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

Abstract

This bright genus mobile communication technology field, is specifically related to a kind of minimum Mean Square Error Linear equalization methods eliminating impulsive noise.The present invention, by deriving the theoretical knowledge of least mean-square error, obtains the tap coefficient computational methods of the minimum Mean Square Error Linear equalizer of impulsive noise under Bernoulli Jacob-Gauss model, thus obtains in the case of lower complexity and preferably detect performance.

Description

Minimum mean square error linear equalization method for eliminating impulse noise

Technical Field

The invention belongs to the technical field of mobile communication, and particularly relates to a minimum mean square error linear equalization method for eliminating impulse noise.

Background

Impulse noise is discontinuous and consists of irregular pulses or noise spikes of short duration and large amplitude. Impulsive noise is intermittent noise with a duration less than 1 second and a peak value at least 10dB greater than the root mean square value and a repetition rate less than 10 Hz. The impulse noise is generated from various reasons, including electromagnetic interference and malfunction and defect of the communication system, and may be generated when the electrical switches and relays of the communication system change states. In digital data communication, impulse noise is a major cause of errors. One commonly used model of impulse noise is the bernoulli-gaussian model.

Minimum mean-square error (MMSE) Linear Equalization (LE) is a method of minimizing the mean-square error between the estimated and transmitted symbols output by an equalizer. MMSE equalization considers the factors of signal-to-noise ratio, realizes a balance between eliminating Intersymbol Interference (ISI) and not amplifying noise, has low complexity and is a better equalization algorithm.

Disclosure of Invention

For convenience of describing the contents of the present invention, terms used in the present invention will be first explained:

bernoulli-gaussian model a commonly used model of impulse noise, the random variable under which can be expressed as η ═ w + b · g, where w, g obey a mean of 0 and variances of 0, respectivelyB obeys a bernoulli distribution, and P (b-1) P, P (b-0) 1-P, P (·) indicates the probability of occurrence of an event in parentheses.

Mathematical expectation: the statistical term, which reflects the average value of the random variables, is also called expectation or mean, and is denoted as E (-).

Variance: the statistical term measures the degree of deviation between a random variable and its mathematical expectation, denoted as D (-).

Prior probability of a bit: the probability that a bit is 0 or 1 is generally obtained from past experience and analysis.

Gaussian distribution: also called normal distribution, is a very important probability distribution in the fields of mathematics, physics, engineering and the like, if the random variable X obeys the mean value of mu and the variance of sigma²Is expressed as X and is defined as N (mu, sigma)²) And has a probability density function of $f\left(x\right)=\frac{1}{\sqrt{2\mathrm{\π}}\mathrm{\σ}}\exp (-\frac{{(x-\mathrm{\μ})}^2}{2{\mathrm{\σ}}^2}).$

The invention provides a minimum mean square error linear equalization method for eliminating impulse noise, which is used for improving the detection performance of a system receiver and eliminating the impulse noise in a communication system.

In order to achieve the purpose, the technical scheme of the invention is as follows:

a minimum mean square error linear equalization method for eliminating impulse noise comprises the following steps:

s1: the received signal at the nth (n > 0) time of the receiver is:

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_nN > 0 is the transmitted symbol at time n of the transmitter, and x is assumed_n＝0,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;

s2: according to the mean value ${\stackrel{\‾}{x}}_n=E\left(x_n\right)=\underset{x\∈\mathrm{\β}}{\mathrm{\Σ}}x\·P(x_n=x),$ Variance (variance) $v_n=\underset{x\∈\mathrm{\β}}{\mathrm{\Σ}}{|x-E\left(x_n\right)|}^2\·P(x_n=x),$ β for modulation symbol set, obtaining each timev_nWherein 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 a transpose of a matrix or vector) as an input to a minimum mean square error linear equalizer, and z is assumed to be_nWhen n is less than or equal to 0, the equalizer outputs x at the nth time_nIs estimated from the symbolsComprises the following steps: ${\hat{x}}_n=E\left(x_n\right)+\mathrm{Cov}(x_n,z_n)\mathrm{Cov}{(z_n,z_n)}^{-1}(z_n-H{\stackrel{\‾}{x}}_n),$ 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, ${\stackrel{\‾}{x}}_n={\left[\begin{array}{cccc}{\stackrel{\‾}{x}}_{n-M-N_2+1}& {\stackrel{\‾}{x}}_{n-M-N_2+2}& ...& {\stackrel{\‾}{x}}_{n+N_1}\end{array}\right]}^T,$

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:

${\hat{x}}_n={\stackrel{\‾}{x}}_n+v_n{s}^H{[({\mathrm{\σ}}_w^2+{\mathrm{p\σ}}_i^2)I_N+{\mathrm{HV}}_n{H}^H]}^{-1}(z_n-H{\stackrel{\‾}{x}}_n),$

wherein, $s=H{\left[\begin{array}{ccc}{0}_{1\×(N_2+M-1)}& 1& 0_{1\×N_1}\end{array}\right]}^T,$ I_Nis an identity matrix of N × N,

$V_n=\mathrm{Diag}\left(\begin{array}{cccc}{v}_{n-M-N_2+1}& v_{n-M-N_2+2}& ...& v_{n+N_1}\end{array}\right),$ 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 is $c_n={\left[\begin{array}{cccc}{c}_{n,N_2}^{*}& c_{n,N_2-1}^{*}& ...& c_{n,-N_1}^{*}\end{array}\right]}^T,$ Then

$c_n={[({\mathrm{\σ}}_w^2+p{\mathrm{\σ}}_i^2)I_N+{\mathrm{HV}}_n{H}^H]}^{-1}s;$

S5: suppose thatObeys a mean value of mu for the probability density function_n,x，μ_n,xIs defined asVariance ofIs defined as $\mathrm{Cov}({\hat{x}}_n,{\hat{x}}_n|x_n=x)$ The gaussian distribution of (a), then:

${\mathrm{\μ}}_{n,x}=c_n^H(E\left(z_n\right|x_n=x)-{H\stackrel{\‾}{x}}_n+{\stackrel{\‾}{x}}_{n}s)=x\·c_n^{H}s$

${\mathrm{\σ}}_{n,x}^2=c_n^H\mathrm{Cov}(z_n,z_n|x_n=x)c_n$

$=c_n^H({\mathrm{\σ}}_w^2{I}_N+{\mathrm{HV}}_n{H}^H-v_n{\mathrm{ss}}^H)c_n$

$=c_n^{H}s(1-s^H{c}_n)$

the probability density function of the Gaussian distribution can be calculated；

S6: according to ${\stackrel{\‾}{x}}_n=\underset{x\∈\mathrm{\β}}{\mathrm{\Σ}}x\·P(x_n=x),$ $v_n=\underset{x\∈\mathrm{\β}}{\mathrm{\Σ}}{|x-{\stackrel{\‾}{x}}_n|}^2\·P(x_n=x),$ Substitution into $P(x_n=x)=p\left({\hat{x}}_n\right|x_n=x)$ A 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.

The invention has the beneficial effects that: the tap coefficient calculation method of the minimum mean square error linear equalizer of the impulse noise under the Bernoulli-Gaussian model is obtained by deducing the theoretical knowledge of the minimum mean square error, so that the better detection performance is obtained under the condition of lower complexity.

Drawings

FIG. 1 is a schematic diagram of an implementation process of the minimum mean square error linear equalization method for eliminating impulse noise at time n according to the present invention;

figure 2 is a model of the structure of a minimum mean square error linear equalizer.

Detailed Description

The following description of the embodiments of the invention refers to the accompanying drawings:

setting the nth time of transmitter in communication system to transmit modulation symbol x_nL modulation symbols x ═ x (x) are transmitted in total₁x₂... x_L)^TAnd the channel passed by the communication system has M time delay paths, and the fading coefficient of the k time delay path is h_k(k ∈ {0, 1.., M-1}) the received signal at time n of the receiver is z_nA total of L symbols z ═ (z) is received₁z₂... z_L)^T。

Fig. 1 is a schematic diagram of a specific implementation process of the present invention at the nth time of the minimum mean square error linear equalization method for eliminating impulse noise. As shown in fig. 1, the minimum mean square error linear equalization method for eliminating impulse noise of the present invention includes the following steps:

s1, the received signal z of the receiver at the nth (n ∈ {1, 2.. gtoreq.L }) time_nIs composed of

$z_n=\underset{k=0}{\overset{M-1}{\mathrm{\Σ}}}{h}_k{x}_{n-k}+w_n+b_n\·g_n,$

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.. multidata., L }) is the transmission symbol at time instant n of the transmitter, and x is assumed to be_n0 (n.ltoreq.0 or n > L), w_n,g_nObedience mean value 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;

s2: according to ${\stackrel{\‾}{x}}_n=E\left(x_n\right)=\underset{x\∈\mathrm{\β}}{\mathrm{\Σ}}x\·P(x_n=x),$ $v_n=\underset{x\∈\mathrm{\β}}{\mathrm{\Σ}}{|x-E\left(x_n\right)|}^2\·P(x_n=x),$ β for modulation symbol set, obtaining each timev_nWherein E (-) represents the mathematical expectation of the random variable;

s3: let c be the tap coefficient of the linear equalizer with minimum mean square error at the nth time_n,k,k＝-N₁,1-N₁,...,N₂Total length N ═ N₁+N₂+1, fig. 2 is a structural model of the minimum mean square error linear equalizer; at the same time, N received symbols are taken(wherein (·)^TRepresenting a transpose of a matrix or vector) as an input to a minimum mean square error linear equalizer, and z is assumed to be_n0(n ≦ 0 or n > L), the equalizer outputs the pair x at the nth time instant_nIs estimated from the symbolsIs composed of

${\hat{x}}_n=E\left(x_n\right)+\mathrm{Cov}(x_n,z_n)\mathrm{Cov}{(z_n,z_n)}^{-1}(z_n-H{\stackrel{\‾}{x}}_n)$

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,

${\stackrel{\‾}{x}}_n={\left[\begin{array}{cccc}{\stackrel{\‾}{x}}_{n-M-N_2+1}& {\stackrel{\‾}{x}}_{n-M-N_2+2}& ...& {\stackrel{\‾}{x}}_{n+N_1}\end{array}\right]}^T,$

s4: for the equalizer output symbol at the nth timeIndependent of P (x)_nX), makingv_n=1, and theoretical derivation is possibleThe estimated symbol output by the equalizer at time nBecome into

${\hat{x}}_n={\stackrel{\‾}{x}}_n+v_n{s}^H{[({\mathrm{\σ}}_w^2+{\mathrm{p\σ}}_i^2)I_N+{\mathrm{HV}}_n{H}^H]}^{-1}(z_n-H{\stackrel{\‾}{x}}_n)$

Wherein, $s=H{\left[\begin{array}{ccc}{0}_{1\×(N_2+M-1)}& 1& 0_{1\×N_1}\end{array}\right]}^T$ ，I_Nis an identity matrix of N × N,

$V_n=\mathrm{Diag}\left(\begin{array}{cccc}{v}_{n-M-N_2+1}& v_{n-M-N_2+2}& ...& v_{n+N_1}\end{array}\right),$ diag (-) denotes changing a vector of length l to a square matrix of l × l with the vector elements lying on the diagonal of the square matrix;

and, assume the tap coefficient vector c of the equalizer_nIs defined as $c_n={\left[\begin{array}{cccc}{c}_{n,N_2}^{*}& c_{n,N_2-1}^{*}& ...& c_{n,-N_1}^{*}\end{array}\right]}^T,$ Then

$c_n={[({\mathrm{\σ}}_w^2+p{\mathrm{\σ}}_i^2)I_N+{\mathrm{HV}}_n{H}^H]}^{-1}s;$

S5: suppose thatObeys a mean value of mu for the probability density function_n,x，μ_n,xIs defined asVariance ofIs defined as $\mathrm{Cov}({\hat{x}}_n,{\hat{x}}_n|x_n=x)$ A Gaussian distribution of

${\mathrm{\μ}}_{n,x}=c_n^H(E\left(z_n\right|x_n=x)-{H\stackrel{\‾}{x}}_n+{\stackrel{\‾}{x}}_{n}s)=x\·c_n^{H}s$

${\mathrm{\σ}}_{n,x}^2=c_n^H\mathrm{Cov}(z_n,z_n|x_n=x)c_n$

$=c_n^H({\mathrm{\σ}}_w^2{I}_N+{\mathrm{HV}}_n{H}^H-v_n{\mathrm{ss}}^H)c_n$

$=c_n^{H}s(1-s^H{c}_n)$

The probability density function of the Gaussian distribution can be calculated；

S6: according to ${\stackrel{\‾}{x}}_n=\underset{x\∈\mathrm{\β}}{\mathrm{\Σ}}x\·P(x_n=x),$ $v_n=\underset{x\∈\mathrm{\β}}{\mathrm{\Σ}}{|x-{\stackrel{\‾}{x}}_n|}^2\·P(x_n=x),$ Substitution into $P(x_n=x)=p\left({\hat{x}}_n\right|x_n=x)$ A new nth moment can be obtainedAnd v_nValue, which can be used to update the equalization at time n +1Tap coefficients of the tap;

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

Claims (1)

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.