CN102122972A - Transform-domain-based narrowband interference inhibiting method in shortwave spread spectrum communication - Google Patents

Abstract

The invention relates to a transform-domain-based narrowband interference inhibiting method in shortwave spread spectrum communication. The method comprises the following steps of performing chip matching filtering, sampling, wavelet packet transformation and filtering on a spread spectrum signal with narrowband interference in shortwave communication; then limiting an amplitude of the transform domain, and performing wavelet packet inverse transformation and lease mean square (LMS) filtering on the signal which is subjected to amplitude limiting; and judging the filtered data to obtain an estimated transmission symbol. By the method, the algorithm offset error is reduced, the algorithm has lower complexity, and the narrowband interference can be effectively located in a certain frequency domain range; and the located 'polluted' frequency range is removed, and then the signal which is subjected to interference inhibition-recovery is synthesized by a wavelet packet. The narrowband interference inhibiting effect is improved in terms of real time property and accuracy.

Description

In the shortwave spread spectrum communication based on the narrow-band interference rejection method of transform domain

Technical field

The invention belongs to the ICT (information and communication technology) field, in particularly a kind of shortwave spread spectrum communication based on the narrow-band interference rejection method of transform domain.

Background technology

Spread spectrum communication system has inherent antijamming capability, but when the power of interference during much larger than the power of signal, such as being subjected to when signal declining significantly or when being subjected to artificial disturbance, for the communication quality that improves system must be by means of other signal processing measure.Since DSSS (direct sequence spread spectrum) technology begins to use, just the begin one's study Anti-Jamming Technique of DSSS system of people, and a lot of effectively disturbance restraining methods have been proposed, these technology generally all are to adopt different signal processing methods interference to be predicted or interference is positioned before despreading, thereby interference are eliminated again.Interference mitigation technology generally can be divided into time domain interference mitigation technology, transform domain interference mitigation technology, based on several classes such as interference mitigation technology of time frequency analysis.

The invention belongs to transform domain interference mitigation technology category, the transform domain interference mitigation technology is corresponding with the time domain interference mitigation technology.The transform domain here is meant orthogonal transform domain, such as Fourier transform, sine transform, cosine transform.The interference inhibiting structure of transform domain is before disturbing inhibition and done orthogonal transform and inverse transformation afterwards respectively, and therefore the implementation complexity of this method disturbs the method height that suppresses than time domain.The transform domain Suppression of narrow band interference is to utilize wideband spread-spectrum signal and narrow band interference at the different qualities of transform domain interference to be rejected, such as at frequency domain, the energy of spread-spectrum signal and noise can be distributed in the whole frequency band, and narrow band interference only is present in a certain section narrow frequency band, and when interference strength is bigger, tangible peak value can appear in this narrower frequency band, this has just made things convenient for the processing to disturbing, narrow band interference for dash area, be easy to it is detected, and with the method for thresholding filtering it is eliminated, this algorithm does not need adaptive process, and is therefore good than the Time-Domain algorithm real-time based on linear prediction.

Yet owing to carry out windowing before the conversion, and data truncation, narrow band interference has certain spectrum energy at transform domain and leaks, and causes the contaminated frequency range of signal to increase.In the research at home and abroad, in order to reduce the leakage of spectrum energy, traditional way is to add the non-rectangle window to the data of blocking, and such as for Hamming window etc., has so just reduced the leakage of interfering energy greatly.But the windowing meeting brings the another one problem, and promptly the energy of useful signal is lost.

What at present, be subjected to that domestic and international correlative study mechanism pays close attention to is: the adaptive-filtering of transform domain can further reduce the destruction to signal, and this is because this technology adopts least-mean-square-error criterion, can make the distortion of signal reach minimum.On the other hand, when signal carries out orthogonal transform, be equivalent to signal has been carried out decorrelation computing to a certain degree, angle from filter, originally the single channel transversal filter in time domain has become multi-channel filter of equal value, the input signal of former time domain then is equivalent to and has passed through a bank of filters, therefore compares with time domain, and it is fast that the convergence rate of transform domain adaptive algorithm is wanted.

In numerous adaptive algorithms, LMS (Lease Mean Square) algorithm is favored owing to computation complexity is little, but LMS convergence of algorithm speed is slow, the RLS algorithm is the quickish algorithm of a kind of convergence rate, but its computation complexity is too high again, be an implacable contradiction between the computation complexity of adaptive algorithm and the convergence rate in a word, we need propose the technology of the present invention according to actual conditions.

Summary of the invention

The object of the present invention is to provide in a kind of shortwave spread spectrum communication narrow-band interference rejection method, reduce the algorithm offset error, and make it have lower complexity based on transform domain.

The technical scheme that realizes the object of the invention is: in a kind of shortwave spread spectrum communication based on the narrow-band interference rejection method of transform domain, after the spread-spectrum signal that has narrow band interference in the short wave communication carries out chip matched filtering, sampling, wavelet package transforms, filtering, carry out the transform domain amplitude limit, carry out the wavelet packet inverse transformation again and LMS filtering gets for the signal behind the amplitude limit

$\hat{d}\left(m\right)=w^H\tilde{R}\left(m\right)$

$e\left(m\right)=d\left(m\right)-\hat{d}\left(m\right)$

$w(m+1)=w\left(m\right)+\mathrm{\μ}\tilde{R}\left(m\right)e^{*}\left(m\right)$

For filtered data adjudicate again the transmission symbol estimated

The present invention compared with prior art, its remarkable advantage is: at first can be when guaranteeing algorithm the convergence speed, reduce the algorithm offset error, and make it have lower complexity, and narrow band interference can be positioned at effectively certain frequency domain scope, ' pollution ' frequency range that is subjected to that will locate is then removed, then by the synthetic signal that recovers to disturb after suppressing of wavelet packet, aspect real-time and accuracy, improve the Suppression of narrow band interference effect.A kind of improvement algorithm is promptly proposed on the basis of transform domain Suppression of narrow band interference LMS algorithm, it is the AL_TRLMS algorithm, this algorithm is to have increased a kind of transform domain amplitude limit process before former algorithm carries out adaptive-filtering, imbalance and the characteristic value diffusance of improving algorithm have obtained reduction, and this improvement algorithm has obtained and other two kinds of algorithms with lower computation complexity: based on from the SO_TRLMS of orthogonalized transform domain algorithm, upgrade the TDVSS algorithm of considering error, identical convergence rate with step-length; And in the processing procedure of transform domain, handle (TDP) Suppression of narrow band interference technology based on the adaptive transformation territory that wavelet packet changes, narrow band interference can be positioned at effectively certain frequency domain scope, ' pollution ' frequency range that is subjected to that to locate is then removed, then by the synthetic signal that recovers to disturb after suppressing of wavelet packet.Aspect real-time and accuracy, improve the Suppression of narrow band interference effect.In a word, imbalance of improving one's methods and characteristic value diffusance have obtained reduction, and have obtained and other two kinds of convergence rates that algorithm is identical with lower computation complexity.

Below in conjunction with accompanying drawing the present invention is described in further detail.

Description of drawings

Fig. 1 is based on the receiver structure figure of wavelet package transforms and AL_TRLMS algorithm.

The WAVELET PACKET DECOMPOSITION schematic diagram of Fig. 2 signal.

Fig. 3 improves algorithm to compare analogous diagram with the degree of algorithm imbalance in the past.

Fig. 4 improves algorithm and the analogous diagram of algorithm characteristics value diffusance comparison in the past.

Fig. 5 is the ber curve of AL_TRLMS algorithm.

Embodiment

The spreading code that the utilization of DSSS system has a good correlation is in the frequency band of the energy dissipation to of signal of communication than broad, with spreading code signal energy is put together again at receiving terminal, correlation properties according to spreading code, very important characteristics of spread-spectrum signal frequency spectrum be exactly it in a very wide frequency band, be smooth, when in the Direct Sequence Spread Spectrum Signal strong jamming being arranged, can very clearly distinguish at transform domain.But in the specific implementation process, can run into the contradiction of energy loss in the conversion, algorithm real-time and accuracy, provide following solution for this reason, in the shortwave spread spectrum communication based on the narrow-band interference rejection method of transform domain, after the spread-spectrum signal that has narrow band interference in the short wave communication carries out chip matched filtering, sampling, wavelet package transforms, filtering, carry out the transform domain amplitude limit, carry out the wavelet packet inverse transformation again and LMS filtering gets for the signal behind the amplitude limit, for filtered data adjudicate again the transmission symbol estimated

When signal is carried out the territory conversion, the brachymemma of data is that inevitably the data brachymemma is equivalent to signal multiplies each other in time domain and a rectangular window, is exactly the convolution of corresponding transform domain spectrum on transform domain, can cause the spectrum diffusion of transform domain like this, thereby increase the weight of damage useful signal.Add the secondary lobe that the non-rectangle window can reduce window function to a certain extent, thereby reduce the frequency spectrum diffusion of interference signal, but windowing still can make the portion of energy of useful signal be lost, for this reason, the present invention adopts a kind of new real-time self adaptation TDP Suppression of narrow band interference technology based on wavelet package transforms that is used for the DSSS system.It has utilized power spectral density (PSD) characteristic distributions and energy accumulating degree (ECM) criterion of spread-spectrum signal, earlier narrow band interference is positioned at certain frequency domain scope effectively by WAVELET PACKET DECOMPOSITION, ' pollution ' frequency range that is subjected to that to locate is then removed, at last by the synthetic signal that recovers to disturb after suppressing of wavelet packet.

The wavelet package transforms technology of the present invention's utilization mainly is based on following thought: the spread-spectrum signal frequency spectrum is smooth in a very wide frequency band, the power spectral density (PSD) that is PN sign indicating number sequence is a relatively flat, and narrow band interference demonstrates the spike characteristic in a certain local frequency range.Therefore utilize this character and energy accumulating degree criterion (ECM) just useful signal might be separated with being subjected to the interference signals component.Good time-frequency locality and multiresolution analysis characteristic that wavelet package transforms had make it can become input signal at any time and adjust corresponding time frequency resolution flexibly, promptly choose the set with Signal Matching adaptively from wavelet packet functions.Therefore, this system at first estimates the position of interference by PSD and ECM.In WPT (wavelet transform) analytic process of the present recurrence of disturbing of estimated body, WPT is as the effective tool that separates useful signal and interference signal rapidly here.

In this programme, WPT mainly depends on bandwidth, position and its intensity of NBI (Suppression of narrow band interference).The process of wavelet package transforms is exactly rapid tracking and the accurate location process of NBI.Owing to its generation that is presented as tree (binary tree) on the data structure that is implemented in, so we are referred to as the tree structure algorithm (tree structurealgorithms-TSA) [5] based on WPT.On the other hand, the QMF of self adaptation WPT noise killer correspondence has good amplitude-frequency characteristic (referring to Fig. 1), so it can remove disturbed frequency band more up hill and dale.By top method narrow band interference reasonably is positioned at some or several frequency bands, especially when disturbing two frequency bands that just in time are in the QMF overlapping, this two frequency band can be continued to decompose to certain degree of depth, then when wavelet packet is comprehensive with adjacent two or more sub-bands of correspondence reject together.Multiresolution analysis advantage that so-called wavelet transformation has that Here it is.When the collection of functions coupling of the frequency spectrum of input signal and wavelet packet, its corresponding energy accumulating degree will sharply increase.

Therefore the present invention has utilized power spectral density (PSD) characteristic distributions and energy accumulating degree (ECM) criterion of spread-spectrum signal, earlier narrow band interference is positioned at certain frequency domain scope effectively by WAVELET PACKET DECOMPOSITION, ' pollution ' frequency range that is subjected to that to locate is then removed, at last by the synthetic signal that recovers to disturb after suppressing of wavelet packet.

In addition, in the processing procedure of transform domain, in the transform domain obvious characteristics, can eliminate the peak value that disturbs earlier, and then carry out adaptive-filtering, can expect that like this convergence of algorithm speed obtains accelerating according to narrow band interference.The adaptive filter algorithm of transform domain is criterion with MMSE, to being subjected to interference signals reparation, because it can make the extent of damage of signal reach minimum by the self adaptation computing.But this method needs convergence process, so real-time decreases, when especially using the LMS adaptive algorithm.In order to improve LMS convergence of algorithm performance, document [96] has been introduced a kind of based on from orthogonalized transform domain LMS algorithm, i.e. SO_TRLMS, and this algorithm is by carrying out having accelerated LMS convergence of algorithm speed from the quadrature adjustment to transform-domain signals.In the algorithm research before, there is the people that the error of filter output is considered to propose a kind of TDVSS algorithm, thereby further improved convergence of algorithm speed in the renewal process of step-length.The present invention has increased a transform domain amplitude limit process on the basis of TRLMS algorithm, propose a kind of AL (Amplitude Limiting) _ TRLMS algorithm.

Comprehensive wavelet package transforms and based on two kinds of technology of TRLMS algorithm of amplitude limit (Amplitude Limiting), we provide structure such as Fig. 1 of following receiving terminal.

Wherein, the Suppression of narrow band interference tree structure algorithm based on the adaptive wavelet packet transform comprises three steps.It is described below:

(1) WAVELET PACKET DECOMPOSITION process

1. calculate the variance δ of input signal _In, supposing the system is with the chip speed sampling of PN sign indicating number sequence here;

2. carry out the decomposition that WPT (as Fig. 2) obtains two bank of filters, see formula 1, the signal variance δ of the corresponding sequence of two child nodes about calculating then _(l)With δ _(r), so just can obtain its energy accumulating degree value G.

$\left\{\begin{array}{c}{a}_l^{2n}\left(i\right)={\mathrm{\Σ}}_{k\∈Z}h(k-2i)a_{l+1}^n\left(k\right)\\ a_l^{2n+1}\left(i\right)={\mathrm{\Σ}}_{k\∈Z}g(k-2i)a_{l+1}^n\left(k\right)\end{array}\right.---\left(1\right)$

$a_{l+1}^n\left(i\right)={\mathrm{\Σ}}_{k\∈Z}h(i-2k)a_l^{2n}\left(k\right)+{\mathrm{\Σ}}_{k\∈Z}g(i-2k)a_l^{2n+1}\left(k\right)---\left(2\right)$

The ECM of each node of tree has delineated its uniformity, and it is defined as: w _h

σ wherein _p, σ _lAnd σ _rBe respectively father node, the variance of left child node and right child node signal component,

${\mathrm{\σ}}^2=\frac{1}{\mathrm{\π}}{\∫}_{w_i}^{w_h}{P}_{\mathrm{xx}}\left(w\right)\mathrm{dw}$

P _Xx(w) be the power spectral density of input signal; w _lAnd w _hBe based on the low side and the high-end cut-off frequency of the band pass filter of wavelet packet respectively.Relation between father node and the child node is seen formula (1) and formula (2).

If G＜T _(l), T wherein _(l)Be the threshold value of l layer, this time-frequency spectrum uniformity occurs.Can think and perhaps mainly constitute corresponding father node frequency band or mainly constitute by the component that does not comprise NBI by NBI (Suppression of narrow band interference).At this moment the decomposition procedure termination of this node; Otherwise continuing to decompose current node is two sub-frequency bands components, as long as the degree of depth of decomposing is no more than peak frequency resolution;

3. repetitive process 2 has reached the highest up to the frequency resolution of appearance of the frequency spectrum uniformity or hope, has so just generated non-regular binary tree.Just can determine the position of NBI from the transition situation of the uniformity.

(2) NBI removes process

Here introduce another variable The ratio of representing the power of two subbands.Only observe relatively all leafy nodes of binary tree,

1. if η＞T _η, T wherein _ηBe the removal thresholding of subband signal, then right child node is subjected to ' pollution ', and the narrow band interference component is its chief component.Therefore with the whole zero setting of subband sequence of correspondence;

2. if 1/ η＞T _η, with the zero clearing of left child node sequence.

(3) wavelet packet combined process

Recover the information sequence that needs by the filtered data of IWPT (seeing formula 2) reconstruct.Then it is carried out relevant despreading with the PN sign indicating number, can alleviate the influence of interference greatly.

TRLMS algorithm based on amplitude limit (Amplitude Limiting)---AL_TRLMS method, process is as follows:

If become behind R (m) amplitude limit

Then the amplitude limit process is as follows:

At first we ask the average power S of R (m) to get

$S=[\underset{k=1}{\overset{G}{\mathrm{\Σ}}}R(m,k)\×R^{*}(m,k)]/G---\left(3\right)$

Wherein (m, k), k=1 .., G are k the element of R (m) to R, and R (m) is done following amplitude limit computing

$\tilde{R}(m,k)=\left\{\begin{array}{cc}R(m,k)& {\left|R(m,k)\right|}^2<\mathrm{\&alpha;S}\\ \mathrm{\&beta;S}& {\left|R(m,k)\right|}^2\&GreaterEqual;\mathrm{\&alpha;S}\end{array}\right.---\left(4\right)$

Wherein

K=1 .., G are that (m k) presses the filtered value of thresholding α S to X, and α and β are for optimizing coefficient.Choosing of α can be subjected to certain restriction, because if the narrow band interference behind the excessive amplitude limit so of α value is still more intense, if and the α value is too small, can be filtered by some useful signal so, either way can influence the performance of algorithm, learn that by emulation α answers value between 0.5～1.5.

Carrying out LMS filtering again for the signal behind the amplitude limit gets

$\hat{d}\left(m\right)=w^H\tilde{R}\left(m\right)---\left(5\right)$

$e\left(m\right)=d\left(m\right)-\hat{d}\left(m\right)---\left(6\right)$

$w(m+1)=w\left(m\right)+\mathrm{\&mu;}\tilde{R}\left(m\right)e^{*}\left(m\right)---\left(7\right)$

For filtered data adjudicate again the transmission symbol estimated

The imbalance and the characteristic value diffusance of AL_TRLMS algorithm among the present invention and traditional TRLMS algorithm are analyzed, and the computation complexity of several algorithms is compared.

1. imbalance is analyzed

By the matrix decomposition theory as can be known, matrix trace equal it characteristic value and.Therefore, the expression formula of imbalance can be written as

$\mathrm{Ms}=\frac{\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="HSA00000444084600013.png,HSA00000444084600021.png"\; state="\{\{state\}\}">u</figure-callout>}}{2}\mathrm{tr}\left[X_R\right]---\left(8\right)$

Because X _RBe the nonnegative definite matrix with Toeplitz structure, its leading diagonal element equals x (0), and x (0) itself equals the mean-square value of each tap input, therefore has

$\mathrm{tr}\left[X_R\right]=G\&times;x\left(0\right)$

$=\underset{k=1}{\overset{G}{\mathrm{\&Sigma;}}}E\left[{\left|R(m,k)\right|}^2\right]---\left(9\right)$

Wushu (9) substitution formula (8)

$\mathrm{Ms}=\frac{\mathrm{\&mu;}}{2}\underset{k=1}{\overset{G}{\mathrm{\&Sigma;}}}E\left[{\left|R(m,k)\right|}^2\right]---\left(10\right)$

I.e. imbalance is to be determined by total tap input power and step-length.Suppose to adopt the TRLMS algorithm, can get according to the definition of several orthogonal transforms

$\mathrm{Ms}=\frac{\mathrm{\&mu;G\&rho;}}{2}E\left[{\left|r\left(m\right)\right|}^2\right]---\left(11\right)$

ρ=G (DFT) wherein, ρ=1 (DCT), ρ=(G+1)/2 (DST).(4.1) formula substitution (11) formula is got

$\mathrm{Ms}=\frac{\mathrm{\&mu;G\&rho;}}{2}E{[\underset{l=0}{\overset{L-1}{\mathrm{\&Sigma;}}}h\left(l\right)d(m-l)+I\left(m\right)+N\left(m\right)]}^2$

$\&ap;\frac{\mathrm{\&mu;G\&rho;}}{2}[\underset{l=0}{\overset{L-1}{\mathrm{\&Sigma;}}}{\left|h\left(l\right)\right|}^2+P_I+{\mathrm{\&sigma;}}^2]---\left(12\right)$

As seen because the existence of narrow band interference makes the imbalance of algorithm bigger,, then be equivalent to before LMS filtering, suppress a part of narrow band interference, promptly reduced P in (12) formula if we adopt the AL_TRLMS algorithm _IValue, so its imbalance has also obtained reduction.

2. the characteristic value diffusance is analyzed

The characteristic value diffusance is meant the degree of scatter of filter input signal autocorrelation matrix characteristic value, and the big more convergence of algorithm speed of characteristic value diffusance is just slow more.Suppose that algorithm is the TRLMS algorithm, then the input signal of filter is R (m), establishes λ _MinAnd λ _MaxBe X _RMinimum and eigenvalue of maximum, the characteristic value diffusance is [99] so

κ＝λ _max/λ _min (13)

Can get X by document [38] _RThe instantaneous estimation formula be

${\hat{X}}_R\left(m\right)\stackrel{\mathrm{\&Delta;}}{=}\mathrm{diag}({\mathrm{\&delta;}}_{m,1},{\mathrm{\&delta;}}_{m,2},\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&delta;}}_{m,G})---\left(14\right)$

δ wherein _{M, k}=β δ _{M-1, k}+ (1-β) R (m, k) R ^*(n, k), 1≤k≤G, and β is a forgetting factor.Can get thus

$X_R=E\left[{\hat{X}}_R\left(m\right)\right]$

$\stackrel{\mathrm{\&Delta;}}{=}\mathrm{diag}(E\left({\mathrm{\&delta;}}_{m,1}\right),E\left({\mathrm{\&delta;}}_{m,2}\right),\&CenterDot;\&CenterDot;\&CenterDot;,E\left({\mathrm{\&delta;}}_{m,G}\right))---\left(15\right)$

Because E is (δ _{M, k})=E (δ _{M-1, k}), so E (δ _{M, k})=E[R (m, k) R ^*(m, k)], at this moment

$X_R\stackrel{\mathrm{\&Delta;}}{=}\mathrm{diag}\left(E\right[{\left|R(m,1)\right|}^2],E[{\left|R(m,2)\right|}^2]---\left(16\right)$

$,\&CenterDot;\&CenterDot;\&CenterDot;,E\left[{\left|R(m,G)\right|}^2\right])$

Because X _RBe diagonal matrix, so its characteristic value equals its diagonal element, establishes P=[E[|R (m, 1) | ²] ..., E[|R (m, G) | ²]], then

κ＝max(P)/min(P) (17)

If

The characteristic value diffusance that in like manner can get the AL_TRLMS algorithm is

${\mathrm{\&kappa;}}^{\&prime;}=\max \left(\tilde{P}\right)/\min \left(\tilde{P}\right)---\left(18\right)$

Get by (4) formula,

So κ '≤κ, the characteristic value diffusance that is the AL_TRLMS algorithm is less than or equal to the TRLMS algorithm, so AL_TRLMS convergence of algorithm speed is more than or equal to the TRLMS algorithm, particularly when the power ratio of narrow band interference was big, AL_TRLMS convergence of algorithm speed will be obviously more than or equal to the TRLMS algorithm.

3. computation complexity analysis

Can be calculated by above proof, the imbalance of AL_TRLMS algorithm and characteristic value diffusance be all less than the TRLMS algorithm, the computation complexity of algorithm compared here again, provided SO_TRLMS and TDVSS computation complexity simultaneously, and be as shown in table 1.Because four kinds of algorithms are the IncFlds transform operation all, the therefore required operand of IncFlds conversion not here.As can be seen from Table 1, three kinds of computation complexities that improve algorithms are all than TRLMS algorithm height, improve minimum one of computation complexity in the algorithms but this paper institute is three kinds to algorithm, and its operand is about 1/2 of other two kinds improvement algorithms.

The computation complexity of table 1 algorithm

The simulation parameter of table 2 algorithm

Claims (2)

1. In the shortwave spread spectrum communication based on the narrow-band interference rejection method of transform domain, it is characterized in that: after the spread-spectrum signal that has narrow band interference in the short wave communication carries out chip matched filtering, sampling, wavelet package transforms, filtering, carry out the transform domain amplitude limit, carry out the wavelet packet inverse transformation again and LMS filtering gets for the signal behind the amplitude limit

   $\hat{d}\left(m\right)=w^H\tilde{R}\left(m\right)$

   $e\left(m\right)=d\left(m\right)-\hat{d}\left(m\right)$

   $w(m+1)=w\left(m\right)+\mathrm{\&mu;}\tilde{R}\left(m\right)e^{*}\left(m\right)$

   For filtered data adjudicate again the transmission symbol estimated

   
2. 2. based on the narrow-band interference rejection method of transform domain, it is characterized in that transform domain amplitude limit step is as follows in the shortwave spread spectrum communication according to claim 1: become after establishing R (m) amplitude limit

   

   Then the amplitude limit process is as follows

   At first ask the average power S of R (m) to get

   $S=[\underset{k=1}{\overset{G}{\mathrm{\&Sigma;}}}R(m,k)\&times;R^{*}(m,k)]/G$

   Wherein (m, k), k=1 .., G are k the element of R (m) to R, and R (m) is done following amplitude limit computing

   $\tilde{R}(m,k)=\left\{\begin{array}{cc}R(m,k)& {\left|R(m,k)\right|}^2<\mathrm{\&alpha;S}\\ \mathrm{\&beta;S}& {\left|R(m,k)\right|}^2\&GreaterEqual;\mathrm{\&alpha;S}\end{array}\right.$

   Wherein

   

   K=1 .., G are that (m k) presses the filtered value of thresholding α S to X, and α and β are for optimizing coefficient, and the α value is between 0.5～1.5.

Images