mi>f</mi> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msub> <mover> <mo>&Element;</mo> <mo>~</mo> </mover> <mi>opt</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mn>4</mn> <msup> <mrow> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>4</mn> </msup> <msup> <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> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>34</mn> <mo>)</mo> </mrow> </mrow></math>

Taking the above formula as a cost function, and solving the cost function about mu_kAnd equals 0, one can obtain

<math><mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <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> </mtd> </mtr> <mtr> <mtd> <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> <mo>[</mo> <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> <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> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>35</mn> <mo>)</mo> </mrow> </mrow></math>

Instantaneous error at time k is

<math><mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <mover> <mi>e</mi> <mo>~</mo> </mover> <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> <msup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>x</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <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> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>36</mn> <mo>)</mo> </mrow> </mrow></math>

Substituting (36) into (35) to obtain

<math><mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <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> </mtd> </mtr> <mtr> <mtd> <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> <mover> <mi>e</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>37</mn> <mo>)</mo> </mrow> </mrow></math>

By passing

<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>

Then the two sides are simultaneously subjected to conjugate transpose and then right multiplicationAnd two sides simultaneously obtain the expectation

<math><mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <mi>E</mi> <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> <mover> <mi>x</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>]</mo> <mo>=</mo> <mi>E</mi> <mo>[</mo> <msubsup> <mi>s</mi> <mn>0</mn> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>x</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>]</mo> <mo>-</mo> <mi>E</mi> <mo>[</mo> <mover> <mi>x</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msup> <mover> <mi>x</mi> <mo>~</mo> </mover> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>]</mo> <msub> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>opt</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mi>x</mi> </msub> <mo>-</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>x</mi> </msub> <msubsup> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>x</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>38</mn> <mo>)</mo> </mrow> </mrow></math>

Suppose also thatIs irrelevant, it is shown by the above formulaAnd extended input signalIs perpendicular to andthen the probable prior errorAndis also not relevant, i.e.

E [{| | x (k) | |}^{2} {| \tilde{e} (k) |}^{2}] = E [{| | x (k) | |}^{2}] E [{| \tilde{e} (k) |}^{2}] - - - (39)

V is known from (29)₁(k)，v₂(k) Are each independently of x (k), x^*(k) Are not relevant. Is provided withFrom the results of (33) and (38)

<math><mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <mi>E</mi> <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> <mo>=</mo> <mi>E</mi> <mo>{</mo> <mo>[</mo> <mo>-</mo> <msubsup> <mi>v</mi> <mn>1</mn> <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>v</mi> <mn>2</mn> <mi>H</mi> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <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> <mo>}</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mi>E</mi> <mo>[</mo> <mo>-</mo> <msup> <mover> <mi>v</mi> <mo>~</mo> </mover> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>x</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <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> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>40</mn> <mo>)</mo> </mrow> </mrow></math>

The simultaneous expectation of both sides of (37) is obtained, and the results of reuse of (38) and (39) are obtained

The above equation is based on the assumption that

E [ mu ] of_k]As mu_kSubstituting the estimate of (2) into (30) to obtain a generalized linear complex least squares method.

In the formula (41)Can be obtained by

Extended noiseless prior errorMean of squares

Instead of (41)Extending noise-free prior errorCan pass throughComing back to

Then, how the threshold α needs to be selected is based on the fact that both interference and background noise are uncorrelated with the desired signalCan be expressed as

Wherein,is the power of the desired signal. The optimal weight vector at this time is

This result is analogous to the generalized optimal minimum variance undistorted response

Wherein, andonly the constant part is different. When the angle of arrival of the disturbance differs substantially from the angle of arrival of the desired signalWhen it is, then there are

Wherein,whereinIs the initial phase of the ith interferer.Is approximated as

<math><mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <mi>E</mi> <mo>[</mo> <msup> <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> <mi>E</mi> <mo>[</mo> <msup> <mrow> <mo>|</mo> <msub> <mi>s</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>opt</mi> <mi>H</mi> </msubsup> <mover> <mi>x</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>]</mo> </mtd> </mtr> <mtr> <mtd> <mo>≈</mo> <mi>E</mi> <mo>[</mo> <msup> <mrow> <mo>|</mo> <msub> <mi>s</mi> <mn>0</mn> </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> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>opt</mi> <mi>H</mi> </msubsup> <mover> <mi>η</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>]</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <msubsup> <mi>σ</mi> <mi>η</mi> <mn>2</mn> </msubsup> <msup> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>opt</mi> </msub> <mo>|</mo> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>50</mn> <mo>)</mo> </mrow> </mrow></math>

Wherein,due to the fact thatSubstituting this into (50) then yields

<math><mrow> <mi>E</mi> <mo>[</mo> <msup> <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>&GreaterEqual;</mo> <mfrac> <msubsup> <mi>σ</mi> <mi>η</mi> <mn>2</mn> </msubsup> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>51</mn> <mo>)</mo> </mrow> </mrow></math>

Thus, can obtainWhen these are substituted into (3-41), there are

And substituting the result in the step (52) into the step (6) to obtain a weight updating process of the shrunk generalized complex least square method.

As shown in FIG. 1, the inventive systolic linear least squares algorithm comprises the following steps:

1. calculating posterior error in the absence of noise_f(k) The energy of (3-12) is then simultaneously applied to both sides_kAfter derivation equals 0, (3-9) is substituted into (3-13) and (3-15) (3-16) is combined to obtain

2. Since generally E [ | | x (k) | non-woven phosphor²]Is known, E [ | E (k) ] non-combustible²]Is obtained by (3-21) estimation, and can use the shrinkage de-noising method, i.e. the (3-22) is obtained at the minimum

3. E [ | E ] can be obtained by (3-27)_f(k)|²]An estimate of (a);

4. substituting the result of the estimator into [ mu ] in (1)_kObtained in the formula

5. Obtaining the weight update process w (k +1) ═ w (k) + mu_ke^*(k)x(k)。

The shrinkage generalized linear least square algorithm comprises the following steps:

1. calculating the posteriori error in the presence of noise in the instantaneous spreadSquaring, and then simultaneously aligning two sides of (3-34) < mu >_kAfter derivation equals 0, (3-36) is substituted into (3-35) and (3-38) (3-39) is combined to obtain

2. Since generally E [ | | x (k) | non-woven phosphor²]It is known that it is possible to use,is obtained by (3-43) estimation, and can use the shrinkage de-noising method, even if (3-45) is obtained at minimum

3. Can be obtained by (3-24)An estimate of (a);

5. And obtaining a weight updating process:

consider a uniform linear array with an array element number of 4 and an array pitch ofFour equally powered BPSK signals having a non-circular coefficient of 1, initial phases of 0, and an angle of arrival θ at which the desired signal is incident on the array_dAt-45 deg., the S/N ratio is 10dB, and the arrival angles of the other three interference signals are theta₁＝8°、θ₂＝-13°、θ₃The dry-to-noise ratio (INR) was fixed at 10dB, and all simulation results were obtained from 500 monte carlo experiments. At the initial value of the linear least squares method of shrinkageAnd w (0) ═ 0_M×1. Furthermore, the initial value of the generalized linear complex least squares method of shrinkage isw₁(0)＝0_M×1And w₂(0)＝0_M×1. The forgetting factor λ is fixed, λ being 0.95.

Experiment 1 outputs the change of the signal-to-interference-noise ratio with the number of iterations.

In this simulation, the variation of the algorithm proposed by the present invention with the complex least square method and the generalized linear complex least square method along with the number of iterations is compared, and when the step lengths of the complex least square method and the generalized linear complex least square method are 0.001 and 0.0005, respectively, it can be observed that in these two cases, the convergence rate is faster when considering the generalized (non-circularity) than when not considering the non-circularity, and the output signal-to-interference-and-noise ratio is higher when considering the non-circularity, and as can be seen from fig. 2, Q has less influence on the steady-state property of the linearly contracted complex least square method. Furthermore, the generalized linear complex least squares method of shrinkage has improved performance when Q is 2 compared to Q being 1. Fig. 3 shows the complex least squares and generalized linear complex least squares of fig. 3(a) at 0.008 and 0.004, respectively, and fig. 3(b) at 0.0005 and 0.00025, respectively, when Q is fixed. As can be seen from fig. 3(a), when the step size is 8 times larger than that in fig. 2(b), the convergence speed of the complex least square method and the generalized linear complex least square method is larger than that in fig. 2 (b). The convergence speed can be increased due to the larger step size. However, in steady state, the output interference is small when the generalized linear complex least squares method and the complex least squares method are used. If we fix the step size to half μ of fig. 2(b), from fig. 3(b), the generalized linear complex least squares and complex least squares approach to the same output sir as the proposed algorithm in the present invention at steady state. However, the two methods require 200 and 400 iterations to reach steady state, respectively, whereas fig. 2(b) requires only 100 and 200 iterations to reach steady state, respectively. This is because the step size is small, decreasing the convergence speed. It can be seen from fig. 3 that the algorithm of the present invention requires approximately 60, 100 iterations to reach steady state, respectively. Moreover, the signal-to-interference-and-noise ratios output by the algorithm of the invention respectively approximately reach the optimal generalized linear signal-to-interference-and-noise ratio and the optimal signal-to-interference-and-noise ratio. Therefore, compared with the generalized linear complex least square method and the complex least square method, the algorithm of the invention has the advantages of higher convergence speed and higher output signal-to-interference-and-noise ratio.

FIG. 4 is a diagram comparing the contracting linear complex least squares (CMLS) with the standard complex least squares (CNLMS) and the variable step length (VSS) proposed by the present invention, and comparing the contracting generalized linear complex least squares (CMLS) with the generalized linear standard complex least squares (GMLS) proposed by the present invention(WL-CNLMS), generalized Linear variable step Length method (WL-VSS). The step lengths of the complex standard least square method and the generalized linear complex standard least square method are set to be 0.2 and 0.1 respectively. The variable step size parameter is mu_min＝e^-6、μ_max＝3e^-3、Andthe generalized variable step size parameter is mu_min＝5e^-7、μ_max＝1.5e^-3、Andas can be seen from fig. 4, the algorithm of the present invention converges faster than the complex standard least squares method and the generalized complex least squares method. Moreover, compared with the variable step length method and the generalized variable step length method, the algorithm of the invention has higher output signal-to-interference-and-noise ratio. It can also be seen that these several methods when non-circularity is considered are superior to the algorithm when non-circularity is not considered.

Experiment 2 outputs the relationship between the mean square error and the number of iterations.

In the experiment, the value of each variable was set as in the previous experiment. It can be seen from fig. 5 that the algorithm of the present invention has a faster convergence rate than the generalized linear complex least square method and the complex least square method. The contraction linear complex least square method and the complex least square method of the invention approximately have equal mean square errors in a steady state, and similarly, the contraction generalized linear complex least square method and the generalized complex least square method of the invention approximately have equal mean square errors in the steady state. Furthermore, the generalized linear based approach has better properties due to the non-circularity considerations. Since the extended array aperture makes the shrinking generalized complex least squares more sensitive to the selection of α, it can be seen from fig. 5(a) that the shrinking generalized linear complex least squares of the present invention is slightly inferior to the generalized complex least squares at steady state when Q is 1. As can be seen from fig. 5(b), when Q is 2, the shrinking generalized linear complex least squares method of the present invention is slightly better than the generalized complex least squares method at steady state. And unlike linear-based algorithms, the magnitude of the Q value has little effect on the shrinkage linear complex least squares and complex least squares. FIG. 6 is a graph comparing the algorithm of the present invention with complex least squares and generalized complex least squares with unsynchronized long values. The complex least squares and generalized linear complex least squares in fig. 6(a) have faster convergence speed than fig. 5(b) because their step values are 0.008, 0.004, respectively, which is 8 times as large as fig. 5 (b). As can be seen from FIG. 6(a), the generalized complex least squares and the complex least squares converge to-4 dB and 0dB, respectively. However, in fig. 5(b), when μ is 0.001, they converge to-8 dB and-4 dB, respectively. Fig. 6(b) is a graph showing the comparison of the mean square error of the generalized linear complex least squares method and the complex least squares method of the present invention with step sizes of 0.00025 and 0.0005, respectively. Although the complex least squares and the generalized linear complex least squares can achieve the same steady state values as the algorithm of the present invention, they require a larger number of iterations than in fig. 5 (b). The former requires 200 and 400 iterations respectively, and the latter requires 150 and 250 iterations respectively. Therefore, the algorithm of the invention is superior to the complex least square method and the generalized linear complex least square method in both convergence speed and mean square error.

In FIG. 7, the mean square error of the algorithm of the present invention is compared to the complex standard least squares method, the generalized linear complex standard least squares method, the variable step size method, and the generalized linear variable step size method. The parameter settings of the complex standard least square method, the generalized linear complex standard least square method, the variable step length method, and the generalized linear variable step length method were the same as in experiment 1. As can be seen from fig. 7, the contracting linear complex least squares and the contracting generalized linear complex least squares of the present invention require 60 and 50 iterations, respectively, to reach steady state. However, the complex standard least squares method and the quasi-linear complex standard least squares method require 100, 150 iterations to reach steady state, respectively. The convergence rate 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 rates to the algorithm herein, they are highly imbalanced at steady state. Therefore, the algorithm has higher convergence speed and smaller mean square error.

The foregoing is a more detailed description of the invention in connection with specific preferred embodiments and it is not intended that the invention be limited to these specific details. For those skilled in the art to which the invention pertains, several simple deductions or substitutions can be made without departing from the spirit of the invention, and all shall be considered as belonging to the protection scope of the 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.