CN106130689A - A kind of non-linear self-feedback chaotic neural network signal blind checking method - Google Patents

Abstract

The present invention proposes a kind of non-linear self-feedback chaotic neural network signal blind checking method, use nonlinear function as the self feed back item of chaotic neural network, and double Sigmoid functions are applied in blind checking method, every time during iteration, initially enter chaotic neural network, then enter back into second activation primitive.Owing to chaotic neural network has the advantage that can avoid sinking into local optimum, so the present invention inherits this feature of chaotic neural network, improve blind Detecting performance；Further, compared with the chaotic neural network of linear self feed back item, non-linear self-feedback chaotic neural network has increasingly complex dynamic behavior, makes the internal state of network have highly efficient Chaos Search ability and search efficiency.The inventive method, under equal conditions, noiseproof feature is better than traditional Hopfield signal blind checking method.

Description

A kind of non-linear self-feedback chaotic neural network signal blind checking method

Technical field

The invention belongs to wireless communication signals process and nerual network technique field, especially relate to a kind of non-linear reflexive Feedback chaotic neural network signal blind checking method.

Background technology

Data communication and the fast development of sensor network technology, the blind Detecting (Blind to signal of communication Detection) have higher requirement.So-called blind Detecting is just capable of detecting when to send signal merely with accepting signal itself, Thus eliminate intersymbol interference (ISI) to improve the rate of information throughput and reliability.

For solving the problems such as the tradition channel bandwidth utilization ratio that easily causes of adaptive equalization technique is low, many documents start Use Hopfield neutral net that signal blind Detecting problem is studied.Hopfield neutral net (Hopfield Neural Networks, HNN) whether Blind Detect Algorithm is not limited containing Common zero by channel and required transmission data are shorter, unite with second order Metering blind arithmetic is compared with high-order statistic blind arithmetic, more can meet Modern Communication System high speed and reliable transmission requirement.Literary composition Offer [Zhang Yun, Modern Communication System processes [PhD] with signal of communication, Ph.D. Dissertation (Nanjing: Nanjing Univ. of Posts and Telecommunications), 2012.] the existing first-stage success of Blind Detect Algorithm research based on HNN, it was demonstrated that network tends to the necessary and sufficient condition of stable equilibrium.Literary composition Offer [Yang S, Lee C M, HBP:improvement in BP algorithm for an adaptive MLP decision feedback equalizer[J].IEEE Transactions on Circuits and System,2006, 53 (3): 240-244] point out that HNN algorithm is often absorbed in local minimum point, the most even restrain the optimum less than optimization problem Solve or approximate optimal solution.Document [M Mart í n-Valdivia, A Ruiz-Sep ú lveda, F Triguero-Ruiz, Improving local minima of Hopfield networks with augmented Lagrange Multipliers for large scale TSPs [J] .Neural Networks, 2000,13 (3): 283-285] for solving Local minimum point's problem, in algorithm flow, need to select different starting points after evaluation algorithm is absorbed in local minimum separately, To obtain globe optimum.Document [Luonan Chen, Kazuyuki Aihara, Chaotic simulated annealing By a neural network model with transient chaos [J] .Neural Networks, 1995,8 (6): 915 930] pointing out, transient chaotic neural network (Transiently Chaotic Neural Network, TCNN) can be kept away Exempt to be absorbed in local optimum.But, TCNN has negative coupling certainly, and the convergence rate that can cause energy function is slack-off.For this Problem, the present invention proposes a kind of based on double Sigmoid non-linear self-feedback chaotic neural network Blind Detect Algorithm, mixes in transient state The self feed back item of improved chaotic neural network on the basis of ignorant neural network model, the dynamic characteristic that can make full use of chaos is carried out Search, it is to avoid be absorbed in local optimum, and double Sigmoid structure can improve search speed.

Summary of the invention

The technical problem to be solved is to overcome the defect of prior art and deficiency, it is provided that a kind of non-linear Self feed back chaotic neural network signal blind checking method.The inventive method is to change on the basis of transient chaotic neural network model Entering the self feed back item of chaotic neural network, the dynamic characteristic that can make full use of chaos scans for, it is to avoid be absorbed in local optimum, And double Sigmoid structures can improve search speed.This network is it is intended that the signal blind check of unity feedback network of wireless communication networks Surveying provides a kind of and avoids sinking into local optimum and the high algorithm of search precision, provides accurate and quick signal for wireless communication networks Blind checking method.

For solving above-mentioned technical problem, the technical solution adopted in the present invention is:

A kind of non-linear self-feedback chaotic neural network signal blind checking method, its step is as follows:

Step A, structure reception data matrix:

Receiving terminal receives unique user and sends signal, through over-sampling, it is thus achieved that the reception equation of discrete-time channel:

X_N=S Γ^T

In formula, X_NBeing to receive data matrix, S is to send signal battle array, and Γ is to be rung h by channel impulse_jjThe block Toeplitz constituted Matrix；(·)^TRepresenting matrix transposition；

Wherein,

$S={\[s_{L+M}\left(k\right),...,s_{L+M}(k+N-1)\]}_{N\×(L+M+1)}^T={\[s_N\left(k\right),...,s_N(k-M-L)\]}_{N\×(L+M+1)},$

M is channel exponent number, and L is equalizer exponent number, and N is desired data length；

s_L+M(k)=[s (k) ..., s (k-L-M)]^T；Wherein, { ± 1}, moment k are natural number to s ∈；

h_jj=[h₀,…,h_M]_q×(M+1), jj=0,1 ..., M；

Q is oversample factor, and value is positive integer；

X_N=[x_L(k),…,x_L(k+N-1)]^TIt is that N × (L+1) q receives data matrix, wherein x_L(k)=Γ s_L+M(k)；

Step B, reception data matrix singular value decomposition:

$X_N=\[U,U_c\]\·\left[\begin{array}{c}D\\ 0\end{array}\right]\·V^H$

In formula,

(·)^HIt it is Hermitian transposition；

U is the N in singular value decomposition × (L+M+1) basic matrix at the tenth of the twelve Earthly Branches；

0 is (N-(L+M+1)) × (L+1) q null matrix；

V is (L+1) q × (L+1) q basic matrix at the tenth of the twelve Earthly Branches；

U_cIt it is N × (N-(L+M+1)) basic matrix at the tenth of the twelve Earthly Branches；

D is (L+M+1) × (L+1) q singular value battle array；

Step C, arranges weight matrix W=I_N-Q, wherein I_NIt is the unit matrix of N × N-dimensional,

Step D, changes the linear self feed back item in Chen ' s chaotic neural network into non-linear self-feedback item, in order to accelerate System convergence speed uses double Sigmoid structure；

Double Sigmoid non-linear self-feedback chaotic neural network dynamical equations are:

$\frac{{\mathrm{dy}}_i\left(t\right)}{dt}=f\left({\mathrm{\ϵy}}_i\right(t)+\mathrm{\α}\[\underset{j=1}{\overset{n}{\Σ}}{w}_{ij}{x}_j(t)+I_i\]-{\mathrm{\λz}}_i(t\left)g\right(x_i\left(t\right)-I_0\left)\right)$

x_i(t)=σ (y_i(t))

z_i(t+1)=(1-β) z_i(t)

The equation is iterated computing, then the result of each iteration is substituted into double Sigmoid non-linear self-feedback and mix In energy function E (t) of ignorant neutral net, when this energy function E (t) minimizes value, i.e. y_i(t)=y_i(t-1) time, this pair Sigmoid non-linear self-feedback chaotic neural network reaches balance, and iteration terminates；

Wherein,

y_iT () is the internal state of i-th neuron in neutral net；I represents i-th neuron, and j represents jth god Arbitrary integer in unit, i ≠ j, and i, j are [0, N]；T is the time that neutral net iterative process is run, in this neutral net Continuous time t and discrete time k by Euler's formula realize conversion；

ε is the decay factor of this network, and 0≤ε≤1；w_ijFor the interconnection weights of neuron j in this network to neuron i, And w_ij=w_ji；α is the coupling factor of this network；I_iFor the biasing of i-th neuron, I₀It it is the biasing of initial neuron；

z_iT () is the self-feedback connection weights value of i-th neuron, λ is decay factor between neuron, and λ > 0, and β is variable z_i The decay factor of (t)；

x_iT () is the output of i-th neuron；When this neutral net reaches finally to balance, each nerve can be approximately considered The x of unit_i(t)=y_i(t), x_iT () is the transmission signal asked for；

σ (.) is first Sigmoid activation primitive of this neutral net, and f (.) is second of this neutral net Sigmoid activation primitive；

And:

σ (s)=tanh (c s)

$f\left(s\right)=\frac{1}{1+e^{-s}}$

Wherein, s is the input of neutral net, and c is the built-in adjustment parameter of activation primitive, and the derivative of f (.) is much smaller than σ (.) Derivative；

G (.) is the non-linear self-feedback item of this neutral net:

G (s)=tanh (s)

Energy function E (t) of described pair of Sigmoid non-linear self-feedback chaotic neural network is:

E (t)=E_hopfield+E_add

$E_{hopfield}=-\frac{\mathrm{\α}}{2}\underset{i=1}{\overset{n}{\Σ}}\underset{j=1}{\overset{n}{\Σ}}{x}_i\left(t\right)w_{ij}{x}_j\left(t\right)-\mathrm{\ϵ}\underset{i=1}{\overset{n}{\Σ}}{\∫}_0^{x_i\left(t\right)}{\mathrm{\σ}}_i^{-1}\left(\mathrm{\τ}\right)d\mathrm{\τ}-\mathrm{\α}\underset{i=1}{\overset{n}{\Σ}}{I}_i{x}_i\left(t\right)$

$E_{add}=\mathrm{\λ}\underset{i=1}{\overset{n}{\Σ}}{z}_i\left(t\right){\∫}_0^{x_i\left(t\right)}g(\mathrm{\τ}-I_0)d\mathrm{\τ}$

Wherein:

This chaotic neural network is made up of N number of neuron；E (t) is energy function, and this energy function is by E_hopfieldAnd E_add Two parts form, E_hopfieldFor the energy function item of common Hopfield neutral net, E_addNon-linear reflexive for double Sigmoid The additional-energy item of feedback chaotic neural network energy function；

Sigmoid function σ for i-th neuron_i(τ) inverse function.

The invention has the beneficial effects as follows: the present invention proposes a kind of non-linear self-feedback chaotic neural network signal blind Detecting side Method, use nonlinear function is as the self feed back item of chaotic neural network, and double Sigmoid functions are applied to blind checking method In, during each iteration, initially enter chaotic neural network, then enter back into second activation primitive.Due to chaotic neural network There is the advantage that can avoid sinking into local optimum, so the present invention inherits this feature of chaotic neural network, improve blind Detection performance；Further, compared with the chaotic neural network of linear self feed back item, non-linear self-feedback chaotic neural network has more For complicated dynamic behavior, the internal state of network is made to have highly efficient Chaos Search ability and search efficiency.This Bright method, under equal conditions, noiseproof feature is better than traditional Hopfield signal blind checking method.

Accompanying drawing explanation

The double Sigmoid non-linear self-feedback chaotic neural network system construction drawing of Fig. 1 present invention.

Fig. 2 present invention is based on double Sigmoid non-linear self-feedback chaotic neural network Blind Detect Algorithm and Hopfield god Bit error rate comparison diagram through network Blind Detect Algorithm and transient chaotic neural network Blind Detect Algorithm.HNN(Hopfield Neural Network) algorithm is Hopfield neutral net Blind Detect Algorithm, TCNN (Transiently Chaotic Neural Network) algorithm is transient chaotic neural network algorithm, DS-NSCNN (Nonlinear Self-feedback Chaotic Neural Network With Double Sigmoid) algorithm is double Sigmoid non-linear self-feedback chaos god Through network algorithm.

Fig. 3 present invention is based on double Sigmoid non-linear self-feedback chaotic neural network Blind Detect Algorithm and transient chaos god Through network (TCNN) algorithm bit error rate comparison diagram under the conditions of different data lengths respectively.Fig. 3. (a) is based on TCNN (Transiently Chaotic Neural Network) Blind Detect Algorithm, Fig. 3. (b) is based on DS-NSCNN (Nonlinear Self-feedback Chaotic Neural Network With Double Sigmoid) blind check is calculated Method.

Detailed description of the invention

Below in conjunction with the accompanying drawings, the letters based on the double Sigmoid non-linear self-feedback chaotic neural networks present invention proposed Number blind checking method is described in detail:

Signal blind checking methods based on double Sigmoid non-linear self-feedback chaotic neural networks, its implementation process is as follows:

When ignoring noise, the reception equation of discrete-time channel is defined as follows

X_N=S Γ^T (1)

In formula, X_NBeing to receive data matrix, S is to send signal battle array, and Γ is to be rung h by channel impulse_jjThe block Toeplitz constituted Matrix；(·)^TRepresenting matrix transposition；

Wherein, signal battle array is sent:

S=[s_L+M(k),…,s_L+M(k+N-1)]^T=[s_N(k),…,s_N(k-M-L)]_N×(L+M+1),

M is channel exponent number, and L is equalizer exponent number, and N is desired data length；

s_L+M(k)=[s (k) ..., s (k-L-M)]^T；Wherein, { ± 1}, moment k are natural number to s ∈；

h_jj=[h₀,…,h_M]_q×(M+1), jj=0,1 ..., M；

Q is oversample factor, and value is positive integer；

X_N=[x_L(k),…,x_L(k+N-1)]^TIt is that N × (L+1) q receives data matrix,

Wherein x_L(k)=Γ s_L+M(k)；

For formula (1), during the full column rank of Γ, necessarily haveMeet Qs_N(k-d)=0, U_cIt is N × (N-(L+M+1)) The tenth of the twelve Earthly Branches basic matrix, by singular value decompositionIn obtain；

Wherein:

(·)^HIt it is Hermitian transposition；

U is the N in singular value decomposition × (L+M+1) basic matrix at the tenth of the twelve Earthly Branches；

0 is (N-(L+M+1)) × (L+1) q null matrix；

V is (L+1) q × (L+1) q basic matrix at the tenth of the twelve Earthly Branches；

U_cIt it is N × (N-(L+M+1)) basic matrix at the tenth of the twelve Earthly Branches；

D is (L+M+1) × (L+1) q singular value battle array；

Structural behavior function and optimization problem accordingly

$J_0=s_N^H(k-d){\mathrm{Qs}}_N(k-d)=s^{H}Qs---\left(2\right)$

$\hat{s}=\underset{\hat{s}\∈{\{\±1\}}^N}{\arg min}\left\{J_0\right\}---\left(3\right)$

Wherein, s ∈ { ± 1}^NIt is N-dimensional vector, affiliated character set ± 1},Represent the estimated value of signal.Arg min () table Showing variate-value when making object function take minima, d is delay factor, d=0 ..., M+L.So, blind Detecting problem just becomes The globally optimal solution problem of formula (3).

Fig. 1 is the double Sigmoid non-linear self-feedback chaotic neural network system construction drawing of the present invention, comprises weight matrix mould Block, two activation primitives, integrator, decay factor, coupling factor and self feed back item.

A.) dynamical equation of this system is:

$\frac{{\mathrm{dy}}_i\left(t\right)}{dt}=f\left({\mathrm{\ϵy}}_i\right(t)+\mathrm{\α}\[\underset{j=1}{\overset{n}{\Σ}}{w}_{ij}{x}_j(t)+I_i\]-{\mathrm{\λz}}_i(t\left)g\right(x_i\left(t\right)-I_0\left)\right)---\left(4\right)$

x_i(t)=σ (y_i(t)) (5)

z_i(t+1)=(1-β) z_i(t) (6)

Wherein, this chaotic neural network is made up of N number of neuron；T is the time run in network iterative process, this network In continuous time t and discrete time k between can pass through the mutual phase transformation of Euler's formula, y_iT () is i-th god in neutral net Internal state through unit；I represents i-th neuron, and it is the most whole in [0, N] that j represents jth neuron, i ≠ j, and i, j Number；σ (.) is first Sigmoid function of neuron, and f (.) is second Sigmoid function of neuron, and g (.) is this net The non-linear self-feedback item of network；ε is the decay factor of this network, and 0≤ε≤1, w_ijMix for double Sigmoid non-linear self-feedbacks The interconnection weights of neuron j to neuron i in ignorant neutral net, and w_ij=w_ji；α is the coupling factor of this network, I_iIt is i-th The biasing of individual neuron, I₀Biasing for initial neuron；z_iT () is the self-feedback connection weights value of i-th neuron, λ is neural Decay factor between unit, and λ > 0, β is variable z_iThe decay factor of (t)；x_iT () is the output of i-th neuron；This neutral net When reaching finally to balance, the x of each neuron can be approximately considered_i(t)=y_i(t), x_iT () is the transmission signal asked for；

Herein the activation primitive of double Sigmoid non-linear self-feedback chaotic neural networks is designed as:

σ (s)=tanh (c s) (7)

$f\left(s\right)=\frac{1}{1+e^{-s}}---\left(8\right)$

Wherein, c is the built-in adjustment parameter of activation primitive, the derivative of f (.) derivative much smaller than σ (.)；

Non-linear self-feedback item as herein described design g (.) is:

G (s)=tanh (s) (9)

B.) energy function

In double Sigmoid chaotic neural networks that the employing formula (4) shown in Fig. 1, formula (5), formula (6) describe, if this net Network is made up of N number of neuron, w_ijMeet w_ij=w_ji, and w_ii>0；Variable z_iT decay factor β of () meets β > 0, Sigmoid letter The derivative of number σ (.) and f (.) is all respectively greater than zero, then the energy function of this neutral net is expressed as:

E (t)=E_hopfield+E_add (10)

$E_{hopfield}=-\frac{\mathrm{\α}}{2}\underset{i=1}{\overset{n}{\Σ}}\underset{j=1}{\overset{n}{\Σ}}{x}_i\left(t\right)w_{ij}{x}_j\left(t\right)-\mathrm{\ϵ}\underset{i=1}{\overset{n}{\Σ}}{\∫}_0^{x_i\left(t\right)}{\mathrm{\σ}}_i^{-1}\left(\mathrm{\τ}\right)d\mathrm{\τ}-\mathrm{\α}\underset{i=1}{\overset{n}{\Σ}}{I}_i{x}_i\left(t\right)---\left(11\right)$

$E_{add}=\mathrm{\λ}\underset{i=1}{\overset{n}{\Σ}}{z}_i\left(t\right){\∫}_0^{x_i\left(t\right)}g(\mathrm{\τ}-I_0)d\mathrm{\τ}---\left(12\right)$

Wherein: E (t) is energy function, this energy function is by E_hopfieldAnd E_addTwo parts form, E_hopfieldFor commonly The energy function item of Hopfield neutral net, E_addFor double Sigmoid non-linear self-feedback chaotic neural network energy functions Additional-energy item.Energy function is one and the related variable of iteration time,Sigmoid letter for i-th neuron Number σ_i(τ) inverse function.

In sum, this network circulates every time and is all introduced into chaotic neural network structure and has jumped out after local minimum point again Enter second activation primitive, chaotic neural network and second activation primitive and just constitute a double Sigmoid chaos nerve Network, not only ensure that network can be avoided local minimum point but also the convergence rate of network is accelerated, and finally reached the flat of network Weighing apparatus.

For utilizing double Sigmoid non-linear self-feedback chaotic neural network to realize signal blind Detecting, solve formula (2), (3) Signal blind Detecting problem, the minimum point of energy function to be made is corresponding to the minimum point of blind Detecting performance function.Due to Euler Formula can make mutually to change between continuous time and discrete time, when network reaches stable, can be approximately considered x_i(t) =y_iT (), compares Part I and the performance function formula (2) of energy function formula (11), then can be seen that one negative sign of difference, so It is contemplated that the weight matrix of the double non-linear chaotic neural network of Sigmoid of design is projection operator form W=I_N-Q, wherein I_NBe N × The unit matrix of N-dimensional,Thus make the minimum point of energy function E (t) corresponding to blind Detecting performance function (2) Minimum point such that it is able to realize signal blind Detecting with double Sigmoid non-linear self-feedback chaotic neural networks.

Fig. 2 and Fig. 3 is present invention signal blind Detecting based on double Sigmoid non-linear self-feedback chaotic neural networks respectively The emulation experiment figure of method.Here emulation uses the classical documents channel without Common zero, and transmission signal is binary phase-shift Keying signal, noise is white Gaussian noise, and all simulation results all obtain through 100 Monte Carlo Experiments.

Fig. 2 is in the case of condition is identical, inventive algorithm and Hopfield neutral net Blind Detect Algorithm and transient state The bit error rate comparison diagram of chaotic neural network Blind Detect Algorithm.Fig. 3 present invention divides with transient chaotic neural network (TCNN) algorithm The not bit error rate comparison diagram under the conditions of different data lengths.Fig. 3. (a) is transient chaotic neural network (TCNN) blind check measuring and calculating Method bit error rate variation diagram under the conditions of different data lengths, Fig. 3. (b) is double Sigmoid non-linear self-feedback chaos nerve net Network (DS-NSCNN) Blind Detect Algorithm bit error rate variation diagram under the conditions of different data lengths.HNN (Hopfield in figure Neural Network) algorithm is Hopfield neural network algorithm, TCNN (Transiently Chaotic Neural Network) algorithm is Chaotic Neutral Network Algorithm, DS-NSCNN (Nonlinear Self-feedback Chaotic Neural Network With Double Sigmoid) algorithm is that double Sigmoid non-linear self-feedback chaotic neural network is calculated Method.Analogous diagram shows: the double Sigmoid non-linear self-feedback Chaotic Neutral Network Algorithm of the present invention is than Hopfield algorithm and transient state Chaotic neural network Blind Detect Algorithm has more preferable error performance, and the present invention is based on double Sigmoid non-linear self-feedbacks Chaotic Neutral Network Algorithm desired data amount when blind Detecting is more shorter than transient chaotic neural network algorithm.

Claims (1)

1. a non-linear self-feedback chaotic neural network signal blind checking method, it is characterised in that its step is as follows:

Step A, structure reception data matrix:

Receiving terminal receives unique user and sends signal, through over-sampling, it is thus achieved that the reception equation of discrete-time channel:

X_N=S Γ^T

In formula, X_NBeing to receive data matrix, S is to send signal battle array, and Γ is to be rung h by channel impulse_jjThe block Toeplitz matrix constituted； (·)^TRepresenting matrix transposition；

Wherein,

$S={\[s_{L+M}\left(k\right),...,s_{L+M}\left(k+N-1\right)\]}_{N\×\left(L+M+1\right)}^T={\[s_N\left(k\right),...,s_N\left(k-M-L\right)\]}_{N\×\left(L+M+1\right)},$

M is channel exponent number, and L is equalizer exponent number, and N is desired data length；

s_L+M(k)=[s (k) ..., s (k-L-M)]^T；Wherein, { ± 1}, moment k are natural number to s ∈；

h_jj=[h₀,…,h_M]_q×(M+1), jj=0,1 ..., M；

Q is oversample factor, and value is positive integer；

X_N=[x_L(k),…,x_L(k+N-1)]^TIt is that N × (L+1) q receives data matrix, wherein x_L(k)=Γ s_L+M(k)；

Step B, reception data matrix singular value decomposition:

$X_N=\[U,U_c\]\·\left[\begin{array}{c}D\\ 0\end{array}\right]\·V^H$

In formula,

(·)^HIt it is Hermitian transposition；

U is the N in singular value decomposition × (L+M+1) basic matrix at the tenth of the twelve Earthly Branches；

0 is (N-(L+M+1)) × (L+1) q null matrix；

V is (L+1) q × (L+1) q basic matrix at the tenth of the twelve Earthly Branches；

U_cIt it is N × (N-(L+M+1)) basic matrix at the tenth of the twelve Earthly Branches；

D is (L+M+1) × (L+1) q singular value battle array；

Step C, arranges weight matrix W=I_N-Q, wherein I_NIt is the unit matrix of N × N-dimensional,

Step D, changes the linear self feed back item in Chen ' s chaotic neural network into non-linear self-feedback item, in order to accelerate system Convergence rate uses double Sigmoid structure；

Double Sigmoid non-linear self-feedback chaotic neural network dynamical equations are:

$\frac{{\mathrm{dy}}_i\left(t\right)}{dt}=f\left({\mathrm{\ϵy}}_i\left(t\right)+\mathrm{\α}\[\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{w}_{ij}{x}_j\left(t\right)+I_i\]-{\mathrm{\λz}}_i\left(t\right)g\left(x_i\left(t\right)-I_0\right)\right)$

x_i(t)=σ (y_i(t))

z_i(t+1)=(1-β) z_i(t)

The equation is iterated computing, then the result of each iteration is substituted into double Sigmoid non-linear self-feedback chaos god In energy function E (t) of network, when this energy function E (t) minimizes value, i.e. y_i(t)=y_i(t-1) time, this pair Sigmoid non-linear self-feedback chaotic neural network reaches balance, and iteration terminates；

Wherein,

y_iT () is the internal state of i-th neuron in neutral net；I represents i-th neuron, and j represents jth neuron, i ≠ j, and i, j be arbitrary integer in interval [0, N]；T is the time that neutral net iterative process is run, in this neutral net Continuous time t and discrete time k by Euler's formula realize conversion；

ε is the decay factor of this network, and 0≤ε≤1；w_ijFor the interconnection weights of neuron j in this network to neuron i, and w_ij=w_ji；α is the coupling factor of this network；I_iFor the biasing of i-th neuron, I₀It it is the biasing of initial neuron；

z_iT () is the self-feedback connection weights value of i-th neuron, λ is decay factor between neuron, and λ > 0, and β is variable z_i(t) Decay factor；

x_iT () is the output of i-th neuron；When this neutral net reaches finally to balance, the x of each neuron can be approximately considered_i (t)=y_i(t), x_iT () is the transmission signal asked for；

σ (.) is first Sigmoid activation primitive of this neutral net, and f (.) is that second Sigmoid of this neutral net swashs Function alive；

And:

σ (s)=tanh (c s)

$f\left(s\right)=\frac{1}{1+e^{-s}}$

Wherein, s is the input of neutral net, and c is the built-in adjustment parameter of activation primitive, and the derivative of f (.) is led much smaller than σ's (.) Number；

G (.) is the non-linear self-feedback item of this neutral net:

G (s)=tanh (s)

Energy function E (t) of described pair of Sigmoid non-linear self-feedback chaotic neural network is:

E (t)=E_hopfield+E_add

$E_{hopfield}=-\frac{\mathrm{\α}}{2}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{x}_i\left(t\right)w_{ij}{x}_j\left(t\right)-\mathrm{\ϵ}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^{x_i\left(t\right)}{\mathrm{\σ}}_i^{-1}\left(\mathrm{\τ}\right)d\mathrm{\τ}-\mathrm{\α}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{I}_i{x}_i\left(t\right)$

$E_{add}=\mathrm{\λ}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i\left(t\right){\∫}_0^{x_i\left(t\right)}g\left(\mathrm{\τ}-I_0\right)d\mathrm{\τ}$

Wherein:

This chaotic neural network is made up of N number of neuron；E (t) is energy function, and this energy function is by E_hopfieldAnd E_addTwo It is grouped into, E_hopfieldFor the energy function item of common Hopfield neutral net, E_addMix for double Sigmoid non-linear self-feedbacks The additional-energy item of ignorant neural network energy function；

Sigmoid function σ for i-th neuron_i(τ) inverse function.