CN108009347B - Time-frequency analysis method based on synchronous compression joint improvement generalized S transformation - Google Patents

Time-frequency analysis method based on synchronous compression joint improvement generalized S transformation Download PDF

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CN108009347B
CN108009347B CN201711240195.7A CN201711240195A CN108009347B CN 108009347 B CN108009347 B CN 108009347B CN 201711240195 A CN201711240195 A CN 201711240195A CN 108009347 B CN108009347 B CN 108009347B
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芮义斌
严丽萍
谢仁宏
李鹏
郭山红
吕云涛
杜禹
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Nanjing University of Science and Technology
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Abstract

The invention discloses a time-frequency analysis method based on synchronous compression joint improvement generalized S transformation, which introduces synchronous compression transformation on the basis of window function improved generalized S transformation, and rearranges signal energy along the frequency direction on a time-frequency plane so as to lead the energy to be gathered on instantaneous frequency. The specific implementation comprises the following steps: the method comprises the steps of sampling a signal, performing fast Fourier transform, determining a window function and a partial derivative of the window function to time according to the obtained frequency spectrum, performing improved generalized S transform on the signal by using the window function and the partial derivative thereof respectively to obtain instantaneous frequency and time frequency spectrum of the signal, and performing energy rearrangement on the time frequency spectrum according to the instantaneous frequency to obtain a high-resolution time frequency spectrum. The method not only keeps the advantages of wide application and high resolution of the improved generalized S transform, but also contains the characteristic of high energy aggregation of synchronous compression, and is a high-performance time-frequency analysis method.

Description

Time-frequency analysis method based on synchronous compression joint improvement generalized S transformation
Technical Field
The invention belongs to the technical field of signal processing, and particularly relates to a time-frequency analysis method based on synchronous compression joint improvement generalized S transformation.
Background
Non-stationary signals are the most common signals in radar signal processing, and time-frequency analysis is an important tool for analyzing such signals. In order to accurately analyze the local characteristics of the signals, the time-frequency analysis maps the one-dimensional time-domain signals to a two-dimensional time-frequency plane, thereby obtaining the time-frequency distribution of the signals. Currently, common time-frequency analysis methods mainly include short-time fourier transform (STFT), Wavelet Transform (WT), S-transform (ST), and the like.
Dennis Gabor proposed a short-time fourier transform in 1946, the basic idea being to implement a segmented fourier transform of a signal by windowing, resulting in a time-varying characteristic of the signal. However, the window function used for STFT is fixed, independent of time and frequency, and is a single-resolution analysis method. The idea of wavelet transformation is derived from a telescopic and translation method, is a time-frequency localization analysis method with a fixed window area and a changeable shape, and can adaptively adjust a time-frequency window according to characteristics of high and low frequency signals. But the wavelet base design difficulty is higher, the constraint of tolerance conditions is also realized, and the defects of insufficient time-frequency resolution, complex scale frequency conversion and the like exist at the same time. To compensate for the deficiencies of the short-time fourier transform and the wavelet transform, Stockwell proposed an S-transform, introducing a variable gaussian window function, with the time window width inversely proportional to the frequency derivative. The frequency resolution of the time frequency spectrum obtained by the method is high in the low-frequency part, and is low in the high-frequency part, namely the resolution is variable. However, the inverse relationship causes the window function to have the problems of too wide and too narrow window length locally, which results in time positioning failure at low frequency and frequency positioning failure at high frequency.
The patent application number is CN201610946585.5, the name of which is Chinese patent of 'seismic frequency spectrum imaging method based on deconvolution generalized S transformation', and the time frequency spectrum is obtained by performing two-dimensional convolution on original signals and respective Weiganan distribution of a Gaussian window. The method can suppress the generation of cross terms of Wigner-Ville distribution, and simultaneously enables the generalized S transform spectrum to obtain higher time-frequency aggregation, but has the limitation that the problem of insufficient resolution at low frequency and high frequency cannot be solved.
The limited window long-time and long-frequency analysis method for improving S transformation provided previously realizes window length control under the condition of variable window function, has better time-frequency resolution, but is still limited by the Heisenberg uncertainty principle, and the time resolution and the frequency resolution cannot reach the optimum.
From the above, the existing time-frequency analysis method has the defects, and further improvement is needed to realize high-precision time-frequency analysis.
Disclosure of Invention
The invention aims to provide a time-frequency analysis method based on synchronous compression joint improvement generalized S transformation.
The technical solution for realizing the purpose of the invention is as follows: a time-frequency analysis method based on synchronous compression joint improvement generalized S transformation comprises the following steps:
step 1, performing fast Fourier transform on an input signal to obtain a signal frequency spectrum;
step 2, determining values of control factors a, b and c in the improved generalized S transformation according to the signal frequency spectrum and the analysis requirement to obtain a window function expression;
step 3, performing improved generalized S transformation on the signals to obtain time-frequency distribution MGST (t, f);
step 4, carrying out derivation on the window function to obtain a partial derivative corresponding to each frequency point;
step 5, taking the window function partial derivative as a new window function to carry out generalized S transformation on the signal, and calculating the instantaneous frequency v by combining a threshold valueMGST(t,f)；
Step 6, synchronously compressing the time frequency plane signals to obtain high-resolution time frequency distribution;
compared with the prior art, the invention has the following remarkable advantages: 1) the method comprises the steps of improving generalized S transformation, adjusting a window function according to frequency self-adaptation, limiting the change range of the window length through an arc tangent function, and overcoming the defect of insufficient resolution in high and low frequency areas; 2) for different types of signals, the high-resolution time-frequency analysis can be realized by adjusting the value of the control factor, so that the method has strong flexibility; 3) the invention carries out energy rearrangement on the time-frequency signal through synchronous compression, improves the energy aggregation and overcomes the limitation of the Heisenberg uncertain principle.
The present invention is described in further detail below with reference to the attached drawing figures.
Drawings
FIG. 1 is a flow chart of a time-frequency analysis method based on synchronous compression joint improved generalized S-transform according to the present invention.
Fig. 2 is a time-frequency analysis result diagram of signal improved generalized S-transform in embodiment 1 of the present invention.
Fig. 3 is a time-frequency analysis result diagram of the signal synchronous compression combined with the improved generalized S transform in embodiment 1 of the present invention.
Fig. 4 is a time-frequency analysis result diagram of signal improved generalized S-transform in embodiment 2 of the present invention.
Fig. 5 is a time-frequency analysis result diagram of the signal synchronous compression joint improved generalized S transform in embodiment 2 of the present invention.
Fig. 6 is a time-frequency analysis result diagram of signal improved generalized S-transform in embodiment 3 of the present invention.
Fig. 7 is a time-frequency analysis result diagram of the signal synchronous compression combined with the improved generalized S transform in embodiment 3 of the present invention.
Detailed Description
With reference to fig. 1, the time-frequency analysis method based on the synchronous compression joint improved generalized S-transform of the present invention includes the following steps:
step 1, performing fast Fourier transform on an input signal to obtain a signal frequency spectrum;
step 2, determining values of control factors a, b and c in the improved generalized S transformation according to the signal frequency spectrum to obtain a window function expression; the method comprises the following specific steps:
step 2-1, determining a time window value range [ delta t ] according to the signal frequency spectrummin,Δtmax]Where Δ t isminFor minimum time window length, Δ tmaxFor the maximum time window length, the value ranges of a and c are determined by the following inequality:
Figure BDA0001489689580000031
step 2-2, determining the values of a and c in the value range, and taking
Figure BDA0001489689580000032
Wherein
Figure BDA0001489689580000033
fsFor the sampling frequency, and the values of a, b, c are substituted into the window length control function:
Figure BDA0001489689580000034
wherein f is a frequency variable in time-frequency analysis;
step 2-3, a window length control function is brought into the window function to obtain an improved window function expression:
Figure BDA0001489689580000035
wherein t is a time variable in time-frequency analysis, and f is a frequency variable.
And 3, performing improved generalized S transformation on the signals to obtain time-frequency distribution MGST (t, f), wherein the formula is as follows:
Figure BDA0001489689580000036
where τ is the integral variable.
Step 4, derivation is conducted on the window function to obtain a partial derivative corresponding to each frequency point, and the formula is as follows:
Figure BDA0001489689580000037
step 5, taking the window function partial derivative as a new window function to carry out generalized S transformation on the signal, and calculating the instantaneous frequency v by combining a threshold valueMGST(t,f)；
Step 5-1, taking the window function partial derivative as a new window function, and carrying out generalized S transformation on the signal to obtain:
Figure BDA0001489689580000041
step 5-2, setting a reference threshold value gamma, and calculating the instantaneous frequency v according to the standardMGST(t, f), the instantaneous frequency calculation formula is:
Figure BDA0001489689580000042
and step 6, synchronously compressing the time-frequency plane signals to obtain high-resolution time-frequency distribution.
Step 6-1, calculating delta v for the ith frequency point on the instantaneous frequency axisl，Δvl＝vl-vl-1Where l is [1, N ]]N is the total number of signal sampling points, and a rearrangement interval [ v ] is determinedl-Δvl,vl+Δvl]；
Step 6-2, carrying out energy rearrangement to obtain the SSTMGST(t,vl) The calculation formula is as follows:
Figure BDA0001489689580000043
wherein v islFor synchronizing the compressed frequencies, fkTo improve the discretized frequency points on the generalized S-transform spectrum, Δ fk＝fk-fk-1，k∈[1,N]；
And 6-3, repeating the step 6-1 and the step 6-2 until all the instantaneous frequency points are calculated to obtain a time-frequency result.
On one hand, the method comprises the steps of improving generalized S transformation, realizing high-resolution time-frequency analysis by adjusting the value of a control factor, and having strong flexibility; on the other hand, synchronous compression transformation is added, and a high-resolution time-frequency result is realized by performing energy rearrangement on signals on a time-frequency plane.
The present invention will be described in further detail with reference to examples.
Example 1
The simulation signal is the superposition of two sinusoidal signals, the signal frequency is 100Hz and 400Hz respectively, and the analytic formula is as follows:
h(t)＝sin(200πt)+sin(800πt)t∈[0,1]
signal sampling frequency fs1024Hz, fig. 2 shows the result of the improved generalized S transform, and fig. 3 shows the simulation result of the simultaneous compression joint improved generalized S transform, where the threshold γ is 0.001. In FIG. 2, the frequency components are distributed in bands, and after the combination of synchronous compression, the frequency components areThe thin straight line type has higher time-frequency resolution, and the original edge divergence problem is also greatly improved. And comparing with the color bar in the figure, it can be obviously seen that the amplitude of the frequency component in the time-frequency analysis result after synchronous compression is much larger than that without compression, so that the conclusion can be drawn: through synchronous compression, the signal energy gathering performance is obviously improved, and the readability of time-frequency distribution is greatly improved.
Example 2
The simulation signal is a linear frequency modulation signal with the frequency modulation slope k being 400, and the analytic formula is as follows:
Figure BDA0001489689580000051
signal sampling frequency fs1024Hz, fig. 4 shows the result of the improved generalized S transform, and fig. 5 shows the simulation result of the simultaneous compression joint improved generalized S transform, where the threshold γ is 0.001. From the analysis results, it can be seen that the time-frequency analysis based on the window function is always limited by the Heisenberg uncertainty principle, and even if the control factor is adjusted to be optimal and the window function performance is continuously improved, the problem of signal energy divergence cannot be solved, and the frequency curve on the time-frequency distribution always has a certain bandwidth. The time-frequency analysis combined with the synchronous compression well solves the defects of the time-frequency analysis based on the window function, effectively concentrates the energy in a certain frequency range to the instantaneous frequency, and solves the problem that the signal edge is fuzzy in S transformation.
Example 3
The nonlinear frequency modulation signal is a signal with sine variation frequency, and the analytic expression is as follows:
h(t)＝ej2π[6cos(10πt)+260t]t∈[0,1]
signal sampling frequency fs1024Hz, fig. 6 shows the result of the improved generalized S transform, and fig. 7 shows the simulation result of the simultaneous compression joint improved generalized S transform, where the threshold γ is 0.02. For the nonlinear frequency modulation signal, each part of the frequency change has higher resolution, and the problem of serious signal energy dispersion at the original position with violent frequency change is solved better. Therefore, the same asThe time-frequency analysis method of the step compression combined improved generalized S transformation has great advantages when analyzing linear frequency modulation signals or non-linear frequency modulation signals.

Claims (3)

1. A time-frequency analysis method based on synchronous compression joint improvement generalized S transformation is characterized by comprising the following steps:
step 1, performing fast Fourier transform on an input signal x (t) to obtain a signal frequency spectrum;
step 2, determining values of control factors a, b and c in the improved generalized S transformation according to the signal frequency spectrum to obtain a window function expression;
step 3, performing improved generalized S transformation on the signals to obtain time-frequency distribution MGST (t, f), wherein t is a time variable in time-frequency analysis, and f is a frequency variable;
step 4, deriving the window function to obtain a partial derivative g' (t, f) corresponding to each frequency point, which specifically comprises:
Figure FDA0002903549810000011
wherein w (f) is a window length control function;
step 5, taking the window function partial derivative as a new window function to carry out generalized S transformation on the signal, and calculating the instantaneous frequency v by combining a threshold valueMGST(t, f), comprising the steps of:
step 5-1, taking the window function partial derivative as a new window function, and carrying out generalized S transformation on the signal to obtain:
Figure FDA0002903549810000012
step 5-2, setting a reference threshold value gamma, and calculating the instantaneous frequency vMGST(t, f), the instantaneous frequency calculation formula is:
Figure FDA0002903549810000013
step 6, synchronously compressing the time-frequency plane signal to obtain high-resolution time-frequency distribution, which specifically comprises the following steps: step 6-1, calculating delta v for the ith frequency point on the instantaneous frequency axisl，Δvl＝vl-vl-1Where l is [1, N ]]N is the total number of signal sampling points, and a rearrangement interval [ v ] is determinedl-Δvl，vl+Δvl]；
Step 6-2, carrying out energy rearrangement to obtain the SSTMGST(t，vl) The calculation formula is as follows:
Figure FDA0002903549810000014
wherein v islFor synchronizing the compressed frequencies, fkTo improve the discretized frequency points on the generalized S-transform spectrum, Δ fk＝fk-fk-1，k∈[1，N]；
And 6-3, repeating the step 6-1 and the step 6-2 until all the instantaneous frequency points are calculated to obtain a time-frequency result.
2. The time-frequency analysis method based on the synchronous compression joint improvement generalized S transform as claimed in claim 1, wherein in step 2, the values of the control factors a, b, c in the improvement generalized S transform are determined according to the signal spectrum to obtain the window function expression, and the specific steps are as follows:
step 2-1, determining a time window value range [ delta t ] according to the signal frequency spectrummin，Δtmax]Where Δ t isminFor minimum time window length, Δ tmaxFor the maximum time window length, the value ranges of a and c are determined by the following inequality:
Figure FDA0002903549810000021
step 2-2, determining the values of a and c in the value range, andget
Figure FDA0002903549810000022
Wherein
Figure FDA0002903549810000023
fsFor the sampling frequency, and the values of a, b, c are substituted into the window length control function:
Figure FDA0002903549810000024
wherein f is a frequency variable in time-frequency analysis;
and 2-3, substituting the window length control function into the window function to obtain an improved window function, wherein the formula is as follows:
Figure FDA0002903549810000025
wherein t is a time variable in time-frequency analysis, and f is a frequency variable.
3. The time-frequency analysis method based on the synchronous compression joint improvement generalized S transform as claimed in claim 1, wherein the improved generalized S transform is performed on the signal in step 3 to obtain a time-frequency distribution MGST (t, f), and the formula is as follows:
Figure FDA0002903549810000026
where τ is the integral variable and w (f) is the window length control function.
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