A multi-mode blind equalization method based on a foley-schmitz function and a variable fractional step size

Abstract

The application provides a multimode blind equalization method based on a swallow line function and a variable fractional order gradient. The algorithm uses statistical characteristics of an equalizer output signal and a transmitting end signal to construct a multimode cost function based on a swallow line function, so that phase errors in a channel equalization process can be compensated without an additional carrier recovery loop. Subsequently, a variable fractional order gradient method is used to replace a traditional fixed order gradient updating mode to adaptively update equalizer weight vectors, so that optimal weight vectors are obtained. Compared with the prior art, the application can not only realize channel equalization and phase recovery, but also effectively reduce residual inter-symbol interference and bit error rate in a strong impulse noise environment, while the convergence speed is accelerated, and higher equalization efficiency and robustness are shown.

Description

Technical Field

[0001] This invention relates to the field of digital wireless communication technology, and in particular to a multimode channel blind equalization method based on the finfish tongue function and variable fractional gradient in an impulse noise environment. Background Technology

[0002] In wireless communication systems, inter-symbol interference (ISI) occurs due to multipath propagation and bandwidth limitations, affecting communication reliability and leading to a decline in communication quality. Blind equalization techniques that do not require training sequences are commonly used to eliminate ISI and achieve channel equalization. The Constant Modulus Algorithm (CMA) and the Multimodulus Algorithm (MMA) are two representative blind equalization algorithms for Gaussian noise, employing the minimum mean square error criterion and achieving effective equalization under Gaussian noise. In practical communication systems, impulse noise caused by atmospheric noise and human interference also exists. This type of noise exhibits significant short-term spikes and long tails. The alpha-stable distribution is often used to model impulse noise. The alpha-stable distribution not only better characterizes the tail characteristics of impulse noise but is also the only distribution that satisfies the generalized central limit theorem, giving it a natural advantage in modeling additive noise processes with different impact levels.

[0003] In his paper "Maximum versoria criterion-based robust adaptive filtering algorithm" (IEEE Transactions on Circuits and Systems II: Express Briefs, Vol. 64, No. 10, 2017), Fuyi Huang disclosed a constant modulus blind equalization method based on the maximum winnowing loop criterion (MVC). This method improves the cost function of the traditional constant modulus algorithm under non-Gaussian impulse noise environments. Utilizing the rapid change and greater sensitivity to errors of the winnowing loop function, it achieves higher convergence speed and lower computational complexity, effectively solving the problem of equalization performance failure of the traditional constant modulus algorithm under impulse noise. However, this method still has certain limitations. Its cost function is based on an integer-order differential form and only utilizes the amplitude information of the signal. Therefore, the equalization performance degrades significantly under strong impulse noise environments, leading to unstable convergence, high bit error rate, and an inability to effectively overcome the phase rotation problem.

[0004] Xi'an University of Electronic Science and Technology disclosed a constant-mode blind equalization method based on fractional-order correlation entropy and fractional-order gradient in its patent application "Constant-Mode Blind Equalization Method Based on Fractional-Order Correlation Entropy and Fractional-Order Gradient" (Patent Application No. 202111424196.3, Publication No. CN 114143152 A). This method utilizes the fractional-order correlation entropy of the equalizer output signal error to construct a cost function, and replaces the traditional integer-order gradient update rule with a fractional-order gradient method to find the optimal value of the equalizer weight vector. The algorithm achieves stable convergence under weak impulse noise conditions and exhibits low residual inter-symbol interference under strong impulse noise conditions. However, this method still has certain shortcomings: its convergence speed is slow, its bit error rate is high, and it cannot effectively overcome the phase rotation problem. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of the existing technologies by providing a multi-mode blind equalization method based on the finger function and variable fractional gradient, aiming to solve the problems of slow convergence speed, high bit error rate, degraded equalization performance, and phase rotation in the existing technologies under strong impulse noise environments.

[0006] To achieve the above objectives, this invention provides a multi-mode blind equalization method based on the snout line function and variable fractional gradient. A multi-mode cost function based on the snout line function is constructed, and the equalizer weight vector is iteratively updated using the variable fractional gradient method. The equalization method includes the following steps:

[0007] Step 1. Obtain the constants of the signal to be equalized and the signal transmitted;

[0008] (1a) Convolve the transmitter signal with the impulse response of the transmission channel and then add an impulse noise to obtain the signal to be equalized;

[0009] (1b) Calculate the signal constant of the transmitting signal according to the following formula:

[0010] R R =E[|s R (n)| 4 ] / E[|s R (n)| 2 ], R I =E[|s I (n)| 4 ] / E[|s I (n)| 2 ]

[0011] Among them, s R (n) and s I R(n) represents the real and imaginary parts of the transmitted signal s(n), respectively. R and R IIt is a constant determined by the transmitting signal s(n), E(g) represents the desired operation, and |g| represents the modulo operation;

[0012] Step 2. Calculate the balanced data according to the following formula;

[0013] y(n)=w T (n)x(n)

[0014] Where y(n) represents the data after equalization by the equalizer at time n, w(n) represents the tap weight vector of the equalizer at time n, and w(n) = [w0(n), w1(n), ..., w M-1 (n)] T is the equalizer weight vector of order M, (g) Τ Let x(n) represent the transpose, and let x(n) represent the column vector corresponding to the signal at the nth time step in the signal to be equalized.

[0015] Step 3. Construct the multimode blind equilibrium cost function based on the variable fractional gradient of the looper line according to the following formula:

[0016]

[0017] Where J(n) represents the multimode cost function corresponding to the signal data to be equalized at time n; y R (n) and y I (n) represent the real and imaginary parts of y(n) respectively, and τ is the parameter of the loop function, τ=(2a) -p , where a > 0 is the radius of the circle corresponding to the loop line function, and p is a real number greater than 0;

[0018] Step 4. Update the equalizer weight vector using the variable fractional gradient method:

[0019] (4a) Calculate the fractional gradient of the cost function according to the following formula:

[0020]

[0021] Among them, J (λ) (n) represents the λ-th derivative of the cost function of the equalizer output signal at time n, where λ represents the fractional gradient order, j represents the imaginary unit, and sign(g) represents the sign function. * This indicates the complex conjugate operation;

[0022] (4b) Calculate the fractional gradient order at time n according to the following formula:

[0023]

[0024] Where η∈(0,1) represents the attenuation coefficient, and k is the control coefficient;

[0025] (4c) Update the equalizer weight vector using the λ-th derivative of the cost function corresponding to each equalizer output signal:

[0026] w(n+1)=w(n)-μJ (λ) (n)

[0027] Where w(n+1) represents the updated equalizer weight vector, w(n) represents the original equalizer weight vector, and μ represents the step size factor;

[0028] Step 5. Determine whether the equalization of all signals to be equalized has been completed. If yes, proceed to step 6; otherwise, return to step 2.

[0029] Step 6. Complete channel blind equalization.

[0030] Preferably, the impulse noise in step (1a) follows an alpha-stable distribution and has significant short-term spike characteristics and long tail characteristics.

[0031] Preferably, the center of the circle in step 3 is located at (0, a).

[0032] Preferably, in step 3, a = 2 and p = 2.5.

[0033] Preferably, in step (4a), λ takes a value of 0 to 1.

[0034] Preferably, in step (4c), the value of μ is 1×10⁻⁶. -6 ~1×10 -2 .

[0035] Preferably, μ = 5 × 10 -4 .

[0036] Compared with the prior art, the present invention has the following advantages:

[0037] First, this invention utilizes a generalized frustum line function to construct a multimode cost function. The frustum line function reduces computational complexity, accelerates convergence, and suppresses large outliers caused by impulse noise, overcoming the problems of unstable convergence and high bit error rate after equalization in existing technologies under strong impulse noise. Simultaneously, it solves the phase rotation problem in channel equalization in existing technologies, enabling this invention to simultaneously achieve blind channel equalization and carrier phase recovery.

[0038] Second, the present invention uses a variable fractional gradient method to adaptively update the equalizer weight vector. During the iteration process, the fractional order can be dynamically adjusted, overcoming the problem of slow convergence speed or unstable convergence in the existing technology during the equalization process. This enables the present invention to achieve channel equalization with low residual inter-symbol interference and low bit error rate in both weak impulse noise environment and strong impulse noise environment. Attached Figure Description

[0039] Figure 1 This is a flowchart of the present invention;

[0040] Figure 2 The remaining ISI diagram of this invention;

[0041] Figure 3 This is the SER diagram of the present invention; Detailed Implementation

[0042] 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.

[0043] Reference Figure 1 The specific steps for implementing the present invention will be further described in conjunction with the embodiments.

[0044] Step 1: Obtain the constants of the signal to be equalized and the signal at the transmitting end.

[0045] An embodiment of the present invention involves convolving the baseband 16QAM signal received by the receiving antenna with the impulse response of the transmission channel, and then superimposing an impulse noise that follows a stable alpha distribution to obtain the signal to be equalized, wherein the impulse response of the transmission channel is h = [0.1, 0.3, 1, -0.1, 0.5, 0.2].

[0046] Calculate the signal constant of the received transmitter signal using the following formula:

[0047] R R =E[|s R (n)| 4 ] / E[|s R (n)| 2 ], R I =E[|s I (n)| 4 ] / E[|s I (n)| 2 ]

[0048] Among them, s R (n) and s I R(n) represents the real and imaginary parts of the transmitted signal s(n), respectively. R and R IIt is a constant determined by the transmitted signal s(n), which depends only on the modulation type of the transmitted signal and contains the amplitude information of the transmitted signal. E(g) represents the desired operation, and |g| represents the modulus operation.

[0049] Step 2: Calculate the balanced data according to the following formula.

[0050] y(n)=w T (n)x(n)

[0051] Where y(n) represents the data after equalization by the equalizer at time n, w(n) represents the tap weight vector of the equalizer at time n, and w(n) = [w0(n), w1(n), ..., w M-1 (n)] T Let w(0) be the equalizer weight vector of order M, where M = 21 represents the tap length of the equalizer. At time n = 0, the equalizer weight vector is initialized as w(0) = [0,L,0,1,0,L 0]. Τ (g) Τ Let x(n) represent the transpose, and let x(n) represent the column vector corresponding to the signal at the nth time step in the signal to be equalized.

[0052] Step 3. Construct the blind equilibrium cost function based on the variable fractional gradient of the looper line according to the following formula.

[0053]

[0054] Where J(n) represents the multimode cost function corresponding to the signal data to be equalized at time n. R (n) and y I (n) represent the real and imaginary parts of y(n) respectively, and τ is the parameter of the loop function, τ=(2a) -p 'a' is the radius of the circle corresponding to the loop function, and its center is located at (0, a). 'p' is a real number greater than 0, representing the shape parameter in the generalized loop function. When p = 2, it is the standard loop function. In the embodiments of this invention, a = 2 and p = 2.5.

[0055] Step 4. Update the equalizer weight vector using the variable fractional gradient method.

[0056] Calculate the fractional gradient of the cost function using the following formula:

[0057]

[0058] Among them, J (λ) (n) represents the λ-th derivative of the cost function of the equalizer output signal at time n, where λ represents the fractional gradient order, ranging from 0 to 1, j represents the imaginary unit, and sign(g) represents the sign function. *This indicates the complex conjugate operation.

[0059] The fractional gradient order at time n is calculated using the following formula:

[0060]

[0061] Where η∈(0,1) represents the attenuation coefficient, and k is the control coefficient. In the embodiment of the present invention, η=0.99995 and k=0.8.

[0062] The equalizer weight vector is updated using the λ-order derivative of the cost function corresponding to each equalizer output signal:

[0063] w(n+1)=w(n)-μJ (λ) (n)

[0064] Where w(n+1) represents the updated equalizer weight vector, w(n) represents the original equalizer weight vector, and μ represents the step size factor. In this embodiment of the invention, μ = 5 × 10 -4 .

[0065] Step 5. Determine whether the equalization of all signals to be equalized has been completed. If yes, proceed to step 6; otherwise, return to step 2.

[0066] Step 6. Complete channel blind equalization.

[0067] The effects of the present invention will be further explained below with reference to simulation experiments.

[0068] 1. Simulation conditions:

[0069] The computer operating system used for the simulation experiments in this invention is Windows 11 64-bit, and the compilation environment is Matlab R2022a simulation software.

[0070] The simulation parameters of the system of this invention are as follows:

[0071] The transmitted baseband signal s(n) is a 16QAM signal. The impulse response parameters of the transmission channel are set to h = [0.1, 0.3, 1, -0.1, 0.5, 0.2]. The channel noise is a complex additive impulse noise that follows an alpha-stable distribution. The equalizer length M = 21, and the initial tap weight vector of the equalizer is w(0), where w(0) = [0, L, 0, 1, 0, L 0]. Τ (g) Τ This indicates the transpose, with a center tap coefficient of 1, and 50 Monte Carlo simulations.

[0072] 2. Simulation Experiment Content and Result Analysis:

[0073] The simulation experiment of this invention compares the equalization performance of transmitted baseband signals using the method of this invention and existing blind equalization methods. In the experiment, a 16QAM modulated signal is selected for transmission, convolved with the transmission channel impulse response h, and then superimposed with impulse noise before channel equalization is performed. The equalization performance of the two methods is then compared.

[0074] The existing technology refers to the constant mode blind equalization method based on the maximum versoria criterion (MVC), which Fuyi Huang disclosed in his paper "Maximum versoria criterion-based robust adaptive filtering algorithm" (IEEE Transactions on Circuits and Systems II: Express Briefs, Vol. 64, No. 10, 2017), abbreviated as MVC-CMA. The method of this invention is abbreviated as VFOMVC-MMA.

[0075] To verify the effectiveness of the present invention, residual inter-symbol interference (ISI) and bit error rate (SER) were used to evaluate the channel equalization capability of the two methods.

[0076] In the calculation of ISI and SER, the prior art sets the step size parameter to μ = 5 × 10. -4 a = 2, p = 2.5. The step size parameter of this invention is set to μ = 5 × 10. -4 a = 2, p = 2.5, attenuation coefficient η = 0.99995, control coefficient k = 0.8.

[0077] ISI is calculated using the following formula:

[0078]

[0079] Where C(n) represents the joint impulse response of the transmission channel h and the equalizer weights w(n). in It's a convolution operator. Σ represents the summation operation, |g| represents the modulus operation, and |C... j (n)| represents the module value of the j-th element in |C(n)|. The smaller the value of ISI, the stronger the balancing ability of the algorithm.

[0080] Bit error rate (SER) is the ratio of all erroneous symbols to all symbols in the output signal of an equalizer.

[0081] The following is combined with Figure 2 The effects of the present invention will be further described below.

[0082] Figure 2The performance comparison between the method of this invention and existing methods in different intensity impulse noise environments is shown under a generalized signal-to-noise ratio (GSNR) of 25 dB. Channel equalization was performed on 50,000 sampling points in the received signal. Figure 2 The figure shows the curves of the residual inter-symbol interference (ISI) of the two algorithms as a function of the number of iterations during the equalization process. The horizontal axis represents the number of iterations, and the vertical axis represents the residual ISI value after each iteration, in decibels (dB). Figure 2 The solid line marked with a circle represents the curve of the remaining inter-symbol interference (ISI) of the prior art changing with the number of iterations; the solid line marked with a diamond represents the curve of the remaining ISI of the present invention changing with the number of iterations. Figure 2 In this context, α represents the characteristic parameter of the stable distribution of alpha. The larger α is, the weaker the impulse noise intensity; the smaller α is, the stronger the impulse noise intensity.

[0083] Depend on Figure 2 It can be seen that, under weak impulse noise conditions, the algorithm of this invention converges after approximately 10,000 iterations, with a steady-state residual inter-symbol interference (ISI) of approximately -27 dB; while existing technologies converge after approximately 20,000 iterations, with a steady-state residual ISI of approximately -26 dB. Under strong impulse noise conditions, this invention still maintains a relatively fast convergence speed and low residual ISI. Figure 2 As shown in (d), the present invention converges at approximately 10,000 iterations, with a steady-state residual inter-symbol interference (ISI) of approximately -27 dB. In contrast, the prior art is significantly affected by strong impulse noise, resulting in a significantly slower convergence speed (approximately 25,000 iterations) and an increased steady-state ISI of approximately -24 dB, along with noticeable jitter in the later stages of convergence. In summary, the present invention offers faster convergence, lower residual ISI, and better stability compared to the prior art.

[0084] The following is combined with Figure 3 The effects of the present invention will be further described below.

[0085] Figure 3 The bit error rate curves under different generalized signal-to-noise ratio (SNR) conditions are presented. In the experiment, channel equalization was performed on 200,000 sampling points in the received signal. The bit error rate curves were obtained by statistically analyzing the ratio of the number of erroneous symbols to the total number of symbols after equalization using both the method of this invention and existing technologies. Figure 3 The horizontal axis represents the generalized signal-to-noise ratio (SNR) in dB, and the vertical axis represents the corresponding bit error rate (BER) in dB. In a weak impulse noise environment with a SNR ranging from 16 dB to 30 dB in 1 dB increments, the solid lines marked with circles and diamonds respectively represent the BER variation curves when using existing technology and the method of this invention for channel blind equalization.

[0086] Depend on Figure 3It can be seen that, under different generalized signal-to-noise ratio (SNR) conditions, the present invention exhibits a lower bit error rate than the prior art. When the generalized SNR is 30dB, the bit error rate of the present invention is reduced by about 2dB compared with the prior art, further verifying its equalization performance advantage in impulse noise environments.

[0087] The specific embodiments described herein are merely illustrative of the spirit of the invention. Those skilled in the art to which this invention pertains may make various modifications, additions, or similar substitutions to the described specific embodiments without departing from the spirit of the invention or exceeding the scope defined by the appended claims.

Claims

1. A multi-mode blind equalization method based on the snipeline function and variable fractional gradient, characterized in that, A multi-mode cost function based on the looper line function is constructed, and the equalizer weight vector is iteratively updated using the variable fractional gradient method. This equalization method includes the following steps: Step 1. Obtain the constants of the signal to be equalized and the signal transmitted; (1a) Convolve the transmitter signal with the impulse response of the transmission channel and then add an impulse noise to obtain the signal to be equalized; (1b) Calculate the signal constant of the transmitting signal according to the following formula: R R =E[|s R (n)| 4 ] / E[|s R (n)| 2 ],R I =E[|s I (n)| 4 ] / E[|s I (n)| 2 ] Among them, s R (n) and s I R(n) represents the real and imaginary parts of the transmitted signal s(n), respectively. R and R I It is a constant determined by the transmitting signal s(n), E(·) represents the desired operation, and |·| represents the modulus operation; Step 2. Calculate the balanced data according to the following formula; y(n)=w T (n)x(n) Where y(n) represents the data after equalization by the equalizer at time n, w(n) represents the tap weight vector of the equalizer at time n, and w(n) = [w0(n), w1(n), ..., w M-1 (n)] T It is the equalizer weight vector of order M, (·) Τ Let x(n) represent the transpose, and let x(n) represent the column vector corresponding to the signal at the nth time step in the signal to be equalized. Step 3. Construct the multimode blind equilibrium cost function based on the variable fractional gradient of the looper line according to the following formula: Where J(n) represents the multimode cost function corresponding to the signal data to be equalized at time n; y R (n) and y I (n) represent the real and imaginary parts of y(n) respectively, and τ is the parameter of the loop function, τ=(2a) -p , where a > 0 is the radius of the circle corresponding to the loop line function, and p is a real number greater than 0; Step 4. Update the equalizer weight vector using the variable fractional gradient method: (4a) Calculate the fractional gradient of the cost function according to the following formula: Among them, J (λ) (n) represents the λ-th derivative of the cost function of the equalizer output signal at time n, where λ represents the fractional gradient order, j represents the imaginary unit, and sign(·) represents the sign function. * This indicates the complex conjugate operation; (4b) Calculate the fractional gradient order at time n according to the following formula: Where η∈(0,1) represents the attenuation coefficient, and k is the control coefficient; (4c) Update the equalizer weight vector using the λ-th derivative of the cost function corresponding to each equalizer output signal: w(n+1)=w(n)-µJ (λ) (n) Where w(n+1) represents the updated equalizer weight vector, w(n) represents the original equalizer weight vector, and μ represents the step size factor; Step 5. Determine whether the equalization of all signals to be equalized has been completed. If yes, proceed to step 6; otherwise, return to step 2. Step 6. Complete channel blind equalization.

2. The multi-mode blind equalization method based on the snipeline function and variable fractional gradient as described in claim 1, characterized in that, The impulse noise described in step (1a) follows an alpha-stable distribution and has significant short-term spike characteristics and long tail characteristics.

3. The multi-mode blind equalization method based on the snipeline function and variable fractional gradient as described in claim 1, characterized in that, The center of the circle mentioned in step 3 is located at (0, a).

4. The multi-mode blind equalization method based on the snipeline function and variable fractional gradient as described in claim 1, characterized in that, In step 3, a = 2 and p = 2.

5.

5. The multi-mode blind equalization method based on the snipeline function and variable fractional gradient as described in claim 1, characterized in that, In step (4a), λ takes a value of 0 to 1.

6. The multi-mode blind equalization method based on the snipeline function and variable fractional gradient as described in claim 1, characterized in that, In step (4c), the value of μ is 1×10 -6 ~1×10 -2 .

7. The multi-mode blind equalization method based on the snipeline function and variable fractional gradient as described in claim 6, characterized in that, μ=5×10 -4 。

Images