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

Abstract

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.

Description

A Minimum Mean Square Error Linear Equalization Method for Eliminating Impulse Noise

technical field

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.

Background technique

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.

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.

Contents of the invention

In order to describe content of the present invention conveniently, at first the terms used in the present invention are explained:

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.

Mathematical expectation: A statistical term that reflects the average value of a random variable, also known as expectation or mean, denoted as E(·).

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

The prior probability of a bit: the probability that a bit is 0 or 1, generally based on previous experience and analysis.

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 $f\left(x\right)=\frac{1}{\sqrt{2\mathrm{\π}}\mathrm{\σ}}\exp (-\frac{{(x-\mathrm{\μ})}^2}{2{\mathrm{\σ}}^2}).$

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

For realizing the above-mentioned purpose of the invention, the technical scheme of the present invention is as follows:

A minimum mean square error linear equalization method for eliminating impulse noise, comprising the steps of:

S1: The received signal of the receiver at the nth (n>0) moment is:

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: According to the mean ${\stackrel{\‾}{x}}_{\mathrm{no}}=\mathrm{E.}\left(x_{\mathrm{no}}\right)=\underset{x\∈\mathrm{\β}}{\mathrm{\Σ}}x\&Center\; Dot;P(x_{\mathrm{no}}=x),$ variance $v_{\mathrm{no}}=\underset{x\∈\mathrm{\β}}{\mathrm{\Σ}}{|x-\mathrm{E.}\left(x_{\mathrm{no}}\right)|}^2\&Center\; Dot;P(x_{\mathrm{no}}=x),$ β is a set of modulation symbols, to obtain the v _n , where E(·) represents the mathematical expectation of the random variable;

S3: 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: ${\hat{x}}_{\mathrm{no}}=\mathrm{E.}\left(x_{\mathrm{no}}\right)+\mathrm{Cov}(x_{\mathrm{no}},z_{\mathrm{no}})\mathrm{Cov}{(z_{\mathrm{no}},z_{\mathrm{no}})}^{-1}(z_{\mathrm{no}}-h{\stackrel{\‾}{x}}_{\mathrm{no}}),$ 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, ${\stackrel{\‾}{x}}_{\mathrm{no}}={\left[\begin{array}{cccc}{\stackrel{\‾}{x}}_{\mathrm{no}-m-N_2+1}& {\stackrel{\‾}{x}}_{\mathrm{no}-m-N_2+2}& ...& {\stackrel{\‾}{x}}_{\mathrm{no}+N_1}\end{array}\right]}^T,$

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:

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}{<span\; class="notranslate">\; [</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; p\&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; HV</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; ]</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; h</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

in, $\mathrm{the\; s}=h{\left[\begin{array}{ccc}{0}_{1\&times;(N_2+m-1)}& 1& 0_{1\&times;N_1}\end{array}\right]}^T,$ I _N is the identity matrix of N×N,

$V_{\mathrm{no}}=\mathrm{Diag}\left(\begin{array}{cccc}{v}_{\mathrm{no}-m-N_2+1}& v_{\mathrm{no}-m-N_2+2}& ...& v_{\mathrm{no}+N_1}\end{array}\right),$ 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 $c_{\mathrm{no}}={\left[\begin{array}{cccc}{c}_{\mathrm{no},N_2}^{*}& c_{\mathrm{no},N_2-1}^{*}& ...& c_{\mathrm{no},-N_1}^{*}\end{array}\right]}^T,$ but

${\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; [</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; p</span>}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; HV</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; ]</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; ;</span>$

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 $\mathrm{Cov}({\hat{x}}_{\mathrm{no}},{\hat{x}}_{\mathrm{no}}|x_{\mathrm{no}}=x)$ Gaussian distribution of , then:

${\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; h</span>}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; +</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}$

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; Cov</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}$

$<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; HV</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; ss</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}$

$<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span>$

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

S6: According to ${\stackrel{\&OverBar;}{x}}_{\mathrm{no}}=\underset{x\&Element;\mathrm{\&beta;}}{\mathrm{\&Sigma;}}x\&Center\; Dot;P(x_{\mathrm{no}}=x),$ $v_{\mathrm{no}}=\underset{x\&Element;\mathrm{\&beta;}}{\mathrm{\&Sigma;}}{|x-{\stackrel{\&OverBar;}{x}}_{\mathrm{no}}|}^2\&CenterDot;P(x_{\mathrm{no}}=x),$ substitute $P(x_{\mathrm{no}}=x)=p\left({\hat{x}}_{\mathrm{no}}\right|x_{\mathrm{no}}=x)$ 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: Estimated sign of equalizer output for each time instant Demodulate to recover the original binary bit information sequence.

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.

Description of drawings

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;

Fig. 2 is the structural model of the minimum mean square error linear equalizer.

detailed description

The specific embodiment of the present invention is described below in conjunction with accompanying drawing:

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.

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: The received signal z _n of the receiver at the nth (n∈{1,2,...,L}) moment is

${\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; ,</span>$

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: According to ${\stackrel{\&OverBar;}{x}}_{\mathrm{no}}=\mathrm{E.}\left(x_{\mathrm{no}}\right)=\underset{x\&Element;\mathrm{\&beta;}}{\mathrm{\&Sigma;}}x\&CenterDot;P(x_{\mathrm{no}}=x),$ $v_{\mathrm{no}}=\underset{x\&Element;\mathrm{\&beta;}}{\mathrm{\&Sigma;}}{|x-\mathrm{E.}\left(x_{\mathrm{no}}\right)|}^2\&Center\; Dot;P(x_{\mathrm{no}}=x),$ β is a set of modulation symbols, to obtain the v _n , where E(·) represents the mathematical expectation of the random variable;

S3: 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

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; Cov</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; Cov</span>}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; h</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span>$

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,

${\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>{\left[\begin{array}{cccc}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}& {\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}}& <span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span>& {\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}\end{array}\right]}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; ,</span>$

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

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}{<span\; class="notranslate">\; [</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; p\&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; HV</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; ]</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; h</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span>$

in, $\mathrm{the\; s}=h{\left[\begin{array}{ccc}{0}_{1\&times;(N_2+m-1)}& 1& 0_{1\&times;N_1}\end{array}\right]}^T$ , I _N is the identity matrix of N×N,

$V_{\mathrm{no}}=\mathrm{Diag}\left(\begin{array}{cccc}{v}_{\mathrm{no}-m-N_2+1}& v_{\mathrm{no}-m-N_2+2}& ...& v_{\mathrm{no}+N_1}\end{array}\right),$ 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, suppose the tap coefficient vector c _n of the equalizer is defined as $c_{\mathrm{no}}={\left[\begin{array}{cccc}{c}_{\mathrm{no},N_2}^{*}& c_{\mathrm{no},N_2-1}^{*}& ...& c_{\mathrm{no},-N_1}^{*}\end{array}\right]}^T,$ but

${\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; [</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; p</span>}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; HV</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; ]</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; ;</span>$

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 $\mathrm{Cov}({\hat{x}}_{\mathrm{no}},{\hat{x}}_{\mathrm{no}}|x_{\mathrm{no}}=x)$ Gaussian distribution of , then

${\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; h</span>}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; +</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}$

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; Cov</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}$

$<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; HV</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; ss</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}$

$<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span>$

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

S6: According to ${\stackrel{\&OverBar;}{x}}_{\mathrm{no}}=\underset{x\&Element;\mathrm{\&beta;}}{\mathrm{\&Sigma;}}x\&Center\; Dot;P(x_{\mathrm{no}}=x),$ $v_{\mathrm{no}}=\underset{x\&Element;\mathrm{\&beta;}}{\mathrm{\&Sigma;}}{|x-{\stackrel{\&OverBar;}{x}}_{\mathrm{no}}|}^2\&Center\; Dot;P(x_{\mathrm{no}}=x),$ substitute $P(x_{\mathrm{no}}=x)=p\left({\hat{x}}_{\mathrm{no}}\right|x_{\mathrm{no}}=x)$ 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: Estimated sign of equalizer output for each time instant Demodulate 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.