A Variable-Parameter Zero-Attraction Least Mean Square Sparse System Identification Method

Abstract

The present invention relates to a variable-parameter zero-attraction least mean square sparse system identification method, and the steps are as follows: obtaining an input signal and forming an input vector, inputting the input vector into an adaptive filter, thereby obtaining an output signal of the adaptive filter and initializing each iteration parameter; adding a zero mean to the output signal to further obtain an expected signal at the moment of the system to be estimated; calculating the error between the expected signal and the output signal, as well as the instantaneous approximation of the error; adding a penalty term to the ZA-LMS algorithm and obtaining an update equation of the weight coefficient of the VP-LZA-LMS algorithm through the gradient descent method; substituting the error and the instantaneous approximation of the error into the adaptive update of the step size and the regularization parameter. The present invention has a faster convergence speed, a lower steady-state error, and good tracking performance.

Description

Technical Field

[0001] The present invention relates to the technical field of adaptive filtering, and particularly to a variable parameter zero attraction least mean square sparse system identification method. Background Art

[0002] Sparse adaptive filtering algorithms are widely used in fields such as echo cancellation, underwater communication, and channel estimation. In these practical applications, the impulse response of the system to be estimated often has sparse characteristics, that is, only a small number of non-zero values in the time domain. Making full use of this sparse characteristic can effectively improve the performance of the algorithm.

[0003] Among many sparse adaptive filtering algorithms, zero attraction algorithms are a very important type of algorithm. Influenced by the LASSO algorithm and compressive sensing theory, this type of algorithm uses the norm of the weight coefficient as a penalty term in the cost function, so that the weight coefficient update equation derived from the gradient descent method contains a zero attraction term, making the real-time updated weight coefficient estimation error continuously approach zero and accelerating the update speed of the algorithm. As the basis of this type of algorithm, some literature uses the l1 norm of the weight coefficient as a penalty term and proposes the zero-attracting LMS algorithm (ZA-LMS). However, during the iteration process, the ZA-LMS algorithm cannot distinguish between zero-valued and non-zero-valued weight coefficients, and applies the same degree of zero attraction to all weight coefficients, resulting in a slow convergence speed and an increase in the steady-state error of the algorithm. For this reason, some literature further improves the penalty term of the ZA-LMS algorithm and proposes the weighted ZA-LMS (RZA-LMS) algorithm. The penalty term of this algorithm uses a logarithmic sum (log-sum) function based on the absolute value of the weight coefficient, that is, it distinguishes between non-zero and zero weight coefficients through a preset threshold and applies different attractions according to the magnitude of the weight coefficient value. After that, scholars proposed to combine the l p (0 < p < 1) norm with the LMS algorithm and control the magnitude of the attraction by adjusting the value of p. For example, some literature makes the algorithm performance reach the best by finding the optimal value of p. Since the minimization problem of parameter p is a non-convex optimization problem and the complexity of the algorithm is high, this idea has certain limitations in practical applications. To avoid the limitations of the fixed step size and fixed regularization parameter on the algorithm performance, some literature proposes the variable parameter ZA-LMS (VP-ZA-LMS) and variable parameter weighted ZA-LMS (VP-RZA-LMS) algorithms by minimizing the mean square deviation. The step size and regularization parameter in both algorithms can be adaptively adjusted according to the environment, effectively solving the problem of pre-setting parameters. However, the steady-state error is large, and the convergence performance and tracking ability are poor. Summary of the Invention

[0004] Objective of the Invention: The present invention provides a variable-parameter zero-attraction least mean square sparse system identification method, aiming to solve the problems of poor convergence performance and tracking ability of existing methods.

[0005] Technical Solution:

[0006] A variable-parameter zero-attraction least mean square sparse system identification method, the steps are as follows:

[0007] Step 1: Obtain an input signal with variance and form an input vector x(n). Input the input vector x(n) into an adaptive filter to obtain the output signal y(n) of the adaptive filter, and initialize each iteration parameter.

[0008] Step 2: Add a zero-mean z(n) to the output signal y(n) to further obtain the desired signal d(n) at time n of the system to be estimated.

[0009] Step 3: Calculate the error e(n) between the desired signal d(n) and the output signal y(n), and the instantaneous error approximation

[0010] Step 4: Add a penalty term to the ZA-LMS algorithm and obtain the update equation of the weight coefficient of the VP-LZA-LMS algorithm through the gradient descent method.

[0011] Step 5: Substitute the error e(n) and the instantaneous error approximation into the adaptive update of the step size μ(n) and the regularization parameter ρ(n). The penalty term of the ZA-LMS algorithm in Step 4 is:

[0012] -γ||w(n)||1 ln B(n)

[0013] where w(n) = [w(n), w(n - 1), …, w(n - L + 1)] T represents the weight coefficient estimation vector obtained by the adaptive filtering algorithm at time n, and w i (n) is the i-th element in w(n) ||·|| ∞ represents the infinity norm, that is, taking the maximum value of w(n), L is the filter length, and ln(·) is the logarithmic function.

[0014] The update equation of the weight coefficient of the VP-LZA-LMS algorithm in Step 4 is:

[0015] w(n + 1) = w(n) - μg w (n)

[0016] = w(n) + μ(n)x(n)e(n) + ρ(n)G LZA [w(n)]

[0017] where \(x(n)=[x(n),x(n - 1),\cdots,x(n - L+1)]\) T represents the input signal vector, the superscript \(T\) represents the matrix transpose, \(w(n)=[w(n),w(n - 1),\cdots,w(n - L+1)]\) T represents the weight coefficient estimation vector obtained by the adaptive filtering algorithm, \(\mu(n)\) represents the step size factor at time \(n\), \(\rho(n)\) represents the regularization parameter at time \(n\), is the zero attraction term. Further, the update formulas for the step size \(\mu(n)\) and the regularization parameter \(\rho(n)\) are:

[0018] \(\mu(n)=\min\{\theta\mu(n - 1)+(1 - \theta)\mu'(n),\mu\}\) max \(

[0019] \(\rho(n)=\theta\rho(n - 1)+(1 - \theta)\rho'(n)\)

[0020] where \(\mu'(n)=\max\{\mu\) * (n),0\}, \rho'(n)=\max\{p\) * (n),0\}, and \(0\lt\theta\lt1\) is the smoothing factor.

[0021] Beneficial effects: Based on the characteristic that the system impulse response is sparse, this application proposes a new zero attraction least mean square algorithm - the variable parameter zero attraction least mean square algorithm based on logarithm (VP - LZA - LMS algorithm): modifying the \(l_1\) norm penalty term of the weight coefficient of the ZA - LMS algorithm into a penalty term in the form of a logarithmic function of the weight coefficient; deriving variable step size and variable regularization parameter by minimizing the MSD, which alleviates the contradiction between the steady - state error and the convergence speed. A large number of simulation results show that in the case of correlated signals and uncorrelated signals as input signals, the algorithm of this application can achieve good results, and has a faster convergence speed, lower steady - state error and good tracking performance compared with some existing algorithms. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Figure 1 is the zero attraction term of the ZA - LMS algorithm;

[0023] Figure 2 is the zero attraction term of the algorithm of the present invention;

[0024] Figure 3 is the system to be estimated with sparsity SR = 0.9177;

[0025] Figure 4 is the system to be estimated with sparsity SR = 0.5;

[0026] Figure 5 The system to be estimated with sparsity SR = 0;

[0027] Figure 6 Performance curves of each algorithm when the input in the sparse system is uncorrelated signal (α = 0);

[0028] Figure 7 Performance curves of each algorithm when the input in the sparse system is correlated signal (α = 0.5);

[0029] Figure 8 Variable step size μ(n) of this algorithm when the input is white signal (α = 0) and when the input is correlated signal (α = 0.5);

[0030] Figure 9 Regularization parameter ρ(n) of this algorithm when the input is white signal (α = 0) and when the input is correlated signal (α = 0.5);

[0031] Figure 10 Learning curves of each algorithm in the case of system mutation and the input signal being a correlated signal;

[0032] Figure 11 Variable step size μ(n) of this algorithm in the case of system mutation and the input signal being a correlated signal;

[0033] Figure 12 Variable regularization parameter ρ(n) of this algorithm in the case of system mutation and the input signal being a correlated signal. Detailed implementation manner

[0034] The present invention will be described in more detail below with reference to the accompanying drawings of the specification.

[0035] This application comprehensively considers three factors, namely zero attraction, step size, and regularization parameter, in the zero attraction class of algorithms, and proposes a variable parameter zero attraction LMS algorithm based on logarithmic function (Variable Parameters and Logarithmicfunction based on ZA-LMS, VP-LZA-LMS algorithm). First, aiming at the problem that the ZA-LMS algorithm applies the same degree of attraction to the weight coefficients, the penalty term of the algorithm is modified, that is, the penalty term of the ZA-LMS algorithm is modified into the form of a logarithmic function, so that the weight coefficient estimation approaches zero more quickly, accelerating the convergence speed of the algorithm. Second, aiming at the problems of fixed step size and fixed regularization parameter, variable step size and regularization parameter are derived by optimizing the Mean Square Deviation (MSD), so that the algorithm no longer needs to preset parameters, and the parameters are adaptively updated during the iteration process. The simulation results show that for sparse systems, in the case of white signals and correlated signals as input signals, the new algorithm of this application has better convergence performance and tracking ability compared with the existing LMS algorithm, ZA-LMS algorithm, RZA-LMS algorithm, VP-ZA-LMS algorithm, and VP-RZA-LMS algorithm.

[0036] A method for identifying a variable parameter zero attraction least mean square sparse system, the steps are as follows:

[0037] Step 1: Obtain an input signal with a variance of and form an input vector x(n), and input the input vector x(n) into an adaptive filter, so as to obtain an output signal y(n) of the adaptive filter; and initialize each iteration parameter;

[0038] Taking the system identification problem as an example, the output signal y(n) of the adaptive filter is expressed as:

[0039] y(n) = x T (n)w(n) (1)

[0040] where x(n) = [x(n), x(n - 1),..., x(n - L + 1)] T represents the input signal vector, w(n) = [w(n), w(n - 1),..., w(n - L + 1)] T represents the weight coefficient estimation vector obtained by the adaptive filtering algorithm, where L is the filter length, n is the signal discrete time series label, and the superscript T represents the matrix transpose.

[0041] Initialize parameters: w = 0 L , ρ(0) = 0, μ(0) = 0.01, θ = 0.95, β = 0.95,

[0042] Step 2: Add additive white Gaussian noise with a mean of 0 and a variance of to the output signal y(n) to further obtain the desired signal d(n) at time n of the system to be estimated;

[0043] The desired signal d(n) at time n of the system to be estimated is:

[0044] d(n) = x T (n)w * + z(n) (2)

[0045] where w * represents the impulse response vector of the system to be estimated, and z(n) is additive white Gaussian noise with a mean of 0 and a variance of .

[0046] Step 3: Calculate the error e(n) between the desired signal d(n) and the output signal y(n), and the instantaneous approximation of the error

[0047] The error e(n) between the desired signal d(n) and the output signal y(n) is:

[0048] e(n) = d(n) - y(n) (3)

[0049] The instantaneous approximation of the error is:

[0050]

[0051] where β is a smoothing factor in the interval [0, 1);

[0052] Step 4: Add a penalty term to the ZA-LMS algorithm and obtain the update equation of the weight coefficients of the VP-LZA-LMS algorithm through the gradient descent method;

[0053] The cost function of the LMS algorithm is:

[0054]

[0055] To make full use of the sparsity of the system, the ZA-LMS algorithm adds the l1 norm of the weight coefficient estimation vector as a penalty term to the cost function of the LMS algorithm:

[0056]

[0057] where the coefficient is for the sake of simple operation, and the constant γ ≥ 0. Using the gradient descent method, the update equation of the ZA-LMS algorithm:

[0058] w(n + 1) = w(n) + μx(n)e(n) + ρG ZA [w(n)] (6)

[0059] where G ZA [w(n)] = -sign[w(n)] is called the zero attractor term and can be represented by Figure 1 , μ is the step size greater than 0, ρ = μγ > 0 is called the regularization parameter, which is used to control the range and strength of the attraction effect of the algorithm. It can be seen from Figure 1 that when the weight coefficient w(n) > 0, the zero attractor term is -1, and a positive value ρ will be subtracted during the update; when w(n) < 0, ρ will be added during the weight coefficient update. This is equivalent to having an attraction at the origin, which "attracts" the error closer to zero during each coefficient iteration.

[0060] It should be noted that during the above "attraction" process, regardless of the value of the weight coefficient, the attraction does not change with the weight coefficient. In addition, it can be seen from Equation (6) that in this algorithm, both the step size μ and the regularization parameter ρ are fixed, which limits the performance of the algorithm to some extent. Therefore, in this application, the penalty term of the ZA - LMS algorithm is first modified to a penalty term in the form of a logarithmic function to solve the problem that the ZA - LMS algorithm has the same attraction to all weight coefficients, resulting in a decline in algorithm performance; then, by using the method of minimizing the mean square deviation, the step size and regularization parameter that can vary with the error during the weight coefficient update process are derived to further improve the robustness of the algorithm.

[0061] Improvement of the zero attractor term in the present invention:

[0062] In a sparse system, the proportion of weight coefficients close to zero is relatively large. Increasing the attraction to them can accelerate the overall convergence speed of the algorithm. Therefore, for the problem that the zero attractor of the ZA - LMS algorithm exerts the same attraction to the weight coefficients, the solution idea of this application is: when the weight coefficient is zero or close to zero, increase the attraction to it, while for non - zero weight coefficients, try to reduce the attraction exerted on it. Therefore, the penalty term in Equation (6) is replaced in this application with:

[0063] -γ||w(n)||1 ln B(n) (7)

[0064] where ||·|| ∞ represents the infinity norm, that is, taking the maximum value of w(n), and L is the filter length.

[0065] Thus, the cost function of the new algorithm is:

[0066]

[0067] Using the gradient descent method, the gradient of Equation (8) is obtained:

[0068]

[0069] Here, the constant term with a relatively small value in the gradient is ignored. Thus, the update equation for the weight coefficients of the new algorithm is:

[0070] where μ(n) > 0 is the step size, ρ(n) = μ(n)γ is the regularization parameter, and G LZA [w(n)] is defined as the zero attraction term of the new algorithm, that is

[0071]

[0072] Using Figure 2 to represent the new attraction term, it can be seen that when the weight coefficient is greater than zero, the value of the sign function is positive, and the ln B(n) included in the last term of Equation (10) is always negative, so it is equivalent to subtracting a positive value from the weight coefficient; similarly, when the weight coefficient is less than zero, the value of the sign function is negative, which is equivalent to adding a positive value to the weight coefficient, and the closer the weight coefficient is to zero, the greater the amplitude of the zero attraction term, that is, the attractor of each weight coefficient is related to its own size and is no longer a constant value.

[0073] Step 5: Substitute the error e(n) and the instantaneous approximation of the error into the minimum mean square deviation MSD, and the minimum mean square deviation MSD is minimized to derive the optimal step size and the optimal regularization parameter, and the step size and the regularization parameter are adaptively updated during the iteration process.

[0074] There is a problem of reasonable selection of the value of ρ(n) in the iteration equation of the new algorithm shown in Equation (10). If the parameter ρ is too small, the force of the zero attraction term ρ(n)ln B(n) is insufficient and the convergence speed decreases; if ρ(n) is too large, the zero attraction term becomes larger and it will affect the steady-state error. Therefore, this application further realizes the variable parameters of the proposed algorithm based on the MSD criterion.

[0075] First, define as the difference between the estimated weight coefficient w(n) and the optimal weight coefficient w * :

[0076]

[0077] Obtain the update equation of :

[0078]

[0079] where e(n) can be further written as in Equation (3):

[0080]

[0081] Assume that is independent of x(n), then the minimum mean square error (MSE) of the algorithm can be written as:

[0082]

[0083] where, Define the trace of F(n) as the MSD at time n:

[0084] ξ(n) = tr[F(n)] (16)

[0085] Therefore, both the step size and the regularization parameter are included in ξ(n). If the MSD at time n shown in Equation (16) is known, then the MSD at time n+1 is expressed as:

[0086] ξ(n+1) = tr[F(n+1)] (17)

[0087] Minimizing the MSD at time n+1 can obtain the optimal step size and the optimal regularization parameter, that is:

[0088]

[0089] In the formula, tr[·] represents the rank operation, is the variance of the noise z(n), is the variance of the signal x(n);

[0090] where the derivation processes of a(n), b(n), c(n), p1(n), p2(n) are:

[0091] The iterative formula of

[0092]

[0093] Define F(n) as

[0094]

[0095] Substitute Equation (19) into Equation (20), we get

[0096]

[0097] Assume that both z(n) and κ(n) are independent of So z(z(n)) in the last term of Equation (21) is:

[0098]

[0099] Assume that the noise z(n) is independent of any signal, so E[z(z(n))] = 0. Thus, tr[F(n + 1)] in Equation (18) can be written as:

[0100] tr[F(n + 1)] = tr[F(n)] + μ 2 (n)a(n) + ρ 2 (n)b(n) - 2μ(n)p1(n) + 2ρ(n)p2(n) + 2μ(n)ρ(n)c(n) (23)

[0101] where,

[0102]

[0103]

[0104]

[0105]

[0106]

[0107] Next, complete the calculation of .

[0108] For Equations (26) and (28), w * needs to be known, so it is difficult to estimate. Here, a local single-step approximation is used for w * :

[0109]

[0110] where κ(n) is a positive step size to be determined, and J[w(n)] is the gradient at w(n). By given ξ(n) at time n, κ(n) can be obtained by solving the minimization of ξ(n + 1). Subtract w * from both sides of Equation (29), that is

[0111]

[0112] Define and rewrite Equation (30) according to Equation (12) as:

[0113]

[0114] The true gradient in Equation (31) cannot be determined and is approximated by the instantaneous value -e(n)x(n). Thus, Equation (31) can be written as:

[0115]

[0116] Define \(T(n)=E(D(n)D T (n))\), and using the assumption that the noise \(z(n)\) is independent of any signal, we have

[0117]

[0118] Minimizing \(tr[T(n)]\) gives \(\kappa(n)\), that is

[0119]

[0120] Therefore,

[0121]

[0122] Thus

[0123]

[0124] where

[0125]

[0126] So

[0127]

[0128] Substitute Equation (38) into Equation (26) and simplify to get \(c(n)\):

[0129]

[0130] Substitute Equation (38) into Equation (28) and simplify to get \(p2(n)\):

[0131] p2(n)≈-g T (n)G LZA [w(n)] (40)

[0132] In summary, \(a(n)\), \(b(n)\), \(c(n)\), \(p1(n)\), \(p2(n)\) are:

[0133]

[0134]

[0135]

[0136]

[0137] p2(n)≈-g T (n)G LZA [w(n)] (45)

[0138] is the variance of the noise z(n), is the variance of the signal x(n). ξ(n) in Equation (44) is unknown, and the estimated value is used for substitution:

[0139]

[0140] where is the instantaneous approximation value, β is the smoothing factor in the interval [0, 1),

[0141] g(n) in Equation (43) and Equation (45) is:

[0142]

[0143] Thus, ξ(n + 1) can be written as:

[0144] ξ(n + 1) = ξ(n) + [μ(n) ρ(n)]H[μ(n) ρ(n)] T - 2[p1(n) p2(n)][μ(n) ρ(n)] T + x(n)(48)

[0145] where

[0146]

[0147] It can be proved that the matrix H is positive semi - definite

[0148] Proof of the positive definiteness of the matrix H:

[0149] The matrix H is defined as:

[0150]

[0151] Substitute a(n), b(n) and c(n) in Equation (41), (42) and (43) into Equation (50) respectively, and we get

[0152]

[0153] Define H1 and H2 as

[0154]

[0155]

[0156] Thus, the H matrix can be written as:

[0157] H = H1 + H2 (54)

[0158] Next, the positive definiteness of \(H_1\) and \(H_2\) in Equation (54) will be proven separately. According to the property of the trace \(tr(AB') = B'A\), \(H_1\) can be written as:

[0159]

[0160] where

[0161]

[0162] First, let \(A\) be an arbitrary (non - zero) complex - valued vector. Define the variable \(Y\) as

[0163] \(Y = A\) H \(V(n)\) (57)

[0164] Take the conjugate transpose of Equation (57):

[0165] \(Y\) * \(= V\) H \((n)A\) (58)

[0166] where \(*\) represents conjugation.

[0167] Therefore, we have

[0168]

[0169] Also, because

[0170] \(E[|Y| 2 \geq 0\) (60)

[0171] So, we have

[0172] \(A\) H \(H_1A \geq 0\) (61)

[0173] Therefore, the matrix \(H_1\) is positive semi - definite.

[0174] Second, \(H_2\) can be written as:

[0175]

[0176] If the eigenvalues of \(H_2\) are greater than or equal to 0, then \(H_2\) is a positive semi - definite matrix. Since both \(H_1\) and \(H_2\) are positive semi - definite matrices, \(H\) is a positive semi - definite matrix.

[0177] For simplicity, this application assumes that \(H\) is positive definite. Therefore, Equation (18) can be written as:

[0178]

[0179] Subsequently, we obtain

[0180] [\(\mu\) * \((n)\rho\) * \((n)\)]T = H -1 [p1(n) p2(n)] T (64)

[0181] wherein,

[0182]

[0183]

[0184] In specific applications, to ensure that both the step size and the regularization parameter are non - negative values, the present application adopts the forms of Equation (67) and Equation (68):

[0185] μ(n)= min{θμ(n - 1)+(1 - θ)μ′(n), μ max} (67)

[0186] ρ(n)= θρ(n - 1)+(1 - θ)ρ′(n) (68)

[0187] wherein, 0 < θ < 1 is the smoothing factor, and μ max is the preset upper limit of the step size. Define μ′(n) and ρ′(n) as:

[0188] μ′(n)= max{μ * (n), 0} (69)

[0189] ρ′(n)= max{ρ * (n), 0} (70)

[0190] In summary, the implementation process of the VP - LZA - LMS algorithm of the present application is shown in Table 1.

[0191] Table 1 Variable - parameter Zero - attraction LMS Algorithm Based on Logarithmic Function

[0192]

[0193] 1. Convergence analysis:

[0194] The update equation of the VP - LZA - LMS algorithm is:

[0195]

[0196] wherein, e(n)= z(n)- x T (n)w * . According to the weight coefficient error defined above,[[]] Subtract w * from both sides of Equation (36) to obtain:

[0197]

[0198] Taking the expectation on both sides of Equation (72), since it is assumed that z(n) and x(n) are independent of each other and have a mean of 0, we have:

[0199]

[0200] where R x =E[x(n)x T (n)] represents the autocorrelation matrix of x(n), and R x can be decomposed as R x =QΛQ T , Λ=diag(λ1, λ2,..., λ M ) is the diagonal matrix composed of the eigenvalues of R x , M is the number of eigenvalues, and Q=[q1, q2,..., q M is a unitary matrix. Define

[0201]

[0202] Using the property of the unitary matrix Q (i.e., Q T Q=QQ T =I), multiply both sides of Equation (38) on the left by Q T :

[0203] E[r(n + 1)]=[I - μ(n)Λ]E[r(n)] + ρ(n)Q T E{G LZA [w(n)]} (75)

[0204] When n→∞, ρ(n)Q T E{G LZA [w(n)]} is bounded. To make Equation (75) converge, it is necessary that:

[0205] |1 - μ(n)λ i |<1 (76)

[0206] Therefore, the convergence condition of the VP - LZA - LMS algorithm is:

[0207]

[0208] where λ i is the i - th element of the diagonal matrix Λ after the eigenvalue decomposition of the matrix R x , R x =E[x(n)x T (n)] represents the autocorrelation matrix of x(n), and R x can be decomposed as R x =QΛQ T , Λ=diag(λ1, λ2,..., λM ) is the diagonal matrix composed of the eigenvalues of R x , and M is the number of eigenvalues.

[0209] 2. Simulation Experiments

[0210] 2.1 Simulation Conditions and Parameter Settings

[0211] Here, the performance of the proposed VP-LZA-LMS algorithm in this application is compared with the existing LMS, ZA-LMS, RZA-LMS, VP-ZA-LMS, and VP-RZA-LMS algorithms. The parameters used by each algorithm during the algorithm verification process are shown in Table 2.

[0212] Table 2 Parameter Settings Used by Each Algorithm in Performance Comparison

[0213]

[0214] The input signal is generated by a first-order AR filter, and its expression is:

[0215] x(n) = αx(n - 1) + v(n) (78)

[0216] where the mean of the excitation signal v(n) is 0 and the variance is σ v 2 = 1 - α 2 , α is the correlation coefficient of x(n). When α = 0, x(n) is a white input. When α ≠ 0, x(n) is a correlated input.

[0217] In all experiments, it is assumed that the adaptive filter and the unknown system have the same length L = 64. Each simulation is the average result of 30 independent experiments. The signal-to-noise ratio is defined as:

[0218]

[0219] The SNR selected in this application is 20 dB.

[0220] In the experiment, the normalized mean square deviation (NMSD) is used as an index to measure the performance of the algorithm, and its definition is:

[0221]

[0222] where w(n) is the filter weight coefficient after each iteration, and w ＊ is the true value of the unknown system.

[0223] The sparsity of the system to be estimated is characterized by SR (Sparse Rate), and its expression is defined as:

[0224]

[0225] where \(L\) represents the length of the unknown system, \(\|\mathbf{w}\) * \|_1\) is the \(l_1\)-norm of the weight coefficients, \(\|\mathbf{w}\) * \|_2\) is the \(l_2\)-norm of the weight coefficients. The larger the value of \(SR\), the sparser the system, and the fewer the number of non-zero elements in the weight coefficients.

[0226] 2.2 Simulation Results

[0227] 2.2.1 In a sparse system with uncorrelated input signals

[0228] The system to be estimated is as Figures 3 - 5 , with a sparsity \(SR = 0.9177\), and the input signal \(x(n)\) is an uncorrelated signal, i.e., the correlation coefficient \(\alpha = 0\). The performance curves of each algorithm are as Figure 6 shown.

[0229] It can be seen from Figure 6 that for a sparse system, the LMS algorithm does not fully utilize the sparsity of the system. Therefore, the LMS algorithm shows the worst performance. The proposed VP-LZA-LMS algorithm in this application adopts variable step sizes and regularization parameters, so its steady-state error is smaller than those of the LMS, ZA-LMS, RZA-LMS, VP-ZA-LMS, VP-RZA-LMS, and VP-LZA-LMS algorithms, and it has a faster convergence speed.

[0230] 2.2.2 In a sparse system with correlated input signals

[0231] The system to be estimated is still as Figure 3 , with a sparsity \(SR = 0.9177\), but the input signal \(x(n)\) is a correlated signal, i.e., the correlation coefficient \(\alpha = 0.5\). The performance curves of each algorithm are as Figure 7 shown.

[0232] It can be seen from Figure 7 that compared with Figure 6 , when the input signal is correlated, the performance of all algorithms decreases. However, compared with the LMS, ZA-LMS, RZA-LMS, VP-ZA-LMS, and VP-RZA-LMS algorithms, the steady-state performance of the algorithm in this application is better because the variable step sizes and regularization parameters can be adaptively adjusted according to the environment to ensure the convergence speed and steady-state error of the algorithm.

[0233] Figures 8 - 9 shows the variation process of the step sizes and regularization parameters during the iterative process of the algorithm. It can be seen that the algorithm in this application sets the step sizes and regularization parameters to larger values in the initial stage to ensure fast convergence, and then the step sizes and regularization parameters gradually decrease during continuous iteration and maintain almost constant values after reaching a certain level to ensure that the algorithm obtains a lower misadjustment error.

[0234] 3. Tracking Performance Simulation

[0235] The purpose of this experiment is to test the tracking performance of the algorithm under the condition of system change. Let the correlation coefficient α of the input signal x(n) be 0.5. After every 8000 sampling intervals of the unknown system, the positions and values of the non-zero coefficients change (indicating that the system undergoes a jump). The first time period is a sparse system with a sparsity SR = 0.9177 (as Figure 3 shown); the second time period is a semi-sparse system with a sparsity SR = 0.5 (as Figure 4 shown); the third time period is a non-sparse system with a sparsity SR = 0 (as Figure 5 shown).

[0236] As Figure 10 shown, for the first-stage system with a sparsity SR = 0.9177, which is a sparse system, the algorithm of this application has a low steady-state error, indicating that the VP-LZA-LMS algorithm has good steady-state performance for sparse systems; for the second-stage system with a sparsity SR = 0.5, the performance of the VP-ZA-LMS, VP-RZA-LMS, and VP-LZA-LMS algorithms all decreases, but the algorithm of this application can still achieve good results; for the third-stage system with a sparsity SR = 0, which is a non-sparse system, as the sparsity decreases, the performance of the ZA-LMS and RZA-LMS decreases, but the performance of the algorithm of this application is still better than other algorithms. Generally speaking, the algorithm of this application can reach the steady state quickly and has good tracking performance.

[0237] Figures 11 - 12 Shown in Figure 11 and 12 is the variation of the variable step size and regularization parameter of the VP-LZA-LMS algorithm in this experiment. It can be seen from

[0238] that after each jump of the system, the variable step size and regularization parameter can be quickly adjusted, and finally, almost constant parameter values are used to ensure that the algorithm obtains a low misadjustment error. Based on the characteristic that the system impulse response has sparsity, this application proposes a new zero attraction least mean square algorithm - the logarithm-based variable parameter zero attraction least mean square algorithm (VP-LZA-LMS algorithm): modifying the l1 norm penalty term of the weight coefficient of the ZA-LMS algorithm into a penalty term in the form of a logarithmic function of the weight coefficient; using the method of minimizing the MSD to derive the variable step size and variable regularization parameter, which alleviates the contradiction between the steady-state error and the convergence speed. A large number of simulation results show that the algorithm of this application can achieve good results in the case of correlated and uncorrelated input signals, and has a faster convergence speed, lower steady-state error, and good tracking performance compared with some existing algorithms.

Claims

1. A variable-parameter zero-attracting least mean square sparse system identification method, characterized in that: The steps are as follows: Step 1: Obtain an input signal with a variance of and form an input vector x(n). Input the input vector x(n) into an adaptive filter to obtain an output signal y(n) of the adaptive filter and initialize each iteration parameter; Step 2: Add zero-mean z(n) to the output signal y(n) to further obtain the desired signal d(n) at time n of the system to be estimated; Step 3: Calculate the error e(n) between the desired signal d(n) and the output signal y(n), as well as the instantaneous approximation of the error Step 4: Add a penalty term to the ZA-LMS algorithm and obtain the update equation of the weight coefficients of the VP-LZA-LMS algorithm through the gradient descent method; Step 5: Substitute the error e(n) and the instantaneous error approximation into the adaptive updates of the step size μ(n) and the regularization parameter ρ(n); In Step 4, the penalty term of the ZA-LMS algorithm is: -γ||w(n)||1lnB(n) where \(w(n)=[w(n),w(n - 1),\cdots,w(n - L+1)]\) T denotes the estimated weight coefficient vector obtained by the adaptive filtering algorithm at time \(n\), and \(w\) i (n) is the \(i\)-th element in \(w(n)\) \(\|\cdot\|\) ∞ denotes the infinity norm, that is, taking the maximum value of \(w(n)\), and \(L\) is the filter length is the logarithmic function The update equation of the weight coefficients of the VP-LZA-LMS algorithm in Step 4 is: w(n + 1) = w(n) - μg w (n) = w(n) + μ(n)x(n)e(n) + ρ(n)G LZA [w(n)] where \(x(n)=[x(n),x(n - 1),\cdots,x(n - L+1)]\) T denotes the input signal vector, and the superscript \(T\) represents matrix transpose. w(n) = [w(n), w(n - 1),..., w(n - L + 1)] T denotes the weight coefficient estimation vector obtained by the adaptive filtering algorithm, μ(n) denotes the step size factor at time n, and ρ(n) denotes the regularization parameter at time n, is the zero attraction term; In Step 5, the update formulas for the step size μ(n) and the regularization parameter ρ(n) are: μ(n) = min{θμ(n - 1)+(1 - θ)μ'(n), μ max} ρ(n) = θρ(n - 1) + (1 - θ)ρ'(n) where μ'(n) = max{μ * (n), 0}, ρ'(n) = max{ρ * (n), 0} 0 < θ < 1 is the smoothing factor.

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