CN103428130B - A kind of minimum Mean Square Error Linear equalization methods eliminating impulsive noise - Google Patents

本明属移动通信技术领域，具体涉及一种消除脉冲噪声的最小均方误差线性均衡方法。本发明通过对最小均方误差的理论知识推导，得到伯努利-高斯模型下脉冲噪声的最小均方误差线性均衡器的抽头系数计算方法，从而在较低复杂度的情况下获取较优的检测性能。

The invention belongs to the technical field of mobile communication, and in particular relates to a minimum mean square error linear equalization method for eliminating impulse noise. The present invention obtains the tap coefficient calculation method of the minimum mean square error linear equalizer of the impulse noise under the Bernoulli-Gauss model by deriving the theoretical knowledge of the minimum mean square error, thereby obtaining better under the condition of lower complexity Detection performance.

一种消除脉冲噪声的最小均方误差线性均衡方法A Minimum Mean Square Error Linear Equalization Method for Eliminating Impulse Noise

技术领域technical field

本发明属移动通信技术领域，具体涉及一种消除脉冲噪声的最小均方误差线性均衡方法。The invention belongs to the technical field of mobile communication, and in particular relates to a minimum mean square error linear equalization method for eliminating impulse noise.

背景技术Background technique

脉冲噪声是非连续的，由持续时间短、幅度大的不规则脉冲或噪声尖峰组成。脉冲噪声的持续时间小于1秒，且其峰值比均方根值大至少10dB，而其重复频率低于10Hz，是一种间断性的噪声。产生脉冲噪声的原因多种多样，其中包括电磁干扰以及通信系统的故障和缺陷，也可能在通信系统的电气开关和继电器改变状态时产生。在数字式数据通信中，脉冲噪声是出错的主要原因。一种常用的脉冲噪声模型为伯努利-高斯模型。Impulse noise is discontinuous and consists of irregular pulses or noise spikes of short duration and large amplitude. The duration of impulse noise is less than 1 second, and its peak value is at least 10dB greater than the root mean square value, and its repetition frequency is lower than 10Hz, which is a kind of intermittent noise. Impulse noise can be generated for a variety of reasons, including electromagnetic interference, failures and defects in communication systems, or when electrical switches and relays in communication systems change state. Impulse noise is a major source of errors in digital data communications. A commonly used impulse noise model is the Bernoulli-Gaussian model.

最小均方误差(Minimum mean-square error,MMSE)线性均衡(linearequalization,LE)是一种使均衡器输出的估计符号与发送符号之间的均方误差最小化的方法。MMSE均衡考虑了信噪比的因素，在既要消除码间干扰(Intersymbol Interference,ISI)又不放大噪声之间实现了一种平衡，且复杂度较低，是一种较优的均衡算法。Minimum mean-square error (MMSE) linear equalization (linear equalization, LE) is a method to minimize the mean square error between the estimated symbol output by the equalizer and the transmitted symbol. MMSE equalization considers the factor of signal-to-noise ratio, realizes a balance between eliminating intersymbol interference (ISI) and not amplifying noise, and has low complexity, so it is a better equalization algorithm.

发明内容Contents of the invention

为了方便地描述本发明的内容，首先对本发明中所使用的术语进行说明：In order to describe content of the present invention conveniently, at first the terms used in the present invention are explained:

伯努利-高斯模型：一种常用的脉冲噪声模型，该模型下的随机变量可表达为η＝w+b·g，其中w,g服从均值为0、方差分别为的高斯分布，b服从伯努利分布，且P(b＝1)＝p,P(b＝0)＝1-p,，P(·)表示括号内的事件发生的概率。Bernoulli-Gaussian model: a commonly used impulse noise model, the random variable under this model can be expressed as η=w+b g, where w and g obey the mean value of 0 and the variance of Gaussian distribution, b obeys Bernoulli distribution, and P(b=1)=p, P(b=0)=1-p, P(·) indicates the probability of occurrence of the event in brackets.

数学期望：统计学术语，反映随机变量平均取值的大小，又称期望或均值，记为E(·)。Mathematical expectation: A statistical term that reflects the average value of a random variable, also known as expectation or mean, denoted as E(·).

方差：统计学术语，度量随机变量和其数学期望之间的偏离程度，记为D(·)。Variance: A statistical term that measures the degree of deviation between a random variable and its mathematical expectation, denoted as D(·).

比特的先验概率：比特分别为0或1的概率，一般根据以往经验和分析得到的概率。The prior probability of a bit: the probability that a bit is 0 or 1, generally based on previous experience and analysis.

高斯分布：又称正态分布，在数学、物理及工程等领域都非常重要的概率分布，若随机变量X服从均值为μ、方差为σ²的高斯分布，则记为X定义为N(μ,σ²)，且其概率密度函数为 f ( x ) = 1 2 π σ exp ( - ( x - μ ) 2 2 σ 2 ) . Gaussian distribution: Also known as normal distribution, it is a very important probability distribution in the fields of mathematics, physics, and engineering. If a random variable X obeys a Gaussian distribution with a mean of μ and a variance of ^σ2 , it is recorded as X and defined as N(μ ,σ ² ), and its probability density function is f ( x ) = 1 2 π σ exp ( - ( x - μ ) 2 2 σ 2 ) .

本发明提供一种消除脉冲噪声的最小均方误差线性均衡方法，用以提高系统接收机的检测性能，消除通信系统中的脉冲噪声。The invention provides a minimum mean square error linear equalization method for eliminating impulse noise, which is used to improve the detection performance of a system receiver and eliminate impulse noise in a communication system.

为实现上述发明目的，本发明的技术方案如下：For realizing the above-mentioned purpose of the invention, the technical scheme of the present invention is as follows:

一种消除脉冲噪声的最小均方误差线性均衡方法，包括如下步骤：A minimum mean square error linear equalization method for eliminating impulse noise, comprising the steps of:

S1：接收机第n(n＞0)时刻的接收信号为：S1: The received signal of the receiver at the nth (n>0) moment is:

其中，M为时延路径的总长度，h_k(k∈{0,1,...,M-1})为第k条时延路径的衰落系数，x_n,n＞0为发射机第n时刻的发送符号，并且假设x_n＝0,n≤0，w_n服从均值为0、方差为的高斯分布，g_n服从均值为0、方差为的高斯分布，b_n服从伯努利分布，且P(b_n＝1)＝p,P(b_n＝0)＝1-p,，P(·)表示括号内的事件发生的概率； Among them, M is the total length of the delay path, h _k (k∈{0,1,...,M-1}) is the fading coefficient of the kth delay path, x _n ,n>0 is the transmitter The transmitted symbol at the nth moment, and assuming that x _n = 0, n≤0, w _n obeys the mean value of 0 and the variance of The Gaussian distribution of g _n obeys the mean value of 0 and the variance of Gaussian distribution of b _n obeys Bernoulli distribution, and P(b _n =1)=p, P(b _n =0)=1-p, P(·) represents the probability of occurrence of events in brackets;

S2：根据均值 x ‾ n = E ( x n ) = Σ x ∈ β x · P ( x n = x ) , 方差 v n = Σ x ∈ β | x - E ( x n ) | 2 · P ( x n = x ) , β为调制符号集合，获取每个时刻的v_n，其中，E(·)表示随机变量的数学期望；S2: According to the mean x ‾ no = E. ( x no ) = Σ x ∈ β x &Center Dot; P ( x no = x ) , variance v no = Σ x ∈ β | x - E. ( x no ) | 2 &Center Dot; P ( x no = x ) , β is a set of modulation symbols, to obtain the v _n , where E(·) represents the mathematical expectation of the random variable;

S3：设第n时刻最小均方误差线性均衡器的抽头系数为c_n,k,k＝-N₁,1-N₁,...,N₂，总长度为N＝N₁+N₂+1，同时，取N个接收符号，其中(·)^T表示矩阵或向量的转置）作为最小均方误差线性均衡器的输入，并且假设z_n＝0,n≤0，则均衡器第n时刻输出对x_n的估计符号为： x ^ n = E ( x n ) + Cov ( x n , z n ) Cov ( z n , z n ) - 1 ( z n - H x ‾ n ) , 其中，Cov(x,y)表示向量x与y的协方差矩阵，即Cov(x,y)＝E(xy^H)-E(x)E(y^H)，(·)^H表示矩阵或向量的共轭转置，(·)^-1表示矩阵的求逆， x ‾ n = x ‾ n - M - N 2 + 1 x ‾ n - M - N 2 + 2 . . . x ‾ n + N 1 T , S3: Let the tap coefficients of the minimum mean square error linear equalizer at the nth moment be c _n,k ,k=-N ₁ ,1-N ₁ ,...,N ₂ , and the total length is N=N ₁ +N ₂ +1, at the same time, take N received symbols , where (·) ^T represents the transpose of a matrix or vector) as the input of the minimum mean square error linear equalizer, and assuming z _n =0,n≤0, then the equalizer outputs the estimated sign of x _n at the nth moment for: x ^ no = E. ( x no ) + Cov ( x no , z no ) Cov ( z no , z no ) - 1 ( z no - h x ‾ no ) , Among them, Cov(x,y) represents the covariance matrix of vector x and y, that is, Cov(x,y)=E(xy ^H )-E(x)E(y ^H ), ( ) ^H represents the matrix or vector The conjugate transpose of , ( ) ^-1 means the inversion of the matrix, x ‾ no = x ‾ no - m - N 2 + 1 x ‾ no - m - N 2 + 2 . . . x ‾ no + N 1 T ,

S4：为使第n时刻的均衡器输出符号独立于P(x_n＝x)，使v_n＝1，则第n时刻均衡器输出的估计符号变为：S4: In order to make the equalizer output symbols at the nth moment Independent of P(x _n =x), such that v _n =1, then the estimated symbol output by the equalizer at the nth moment becomes:

xx ^^ nno == xx ‾‾ nno ++ vv nno sthe s Hh [[ (( σσ ww 22 ++ pσpσ ii 22 )) II NN ++ HVHV nno Hh Hh ]] -- 11 (( zz nno -- Hh xx ‾‾ nno )) ,,

其中， s = H 0 1 × ( N 2 + M - 1 ) 1 0 1 × N 1 T , I_N为N×N的单位矩阵，in, the s = h 0 1 × ( N 2 + m - 1 ) 1 0 1 × N 1 T , I _N is the identity matrix of N×N,

V n = Diag v n - M - N 2 + 1 v n - M - N 2 + 2 . . . v n + N 1 , Diag(·)表示将长度为l的向量变为l×l的方阵，且向量元素位于方阵的对角线上,并且，假设均衡器的抽头系数向量为 c n = c n , N 2 * c n , N 2 - 1 * . . . c n , - N 1 * T , 则 V no = Diag v no - m - N 2 + 1 v no - m - N 2 + 2 . . . v no + N 1 , Diag( ) means to change the vector with length l into a square matrix of l×l, and the vector elements are located on the diagonal of the square matrix, and assume that the tap coefficient vector of the equalizer is c no = c no , N 2 * c no , N 2 - 1 * . . . c no , - N 1 * T , but

cc nno == [[ (( σσ ww 22 ++ pp σσ ii 22 )) II NN ++ HVHV nno Hh Hh ]] -- 11 sthe s ;;

S5：假设的概率密度函数服从均值为μ_n,x，μ_n,x定义为方差为定义为 Cov ( x ^ n , x ^ n | x n = x ) 的高斯分布，则：S5: Suppose The probability density function of obeys the mean value of μ _n,x , μ _n,x is defined as Variance is defined as Cov ( x ^ no , x ^ no | x no = x ) Gaussian distribution of , then:

μμ nno ,, xx == cc nno Hh (( EE. (( zz nno || xx nno == xx )) -- Hh xx ‾‾ nno ++ xx ‾‾ nno sthe s )) == xx ·· cc nno Hh sthe s

σσ nno ,, xx 22 == cc nno Hh CovCov (( zz nno ,, zz nno || xx nno == xx )) cc nno

== cc nno Hh (( σσ ww 22 II NN ++ HVHV nno Hh Hh -- vv nno ssss Hh )) cc nno

== cc nno Hh sthe s (( 11 -- sthe s Hh cc nno ))

通过高斯分布的概率密度函数可计算得到；The probability density function of the Gaussian distribution can be calculated as ;

S6：根据 x ‾ n = Σ x ∈ β x · P ( x n = x ) , v n = Σ x ∈ β | x - x ‾ n | 2 · P ( x n = x ) , 代入 P ( x n = x ) = p ( x ^ n | x n = x ) 可以获得新的第n时刻的和v_n值，可以用于更新第n+1时刻的均衡器抽头系数；S6: According to x ‾ no = Σ x ∈ β x &Center Dot; P ( x no = x ) , v no = Σ x ∈ β | x - x ‾ no | 2 · P ( x no = x ) , substitute P ( x no = x ) = p ( x ^ no | x no = x ) can get the new nth moment And the value of v _n can be used to update the equalizer tap coefficient at the n+1th moment;

S7：对每个时刻均衡器输出的估计符号进行解调，恢复出原始的二进制比特信息序列。S7: Estimated sign of equalizer output for each time instant Demodulate to recover the original binary bit information sequence.

本发明的有益效果：通过对最小均方误差的理论知识进行推导，得到伯努利-高斯模型下脉冲噪声的最小均方误差线性均衡器的抽头系数计算方法，从而在较低复杂度的情况下获取较优的检测性能。Beneficial effect of the present invention: by deriving the theoretical knowledge of the minimum mean square error, the tap coefficient calculation method of the minimum mean square error linear equalizer of the impulse noise under the Bernoulli-Gaussian model is obtained, thereby in the case of lower complexity to obtain better detection performance.

附图说明Description of drawings

图1是本发明的消除脉冲噪声的最小均方误差线性均衡方法第n时刻的具体实施过程示意图；Fig. 1 is the specific implementation process schematic diagram of the nth moment of the minimum mean square error linear equalization method for eliminating impulse noise of the present invention;

图2是最小均方误差线性均衡器的结构模型。Fig. 2 is the structural model of the minimum mean square error linear equalizer.

具体实施方式detailed description

下面结合附图来说明本发明的具体实施方式：The specific embodiment of the present invention is described below in conjunction with accompanying drawing:

设通信系统的发射机第n时刻发送调制符号x_n，共发送了L个调制符号x＝(x₁ x₂... x_L)^T，且通信系统经过的信道有M条时延路径，第k条时延路径的衰落系数为h_k(k∈{0,1,...,M-1})。接收机第n时刻的接收信号为z_n，共接收了L个符号z＝(z₁ z₂ ... z_L)^T。Assuming that the transmitter of the communication system sends modulation symbol x _n at the nth moment, a total of L modulation symbols x=(x ₁ x ₂ ... x _L ) ^T are sent, and the channel passed by the communication system has M delay paths, The fading coefficient of the kth delay path is h _k (k∈{0,1,...,M-1}). The received signal of the receiver at the nth moment is z _n , and a total of L symbols z=(z ₁ z ₂ ... z _L ) ^T have been received.

图1是本发明消除脉冲噪声的最小均方误差线性均衡方法第n时刻的具体实施过程示意图。如图1所示，本发明消除脉冲噪声的最小均方误差线性均衡方法包括以下步骤：FIG. 1 is a schematic diagram of the specific implementation process at the nth moment of the minimum mean square error linear equalization method for eliminating impulse noise in the present invention. As shown in Figure 1, the minimum mean square error linear equalization method for eliminating impulse noise of the present invention comprises the following steps:

S1：接收机第n(n∈{1,2,...,L})时刻的接收信号z_n为S1: The received signal z _n of the receiver at the nth (n∈{1,2,...,L}) moment is

zz nno == ΣΣ kk == 00 Mm -- 11 hh kk xx nno -- kk ++ ww nno ++ bb nno ·· gg nno ,,

其中M为时延路径的总长度，h_k(k∈{0,1,...,M-1})为第k条时延路径的衰落系数，x_n(n∈{1,2,...,L})为发射机第n时刻的发送符号，并且假设x_n＝0(n≤0或n＞L)，w_n,g_n服从均值为0、方差分别为的高斯分布，b_n服从伯努利分布，且P(b_n＝1)＝p,P(b_n＝0)＝1-p,，P(·)表示括号内的事件发生的概率；where M is the total length of the delay path, h _k (k∈{0,1,...,M-1}) is the fading coefficient of the kth delay path, x _n (n∈{1,2, ...,L}) is the transmitted symbol of the transmitter at the nth moment, and assuming that x _n =0 (n≤0 or n>L), w _n , g _n obey the mean value of 0, and the variance is respectively Gaussian distribution of b _n obeys Bernoulli distribution, and P(b _n =1)=p, P(b _n =0)=1-p, P(·) represents the probability of occurrence of events in brackets;

S2：根据 x ‾ n = E ( x n ) = Σ x ∈ β x · P ( x n = x ) , v n = Σ x ∈ β | x - E ( x n ) | 2 · P ( x n = x ) , β为调制符号集合，获取每个时刻的v_n，其中，E(·)表示随机变量的数学期望；S2: According to x ‾ no = E. ( x no ) = Σ x ∈ β x · P ( x no = x ) , v no = Σ x ∈ β | x - E. ( x no ) | 2 &Center Dot; P ( x no = x ) , β is a set of modulation symbols, to obtain the v _n , where E(·) represents the mathematical expectation of the random variable;

S3：设第n时刻的最小均方误差线性均衡器的抽头系数为c_n,k,k＝-N₁,1-N₁,...,N₂，总长度为N＝N₁+N₂+1，图2为最小均方误差线性均衡器的结构模型；同时，取N个接收符号（其中(·)^T表示矩阵或向量的转置）作为最小均方误差线性均衡器的输入，并且假设z_n＝0(n≤0或n＞L)，则均衡器第n时刻输出对x_n的估计符号为S3: Let the tap coefficients of the minimum mean square error linear equalizer at the nth moment be c _n,k ,k=-N ₁ ,1-N ₁ ,...,N ₂ , and the total length is N=N ₁ +N ₂ +1, Figure 2 is the structural model of the minimum mean square error linear equalizer; at the same time, take N received symbols (where (·) ^T represents the transpose of a matrix or vector) as the input of the minimum mean square error linear equalizer, and assuming z _n =0 (n≤0 or n>L), then the equalizer outputs the pair x at the nth moment estimated sign of _n for

xx ^^ nno == EE. (( xx nno )) ++ CovCov (( xx nno ,, zz nno )) CovCov (( zz nno ,, zz nno )) -- 11 (( zz nno -- Hh xx ‾‾ nno ))

其中，Cov(x,y)表示向量x与y的协方差矩阵，即Cov(x,y)＝E(xy^H)-E(x)E(y^H)，(·)^H表示矩阵或向量的共轭转置，(·)^-1表示矩阵的求逆，Among them, Cov(x,y) represents the covariance matrix of vector x and y, that is, Cov(x,y)=E(xy ^H )-E(x)E(y ^H ), ( ) ^H represents the matrix or vector The conjugate transpose of , ( ) ^-1 means the inversion of the matrix,

xx ‾‾ nno == xx ‾‾ nno -- Mm -- NN 22 ++ 11 xx ‾‾ nno -- Mm -- NN 22 ++ 22 .. .. .. xx ‾‾ nno ++ NN 11 TT ,,

S4：为使第n时刻的均衡器输出符号独立于P(x_n＝x)，使v_n=1,同时，经过理论推导可得则第n时刻均衡器输出的估计符号变为S4: In order to make the equalizer output symbols at the nth moment Independent of P(x _n =x), such that v _n =1, at the same time, after theoretical derivation, we can get Then the estimated symbol output by the equalizer at the nth moment becomes

xx ^^ nno == xx ‾‾ nno ++ vv nno sthe s Hh [[ (( σσ ww 22 ++ pσpσ ii 22 )) II NN ++ HVHV nno Hh Hh ]] -- 11 (( zz nno -- Hh xx ‾‾ nno ))

其中， s = H 0 1 × ( N 2 + M - 1 ) 1 0 1 × N 1 T ，I_N为N×N的单位矩阵，in, the s = h 0 1 × ( N 2 + m - 1 ) 1 0 1 × N 1 T , I _N is the identity matrix of N×N,

V n = Diag v n - M - N 2 + 1 v n - M - N 2 + 2 . . . v n + N 1 , Diag(·)表示将长度为l的向量变为l×l的方阵，且向量元素位于方阵的对角线上； V no = Diag v no - m - N 2 + 1 v no - m - N 2 + 2 . . . v no + N 1 , Diag( ) means to change the vector with length l into a square matrix of l×l, and the vector elements are located on the diagonal of the square matrix;

并且，假设均衡器的抽头系数向量c_n定义为 c n = c n , N 2 * c n , N 2 - 1 * . . . c n , - N 1 * T , 则And, suppose the tap coefficient vector c _n of the equalizer is defined as c no = c no , N 2 * c no , N 2 - 1 * . . . c no , - N 1 * T , but

cc nno == [[ (( σσ ww 22 ++ pp σσ ii 22 )) II NN ++ HVHV nno Hh Hh ]] -- 11 sthe s ;;

S5：假设的概率密度函数服从均值为μ_n,x，μ_n,x定义为方差为定义为 Cov ( x ^ n , x ^ n | x n = x ) 的高斯分布，则S5: Suppose The probability density function of obeys the mean value of μ _n,x , μ _n,x is defined as Variance is defined as Cov ( x ^ no , x ^ no | x no = x ) Gaussian distribution of , then

μμ nno ,, xx == cc nno Hh (( EE. (( zz nno || xx nno == xx )) -- Hh xx ‾‾ nno ++ xx ‾‾ nno sthe s )) == xx ·· cc nno Hh sthe s

σσ nno ,, xx 22 == cc nno Hh CovCov (( zz nno ,, zz nno || xx nno == xx )) cc nno

== cc nno Hh (( σσ ww 22 II NN ++ HVHV nno Hh Hh -- vv nno ssss Hh )) cc nno

== cc nno Hh sthe s (( 11 -- sthe s Hh cc nno ))

通过高斯分布的概率密度函数可计算得到；The probability density function of the Gaussian distribution can be calculated as ;

S6：根据 x ‾ n = Σ x ∈ β x · P ( x n = x ) , v n = Σ x ∈ β | x - x ‾ n | 2 · P ( x n = x ) , 代入 P ( x n = x ) = p ( x ^ n | x n = x ) 可以获得新的第n时刻的和v_n值，可以用于更新第n+1时刻的均衡器抽头系数；S6: According to x ‾ no = Σ x ∈ β x &Center Dot; P ( x no = x ) , v no = Σ x ∈ β | x - x ‾ no | 2 &Center Dot; P ( x no = x ) , substitute P ( x no = x ) = p ( x ^ no | x no = x ) can get the new nth moment And the value of v _n can be used to update the equalizer tap coefficient at the n+1th moment;

S7：对每个时刻均衡器输出的估计符号进行解调，恢复出原始的二进制比特信息序列。S7: Estimated sign of equalizer output for each time instant Demodulate to recover the original binary bit information sequence.

1. A minimum mean square error linear equalization method for eliminating impulse noise is characterized in that: the steps are as follows:

s1: when n is greater than 0, the received signal at the nth moment of the receiver is:

where M is the total length of the delay path, h_kIs the fading coefficient, x, of the kth delay path_nFor the transmitted symbol at the nth time of the transmitter, when n is 0Then x_n＝0，w_nObedience mean of 0 and variance ofGaussian distribution of g_nObedience mean of 0 and variance ofGaussian distribution of b_nObey Bernoulli distribution, and P (b)_n＝1)＝p,P(b_n0 — 1-P, P (·) indicates the probability of occurrence of an event in parentheses, k ∈ {0,1, …, M-1 };

s2: according to the mean valueVariance (variance) For a set of modulation symbols, obtaining for each time instantWherein E (-) represents the mathematical expectation of the random variable;

s3: let the tap coefficient of the linear equalizer with minimum mean square error at the nth time be c_n,k,k＝-N₁,1-N₁,…,N₂Total length N ═ N₁+N₂+1, and N received symbolsWherein, (.)^TRepresenting the transpose of a matrix or vector as input to a minimum mean square error linear equalizer, when n is less than or equal to 0, then z is_nWhen equal to 0, the equalizer outputs the pair x at the nth time_nIs estimated from the symbolsComprises the following steps:wherein Cov (x, y) represents a covariance matrix of vectors x and y, i.e., Cov (x, y) is E (xy)^H)-E(x)E(y^H)，(·)^HRepresenting a conjugate transpose of a matrix or vector, (-)^-1The inverse of the representation matrix is then used,

s4: for the equalizer output symbol at the nth timeIndependent of P (x)_nX), makingv_n1, the estimated symbol output by the equalizer at the nth timeThe following steps are changed:

wherein,I_Nis an identity matrix of N × N,

diag (-) denotes changing a vector of length l to a square matrix of l × l with the vector elements on the diagonal of the square matrix, and assuming that the equalizer tap coefficient vector isThen

S5: suppose thatObeys a mean value of mu for the probability density function_n,x，μ_n,xIs defined asVariance ofIs defined asThe gaussian distribution of (a), then:

the probability density function of the Gaussian distribution can be calculated

S6: according toSubstitution intoA new nth moment can be obtainedAnd v_nA value that may be used to update the equalizer tap coefficients at time n + 1;

s7: estimated symbol output by equalizer for each timeAnd demodulating to recover the original binary bit information sequence.