CN105373708A - Parameter optimization based time frequency analysis method for improved generalized S-transform - Google Patents

Abstract

The invention discloses a parameter optimization based time frequency analysis method for an improved generalized S-transform. The parameter optimization based time-frequency analysis method comprises the steps of firstly selecting the range of parameters in the improved generalized S-transform; for each group of parameters, calculating time frequency distribution of an input signal by using the improved generalized S-transform and carrying out energy normalization processing on the time frequency distribution; then calculating the corresponding time frequency aggregation degree of each group of parameters and employing one group of parameters corresponding to maximum values as optimized parameters; finally introducing the optimized parameters into the improved generalized S-transform to calculate the time frequency distribution of the input signal. According to the parameter optimization based time frequency analysis method for the improved generalized S-transform, a first-order function with frequency as an independent variable is introduced into a window function of the S-transform so that the form of the window function is flexible and adjustable, the parameters of the first-order function are optimized through the time frequency aggregation degree, the energy aggregation of the signal time frequency distribution can be improved, the estimated accuracy of signal instantaneous parameters is enhanced, the calculated amount is small, the parameter optimization based time frequency analysis method is suitable for analysis and processing of communication, radar, earthquake and biomedicine signals.

Description

A kind of Time-Frequency Analysis Method of the improvement generalized S-transform based on parameter optimization

Technical field

The invention belongs to digital processing field, be related specifically to the analytical approach of non-stationary signal, based on the Time-Frequency Analysis Method of the improvement generalized S-transform of parameter optimization, can be used for the analysis and treament of communication, radar, earthquake and biomedicine signals.

Background technology

In Practical Project, extensively there is various types of non-stationary signal, the analysis and treament of non-stationary signal is a study hotspot of digital processing field.Different from stationary signal, the frequency of non-stationary signal changes in time, and therefore analyzing it needs to adopt time domain and frequency-domain combined two-dimentional Time-Frequency Analysis Method.

Conventional Time-Frequency Analysis Method has Short Time Fourier Transform (STFT), Wigner-Ville distribution (WVD), wavelet transformation (WT) and S-transformation (ST).The window function of Short Time Fourier Transform is fixed, and fix in the frequency resolution of different frequency range, time-frequency locality is poor; Wigner-Ville distribution has good time-frequency locality, but there is stronger cross term interference; Wavelet transform dimension and signal frequency do not have direct corresponding relation, and when analyzing for signal frequency, physical significance is indefinite; S-transformation combines the advantage of Short Time Fourier Transform and wavelet transformation, and its window width is inverse change with frequency, and its wavelet need not meet admissibility condition, directly corresponding with signal spectrum, has lossless reciprocal.But, because in S-transformation, the form of window function is fixed, cause the time-frequency locality of signal HFS not to be very desirable, make it be restricted in the application.

In order to improve the time-frequency locality of S-transformation, many scholars improve its window function, propose dissimilar generalized S-transform.The people such as such as Manshiha proposed in 1997 and replace parameter f in window function with f/r, and suitably can regulate window function form, it calculates simple, but regulating power is limited; The people such as Pinnegar proposed in 2003 can window function standard deviation adjustable, and the asymmetric hyperbolic curve window function of window, when its high band, window is narrower, selects the good window function of symmetry, reaches the object improving frequency resolution; During low-frequency range, window is wider, selects asymmetric window function, but wherein relates to the calculating of the Hyperbolic Equation of two parameters, comparatively complicated; Chen Xuehua proposed two parameter lambda and p to introduce window function in 2005, utilize λ | f| ^palternative parameter f, further increase the dirigibility that window function regulates, but wherein relate to the exponent arithmetic of frequency f, operand is also larger.

Improvement window function in above-mentioned generalized S-transform respectively has feature, and its parameter regulates more flexible, and calculated amount is also larger, and the selection of parameter is by the restriction of uncertainty principle, and namely temporal resolution and frequency resolution can not reach optimum simultaneously, needs compromise to consider.Therefore, as how less calculated amount, improvement is carried out to window function and to go forward side by side line parameter Automatic Optimal, for the range of application of expansion S-transformation, improve its time frequency analysis performance significant.

Summary of the invention

The technical problem to be solved in the present invention is, for traditional S-transformation above shortcomings, a kind of Time-Frequency Analysis Method of the improvement generalized S-transform based on parameter optimization is proposed, be optimized by the parameter of time-frequency concentration class to function of first order, improve the time frequency resolution of signal entirety with less calculated amount.

The present invention for solving the problems of the technologies described above adopted technical scheme is:

Based on a Time-Frequency Analysis Method for the improvement generalized S-transform of parameter optimization, comprise the following steps:

S1, input non-stationary signal x (t), wherein t is the time, and non-stationary signal comprises simple component and multi-components mixed signal;

S2, window function for the improvement generalized S-transform of the function of first order containing frequency f:

$w_f\left(t\right)=\frac{a\left|f\right|+b}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{t^2{\left(a\right|f|+b)}^2}{2}}---\left(1\right)$

Wherein a is multiplicative regulating parameter, and b is additivity regulating parameter, and a > 0, b>=0 determines the span R of parameter a and b _aand R _b;

S3, to R _aand R _binterior one group of parameter (a, b), utilizes following formula to calculate the time-frequency distributions of input non-stationary signal x (t):

${\mathrm{GS}}_x^{(a,b)}(\mathrm{\τ},f)={\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}x\left(t\right)\frac{a\left|f\right|+b}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{{(\mathrm{\τ}-T)}^{2}a\left|f\right|+b^2}{2}}{e}^{-2\mathrm{\π}ft}dt---\left(2\right)$

Wherein for the improvement generalized S-transform of signal, in formula, t and τ represents the time, and τ is the central point of window function, and control window function position on a timeline, f represents frequency, and t, τ and f are real number;

S4, corresponding for parameter (a, b) in step S3 carry out energy normalized process:

$\stackrel{\‾}{{\mathrm{GS}}_x^{(a,b)}(\mathrm{\τ},f)}=\frac{{\mathrm{GS}}_x^{(a,b)}(\mathrm{\τ},f)}{\sqrt{{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}{{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}\left|{\mathrm{GS}}_x^{(a,b)}(\mathrm{\τ},f)\right|}^{2}d\mathrm{\τ}df}}---\left(3\right)$

S5, utilize step S4 normalized after time-frequency distributions expression formula calculate the time-frequency concentration class of whole time-frequency distributions:

$CM(a,b)=\frac{1}{{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}\left|\stackrel{\‾}{{\mathrm{GS}}_x^{(a,b)}(\mathrm{\τ},f)}\right|d\mathrm{\τ}df}---\left(4\right)$

S6, repetition step S3 ~ S5, traversal R _aand R _ball parameters (a, b) in span, obtain a series of CM (a, b), get one group of parameter corresponding to wherein maximal value as the most optimized parameter:

${(a,b)}_{opt}=\arg \underset{(a,b)}{max}CM(a,b)---\left(5\right)$

S7, the most optimized parameter (a, b) that step S6 is obtained _optbring formula (2) into, calculate the improvement generalized S-transform of input non-stationary signal x (t):

${\mathrm{GS}}_x^{(a,b)}(\mathrm{\τ},f)={\mathrm{GS}}_x^{{(a,b)}_{opt}}(\mathrm{\τ},f)---\left(6\right)$

Formula (6) is the improvement generalized S-transform after parameter optimization.

Further, in described step S2, improve generalized S-transform and come from basic S-transformation, the basic S-transformation of signal x (t) is:

$S(\mathrm{\τ},f)={\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}x\left(t\right)\frac{\left|f\right|}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{{(t-\mathrm{\τ})}^2{f}^2}{2}}{e}^{-2\mathrm{\π}ft}dt---\left(7\right)$

Its window function is:

$g_f\left(t\right)=\frac{\left|f\right|}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{t^2{f}^2}{2}}---\left(8\right)$

Here introduced by the function of first order of frequency f, the window function of the generalized S-transform that is improved is such as formula described in (1), and wherein the span of a and b is: 0 < a≤2,0≤b≤20.

Further, in described step S3, make t=kT, τ=jT, obtain discrete formula as follows:

${\mathrm{GS}}_x^{(a,b)}(jT,\frac{n}{NT})=\left\{\begin{array}{cc}\underset{k=0}{\overset{N-1}{\&Sigma;}}x\left(kT\right)\frac{1}{\sqrt{2\mathrm{\&pi;}}}\left(a\frac{\left|n\right|}{NT}+b\right)e^{-\frac{{(k-j)}^2{T}^2}{2}{\left(a\frac{\left|n\right|}{NT}+b\right)}^2}{e}^{-i2\mathrm{\&pi;}\frac{nk}{N}},& n\&NotEqual;0\\ \frac{1}{N}\underset{k=0}{\overset{N-1}{\&Sigma;}}x\left(kT\right){\mathrm{be}}^{-\frac{{(k-j)}^2{T}^2}{2}{b}^2},& n=0,b\&NotEqual;0\\ \frac{1}{N}\underset{k=0}{\overset{N-1}{\&Sigma;}}x\left(kT\right),& n=0,b=0\end{array}\right.---\left(9\right)$

Wherein T is signal sampling interval.

Further, described step S5 and S6 utilizes overall time-frequency concentration class to be optimized parameter, namely once utilizes the value in all temporal frequency unit of time frequency plane to carry out the calculating of time-frequency concentration class, the optimum of the corresponding overall time-frequency distributions of the Optimal Parameters obtained.

Principle of work of the present invention: on the basis that basic S-transformation and existing generalized S-transform are analyzed, with the function of first order alternative parameter f of frequency f in window function, be optimized by the parameter of time-frequency concentration class to function of first order, improve the time frequency resolution of signal entirety with less calculated amount.

Beneficial effect of the present invention:

1, the present invention replaces f with the function of first order containing frequency f in window function, introduce two parameter a and b, in order to regulate the width of window function, wherein a plays main regulating action, b plays fine adjustments effect, its calculating is easy, flexible adjustment, be optimized by the parameter of time-frequency concentration class to function of first order, improve the energy accumulating (overall time frequency resolution) of signal time-frequency distributions with less calculated amount, be particularly useful for the analysis and treament of communication, radar, earthquake and biomedicine signals;

2, the present invention utilizes time-frequency concentration class to carry out the value of Optimal Parameters a and b, and each value of all temporal frequency unit of time frequency plane that adopts calculates time-frequency concentration class, to reach the optimum of the overall time-frequency locality energy of signal.

Accompanying drawing explanation

Fig. 1 is the realization flow figure of Time-Frequency Analysis Method of the present invention;

Fig. 2 is the time domain beamformer of signal 1 in embodiment;

Fig. 3 is the time frequency distribution map of signal 1 after basic S-transformation in embodiment;

Fig. 4 is the time frequency distribution map of signal 1 after the improvement generalized S-transform of parameter optimization in embodiment;

Fig. 5 is the time domain beamformer of signal 2 in embodiment;

Fig. 6 is the time frequency distribution map of signal 2 after basic S-transformation in embodiment;

Fig. 7 is the time frequency distribution map of signal 2 after the improvement generalized S-transform of parameter optimization in embodiment.

Embodiment

Below in conjunction with drawings and Examples, technical solution of the present invention is described in detail.

With reference to shown in Fig. 1, the Time-Frequency Analysis Method of the improvement generalized S-transform based on parameter optimization of the present invention, comprises the following steps:

S1, input non-stationary signal x (t), wherein t is the time, and non-stationary signal comprises simple component and multi-components mixed signal;

S2, the window function containing the improvement generalized S-transform of the function of first order of frequency f of expressing for formula (1), determine the span R of parameter a and b _aand R _b;

S3, utilize formula (2) calculate input non-stationary signal x (t) time-frequency distributions;

S4, energy normalized process is carried out to the time-frequency distributions in step S3;

S5, utilize step S4 normalized after time-frequency distributions expression formula (3) calculate the time-frequency concentration class of whole time-frequency distributions;

S6, repetition step S3 ~ S5, traversal R _aand R _ball parameters (a, b) in span, obtain a series of CM (a, b), get one group of parameter corresponding to wherein maximal value as the most optimized parameter (a, b) _opt;

S7, the most optimized parameter (a, b) that step S6 is obtained _optbring formula (2) into, calculate the improvement generalized S-transform after input non-stationary signal x (t) parameter optimization such as formula shown in (6).

The Time-Frequency Analysis Method of the embodiment of the present invention is described for group non-stationary signal of two shown in Fig. 2 and Fig. 5, comprises the steps:

The forms of time and space of S1, input non-stationary signal, wherein 2 signals are respectively:

Signal 1:x ₁(t)=cos20 π t+17 π t ²(10)

Signal 2:x ₂(t)=cos25 π ln10t+1+cos48 π t+8 π t ²(11)

Wherein x ₁t () is simple component signal, x ₂t () is two component signals, and two components intersect mutually, are typical non-stationary signal, and the sampling rate of two signals is 256Hz, and the sampling time is 1s, and namely sampling number is 256 points, x ₁(t) as shown in Figure 2, x ₂t () as shown in Figure 5;

S2, determine the span R improving parameter a and b in window function _aand R _b: 0 < a≤2,0≤b≤20;

S3, the initial value choosing a are 0.05, and stepping is 0.05; The initial value of b is 0, and stepping is 1, and in span, a gets 40 values altogether like this, b gets 21 values altogether, and having 40 × 21=840 group (a, b) value like this needs traversal, for each group (a, b) value, formula (2) is utilized to calculate input signal x ₁(t) and x ₂t the time-frequency distributions of (), is designated as with

S4, utilize formula (3) right with carry out energy normalized process, obtain with

S5, utilize formula (4) to calculate the time-frequency concentration class of whole time-frequency distributions (time frequency plane), obtain with

S6, repetition step S3 ~ S5, traversal R _aand R _ball parameters (a, b) in span, obtain a series of with for each signal, get one group of parameter corresponding to wherein maximal value as the optimized parameter of this signal, obtain ${(a,b)}_{x_{1}opt}=(0.05,15),{(a,b)}_{x_{2}opt}=(0.35,6);$

S7, general with bring formula (2) into, obtain x ₁(t), x ₂improvement generalized S-transform after (t) parameter optimization with respectively as shown in figs. 4 and 7.

Obviously, above-described embodiment is only for example of the present invention is clearly described, and is not the restriction to embodiments of the present invention.For those of ordinary skill in the field, be still among protection scope of the present invention according to spirit institute's apparent change of extending out of the present invention or change.

Claims (4)

1., based on a Time-Frequency Analysis Method for the improvement generalized S-transform of parameter optimization, it is characterized in that, comprise the following steps:

S1, input non-stationary signal x (t), wherein t is the time, and non-stationary signal comprises simple component and multi-components mixed signal;

S2, window function for the improvement generalized S-transform of the function of first order containing frequency f:

$w_f\left(t\right)=\frac{a\left|f\right|+b}{\sqrt{2\mathrm{\&pi;}}}{e}^{-\frac{t^2{\left(a\right|f|+b)}^2}{2}}---\left(1\right)$

Wherein a is multiplicative regulating parameter, and b is additivity regulating parameter, and a > 0, b>=0 determines the span R of parameter a and b _aand R _b;

S3, to R _aand R _binterior one group of parameter (a, b), utilizes following formula to calculate the time-frequency distributions of input non-stationary signal x (t):

${\mathrm{GS}}_x^{(a,b)}(\mathrm{\&tau;},f)={\&Integral;}_{-\mathrm{\&infin;}}^{\mathrm{\&infin;}}x\left(t\right)\frac{a\left|f\right|+b}{\sqrt{2\mathrm{\&pi;}}}{e}^{-\frac{{(t-\mathrm{\&tau;})}^{2}a\left|f\right|+b^2}{2}}{e}^{-2\mathrm{\&pi;}ft}dt---\left(2\right)$

Wherein for the improvement generalized S-transform of signal, in formula, t and τ represents the time, and τ is the central point of window function, and control window function position on a timeline, f represents frequency, and t, τ and f are real number;

S4, corresponding for parameter (a, b) in step S3 carry out energy normalized process:

$\stackrel{\&OverBar;}{{\mathrm{GS}}_x^{(a,b)}(\mathrm{\&tau;},f)}=\frac{{\mathrm{GS}}_x^{(a,b)}(\mathrm{\&tau;},f)}{\sqrt{{\&Integral;}_{-\mathrm{\&infin;}}^{\mathrm{\&infin;}}{\&Integral;}_{-\mathrm{\&infin;}}^{\mathrm{\&infin;}}|{\mathrm{GS}}_x^{(a,b)}(\mathrm{\&tau;},f){|}^{2}d\mathrm{\&tau;}df}}---\left(3\right)$

S5, utilize step S4 normalized after time-frequency distributions expression formula calculate the time-frequency concentration class of whole time-frequency distributions:

$CM(a,b)=\frac{1}{{\&Integral;}_{-\mathrm{\&infin;}}^{\mathrm{\&infin;}}{\&Integral;}_{-\mathrm{\&infin;}}^{\mathrm{\&infin;}}\left|\stackrel{\&OverBar;}{{\mathrm{GS}}_x^{(a,b)}(\mathrm{\&tau;},f)}\right|d\mathrm{\&tau;}df}---\left(4\right)$

S6, repetition step S3 ~ S5, traversal R _aand R _ball parameters (a, b) in span, obtain a series of CM (a, b), get one group of parameter corresponding to wherein maximal value as the most optimized parameter:

${(a,b)}_{opt}=\arg \underset{(a,b)}{max}CM(a,b)---\left(5\right)$

S7, the most optimized parameter (a, b) that step S6 is obtained _optbring formula (2) into, calculate the improvement generalized S-transform of input non-stationary signal x (t):

${\mathrm{GS}}_x^{(a,b)}(\mathrm{\&tau;},f)={\mathrm{GS}}_x^{{(a,b)}_{opt}}(\mathrm{\&tau;},f)---\left(6\right)$

Formula (6) is the improvement generalized S-transform after parameter optimization.

2. the Time-Frequency Analysis Method of the improvement generalized S-transform based on parameter optimization according to claim 1, is characterized in that, in described step S2, improve generalized S-transform and come from basic S-transformation, the basic S-transformation of signal x (t) is:

$S(\mathrm{\&tau;},f)={\&Integral;}_{-\mathrm{\&infin;}}^{\mathrm{\&infin;}}x\left(t\right)\frac{\left|f\right|}{\sqrt{2\mathrm{\&pi;}}}{e}^{-\frac{{(t-\mathrm{\&tau;})}^2{f}^2}{2}}{e}^{-2\mathrm{\&pi;}ft}dt---\left(7\right)$

Its window function is:

$g_f\left(t\right)=\frac{\left|f\right|}{\sqrt{2\mathrm{\&pi;}}}{e}^{-\frac{t^2{f}^2}{2}}---\left(8\right)$

Here introduced by the function of first order of frequency f, the window function of the generalized S-transform that is improved is such as formula described in (1), and wherein the span of a and b is: 0 < a≤2,0≤b≤20.

3. the Time-Frequency Analysis Method of the improvement generalized S-transform based on parameter optimization according to claim 1, is characterized in that, in described step S3, make t=kT, τ=jT, obtain discrete formula as follows:

${\mathrm{GS}}_x^{(a,b)}(jT,\frac{n}{NT})=\left\{\begin{array}{cc}\underset{k=0}{\overset{N-1}{\mathrm{\&Sigma;}}}x\left(kT\right)\frac{1}{\sqrt{2\mathrm{\&pi;}}}(a\frac{\left|n\right|}{NT}+b)e^{-\frac{{(k-j)}^2{T}^2}{2}{(a\frac{\left|n\right|}{NT}+b)}^2}{e}^{-i2\mathrm{\&pi;}\frac{nk}{N}},& n\&NotEqual;0\\ \frac{1}{N}\underset{k=0}{\overset{N-1}{\mathrm{\&Sigma;}}}x\left(kT\right){\mathrm{be}}^{-\frac{{(k-j)}^2{T}^2}{2}{b}^2},& n=0,b\&NotEqual;0\\ \frac{1}{N}\underset{k=0}{\overset{N-1}{\mathrm{\&Sigma;}}}x\left(kT\right),& n=0,b=0\end{array}\right.---\left(9\right)$

Wherein T is signal sampling interval.

4. the Time-Frequency Analysis Method of the improvement generalized S-transform based on parameter optimization according to claim 1, it is characterized in that, described step S5 and S6 utilizes overall time-frequency concentration class to be optimized parameter, namely the value in all temporal frequency unit of time frequency plane is once utilized to carry out the calculating of time-frequency concentration class, the optimum of the corresponding overall time-frequency distributions of the Optimal Parameters obtained.