CN106067868A - DS SS signal pseudo-code blind estimating method based on RBF neural - Google Patents

Abstract

The present invention is claimed a kind of DS SS signal pseudo-code blind estimating method based on RBF neural, belongs to signal processing technology field.As input signal and the desired signal of neutral net after the cycle reception signal sampling that first the method will carry out continuous periodic segment, and regulate neutral net with having supervision according to RBF, the pseudo-code sequence of DS SS signal is represented with the sign function of network second layer weight vector, then by continually entering signal and carrying out repetition training weights until convergence, finally just can rebuild out by the pseudo-code of signal by the sign function of this weight vector.The method overcome the existence when realizing DSSS (DS SS) signal pseudo-code blind estimate of signal autocorrelation matrix-eigenvector-decomposition method to process data vector and oversize and BP neural net method can not there is longer and constringency performance difference time signal to noise ratio the is relatively low problem of pseudo-code.Emulation experiment shows, even if under the conditions of a length of 1000 of 13dB signal to noise ratio or pseudo-code, the method is the most effective.

Description

DS-SS signal pseudo-code blind estimating method based on RBF neural

Technical field

The present invention relates to a kind of conventional Direct Sequence Spread Spectrum Signal Direct Sequence Spread in communication The process of Spectrum (DS-SS), is specially a kind of based on radial direction base Radial Basis Function (RBF) neutral net Pseudo-code sequence blind estimate problem.

Background technology

DSSS (DS-SS) signal is by carrying narrow band information code sequence and the expansion at a high speed of useful information The wideband spread-spectrum signal that frequency sequence (also referred to as PN sequence or pseudo-code sequence) is multiplied and obtains, thus there is excellent anti-interference energy Power and low intercepting and capturing characteristic, be widely used in military communication with commercial communication.It is different from " hiding of Frequency-hopping Communication Technology Keep away " strategy, spread spectrum technic is used " hidden " strategy, is i.e. reduced the power of DS-SS signal by spread-spectrum Spectrum density, until being submerged in noise, thus " hides useful signal " and is difficult to intercept and capture useful letter in noise Zhong Shifei partner Breath.Could be recovered by matched filtering device additionally, the recipient of spread spectrum communication must be known by the pseudo-code sequence that sender used Go out useful signal.Therefore, seek a kind of suitable method to carry out blind estimate and go out the pseudo-code sequence of DS-SS signal and just become to have a mind to very much Justice.

At present, in the case of low signal-to-noise ratio, blind estimate goes out the used pseudo-code sequence of sender is a difficulties, for The Research Literature of this aspect is fewer.Document " Zhang Tianqi. the blind estimating method of low signal-to-noise ratio long pseudo-code direct sequence signal. at signal Reason, 2008 " propose to utilize the method for signal autocorrelation matrix-eigenvector-decomposition to realize the blind estimate of pseudo-code sequence, but at the method The data vector of reason can not be oversize, and the ability following the tracks of non-stationary environment change is the strongest.Document " Zhang T Q.Use APEX neural networks to extract the PN sequence in lower SNR DS-SS Signals.Lecture Notes in Computer Science, 2006 " the Hebbian learning rules of belt restraining are used Neutral net realizes the blind estimate of pseudo-code sequence, but this neutral net is based on " without supervision ".Document " Zhao Defang. based on The direct sequence signal spread spectrum code blind recognition of BP neutral net. telecom technology, 2010 " use based on back-propagation (BP) nerve net Network realizes the blind estimate of pseudo-code sequence, but its constringency performance when pseudo-code sequence is longer or signal to noise ratio is relatively low is poor.Therefore this Bright proposition DS-SS based on RBF neural signal pseudo-code blind estimating method.

Summary of the invention

The technical problem to be solved, is realizing direct sequence for signal autocorrelation matrix-eigenvector-decomposition method During spread spectrum (DS-SS) signal pseudo-code blind estimate, existence processes data vector and oversize and BP neural net method can not there is pseudo-code The problem of constringency performance difference when sequence is longer or signal to noise ratio is relatively low, it is proposed that a kind of based on RBF (RBF) neutral net Method.The method can go out DS-SS signal by accurate blind estimate under the conditions of a length of 1000 of-13dB signal to noise ratio or pseudo-code Pseudo-code sequence.

The present invention solves the technical scheme of above-mentioned technical problem: a kind of DS-SS signal pseudo-code based on RBF neural Sequence blind estimating method, its step is, first according to known periods, the signal received is carried out continuous periodic segment and adopts Sample forms the continuous periodic segment sampling observation vector set receiving signal, secondly will carry out one week of continuous periodic segment sampling Phase receives signal as the input signal of RBF neural and desired signal, and regulates nerve net with having supervision according to RBF Network, represents the pseudo-code sequence of DS-SS signal, then by continually entering letter with the sign function of network second layer weight vector Number carry out repetition training weights until convergence, finally just by the sign function of this weight vector, the pseudo-code sequence of signal can be estimated Meter is out.

Pseudo-code sequence takes longest linear feedback shift register sequence, i.e. m-sequence under normal circumstances.Now, base band DS- The reception signal of SS system can be modeled as: x ' (t)=S (t)+n (t).In formula, n (t) be an average be that zero variance is σ²Additive white Gaussian noise, S (t) base band DS-SS signal, its signal model can be expressed as:In formula, m [k] is kth information code symbol, and N is the cycle of frequency expansion sequence Length, and p [i], i=0 ..., N-1} is PN sequence, and q (t) is that a rectangle cuts general pulse, T_cIt is to cut general cycle, T₀It it is PN sequence Cycle (T₀=NT_c), τ=T_xIt it is random delay.

The present invention uses the method for RBF neural that the pseudo-code sequence of DS-SS signal is carried out blind estimate, analytical derivation By gaussian radial basis function, standard gaussian RBF, anti-S type RBF, Cauchy's RBF, many secondaries footpath RBF neural is regulated with having supervision to realize the pseudo-code sequence of DS-SS signal to basic function and inverse many secondaries RBF Row blind estimate, overcoming process data vector that signal autocorrelation matrix-eigenvector-decomposition method exists can not be oversize and BP is neural There is longer or constringency performance difference time signal to noise ratio the is relatively low problem of pseudo-code in network method, it is achieved that in relatively low signal-to-noise ratio and longer puppet The accurate blind estimate of DS-SS signal pseudo-code sequence under the conditions of code sequence.

Accompanying drawing explanation

Fig. 1 matched filtering of the present invention device block diagram；

Fig. 2 RBF neural of the present invention structure chart；

Fig. 3 RBF neural of the present invention blind estimate DS-SS signal pseudo-code flow chart；

Fig. 4 error performance of the present invention figure；

Fig. 5 pseudo-code of the present invention estimates average performance map；

Fig. 6 pseudo-code of the present invention estimates convergence curve figure；

Fig. 7 RBF neural of the present invention average performance map under different code length；

Fig. 8 BP of the present invention neutral net average performance map under different code length；

Detailed description of the invention

Below in conjunction with accompanying drawing and instantiation, the enforcement to the present invention is further described.

Fig. 1 show matched filtering device block diagram of the present invention, if this figure be intended to indicate that the known frequency expansion sequence of recipient, Then can realize the active despreading of signal with a matched filtering device, it is critical only that of despreading realizes code synchronization, i.e. T_x=0. Fig. 1 inputs x_kT () represents the reception signal in kth information code cycle, T_cIt is to cut the general cycle, and p [i], i=0 ..., N-1} is The PN sequence in a cycle identical with sender, and uncorrelated with n (t), then have

$\begin{array}{c}O\left(t\right)<\underset{i=0}{\overset{N-1}{\&Sigma;}}p\&lsqb;i\&rsqb;,x_k\left(t\right)>\\ =<\underset{i=0}{\overset{N-1}{\&Sigma;}}p\&lsqb;i\&rsqb;,m\&lsqb;k\&rsqb;\underset{i=0}{\overset{N-1}{\&Sigma;}}p\&lsqb;i\&rsqb;q(t-{\mathrm{iT}}_c-{\mathrm{kT}}_s-\mathrm{\&tau;})+n\left(t\right)>\\ =<\underset{i=0}{\overset{N-1}{\&Sigma;}}p\&lsqb;i\&rsqb;,m\&lsqb;k\&rsqb;\underset{i=0}{\overset{N-1}{\&Sigma;}}p\&lsqb;i\&rsqb;q(t-{\mathrm{iT}}_c-{\mathrm{kT}}_s-\mathrm{\&tau;})>\\ \stackrel{\mathrm{\&tau;}=0}{=}\&PlusMinus;N\end{array}---\left(1\right)$

From above formula, as the reception signal x of the kth information code symbol period entering matched filtering device_kThe time delay of (t) τ=T_xWhen=0, will match so that the mould exporting O (t) takes maximum by frequency expansion sequence known with hypothesis.The most just may be used Estimate T taking the moment of maximum according to O (t)_xValue so that output O (t) after narrow-band filtering every an information code Symbol period just provides a lock-out pulse, i.e. obtains a local peaking every an information code symbol period, then uses it Remove to control local pseudo-code generator to obtaining synchronization with sender's pseudo-code, complete despreading.

Being analyzed from above, use matched filtering device to realize premise that DS-SS signal de-spreads is it is to be understood that sender Pseudo-code sequence p [i], i=0 ..., N-1}, but pseudo-code is not for the civilian management in spread spectrum communication or military surveillance side Can know, therefore the supervision to signal be intercepted the most relatively difficult, go out pseudo-code by direct blind estimate from the DS-SS signal received Sequence is the major issue confronted.

Fig. 2 show RBF neural structure chart of the present invention, is a kind of p-h-m version three layers feedforward RBF nerve net Network, this network has p input node unit, h hidden node unit and m output node unit.The input vector of this network is x (n)=[x₁(n) x₂(n) … x_P(n)]^T, weight matrix isφ_i(n) RBF activation primitive for i-th hidden node.In the present invention, RBF can be respectively adopted Gaussian function (as Formula (2)), standard gaussian function (such as formula (3)), anti-S type function (such as formula (4)), Cauchy kernel (such as formula (5)), many quadratic functions (such as formula (6)) and inverse many quadratic functions (such as formula (7)).

${\mathrm{\&phi;}}_i\left(n\right)=\exp (-\frac{\left|\right|x\left(n\right)-c_i\left(n\right)|{|}^2}{{\mathrm{\&delta;}}_i^2\left(n\right)})---\left(2\right)$

${\mathrm{\&phi;}}_i\left(n\right)=\exp (-\frac{\left|\right|x\left(n\right)-c_i\left(n\right)|{|}^2}{2{\mathrm{\&delta;}}_i^2\left(n\right)})---\left(3\right)$

${\mathrm{\&phi;}}_i\left(n\right)=\frac{1}{1+\exp \left(\frac{\left|\right|x\left(n\right)-c_i\left(n\right)|{|}^2}{{\mathrm{\&delta;}}_i^2\left(n\right)}\right)}---\left(4\right)$

${\mathrm{\&phi;}}_i\left(n\right)=\frac{1}{1+\frac{\left|\right|x\left(n\right)-c_i\left(n\right)|{|}^2}{{\mathrm{\&delta;}}_i^2\left(n\right)}}---\left(5\right)$

${\mathrm{\&phi;}}_i\left(n\right)=\sqrt{1+\frac{\left|\right|x\left(n\right)-c_i\left(n\right)|{|}^2}{{\mathrm{\&delta;}}_i^2\left(n\right)}}---\left(6\right)$

${\mathrm{\&phi;}}_i\left(n\right)=\frac{1}{\sqrt{1+\frac{\left|\right|x\left(n\right)-c_i\left(n\right)|{|}^2}{{\mathrm{\&delta;}}_i^2\left(n\right)}}}---\left(7\right)$

Here, c_iN () is data center's vector value of i-th hidden node, | | | | then represent European norm, δ_iN () is The extension constant of i hidden node or width, δ_iN () is the least, the width of RBF is the least, and RBF more has Selectivity.

In figure, ∑ represents that output layer neuron uses linear activation primitive, then this RBF neural is output as

$y_j\left(n\right)=\underset{i=1}{\overset{h}{\&Sigma;}}{\mathrm{\&phi;}}_i\left(n\right)w_{ij}\left(n\right),j=1,2,\mathrm{...},N---\left(8\right)$

The present invention uses this RBF neural structured flowchart, when receiving the reception signal that signal is base band DS-SS system Time, signal will be received and sample in nonoverlapping cycle window, without loss of generality, it is assumed here that the sampling period will be equal to a chip width Degree, i.e. the sampling period is T_s=T₀/ N=T_c.Then system carry out n times sampling obtain one the cycle receive signal be

X ' (t)=x ' (n)=[x (t), x (t-T_c),…,x(t-(N-1T_c))]^T=[x₁(n),x₂(n),…,x_N(n)]^T

(9)

Additionally, before x ' (n) input neural network, be normalized to:

X (n)=x ' (n)/| | x ' (n) | | (10)

In formula:

$\left|\right|x^{\&prime;}\left(n\right)\left|\right|=\sqrt{x^{\&prime;T}\left(n\right)x^{\&prime;}\left(n\right)}---\left(11\right)$

So, pseudo-code sequence will estimate with the form of a more robust.

In the present invention, RBF neural takes N-1-N structure, i.e. input neuron number and is equal to output neuron number Pseudo-code length (or twice of pseudo-code length).When time delay is 0, input neuron number and output neuron number are equal to puppet Code length, input signal is the data vector in a cycle；When time delay is not 0, input neuron number and output neuron Number is equal to the pseudo-code length of twice, and input signal is the data vector in two cycles.The present invention with time delay be 0, RBF As a example by the quasi-Gaussian function of label taking, show that hidden node unit is output as

$\mathrm{\&phi;}\left(n\right)=\exp (-\frac{\left|\right|x\left(n\right)-c\left(n\right)|{|}^2}{2{\mathrm{\&delta;}}^2\left(n\right)})---\left(12\right)$

If taking weight vector is w (n)=[w₁(n) w₂(n)…w_N(n)]^T, then output layer unit is output as y (n)=[y₁ (n) y₂(n) … y_N(n)]^T, wherein:

$y_j\left(n\right)=\mathrm{\&phi;}\left(n\right)w_j\left(n\right)=\exp (-\frac{\left|\right|x\left(n\right)-c\left(n\right)|{|}^2}{2{\mathrm{\&delta;}}^2\left(n\right)})\&CenterDot;w_j\left(n\right),j=1,2,\mathrm{...},N---\left(13\right)$

Next need data center c (n) and extension constant δ (n) are entered with weight vector w (n) these three unknown vector Row determines.For data center c (n), the method generally chosen has several, is random choice method, just from input sample respectively Hand over method of least square, have supervision Selection Center method, self-organizing to choose method and intelligent optimization algorithm etc..Choose number in the present invention According to center equal to input signal, i.e. c (n)=x (n) (and in direct document, additive method chooses the footpath that data center builds The object of the invention can not be realized to basic function, need to study further), and extend constant and weight vector employing gradient instruction Practice method to regulate.Therefore, c (n)=x (n) deduce that error correction signal is

e_j(n)=x_j(n)-y_j(n)=x_j(n)-w_j(n) j=1,2 ... N (14)

The then output y of this neutral net_jThe gradient of extension constant and weight vector is respectively

$\frac{\&part;y_j\left(n\right)}{\&part;\mathrm{\&delta;}}=\frac{\left|\right|x\left(n\right)-c\left(n\right)|{|}^2}{{\mathrm{\&delta;}}^3\left(n\right)}\&CenterDot;\exp (-\frac{\left|\right|x\left(n\right)-c\left(n\right)|{|}^2}{2{\mathrm{\&delta;}}^2\left(n\right)})\&CenterDot;w_j\left(n\right)=0---\left(15\right)$

$\frac{\&part;y_j\left(n\right)}{\&part;w_j}=\exp (-\frac{\left|\right|x\left(n\right)-c\left(n\right)|{|}^2}{2{\mathrm{\&delta;}}^2\left(n\right)})=1---\left(16\right)$

So regulated quantity of extension constant and weight vector is respectively

$\&dtri;{\mathrm{\&delta;}}_j\left(n\right)={\mathrm{\&eta;e}}_j\left(n\right)\frac{\&part;y_j\left(n\right)}{\&part;\mathrm{\&delta;}}=0---\left(17\right)$

$\&dtri;w_j\left(n\right)={\mathrm{\&mu;e}}_j\left(n\right)\frac{\&part;y_j\left(n\right)}{\&part;w_j}={\mathrm{\&mu;e}}_j\left(n\right)---\left(18\right)$

Wherein η and μ is learning rate, and 0 ＜ η, μ ＜ 1, finally show that the more new formula of extension constant and weight vector is

$\mathrm{\&delta;}(n+1)=\mathrm{\&delta;}\left(n\right)+\underset{j=1}{\overset{N}{\&Sigma;}}\&dtri;{\mathrm{\&delta;}}_j\left(n\right)=\mathrm{\&delta;}\left(n\right)---\left(19\right)$

$w_j(n+1)=w_j\left(n\right)+\&dtri;w_j\left(n\right)=w_j\left(n\right)+{\mathrm{\&mu;e}}_j\left(n\right)---\left(20\right)$

The most repeatedly updating weight vector makes it gradually approach the pseudo-code sequence of DS-SS signal, is finally reached convergence, now Just can rebuild out by pseudo-code sequence by weight vector.

Fig. 3 show present invention DS-SS based on RBF neural signal pseudo-code sequence blind estimate flow chart, specifically walks Rapid: (1st) step, the signal received is carried out continuous periodic segment according to known periods and formation of sampling receives the continuous of signal Vector set is observed in periodic segment sampling, and it is neural as RBF that the cycle carrying out continuous periodic segment sampling is received signal The input signal of network and desired signal；(2nd) step, RBF neural extension constant and network weight random initializtion, and set Put end condition, the minimum bit-error rate of permission is i.e. set；(3rd) step, for moment n, inputs new data vector x (n)；The (4) step, calculates the output of hidden neuron(5th) step, calculates defeated according to formula (13) Go out y_j(n), j=1,2 ..., N；(6th) step, calculates error e according to formula (14)_j(n)=x_j(n)-y_j(n), j=1,2 ..., p；(7th) step, updates extension constant according to formula (19)；(8th) step, updates weight vector according to formula (20)；(9th) step, Return (2nd) step to continue, until reaching the minimum bit-error rate allowed.Now, it is possible to by RBF neural second layer weights The sign function of vector is by the PN sequence estimation of signal out.

Utilizing emulation experiment to verify the theoretical derivation of inventive algorithm, experiment parameter is arranged: Monte Carlo number of times For Mncr=200, up-sampling number of times is Sa=8 position/chip (i.e. T_s=T_c=8T_e, T_eFor the sampling period), unless otherwise noted, The quasi-gaussian radial basis function of RBF label taking.

It is-5dB that Fig. 4 (a) show snr of received signal, pseudo-code length L=100 position, time delay T_xError code when=0 Can scheme, wherein top subgraph represents that 100 the original PN sequences produced by PN sequencer, middle subgraph represent in above-mentioned reality When under the conditions of testing, RBF neural reaches convergence, 200 Monte Carlo weighted means of output weight vector value (now export Weight vector a length of 100), lower section subgraph represent by the weights shown in the original PN sequence shown in the subgraph of top and middle graph to The sign function of amount subtracts each other the mould of errors sequence.It is apparent that the sign function value of network weight and original PN sequence Train value is completely contrary, thus, it can be known that the sign function of second layer network weight when can be restrained by neutral net will without time Prolong the original PN rebuilding series under situation out.Under the conditions of Fig. 4 (b) is identical signal to noise ratio, pseudo-code length and RBF, Time delay T_xError performance figure when ≠ 0, wherein top subgraph represents 100 the original PN sequences produced by PN sequencer, Middle subgraph represents that RBF neural reaches 200 Meng Teka of output weight vector value during convergence under these experimental conditions Lip river weighted mean (now output weight vector a length of 200), lower section subgraph represents by the original PN sequence shown in the subgraph of top The error sequence subtracting each other gained from the 21st to 120 sign functions blocking sequence of weight vector shown in row and middle subgraph Mould.In like manner can draw, can by the 21st of network weight on earth 120 sign functions blocking sequence will have time delay Original PN rebuilding series under situation is out.

Fig. 5, Fig. 6 respectively snr of received signal is 0 to-13dB, pseudo-code length L=100 position, time delay T_xWhen=0 Average performance chart and study convergence curve figure, as seen from Figure 5 along with the reduction of signal to noise ratio, RBF neural reaches Average data group number needed for convergence increases, average degradation, and when signal to noise ratio is less than-12dB, average performance is drastically disliked Change.Equally being drawn by Fig. 6, when signal to noise ratio is higher, RBF neural convergence rate is very fast, when signal to noise ratio is relatively low, RBF neural convergence rate is relatively slow, and especially when signal to noise ratio is less than-12dB, constringency performance drastically deteriorates.

Fig. 7, Fig. 8 are respectively RBF neural and the BP neutral net average performance chart under different pseudo-code lengths, As seen from the figure, along with the increase of pseudo-code length, RBF neural the most of the present invention or traditional BP neutral net, reach to receive Holding back required average data group number all can increase, performance all can be deteriorated；But, in identical pseudo-code length and identical signal to noise ratio condition Under, RBF neural of the present invention reaches to restrain required average data group number and is less than when BP neutral net reaches to restrain required Average data group number, performance the most of the present invention will be due to traditional BP neutral net, thus for realize DS-SS signal pseudo-code sequence Blind estimate provides a kind of more excellent method.

The present invention proposes a kind of method of DS-SS signal pseudo-code sequence blind estimate based on RBF neural, overcomes Receive data vector that signal autocorrelation matrix-eigenvector-decomposition method processes can not oversize and BP neural net method in pseudo-code Longer or poor performance time signal to noise ratio is relatively low shortcoming.Analytical derivation gaussian radial basis function, standard gaussian RBF, anti- S type RBF, Cauchy's RBF, many secondaries RBF and inverse many secondaries RBF are for DS-SS The blind estimate of signal pseudo-code sequence, and the property that simulating, verifying RBF neural is under different signal to noise ratios, different pseudo-code length Can, and compare with the BP neutral net under corresponding conditions.Result shows, even if RBF neural of the present invention is in noise Ratio is for still effective under the conditions of-13dB, and reducing or the increase of pseudo-code length along with signal to noise ratio, and RBF neural reaches receipts Hold back required average data group number to increase, average degradation, and under the conditions of identical pseudo-code length with identical signal to noise ratio, this The constringency performance of bright RBF neural is better than the constringency performance of BP neutral net, thus the subsequent treatment (information to this signal Code is estimated) significant.

Claims (4)

1. a DS-SS signal pseudo-code blind estimating method based on RBF neural, its step is, first will receive Vector is observed in the continuous periodic segment sampling that signal carries out continuous periodic segment formation reception signal of sampling according to known periods Collection, secondly receives signal as the input of RBF neural and expectation letter using the cycle carrying out continuous periodic segment sampling Number, and regulate neutral net with having supervision according to RBF, represent DS-SS with the sign function of network second layer weight vector The pseudo-code sequence of signal, then by continually entering signal and carrying out repetition training weights until convergence, finally just can pass through this power The sign function of value vector is by the PN sequence estimation of signal out.

Method of estimation the most according to claim 1, it is characterised in that set up the reception signal model of base band DS-SS system For x ' (t)=S (t)+n (t).In formula, n (t) be average be zero variance be σ²Additive white Gaussian noise, S (t) base band DS-SS signal, its signal model can be expressed as:In formula, m [k] is K information code symbol, N is the Cycle Length of frequency expansion sequence, p [i], i=0 ..., N-1} is PN sequence, and q (t) is a rectangle Cut general pulse, T_cIt is to cut general cycle, T₀It is the cycle (T of PN sequence₀=NT_c), τ=T_xIt it is random delay.

Method of estimation the most according to claim 1, it is characterised in that set up three layers of feedforward of a kind of p-h-m version RBF neural model, wherein p is input node unit number, h and m is respectively hidden node unit number and output node unit number. The input vector of this network is x (n)=[x₁(n) x₂(n) … x_P(n)]^T, weight matrix isφ_iN () is that the RBF of i-th hidden node activates letter Number.And the desirable Gaussian function of differenceStandard gaussian functionAnti-S type functionCauchy kernelMany quadratic functionsAnd against many quadratic functionsHere, c_iN () is data center's vector value of i-th hidden node, | | | | then represent European norm, δ_iN () is extension constant or width, the δ of i-th hidden node_iN () is the least, the width of RBF is the least, RBF more has selectivity.RBF neural is output as

4. according to the method for estimation described in claim 1-3, in combination with RBF neural data center and extension selection of constant Method, makes RBF neural take N-1-N structure, makes data center be equal to input signal, i.e. c (n)=x (n).Now can obtain RBF The hidden node unit that goes out of neutral net is output as

$\mathrm{\&phi;}\left(n\right)=\exp (-\frac{\left|\right|x\left(n\right)-c\left(n\right)|{|}^2}{2{\mathrm{\&delta;}}^2\left(n\right)})=1$

Taking weight vector is w (n)=[w₁(n) w₂(n) … w_N(n)]^T, output layer unit can be obtained and be output as y (n)=[y₁ (n) y₂(n) … y_N(n)]^T, wherein:

$y_j\left(n\right)=\mathrm{\&phi;}\left(n\right)w_j\left(n\right)=\exp (-\frac{\left|\right|x\left(n\right)-c\left(n\right)|{|}^2}{2{\mathrm{\&delta;}}^2\left(n\right)})\&CenterDot;w_j\left(n\right)=w_j\left(n\right),j=1,2,...,N$

Error correction signal is

e_j(n)=x_j(n)-y_j(n)=x_j(n)-w_j(n) j=1,2 ... N

The then output y of this neutral net_jThe gradient of extension constant and weight vector is respectively

$\frac{\&part;y_j\left(n\right)}{\&part;\mathrm{\&delta;}}=\frac{\left|\right|x\left(n\right)-c\left(n\right)|{|}^2}{{\mathrm{\&delta;}}^3\left(n\right)}\&CenterDot;\exp \left(-\frac{\left|\right|x\left(n\right)-c\left(n\right)|{|}^2}{2{\mathrm{\&delta;}}^2\left(n\right)}\right)\&CenterDot;w_j\left(n\right)=0$

$\frac{\&part;y_j\left(n\right)}{\&part;w_j}=\exp (-\frac{\left|\right|x\left(n\right)-c\left(n\right)|{|}^2}{2{\mathrm{\&delta;}}^2\left(n\right)})=1$

So regulated quantity of extension constant and weight vector is respectively

$\&dtri;{\mathrm{\&delta;}}_j\left(n\right)={\mathrm{\&eta;e}}_j\left(n\right)\frac{\&part;y_j\left(n\right)}{\&part;\mathrm{\&delta;}}=0$

$\&dtri;w_j\left(n\right)={\mathrm{\&mu;e}}_j\left(n\right)\frac{\&part;y_j\left(n\right)}{\&part;w_j}={\mathrm{\&mu;e}}_j\left(n\right)$

Wherein η and μ is learning rate, and 0 ＜ η, μ ＜ 1, finally show that the more new formula of extension constant and weight vector is

$\mathrm{\&delta;}(n+1)=\mathrm{\&delta;}\left(n\right)+\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}\&dtri;{\mathrm{\&delta;}}_j\left(n\right)=\mathrm{\&delta;}\left(n\right)$

$w_j(n+1)=w_j\left(n\right)+\&dtri;w_j\left(n\right)=w_j\left(n\right)+{\mathrm{\&mu;e}}_j\left(n\right)$

Then make it gradually approach the pseudo-code sequence of DS-SS signal by renewal weight vector repeatedly, be finally reached convergence, now Just can by weight vector by PN sequence estimation out.