A transform domain robust adaptive filtering method for system identification

Abstract

The application discloses a transform domain robust adaptive filtering method for system identification, which firstly utilizes an orthogonal transform matrix to decorrelate a relevant signal, and then utilizes a maximum entropy criterion to improve robustness of the adaptive filtering method in a non-Gaussian noise environment.Meanwhile, aiming at a contradiction that a step length cannot simultaneously optimize a convergence speed and a steady-state error in the adaptive filtering method, two adaptive filters are combined by utilizing a combination factor, so that the method finally has the fast convergence speed and the small steady-state error.

Description

Technical Field

[0001] This invention relates to the field of digital signal processing technology, specifically to a transform domain robust adaptive filtering method for system identification. Background Technology

[0002] Adaptive filtering, as an important branch of digital signal processing, has been widely applied in radar, communication, electronic countermeasures, and echo cancellation after years of development. The Least Mean Square (LMS) algorithm, with its simple structure and ease of implementation, is a widely used adaptive filtering algorithm. However, the convergence speed of the LMS algorithm depends on the autocorrelation matrix of the input signal, which decreases significantly with eigenvalue diffusion. Narayan et al. proposed the Transform Domain Least Mean Square (TDLMS) algorithm. This algorithm first uses orthogonal transformation to convert the input signal to the transform domain to reduce the correlation between input signals. Then, it uses power normalization to constrain the eigenvalues ​​to around 1, reducing eigenvalue diffusion and ultimately achieving the goal of discorrelation of the input signal, thereby improving the overall convergence speed of the algorithm.

[0003] The traditional TDLMS algorithm establishes its cost function based on the minimum mean square error criterion, which can provide the optimal filtering solution when the system noise follows a Gaussian distribution. However, when the system noise is impulse noise, the performance of the TDLMS algorithm under this criterion deteriorates significantly. Inspired by information learning theory, the maximum correlation entropy criterion has been extensively studied and is considered an effective method for handling non-Gaussian system noise.

[0004] Correlation entropy is a measure of the local similarity between two random variables X and Y in kernel space, defined as follows: ,in for The joint probability density function, It uses the Mercer kernel. Currently, the most widely used Mercer kernel is the Gaussian kernel. .in For the kernel width, and The robust adaptive filtering algorithm under the maximum correlation entropy criterion can be implemented by maximizing the following cost function. Summary of the Invention

[0005] To address the shortcomings of existing adaptive filtering techniques, a transform-domain robust adaptive filtering method for system identification is proposed. Based on the maximum entropy criterion, the robustness of adaptive filtering to non-Gaussian noise can be effectively improved.

[0006] To achieve the above objectives, the present invention is implemented through the following technical solutions;

[0007] A transform-domain robust adaptive filtering method for system identification includes the following steps:

[0008] Step 1: Model the input time series signal using a first-order autoregressive model, where the input signal at time n... ,in For adaptive filter length;

[0009] Step 2: Set two lengths as Furthermore, there are independent and parallel adaptive filters, one of which has a step size of... Its corresponding coefficient vector is The step size of the other adaptive filter is Its corresponding coefficient vector is ,and Then initialize the adaptive filter coefficients;

[0010] Step 3: Utilize orthogonal transformation matrix For input signal Preprocessing is performed to obtain the converted input signal. Its expression is:

[0011] ;

[0012] Step 4: Input signal Convolve the values ​​of the two adaptive filters with their coefficient vectors to obtain the corresponding adaptive filter outputs:

[0013]

[0014]

[0015] Using the expected signal The error signals corresponding to the two filters are obtained:

[0016]

[0017]

[0018] Step 5: Based on the error signal obtained in Step 4, construct the cost function based on the maximum entropy correlation entropy criterion, and use the gradient descent method to obtain the expression for updating the adaptive filter coefficient vector:

[0019]

[0020]

[0021] In the formula , The diagonal matrix for estimating the input signal power:

[0022]

[0023] in Represents a diagonal matrix. For the first Input signal Power estimation, , ,in The expression is

[0024] ,

[0025] in A smoothing factor with constant value and ;

[0026] Step 6: Set a combination factor , The two adaptive filters are convexly combined, and the coefficient vector of the combined adaptive filter is calculated based on the coefficient vectors of the two adaptive filters from step five. ,

[0027]

[0028] Using the same convex combination method, the total output signal and error signal of the system can be expressed as follows:

[0029]

[0030]

[0031] Combination factor ,and For about sigmoid activation function

[0032]

[0033]

[0034] In the formula, for The step size for iterative updates.

[0035] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0036] This invention provides a transform domain robust adaptive filtering method for system identification. First, to address the problem of decreased convergence speed of the LMS algorithm for correlated signals, an orthogonal transformation matrix is ​​used to decorrelate the correlated signals, thereby reducing the impact of correlated signals on the convergence speed.

[0037] Then, to address the issue of degraded convergence performance of non-Gaussian signals and noisy systems encountered in practical applications of adaptive filtering techniques, the maximum entropy criterion is used to improve the robustness of the adaptive filtering method in non-Gaussian noise environments.

[0038] Meanwhile, to address the contradiction in adaptive filtering methods where the step size prevents simultaneous optimization of convergence speed and steady-state error, a combination factor is used to combine two adaptive filters, enabling the filtering method to simultaneously optimize both convergence speed and steady-state error.

[0039] Furthermore, the combination factor in this invention Parameters in The symbolic function update method, unlike existing convex combination update methods, can overcome the limitations of traditional methods. The problem of difficult parameter setting is addressed by actual simulations, which show that this method can quickly adjust the weight vector of the self-used filter to the optimal value. Attached Figure Description

[0040] Figure 1 This is a diagram of the adaptive filtering algorithm system identification structure of the present invention.

[0041] Figure 2 This is a schematic diagram of the principle of combining two filters using the combination factor of the present invention.

[0042] Figure 3 This is a simulation comparison diagram of the present invention and the TDLMS algorithm under non-Gaussian noise environment in system identification. Detailed Implementation

[0043] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0044] like Figure 1 As shown, in the system identification model, the desired signal and input signal The linear relationship between them can be expressed as

[0045]

[0046] For the weight vector of the unknown system, For input signal, For the adaptive filter length, This is additive noise. The adaptive filtering method in this invention includes two lengths of... Adaptive filters, such as Figure 2As shown, the output of the entire adaptive filtering system is actually obtained by convex combination of two adaptive filters through a combination factor. One of the filters has a step size of... The coefficient vector is The output signal is Error signal The step size of the other filter is The coefficient vector is The output signal is The error signal is , The cost function for the two filters is established using the maximum entropy criterion. By combining the error signal, the update equation (TD-MCC) for each adaptive filter under the maximum correlation entropy criterion can be obtained.

[0047]

[0048]

[0049] In the formula The diagonal matrix for estimating the input signal power:

[0050] in Represents a diagonal matrix. For the first Input signal Power estimation, , ,in The expression is ,in A smoothing factor with constant value and .

[0051] A large-step filter and a small-step filter are connected in parallel, and the output signal of the entire adaptive filter is: This refers to the TD-CMCC algorithm. Combination factor. ,and For about sigmoid activation function , In the formula, for The step size for iterative updates.

[0052] The principle of the convex combination scheme is to connect two independent, parallel TDLMS adaptive filters in parallel. A large-step TDLMS adaptive filter improves the convergence speed of the combined filter, while a small-step TDLMS self-adaptive filter ensures a smaller steady-state error. The entire system adjusts the combination factor based on error changes, thereby regulating the weight of the two adaptive filters, allowing the convex combination to simultaneously optimize both steady-state error and convergence speed.

[0053] like Figure 3 The simulation results shown depict an impulse noise environment simulated using mixed noise. It is assumed that the number of taps in the unknown system is the same as the number of taps L in the adaptive filter. Mean square deviation is used. As a performance evaluation criterion, the convergence speed and steady-state error of the algorithm were estimated by taking the mean of 100 independent trials. The probability of impulse noise was set to 0.0001, and the sample size (number of iterations) was 40,000. The input signal used in the experiment was obtained by filtering a white Gaussian signal with a mean of 0 and a variance of 1 through a first-order autoregressive model. Figure 3 As can be seen, in mixed noise environments, the TDLMS algorithm is unstable in convergence due to impulse interference, while the TDLMS (TD-CMCC and TD-MCC) algorithms under the maximum entropy criterion can effectively suppress impulse interference and have better robustness than the traditional TDLMS algorithm. Compared with the TD-MCC algorithm, the TD-CMCC algorithm can achieve a steady-state error as small as the TD-MCC algorithm with a small step size and a convergence speed as fast as the TD-MCC algorithm with a large step size. It can overcome the contradiction between the algorithm's convergence speed and steady-state error. The convergence performance of the TD-CMCC algorithm is significantly better than that of the TDLMS and TD-MCC algorithms in impulse noise environments.

[0054] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A transform-domain robust adaptive filtering method for system identification, used to effectively improve the robustness of adaptive filtering under non-Gaussian noise, characterized in that... Includes the following steps: Step 1: Model the input time series signal using a first-order autoregressive model, where the input signal at time n... ,in For adaptive filter length; Step 2: Set two lengths as An adaptive filter, wherein the step size of one of the adaptive filters is... Its corresponding coefficient vector is The step size of the other adaptive filter is Its corresponding coefficient vector is ,and Then initialize the adaptive filter coefficients; Step 3: Utilize orthogonal transformation matrix For input signal Preprocessing is performed to obtain the converted input signal. Its expression is: ， Step 4: Input signal Convolve the values ​​of the two adaptive filters with their coefficient vectors to obtain the corresponding adaptive filter outputs: ， ， Using the expected signal The error signals corresponding to the two filters are obtained: ， ， Step 5: Based on the error signal obtained in Step 4, construct the cost function based on the maximum entropy correlation entropy criterion, and use the gradient descent method to obtain the expression for updating the adaptive filter coefficient vector: ， ， In the formula The diagonal matrix for estimating the input signal power: ， in Represents a diagonal matrix. For the first Input signal Power estimation, , ,in The expression is ， in A smoothing factor with constant value and , Step 6: Set a combination factor , The two adaptive filters are convexly combined, and the coefficient vector of the combined adaptive filter is calculated based on the coefficient vectors of the two adaptive filters from step five. , ， Using the same convex combination method, the total output signal and error signal of the system can be expressed as follows: ， 。 2. The transform domain robust adaptive filtering method for system identification according to claim 1, characterized in that... Combination factor ,and For about The sigmoid activation function, ， ， In the formula, for The step size for iterative updates.

3. The transform domain robust adaptive filtering method for system identification according to claim 1, characterized in that... The two adaptive filters operate independently and in parallel.

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