CN103136443A - Method for estimating weak signal amplitude under alpha noise background - Google Patents

Abstract

The invention relates to a method for estimating weak signal amplitude under an alpha noise background and belongs to the field of nonlinear signal processing. The method comprises the steps of generating a signal to be tested in a simulation mode, establishing a Duffing chaotic system, setting parameters of the Duffing system, using a 4-order Range-Kuttle algorithm to solve a chaotic solution of the Duffing system when the signal to be tested is not added, using the almost periodicity to judge a chaotic threshold of the system, inputting the signal to be tested into the Duffing chaotic system, obtaining a chaotic threshold of the system by the same method and recording the chaotic threshold as Fs, and subtracting Fs from the obtained system chaotic threshold Fo, so that the amplitude B of the signal to be tested can be calculated. The method is simple in algorithm, low in calculated amount and capable of guaranteeing the validity and the accuracy of amplitude estimation and improving the working efficiency.

Description

A kind of feeble signal Amplitude Estimation method under alpha noise background

Technical field

The invention belongs to the nonlinear properties disposal route, what relate to is a kind of feeble signal Amplitude Estimation method, specifically a kind of method of utilizing the Duffing chaos system that the feeble signal amplitude is estimated under the stable non-Gaussian noise background that distributes of alpha.

Background technology

In signals transmission, inevitably will introduce noise, what usually run into is all the additivity non-Gaussian noise, sea, the ground clutter that receives as airborne early warn ing radar, the man made noise in communication and work in sonar system in high reverberation etc., these noises can be described with stable distribution of alpha, are called again the alpha noise.

The stable distribution of α described with its fundamental function, as the formula (7).When having parameter 0＜α≤2, δ〉0 ,-1≤β≤1 and real number μ make stochastic process X satisfy following formula:

Wherein

$\mathrm{\&omega;}(t,\mathrm{\&alpha;})=\left\{\begin{array}{cc}\tan (\mathrm{\&alpha;\&pi;}/2)& \mathrm{\&alpha;}\&NotEqual;1\\ (2/\mathrm{\&pi;}\log |t\left|\right)& \mathrm{\&alpha;}=1\end{array},\right.$ $\mathrm{sgn}\left(t\right)=\left\{\begin{array}{cc}1& t>0\\ 0& t=0\\ -1& t<0\end{array}\right.$

Claim X to obey the stable distribution of α, be designated as X～S _α(β, δ, μ).

Stable 4 parameters that distribute of α have own clear and definite implication:

1, characterization factor α also referred to as characteristic exponent, is well-determined, is used for weighing the hangover thickness situation of distribution function.Obey the stable stochastic process that distributes of α for one, when α is less, illustrate that its corresponding distribution has heavier hangover, so it to depart from the sample point of intermediate value or average just more, when α≤1, this stable distribution has unlimited average and variance.And when α larger, its corresponding distribution more trends towards Gaussian process, when α=2, this stable distribution is Gaussian distribution, the distribution when α=1 becomes Cauchy and distributes.Only having when 0＜α＜2, is just non-Gaussian distribution.

2, scale parameter δ also referred to as dispersion coefficient, is to describe sample data with respect to the degree of scatter of sample median or average, is equivalent to the variance in Gaussian distribution, and when α=2, the value of δ is half of variance yields;

3, deflection parameter beta is used for weighing the gradient of distribution.Represent to be distributed as symmetrical when β=0, note the S into S α by abridging, as β representing to distribute 0 the time is tilted to the right, representing to distribute when β＜0 is tilted to the left, and this paper mainly studies symmetrical α and stablizes the distribution stochastic process;

4, location parameter μ distributes for the α of a symmetry is stable, and when 0＜α≤1, μ represents the intermediate value of this distribution, and when 1＜α≤2, μ represents the average of this distribution.α is stable is distributed with dividing of symmetrical (S α S) and asymmetric distribution two large classes.When β=0, S _α(β, δ, μ) is symmetrical about μ, when μ=0, and S _α(β, δ, μ) is called Standard Symmetric Multivariate and distributes.

The stable distribution of Alpha has the character after the spike tail.Therefore be necessary to study the algorithm that feeble signal is estimated under the alpha noise background.

Detection of Weak Signals has a wide range of applications in every field such as radar, fault diagnosis, communication, System Discriminations.We have following two kinds of implications usually for the understanding of " feeble signal ":

A kind ofly refer to that the power of useful signal is very faint with respect to noise, usually recently describe with noise.This is a relative concept, when being mixed with the random noise of varying strength in the useful signal of same intensity, can produce different effects.In general when S/N be that 1:1(is 0dB) time, roughly can see the overall picture of useful signal; S/N is approximately 6dB of 2:1() time, be quality preferably; S/N is that 1:2(useful signal approximately-6dB) is weak signal, is feeble signal and S/N is 1:4(useful signal approximately-12dB).This order of magnitude may some variation in different field;

Another kind refers to that the amplitude of useful signal is minimum, as the signal of amplitude on the order of magnitude of nV, pV.

Here said feeble signal is under the first implication, and namely signal to noise ratio (S/N ratio) is lower than the signal of-12dB.In the information age, it is an important means of people's obtaining information that signal is detected with Estimation of Parameters.The signal parameter estimation technique has application widely in many fields such as biomedicine, geoscience, galvanochemistry at present.Along with the development of science and technology, the demand of feeble signal being carried out Estimation of Parameters is also day by day urgent, and is significant to the development that promotes association area.

At present, the proposition of the feeble signal algorithm for estimating under various noise conditions provides strong reference and reference for carrying out of this research work.Wherein the structure thought of a class algorithm comes from design of filter, namely by Optimal Filter, extracts immediate signal trajectory from noise cancellation signal is arranged, but under the non-Gaussian noise background, the feeble signal parameter is estimated, the performance of wave filter is significantly degenerated; Another kind of algorithm is to extract useful parameter in signal by the conversion of signal, and modal in engineering is exactly Fast Fourier Transform (FFT) (FFT) method.

Fourier transform is the important tool of analytic signal and system performance, has a wide range of applications in the signal process field, and it has provided the frequency spectrum concept of signal, and can be used to system is carried out frequency-domain analysis.Most of signal all comprises a plurality of different frequency contents in actual life, and these signal frequencies can be along with the times or changed soon or slowly.Fourier analysis and Fourier transform are the mathematical tools of the frequency characteristic of analytical cycle or nonperiodic signal.

From the time angle, Fourier transform comprises the Fourier transform of continuous time and discrete time.Continuous fourier transform is difficult for by sequencing, so our said Fourier transform all refers to discrete Fourier transformation usually.The weak point of discrete Fourier transformation is that its calculated amount is too large, is difficult to realize real-time processing.If want to calculate the discrete Fourier transformation that a N is ordered, generally need N ²The inferior complex addition of inferior complex multiplication and N (N-1).Therefore, when N is larger, carry out real-time processing operation to signal, its arithmetic speed often is difficult to meet the demands.In nineteen sixty-five, J.W.Cooly and J.W.Tukey have found a kind of method of quick calculating discrete Fourier transformation, after again through other scholars' further improvement, formed rapidly an effective operational method of cover, the Fast Fourier Transform (FFT) that Here it is commonly uses now is called for short FFT (The Fast Fourier Transform).The essence of FFT is to utilize weight function in formula (8)

Symmetry and periodically, leaf transformation in the N point discrete Fourier is carried out a series of decomposition and combination, make whole discrete Fourier transformation computation process become a series of interative computation process, the operand of discrete Fourier transformation is greatly simplified, for good condition has been created in real-time processing and the application of discrete Fourier transformation and digital signal.

$x\left(k\right)=\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}x\left(n\right)\exp [-\frac{2\mathrm{\&pi;kn}}{N}],(k=0,1...N-1)---\left(8\right)$

In actual applications, using FFT to carry out Estimation of Parameters to signal is one of method the most widely, under certain signal to noise ratio (S/N ratio), estimates all effective for the signal parameter of Gaussian noise and non-Gaussian noise background.The integrated module of fft algorithm in present digital processing chip makes and uses FFT on engineering signal analysis is become simple.But there is certain error in the method to the Amplitude Estimation of feeble signal under the non-Gaussian noise background, and precision is not high.

In addition, due to chaos system to the susceptibility of starting condition with to the immunity of noise, for detection and the Estimation of Parameters of feeble signal in the alpha non-Gaussian noise provides new method.Chaos detection is based on state of chaotic system and changes and come detection signal, so the differentiation for the chaos system threshold value becomes key issue, the inherent driving force of corresponding system when the Chaotic Threshold here refers to that chaos system is changed to cycle status by chaos state, this moment, system still was in chaos state, but as long as the inherent driving force of system increases the change that will cause system state a little.Chaos system threshold value criterion method is divided into two classes substantially: intuitive analysis method and quantitative analysis method.Wherein, the intuitive analysis method comprises time history method, phase-plane diagram method, power spectrum method etc., has the characteristics such as simple, that calculated amount is little, differentiation is convenient, but can't realize automatic identification, need the people for defining, be vulnerable to the subjective impact of experimenter, error is larger; Quantitative analysis method comprises Lyapunov performance index, Andrei Kolmogorov entropy, fractal dimension etc., has advantages of that degree of accuracy is high, and is easy to realize the automatic search threshold value, yet the sizing technique calculation of complex, calculated amount is larger, the programming difficulty, real-time is poor, is unfavorable for Project Realization.

Summary of the invention

The invention provides a kind of feeble signal Amplitude Estimation method under alpha noise background, have certain error to solve Amplitude Estimation, precision is not high, and calculated amount is larger, the programming difficulty, and real-time is poor, is unfavorable for the problem of Project Realization.Purpose is under the alpha noise background, on the basis that guarantees accuracy of detection, by introducing the almost periodic characteristic of Duffing chaos system, the real-time and the dirigibility that further improve feeble signal amplitude detection system.

The technical scheme that the present invention takes is:

1., produce the Weak Sinusoidal Signal that is flooded by the alpha noise, as measured signal;

2., build the Duffing chaos system;

3., the Duffing systematic parameter is set;

4., when not adding measured signal, adopt 4 rank Runge-Kutta algorithms to ask the Chaotic Solution of Duffig system, utilize its Almost Periodic criterion chaotic systems threshold value, be designated as F _o

5., measured signal is inputted in the Duffing chaos system, the Chaotic Threshold of the system that obtains that uses the same method also is designated as F _s

6., with the chaotic systems threshold value F that obtains _oAnd F _sSubtract each other, can calculate the amplitude B of measured signal.

Step of the present invention 1. in:

The α noise sequence generates in numerical simulation, at first wants given 4 parameters; Suppose that the characteristic exponent that will produce is α (0＜α＜2), symcenter is β (1＜β＜1), and scale parameter is 1, and location parameter is 0 α noise

Be expressed as follows:

$\stackrel{\&OverBar;}{x}=\left\{\begin{array}{cc}S\&CenterDot;\frac{\sin \mathrm{\&alpha;}(U+B)}{{\left(\cos U\right)}^{\mathrm{\&alpha;}/2}}{\left[\frac{\cos (U-\mathrm{\&alpha;}(U+B))}{W}\right]}^{\frac{1-\mathrm{\&alpha;}}{\mathrm{\&alpha;}}}& \mathrm{\&alpha;}\&NotEqual;1\\ \frac{2}{\mathrm{\&pi;}}[(\frac{\mathrm{\&pi;}}{2}+\mathrm{\&beta;U})\tan U-\mathrm{\&beta;}\ln \frac{\frac{\mathrm{\&pi;}}{2}W\cos U}{\frac{\mathrm{\&pi;}}{2}+\mathrm{\&beta;U}}]& \mathrm{\&alpha;}=1\end{array}\right.---\left(1\right)$

Wherein, W obeys the exponential distribution of λ=1, and U obeys Even distribution,

$S={(1+{\mathrm{\&beta;}}^2{\tan }^2\left(\frac{\mathrm{\&pi;\&alpha;}}{2}\right))}^{\frac{1}{2\mathrm{\&alpha;}}},$ $B=\frac{1}{\mathrm{\&alpha;}}\arctan \left(\mathrm{\&beta;}\tan \left(\frac{\mathrm{\&pi;\&alpha;}}{2}\right)\right);$

For the α noise x of any scale parameter δ and location parameter μ as shown in the formula:

$x=\mathrm{\&delta;}\stackrel{\&OverBar;}{x}+\mathrm{\&mu;}---\left(2\right)$

So loop and find the solution, can obtain the stable sequence x (n) that distributes of α;

Weak Sinusoidal Signal is joined in the alpha noise that emulation produces be measured signal.

Step of the present invention 2. in, the step of building the Duffing chaos system is:

The expression formula of Duffing equation is

x″(t)+kx′(t)+f(x,t)=Fcos(wt) （3）

In formula, k represents damping ratio, 0＜k＜1;

F (x, t) represents nonlinear restoring force;

Fcos (wt) is built-in signal, and F is built-in driving force;

If nonlinear terms f (x, t)=-α x ³+ β x ⁵, can obtain better input effect this moment, and for convenience of calculation is established α=β=1, substitution formula (3) obtains formula (4):

x″+kx′-x ³+x ⁵=Fcos(wt) （4）

When setting up the system emulation pattern, need the state equation of Duffing equation, be expressed as

$\left\{\begin{array}{c}{x}^{\&prime;}=\mathrm{wy}\\ y^{\&prime;}=w(-\mathrm{ky}+x^3-x^5+F\cos \left(\mathrm{wt}\right))\end{array}---\left(5\right)\right.$

Add the mathematical model after measured signal to be

$\left\{\begin{array}{c}{x}^{\&prime;}=\mathrm{wy}\\ y^{\&prime;}=w(-\mathrm{ky}+x^3-x^5+F\cos \left(\mathrm{wt}\right)+s)\end{array}---\left(6\right)\right.$

In formula (6), s is the measured signal of input, and s (t)=Bcos (wt)+n (t), B are the amplitude of measured signal, and w is and the measured signal frequency of system same frequency that n (t) represents the alpha noise.

Step of the present invention arranges the Duffing systematic parameter in 3.: system angle frequency w=1 is set, and inherent driving force F=0, step-length is Initial value is (x ₀, y ₀)=(0,0), simulation time length is

Step of the present invention 4. in: regulate by turn F value in the Duffing chaos system, obtain corresponding chaos system solution, utilize its Almost Periodic to judge whether system reaches the chaos critical conditions, and Almost Periodic is defined as:

When f:R → X is a continuous function, if for ε arbitrarily〉0, all exist set T (f, ε)=τ: || f (t+ τ)-f (t) ||＜ε, t ∈ R} is relatively intensive, we claim that f is almost periodic so, that is, for all ε〉0, can find a real number l=l (ε)〉0, be to have a number τ=τ (ε) in l (ε) in each length, satisfy || f (t+ τ)-f (t) ||＜ε

As described in this definition, x (t) is divided into the n group, remember that every group is X (i), i=1,2,3 ..., n, wherein every group of integral multiple that data volume must be the cycle.Calculate difference d (i) between every group of data=|| X (i+1)-X (i) ||, i=1,2,3 ..., n-1; When the Duffing equation entered cycle status, d (i) should be very little and stable, and when being in chaos state, d (i) should be larger; If d (i) ≈ d (i-1), system enters cycle status so, records threshold value, if system does not reach requirement, changes by turn the F value, until meet the demands, note chaotic systems threshold value at this moment is F _s

The invention has the beneficial effects as follows: first produce the feeble signal that contains the non-Gaussian distribution noise of alpha, as measured signal input Duffing chaos system, the Almost Periodic of recycling Chaotic Solution is processed the output data, respectively two groups of data before and after the input measured signal are carried out Chaotic Threshold and judge, two Chaotic Threshold that obtain are subtracted each other the amplitude that can obtain feeble signal.Adopt the method criterion Chaotic Threshold of the Almost Periodic of Chaotic Solution, its algorithm is simple, and calculated amount is little, and can guarantee validity and the accuracy of Amplitude Estimation, adopt the mode of search by turn when the search Chaotic Threshold, can reduce search time like this, increase work efficiency.

Description of drawings

Fig. 1 is that standard S α S stablizes distribution series figure;

Fig. 2 is signal and noise sequential chart;

Fig. 3 is the graph of a relation of F-d.

Embodiment

Below in conjunction with accompanying drawing and for example technical scheme of the present invention being described in further detail, key step is as follows:

1, emulation produces measured signal

The α noise sequence often generates in numerical simulation need to utilize specific algorithm, at first wants given 4 parameters.Suppose that the characteristic exponent that will produce is α (0＜α＜2), symcenter is β (1＜β＜1), and scale parameter is 1, and location parameter is 0 α noise

Be expressed as follows:

$\stackrel{\&OverBar;}{x}=\left\{\begin{array}{cc}S\&CenterDot;\frac{\sin \mathrm{\&alpha;}(U+B)}{{\left(\cos U\right)}^{\mathrm{\&alpha;}/2}}{\left[\frac{\cos (U-\mathrm{\&alpha;}(U+B))}{W}\right]}^{\frac{1-\mathrm{\&alpha;}}{\mathrm{\&alpha;}}}& \mathrm{\&alpha;}\&NotEqual;1\\ \frac{2}{\mathrm{\&pi;}}[(\frac{\mathrm{\&pi;}}{2}+\mathrm{\&beta;U})\tan U-\mathrm{\&beta;}\ln \frac{\frac{\mathrm{\&pi;}}{2}W\cos U}{\frac{\mathrm{\&pi;}}{2}+\mathrm{\&beta;U}}]& \mathrm{\&alpha;}=1\end{array}\right.---\left(1\right)$

Wherein, W obeys the exponential distribution of λ=1, and U obeys

Even distribution,

$S={(1+{\mathrm{\&beta;}}^2{\tan }^2\left(\frac{\mathrm{\&pi;\&alpha;}}{2}\right))}^{\frac{1}{2\mathrm{\&alpha;}}},$ $B=\frac{1}{\mathrm{\&alpha;}}\arctan \left(\mathrm{\&beta;}\tan \left(\frac{\mathrm{\&pi;\&alpha;}}{2}\right)\right).$

For the α noise x of any scale parameter δ and location parameter μ as shown in the formula:

$x=\mathrm{\&delta;}\stackrel{\&OverBar;}{x}+\mathrm{\&mu;}---\left(1\right)$

So loop and find the solution, can obtain the stable noise sequence x (n) that distributes of α.

Setting parameter α=0.5, β=0, δ=1, μ=0 o'clock, emulation produces the α noise sequence in the Matlab platform, by the kurtosis that calculates it be about 797.12, the degree of bias is about 26.52.

Weak Sinusoidal Signal is joined in the alpha noise that emulation produces be measured signal.

2, build the Duffing chaos system

The general expression of Duffing equation is

x″(t)+kx′(t)+f(x,t)=Fcos(wt) （3）

In formula, k represents damping ratio, 0＜k＜1;

F (x, t) represents nonlinear restoring force;

Fcos (wt) is built-in signal, and F is built-in driving force.

If nonlinear terms f (x, t)=-α x ³+ β x ⁵, can obtain better input effect this moment.For convenience of calculation is established α=β=1, substitution formula (5) obtains formula (6):

x″+kx′-x ³+x ⁵=Fcos(wt) （4）

When setting up the system emulation pattern, need the state equation of Duffing equation, be expressed as

$\left\{\begin{array}{c}{x}^{\&prime;}=\mathrm{wy}\\ y^{\&prime;}=w(-\mathrm{ky}+x^3-x^5+F\cos \left(\mathrm{wt}\right))\end{array}---\left(5\right)\right.$

Add the mathematical model after measured signal to be

$\left\{\begin{array}{c}{x}^{\&prime;}=\mathrm{wy}\\ y^{\&prime;}=w(-\mathrm{ky}+x^3-x^5+F\cos \left(\mathrm{wt}\right)+s)\end{array}---\left(6\right)\right.$

In formula (6), s is the measured signal of input, and s (t)=Bcos (wt)+n (t), B are the amplitude of measured signal, and w is and the measured signal frequency of system same frequency that n (t) represents the alpha noise.

3, the Duffing systematic parameter is set

System angle frequency w=1 is set, inherent driving force F=0, step-length is

Initial value is (x ₀, y ₀)=(0,0), simulation time length is And the degree of accuracy of Chaotic Threshold (setting up on their own as required).

Find the solution Duffing chaos system threshold value when 4, not adding measured signal

When not adding measured signal, adopt 4 rank Runge-Kutta algorithms that the Duffing equation is found the solution.Then adopt the method solving system threshold value based on Almost Periodic criterion chaotic systems threshold value, it is characterized in that: regulate by turn F value in the Duffing chaos system, obtain corresponding chaos system solution x (t), utilize its Almost Periodic to judge whether system reaches the chaos critical conditions, and Almost Periodic is defined as:

When f:R → X is a continuous function, if for ε arbitrarily〉0, all exist set T (f, ε)=τ: || f (t+ τ)-f (t) ||＜ε, t ∈ R} is relatively intensive, we claim that f is almost periodic so.That is, for all ε〉0, can find a real number l=l (ε)〉0, be to have a number τ=τ (ε) in l (ε) in each length, satisfy || f (t+ τ)-f (t) ||＜ε,

As described in this definition, it is divided into the n group, remember that every group is X (i), i=1,2,3 ..., n, wherein every group of integral multiple that data volume must be the cycle.If between every group of data, difference is d (i),

D (i)=|| X (i+1)-X (i) ||, i=1 wherein, 2,3 ..., n-1;

According to following formula, obtain d (i), i=1,2,3 ..., n-1, its mean value is

When

The time, decision-making system is in chaos state, and the F value increases; Otherwise decision-making system enters cycle status, and the F value reduces.If the initial value of F is 0, adopt control method by turn, namely first regulate 1/10th, regulate again one of percentage position after determining, by that analogy, until satisfy system requirements.The former threshold value of note chaotic systems is F _o

5, add measured signal, again search for Duffing chaos system Chaotic Threshold

The feeble signal that will be under the alpha noise background joins in the Duffing chaos system, again searches for the Chaotic Threshold of Duffing chaos system, and the initial value that establish F this moment is F _o, according to the described method criterion of step 4 Chaotic Threshold, note chaotic systems threshold value this moment is F _s

6, calculate measured signal amplitude B=F _o-F _s

Feeble signal Amplitude Estimation method provided by the invention, employing is based on the algorithm for estimating of the Almost Periodic of Duffing chaos system solution, guaranteed to detect exactly and Estimation of Parameters under the alpha noise background, adopt the algorithm of Almost Periodic to avoid complicated calculating, accuracy, fitness and the speed of method is merged preferably.

It should be noted that at last, above embodiment is only in order to describe technical scheme of the present invention rather than the present technique method is limited, the present invention not only can estimate effectively the feeble signal under the alpha noise, effective too to the feeble signal under other non-Gaussian noise backgrounds, therefore can extend to other modification, variation, application and embodiment on using, and think that all such modifications, variation, application, embodiment are in spirit of the present invention and teachings.

Experimental example:

At first produce with chaos system with measured signal s (t) frequently=0.03cos (t)+n (t), wherein n (t) is parameter alpha=0.77, β=0, δ=0.5, μ=0, the degree of bias is 54.056 the stable non-Gaussian noise that distributes of alpha for-1.489., kurtosis, and the sequential chart of signal and α noise as shown in Figure 2.Figure center line 1 represents the alpha noise, and line 2 represents that angular frequency is 1, and amplitude is 0.03 sinusoidal signal.

Then set up the chaos system realistic model, system angle frequency w=1 is set, inherent driving force F=0, step-length is

Initial value is (x ₀, y ₀)=(0,0), simulation time length is

And the degree of accuracy of Chaotic Threshold (it is 6 that this example arranges significant figure).

When not adding measured signal, obtain Chaotic Threshold Fc=0.733616;

Add afterwards measured signal, ask the difference of Chaotic Solution

When The time, decision-making system is in chaos state, and the F value increases; Otherwise decision-making system enters cycle status, and the F value reduces.If the initial value of F is 0, adopt control method by turn, namely first regulate 1/10th, regulate again one of percentage position after determining, by that analogy, if reached the significant figure of setting calculated the Amplitude Estimation value this moment, otherwise regulate the F value, continue the search Chaotic Threshold.

Under this noise background environment, utilize the threshold value F of search Duffing chaos system _nObtain the graph of a relation of F-d, as shown in Figure 3.Find new threshold value F _n=0.703616, can obtain thus the Amplitude Estimation value of measured signal $\hat{B}=0.733616-0.703616=0.030000,$ Relative error is 9.25186e-014%.

According to experimental result as can be known, utilize the detection method of chaology under the non-Gaussian noise background, the amplitude of Weak Sinusoidal Signal to be estimated it is feasible.Because there is not limited second moment in the α noise, cause the variance of noise to become nonsensical, therefore can't recently weigh method for detecting chaotic oscillator to the effect of feeble signal Estimation of Parameters by noise, so this example adopts with the method for the maximum fast Fourier method estimated amplitude of use on engineering and compares, and weighs Duffing oscillator estimation effect to the feeble signal amplitude under the alpha noise background.

When the parameter alpha of alpha noise=0.7, β=0, μ=0 remains unchanged, δ changes since 0.1, be 0.1 to amplitude respectively with chaotic oscillator method and Fast Fourier Transform (FFT) method, angular frequency is that the Weak Sinusoidal Signal of 1rad/s carries out the amplitude Estimation of Parameters, and estimated result is as shown in table 1.

Table 1 chaotic oscillator method and FFT estimated result contrast table

The value of kurtosis K and degree of bias S can find out that the α noise has very strong pulse character from table 1, under the interference of this ground unrest, FFT method commonly used is general less than normal and relative error is larger to the Amplitude Estimation of Weak Sinusoidal Signal, and the estimated value of Duffing chaotic oscillator is all bigger than normal and relative error is less, this shows, the Amplitude Estimation method of Duffing chaotic oscillator is better than the FFT method.

Claims (5)

1. feeble signal Amplitude Estimation method under an alpha noise background, is characterized in that comprising the following steps:

1., produce the Weak Sinusoidal Signal that is flooded by the alpha noise, as measured signal;

2., build the Duffing chaos system;

3., the Duffing systematic parameter is set;

4., when not adding measured signal, adopt 4 rank Runge-Kutta algorithms to ask the Chaotic Solution of Duffing system, utilize its Almost Periodic criterion chaotic systems threshold value, be designated as F _o

5., measured signal is inputted in the Duffing chaos system, the Chaotic Threshold of the system that obtains that uses the same method also is designated as F _s

6., with the chaotic systems threshold value F that obtains _oAnd F _sSubtract each other, can calculate the amplitude B of measured signal.

2. feeble signal Amplitude Estimation method under a kind of alpha noise background according to claim 1 is characterized in that: step 1. in:

The α noise sequence generates in numerical simulation, at first wants given 4 parameters; Suppose that the characteristic exponent that will produce is α (0＜α＜2), symcenter is β (1＜β＜1), and scale parameter is 1, and location parameter is 0 α noise

Be expressed as follows:

$\stackrel{\&OverBar;}{x}=\left\{\begin{array}{cc}S\&CenterDot;\frac{\sin \mathrm{\&alpha;}(U+B)}{{\left(\cos U\right)}^{\mathrm{\&alpha;}/2}}[\frac{\cos (U-\mathrm{\&alpha;}(U+B))}{W}{]}^{\frac{1-\mathrm{\&alpha;}}{\mathrm{\&alpha;}}}& \mathrm{\&alpha;}\&NotEqual;1\\ \frac{2}{\mathrm{\&pi;}}[(\frac{2}{\mathrm{\&pi;}}+\mathrm{\&beta;U})\tan U-\mathrm{\&beta;}\ln \frac{\frac{\mathrm{\&pi;}}{2}W\cos U}{\frac{\mathrm{\&pi;}}{2}+\mathrm{\&beta;U}}]& \mathrm{\&alpha;}=1\end{array}\right.---\left(1\right)$

Wherein, W obeys the exponential distribution of λ=1, and U obeys

Even distribution,

$S={(1+{\mathrm{\&beta;}}^2{\tan }^2\left(\frac{\mathrm{\&pi;\&alpha;}}{2}\right))}^{\frac{1}{2\mathrm{\&alpha;}}},$ $B=\frac{1}{\mathrm{\&alpha;}}\arctan \left(\mathrm{\&beta;}\tan \left(\frac{\mathrm{\&pi;\&alpha;}}{2}\right)\right);$

For the α noise x of any scale parameter δ and location parameter μ as shown in the formula:

$x=\mathrm{\&delta;}\stackrel{\&OverBar;}{x}+\mathrm{\&mu;}---\left(2\right)$

So loop and find the solution, can obtain the stable sequence x (n) that distributes of α;

Weak Sinusoidal Signal is joined in the alpha noise that emulation produces be measured signal.

3. feeble signal Amplitude Estimation method under a kind of alpha noise background according to claim 1 is characterized in that: step 2. in, the step of building the Duffing chaos system is:

The expression formula of Duffing equation is

x″(t)+kx′(t)+f(x,t)=Fcos(wt) （3）

In formula, k represents damping ratio, 0＜k＜1;

F (x, t) represents nonlinear restoring force;

Fcos (wt) is built-in signal, and F is built-in driving force;

If nonlinear terms f (x, t)=-α x ³+ β x ⁵, can obtain better input effect this moment, and for convenience of calculation is established α=β=1, substitution formula (3) obtains formula (4):

x″+kx′-x ³+x ⁵=Fcos(wt) （4）

When setting up the system emulation pattern, need the state equation of Duffing equation, be expressed as

$\left\{\begin{array}{c}{x}^{\&prime;}=\mathrm{wy}\\ y^{\&prime;}=w(-\mathrm{ky}+x^3-x^5+F\cos \left(\mathrm{wt}\right))\end{array}---\left(5\right)\right.$

Add the mathematical model after measured signal to be

$\left\{\begin{array}{c}{x}^{\&prime;}=\mathrm{wy}\\ y^{\&prime;}=w(-\mathrm{ky}+x^3-x^5+F\cos \left(\mathrm{wt}\right)+s)\end{array}---\left(6\right)\right.$

In formula (6), s is the measured signal of input, and s (t)=Bcos (wt)+n (t), B are the amplitude of measured signal, and w is and the measured signal frequency of system same frequency that n (t) represents the alpha noise.

4. feeble signal Amplitude Estimation method under a kind of alpha noise background according to claim 1 is characterized in that: step arranges the Duffing systematic parameter in 3.: system angle frequency w=1 is set, and inherent driving force F=0, step-length is

Initial value is (x ₀, y ₀)=(0,0), simulation time length is

5. feeble signal Amplitude Estimation method under a kind of alpha noise background according to claim 1, it is characterized in that: step 4. in, regulate by turn F value in the Duffing chaos system, obtain corresponding chaos system solution, utilize its Almost Periodic to judge whether system reaches the chaos critical conditions, and Almost Periodic is defined as:

When f:R → X is a continuous function, if for ε arbitrarily〉0, all exist set T (f, ε)=τ: || f (t+ τ)-f (t) ||＜ε, t ∈ R} is relatively intensive, we claim that f is almost periodic so, that is, for all ε〉0, can find a real number l=l (ε)＞0, be to have a number τ=τ (ε) in l (ε) in each length, satisfy || f (t+ τ)-f (t) ||＜ε

As described in this definition, x (t) is divided into the n group, remember that every group is X (i), i=1,2,3 ..., n, wherein every group of integral multiple that data volume must be the cycle; Calculate difference d (i) between every group of data=|| X (i+1)-X (i) ||, i=1,2,3 ..., n-1; When the Duffing equation entered cycle status, d (i) should be very little and stable, and when being in chaos state, d (i) should be larger; If d (i) ≈ d (i-1), system enters cycle status so, records threshold value, if system does not reach requirement, changes by turn the F value, until meet the demands, note chaotic systems threshold value at this moment is F _s

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