CN104268125A - Method of Chirp time-frequency atoms denoted with three parameters - Google Patents

Abstract

The invention discloses a simplified representation method of Chirp time-frequency atoms in self-adaption signal decomposition. For representation of time-frequency characteristics of non-stationary signals, time-frequency atoms of a better local time-frequency structure are used for replacing orthogonal basis functions, and linear combination of the best time-frequency atoms is used for representing signals. The time-frequency atoms are obtained through calculation by applying various operators on basic functions, the more operators applied on the basic functions, the more parameters the time-frequency atoms obtain, the stronger local capacity of the time-frequency atoms matching or approaching signals, while the harder the finding of optimal time-frequency atoms. The Chirp time-frequency atoms can well match frequency linear variation elements in the signals and approach nonlinear frequency variation elements. The Chirp time-frequency atoms are commonly represented by 4 parameters of proportion, time shift, frequency shift and linear frequency modulation. The simplified representation method of the Chirp time-frequency atoms in the self-adaption signal decomposition introduces revolve-radial shift operators and composition operators through fractional order Fourier transform to obtain the Chirp time-frequency atoms represented by the three parameters of proportion, revolve and radial shift. According to the simplified representation method of the Chirp time-frequency atoms in the self-adaption signal decomposition, advantages of more concise representation, more definite physical significance of each parameter, great time reduction for searching the optimal time-frequency atoms can be obtained.

Description

The method of the Chirp time-frequency atom of a kind of use three Parametric Representations

Technical field

The present invention relates to a kind of reduced representation method of Chirp time-frequency atom in adaptive signal decomposition.Chirp time-frequency atom generally uses ratio, time shift, frequency displacement, linear frequency modulation four Parametric Representations.The present invention introduces rotation operator, radial displacement operator by Fourier Transform of Fractional Order, can obtain by the Chirp time-frequency atom of ratio, rotation, radial displacement three Parametric Representations.

Background technology

For portraying the local time-frequency characteristics of non-stationary signal, in adaptive signal decomposition, often using the time-frequency atom with better local time-frequency structure to replace the basis function of Orthogonal Decomposition, and describing non-stationary signal with the linear combination of best time-frequency atom.Time-frequency atom carries out various operator operation to basic function to obtain, and makes time-frequency atom mate the local time-frequency characteristic of signal to be portrayed better by these operators, as time-displacement operator and frequency displacement operator determination time-frequency regional area; Ratio operator makes the time Support of basis function and signal and frequency Support mate; Frequency shear operator analytic signal medium frequency variation characteristic in time; Time-shearing operator is then produce different time delays to the signal content of different frequency.The operator acting on basic function is more, and the ability of the signal local time-frequency structure that time-frequency atom can mate or approach is stronger, but also makes the parameter of time-frequency atom increase simultaneously, thus makes the best time-frequency atom of searching more difficult.

Conventional time-frequency atom has Gabor atom, and it is that its expression formula is by ratio operator, time-displacement operator, frequency displacement operator are acted on unit Gaussian function and obtain:

In formula (1), g (t) is unit Gaussian function, and a is scale parameter, shifting parameter when u is, ξ is frequency shift parameters.Gabor atom can represent many non-stationary signals, but because its frequency is constant, low with the matching degree of signal intermediate frequency rate time dependent Chirp composition, use multiple Gabor time-frequency atom could represent a Chirp composition, namely a complete Chirp composition is decomposed and permeates on multiple Gabor time-frequency atom, and this is unfavorable for signal characteristic abstraction and sparse signal representation.

Chirp signal is the time dependent class non-stationary signal of frequency, it is described that a lot of real process and physical phenomenon, and its parameter reflects certain state and the attribute of objective objects.In order to analyze the signal containing a large amount of Chirp composition of outwardness, Chinese scholars adds linear frequency modulation parameter in Gabor time-frequency atom parameter set, proposes the Chirp time-frequency atom of four Parametric Representations.Compared with Gabor time-frequency atom, Chirp time-frequency atom can the composition of frequency linearity change well in matched signal, also can approach the composition of non-linear frequency change.With the Chirp atom of ratio, time shift, frequency displacement, linear frequency modulation four Parametric Representations be:

Here, σ _kfor scale parameter, t _kfor time shifting parameter, ω _kfor frequency shift parameters, β _kfor linear frequency modulation parameter.Chirp time-frequency atom can also use ratio, time shift, frequency displacement, Time-shearing, frequency shear five Parametric Representations.

Summary of the invention

The object of the invention is exactly with the least possible Parametric Representation Chirp time-frequency atom, to reduce the computational complexity of searching for best Chirp atom, shortens search time.

First rotation operator is defined.Signal rotates at time-frequency plane can by after the fraction Fourier conversion that signal carried out to respective angles, then carries out Wei Gena-Willie and distribute (WVD:Wigner Villy Distribution) and reach.By means of fraction Fourier conversion, definition rotation operator is:

Here R φ and be rotation operator and fraction Fourier conversion operator respectively, φ is the anglec of rotation, and fractional order Fourier domain variable r represents, rotates and signal is represented from time domain transform to fractional order Fourier domain, see Fig. 1 (a).

Then radial displacement operator is defined.Postrotational signal is made to move radially along the direction rotated, namely a segment distance ρ is moved at fractional order Fourier domain along r direction, see Fig. 1 (b), represent radial displacement operator with P ρ, then rotation-radial displacement composition operators (see Fig. 1 (c)) can be expressed as:

$\left(P_{\mathrm{\ρ}}{R}_{\mathrm{\φ}}\right)\left(t\right)=\left(P_{\mathrm{\ρ}}{X}^{\mathrm{\φ}}\right)\left(r\right)=X^{\mathrm{\φ}}(r-\mathrm{\ρ})=\sqrt{1-j\cot \mathrm{\φ}}{e}^{\mathrm{j\π}{(r-\mathrm{\ρ})}^2\cot \mathrm{\φ}}\∫x\left(t\right)e^{\mathrm{j\π}{t}^2\cot \mathrm{\φ}-2\mathrm{j\φt}(r-\mathrm{\ρ})\csc \mathrm{\φ}}\mathrm{dt}---\left(4\right)$

As fraction Fourier conversion φ=0 He time to deteriorate to identical transformation respectively the same with common Fourier transform, rotation-radial displacement composition operators P _ρr _φφ=0 He time also deteriorate to time-displacement operator T respectively _ρwith frequency displacement operator F _ρ, see Fig. 1 (d) and Fig. 1 (e), that is:

(P _ρR _φx)(t)＝(T _ρR ₀x)(t)＝(T _ρx)(t)＝x(t-ρ) (5)

$\left(P_{\mathrm{\ρ}}{R}_{\mathrm{\φ}}x\right)\left(t\right)=\left(F_{\mathrm{\ρ}}{R}_{\frac{\mathrm{\π}}{2}}x\right)\left(t\right)=\left(F_{\mathrm{\ρ}}X\right)\left(f\right)=X(f-\mathrm{\ρ})---\left(6\right)$

More special a kind of situation is, just move radially along arbitrary orientation at time-frequency plane non rotating, this is equivalent to carry out respectively time shift that amount of movement is ρ cos φ and ρ sin φ and frequency displacement, sees Fig. 1 (f), at this moment rotates-move radially operator and deteriorate to:

Therefore rotation-radial displacement composition operators is the generalized form of time-displacement operator, frequency displacement operator, fractional order shift operator, produce time shift when the anglec of rotation is 0; Frequency displacement is produced when the anglec of rotation is pi/2.In addition, rotation-radial displacement composition operators can also rotate arbitrarily simultaneously at time-frequency plane, produces to move radially effect on gyrobearing.

Finally, ratio operator and rotation-radial displacement composition operators are acted on unit Gaussian function, obtain the Chirp atom with ratio, rotation, radial displacement three Parametric Representations, its expression formula is as follows:

Here, α represents scale parameter, and φ represents rotation parameter, and ρ represents radial displacement parameter.

Compared to the Chirp time-frequency atom with four Parametric Representations, with the Chirp time-frequency atom of three Parametric Representations, represent more succinct, the explicit physical meaning of each parameter, the time of searching for best time-frequency atom is greatly fewer.

Accompanying drawing explanation

Rotation-radial displacement composition operators that Fig. 1 the present invention proposes and special circumstances schematic diagram thereof

Fig. 2 is the Chirp time-frequency atom of three Parametric Representations and the comparing of the Chirp time-frequency atom of four Parametric Representations

The Chirp time-frequency atom of three Parametric Representations is as shown in Fig. 2 (a), and in Fig. 2 (a), (1) is the result that ratio operator acts on Gaussian function; (2) be effect after rotation operator effect; (3) be the effect of radial displacement.In order to contrast, Fig. 2 (b) is the Chirp time-frequency atom of four Parametric Representations.

Fig. 3 is the step of the Chirp time-frequency atom obtaining three Parametric Representations and the result of each step

5, embodiment

Step 1: ratio operator is acted on Gaussian function, obtains elongated Gabor time-frequency atom;

Step 2: rotation operator is acted on Gabor time-frequency atom, namely carries out fraction Fourier conversion to signal, makes signal rotation to fractional order territory;

Step 3: apply radial displacement operator, even postrotational signal moves radially along the direction rotated;

Step 4: obtain the Chrip atom with scale parameter, rotation parameter, radial displacement parameter three Parametric Representations.

Claims (4)

1., by a Chirp time-frequency atom method for ratio, rotation and radial displacement three Parametric Representations, it is characterized in that:

Rotation operator is introduced by carrying out Fourier Transform of Fractional Order to signal, at this fractional order territory definition radial displacement operator, finally carry out scale operation again, thus obtain by the Chirp time-frequency atom of ratio, rotation and radial displacement three Parametric Representations, replace original Chirp time-frequency atom by ratio, time shift, frequency displacement, linear frequency modulation four Parametric Representations, replace the Chirp time-frequency atom with ratio, time shift, frequency displacement, Time-shearing, frequency shear five Parametric Representations.

2. a kind of Chirp time-frequency atom method by ratio, rotation and radial displacement three Parametric Representations as claimed in claim 1, is characterized in that described signal carries out the method that Fourier Transform of Fractional Order introduces rotation operator and is:

Owing to being equivalent to the combination of frequency shear and Time-shearing in the rotation of time-frequency plane, rotate and not only can obtain the effect of two kinds of shears but also the distortion of the time-frequency Support that two kinds of shears cause can be overcome, and signal rotates at time-frequency plane, by after the fraction Fourier conversion that signal carried out to respective angles, then Wei Gena-Willie distribution (WVD:Wigner Villy Distribution) can be carried out and reaches.By means of fraction Fourier conversion, definition rotation operator is:

Here R _φwith be rotation operator and fraction Fourier conversion operator respectively, φ is the anglec of rotation, and fractional order Fourier domain variable r represents, rotation makes signal represent from time domain and transforms to fractional order Fourier domain.

3. a kind of Chirp time-frequency atom method by ratio, rotation and radial displacement three Parametric Representations as claimed in claim 1, is characterized in that fractional order territory definition radial displacement Operator Method is:

Make postrotational signal move radially along the direction rotated, namely move a segment distance ρ at fractional order Fourier domain along r direction, represent radial displacement operator with P ρ, then rotation-radial displacement composition operators can be expressed as:

$\left(P_{\mathrm{\ρ}}{R}_{\mathrm{\φ}}x\right)\left(t\right)=\left(P_{\mathrm{\ρ}}{X}^{\mathrm{\φ}}\right)\left(r\right)=X^{\mathrm{\φ}}(r-\mathrm{\ρ})=\sqrt{1-j\cot \mathrm{\φ}}{e}^{\mathrm{j\π}{(r-\mathrm{\ρ})}^2\cot \mathrm{\φ}}\∫x\left(t\right)e^{\mathrm{j\π}{t}^2\cot \mathrm{\φ}-2\mathrm{j\πt}(r-\mathrm{\ρ})\csc \mathrm{\φ}}\mathrm{dt}---\left(2\right)$

As fraction Fourier conversion φ=0 He time to deteriorate to identical transformation respectively the same with common Fourier transform, rotation-radial displacement composition operators P _ρr _φφ=0 He time also deteriorate to time-displacement operator T respectively _ρwith frequency displacement operator F _ρ, that is:

(P _ρR _φx)(t)＝(T _ρR ₀x)(t)＝(T _ρx)(t)＝x(t-ρ) (3)

$\left(P_{\mathrm{\ρ}}{R}_{\mathrm{\φ}}x\right)\left(t\right)=\left(F_{\mathrm{\ρ}}{R}_{\frac{\mathrm{\π}}{2}}x\right)\left(t\right)=\left(F_{\mathrm{\ρ}}X\right)\left(f\right)=X(f-\mathrm{\ρ})---\left(4\right)$

More special a kind of situation just moves radially along arbitrary orientation at time-frequency plane non rotating, and this is equivalent to carry out respectively time shift that amount of movement is ρ cos φ and ρ sin φ and frequency displacement, at this moment rotates-moves radially operator and deteriorate to:

$\left(T_{\mathrm{\ρ}\cos \mathrm{\φ}}{F}_{\mathrm{\ρ}\sin \mathrm{\φ}}x\right)\left(t\right)=x(t-\mathrm{\ρ}\cos \mathrm{\φ})e^{-\mathrm{j\π}{\mathrm{\ρ}}^2\sin \mathrm{\φ}\cos \mathrm{\φ}+j2\mathrm{\πt\ρ}\sin \mathrm{\φ}}---\left(5\right)$

Therefore rotation-radial displacement composition operators is the generalized form of time-displacement operator, frequency displacement operator, fractional order shift operator, produce time shift when the anglec of rotation is 0; Frequency displacement is produced when the anglec of rotation is pi/2.In addition, rotation-radial displacement composition operators can also rotate arbitrarily simultaneously at time-frequency plane, produces to move radially effect on gyrobearing.

4. a kind of Chirp time-frequency atom method by ratio, rotation and radial displacement three Parametric Representations as claimed in claim 1, is characterized in that the Chirp time-frequency atom method of three Parametric Representations is:

Act on unit Gaussian function by ratio operator and rotation-radial displacement composition operators, obtain the Chirp atom with ratio, rotation, radial displacement three Parametric Representations, its expression formula is as follows:

$\{g_{\mathrm{\α},\mathrm{\φ},\mathrm{\ρ}}\left(t\right)={\left(\frac{\mathrm{\α}}{\mathrm{\π}}\right)}^{\frac{1}{4}}\exp \{-\frac{\mathrm{\α}}{2}{(t-\mathrm{\ρ}\cos \mathrm{\φ})}^2+j(\mathrm{\ρ}\sin \mathrm{\φ}(t-\mathrm{\ρ}\cos \mathrm{\φ})+\frac{\cot \mathrm{\φ}}{2}{(t-\mathrm{\ρ}\cos \mathrm{\φ})}^2)$

Here, α represents scale parameter, and φ represents rotation parameter, and ρ represents radial displacement parameter.