CN104679976B - Contraction for signal transacting is linear and shrinks the multiple least-squares algorithm of generalized linear - Google Patents

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

technical field

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.

Background technique

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.

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.

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.

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.

Contents of the invention

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.

The present invention realizes through following technical scheme:

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

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 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. Reuse the method of contraction. The specific implementation steps are as follows:

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, 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 restored. 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.

Description of drawings

Fig. 1 is contraction linear and contraction generalized linear complex least squares algorithm A estimation method flowchart of the present invention;

Fig. 2 (a) and Fig. 2 (b) are when Q is different, the output signal-to-noise ratio of algorithm of the present invention and complex least squares method, generalized linear complex least squares method changes the curve figure with number of iterations;

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 ;

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;

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;

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 change graph of number of iterations;

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.

detailed description

The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

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 )s ₀ (k)+n(k)

in, 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:

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 neither the desired signal nor the interference.

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)

Among them, s _d (k) = s ₀ (k), the optimal weight can be solved by some operations:

in, Minimize the power J[w(k)] of the instantaneous variance

So as to get the update process of the 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., 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

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

in, is the extended instantaneous error, represents the output of the expanded beamformer.

at this time, The update process of the generalized weight vector is

Make Minimize to get the optimal generalized weight vector

in,

Replace μ in formula (3) with variable step size μ _k , then the update process of the weight vector at this time is

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 w _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

in, At time k, the error between the output of the beamformer and the desired signal s ₀ (k) is

in, 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

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)

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

After deriving both sides of formula (12) with respect to μ _k , it is equal to 0, then we get

Substituting formula (9) into formula (13), we get

right After the conjugate is multiplied to the right by x(k), both sides are expected at the same time, then

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)

Observe that when j<k, when the input sequence is independent, v(k) is only related to {x(j), s ₀ (j), and independent of the current input signal x(k), we can get

Finding the expectation on both sides of formula (14) and combining the results of formulas (15) to (17), we get

Among them, E[μ _k ||x(k)|| ² |e(k)| ² ]＝E[μ _k ]E[||x(k)|| ² |e(k)| ² ] (19 )

When μ _k is a constant, the above formula must be established. 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.

Since E[| _ef (k)| ² ] is the redundant mean square error caused by the error between the weight vector and the optimal weight vector.

In formula (3), replace μ by μ _k , and the weight update process of the complex least squares method can be obtained.

In practical applications, usually E[||x(k)|| ² ] is known, and E[|e(k)| ² ] is estimated by the following formula

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 e _f (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)

Minimizing the above formula with respect to e _f (k), we can get

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

Through (9) and (24) can get

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

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

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

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.

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:

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

in, 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 have to

Due to the extended noiseless posterior error and prior error are respectively

Similar to the contracted linear complex least squares method, the square of the instantaneously expanded noise-free posterior error is

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

The instantaneous error at time k is

Substitute (36) into (35) to get

pass 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

Suppose also with irrelevant, it can be seen from the above formula with extended input signal vertical and a priori error is also uncorrelated with x(k), ie

It is known from (29) that v ₁ (k), v ₂ (k) are irrelevant to x(k), x ^* (k) respectively. Assume From the results of (33) and (38) we know

Find the expectation on both sides of (37) at the same time, and then use the results of (38) and (39) to get

The above formula is based on the assumption that

Substituting E[μ _k ] as the estimate of μ _k into (30), the generalized linear complex least squares method is obtained.

In formula (41) can be estimated by the following formula

Extended noise-free prior error mean of squares

to replace in (41) Extended noise-free prior error accessible recovered

Then, how to choose the threshold α, since the interference and background noise are irrelevant to the desired signal, then can be expressed as

in, is the power of the desired signal. Then the optimal weight vector at this time is

This result is similar to the generalized optimal minimum variance distortion-free response

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

in, i=1, 2, . . . , P. in is the initial phase of the i-th disturbance. The approximate result of

in, because Substituting this into (50) gives

From this, it can be obtained Substituting these into (3-41), we have

Substituting the result in (52) into (6), the weight update process of the contracted generalized complex least squares method can be obtained.

As shown in accompanying drawing 1, shrinkage linear least square algorithm of the present invention comprises the following steps:

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

2. Since E[||x(k)|| ² ] is usually known, E[|e(k)|2] is estimated by (3-21), and the shrinkage denoising method can be used, even (3-22) The minimum is obtained

3. The estimator of E[|e _f (k)| ² ] can be obtained through (3-27);

4. Substituting the result of the estimator into the μ _k formula in (1) can be obtained

5. Obtain the weight update process w(k+1)=w(k)+μ _k e ^* (k)x(k).

Shrinkage generalized linear least squares algorithm of the present invention comprises the following steps:

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

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

3. Can be obtained through (3-24) the estimate of

4. Substituting the result of the estimator into the μ _k formula in (1) can be obtained

5. Get the weight update process:

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.

Experiment 1 output signal to noise ratio changes with the number of iterations.

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 rate is faster when considering the generalized (non-circularity) than when not 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. Also, the contracted generalized linear complex least squares method improves performance at Q=2 over 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 signal-to-interference-noise ratio output by the algorithm of the present invention approximately reaches the optimal generalized linear signal-to-interference-to-noise ratio and the optimal signal-to-interference-to-noise ratio 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.

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 , with The generalized variable step size parameters are μ _min ＝5e ^-7 , μ _max ＝1.5e ^-3 , with 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.

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

In the experiment, the value setting of each variable is the same as the previous experiment. It can be seen from accompanying drawing 5 that the algorithm of the present invention has a faster convergence speed than the generalized linear complex least squares method and the complex least squares method. 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.

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.

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 (2)

1. A shrinkage linear complex least squares method for signal processing, said method is applied in beamforming, it is characterized in that: said method comprises the following steps: 1) Consider a uniform linear array with M array elements, receive a far-field narrowband signal s ₀ (k), and the corresponding angle of arrival is θ _d , let x(k) represent the sample data matrix it receives, and get x(k )＝a(θ _d )s ₀ (k)+n(k), where is the steering vector of the desired signal, Δ represents the array spacing between adjacent array elements, λ represents the wavelength, where η(k) is the background noise uncorrelated with the desired signal and interference; 2) Calculate the output of the array y=ω(k) ^H x(k), obtained by the minimum mean square error criterion Make J[ω(k)] the smallest, then get ω(k+1)=ω(k)+μe ^* (k)x(k); 3) Substituting the variable step size μ _k for the value of μ above, we get Let the weight error vector be v(k)=ω(k)-ω _opt , and ω _opt (k) is the optimal weight vector, then the update process of the weight error vector is obtained in, is the error between the waveform output of the optimal weight vector and the desired signal; 4) At time k, the prior error between the beamer output and the expected signal is The prior error when there is no noise is 5) Similarly, the posterior error is ε(k) = ε _opt (k) + ε _f (k), where the posterior error when there is no noise is 6) Through calculation, the relationship between ε _f (k) and e _f (k) can be obtained as: ε _f (k)＝(1-μ _k ||x(k)|| ² )e _f (k)-μ _k ε _opt (k)||x(k)|| ² , Taking the derivative of μ _k on both sides of its square and making the derivative equal to 0 gives 7) According to the above formula, step (6) is simplified to get When μ _k is a constant, the above formula must be true; in fact, in the steady state, μ _k changes slowly compared with x(k) and e(k), so if they are considered to be approximately uncorrelated, then we can have to 8) In practical applications, usually E[||x(k)|| ² ] is known, and E[|e(k)| ² ] is obtained by Estimated, however, e _f (k) is unknown, it is difficult to solve the above step size μ _k by solving E[ _ef (k)| ² ]; 9) Restore e(k) to e _f (k) by shrinkage denoising method: that is, use f[e _f (k)]=0.5|e _f (k)-e(k)| ² +α|e _f (k)|, To minimize this formula, we can get thus substituting into Get the estimator of E[e _f (k)| ² ] in; 10) Substitute the estimators of E[e _f (k)| ² ] and E[|e(k)| ² above into Then get the step size value of the contracted linear least squares method Thereby replacing μ in step 2 to obtain the weight update process of the contracted linear complex least squares method. 2. A contraction generalized linear complex least squares method for signal processing, said method is applied in the beam forming, it is characterized in that: comprise the following steps: 1) Consider a uniform linear array with M array elements, receive a far-field narrowband signal s ₀ (k), and the corresponding angle of arrival is θ _d , let x(k) represent the sample data matrix it receives, and get x(k )＝a(θ _d )s ₀ (k)+n(k), where is the steering vector of the desired signal, Δ represents the array spacing between adjacent array elements, λ represents the wavelength, where η(k) is the background noise uncorrelated with the desired signal and interference; 2) When C _x ＝E[x(k)x ^T (k)]≠0 _M×M , the non-circularity of the signal needs to be considered at this time, and the expanded data at this time is: Calculate the output of the array 3) According to the minimum mean square error criterion, minimum, and Then the weight update process at this time is 4) Suppose v ₁ (k)=ω ₁ -ω _opt , v ₂ (k)=ω ₂ -ω _opt , where, η(k) is the background noise uncorrelated with the desired signal and interference, then the weight vector error update process Then it can be written in matrix form as: in, is the error between the waveform output of the expanded weight vector and the desired signal; 5) The relationship between the extended noise-free posterior error and the prior error obtained by the above formula is And the noiseless posterior error and prior error are: 6) Square the instantaneous posterior error, and then calculate its derivative with respect to μ _k to be 0, and obtain 7) According to the above formula and simplified to get When μ _k is a constant, the above formula must be true; in fact, in steady state, μ _k and x(k), Changes are slower than , therefore, considering them to be approximately uncorrelated, we can get 8) Among them, accessible beg, but, is unknown, it is difficult to solve by to solve the above step size μ _k ; through the shrinkage denoising method, we get Substitute it into the following formula to get Estimated value of : 9) Put the above Substituting the estimate of Then get the step size value of contracted generalized linear complex least squares method Thus, it is substituted into the update process of obtaining the weight value.