CN111198357A - S-transform time-frequency analysis method based on adjustable window function - Google Patents

S-transform time-frequency analysis method based on adjustable window function Download PDF

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CN111198357A
CN111198357A CN201911320328.0A CN201911320328A CN111198357A CN 111198357 A CN111198357 A CN 111198357A CN 201911320328 A CN201911320328 A CN 201911320328A CN 111198357 A CN111198357 A CN 111198357A
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芮义斌
焦碧璇
谢仁宏
李鹏
高猛
费海凤
于晴
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Nanjing University of Science and Technology
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Abstract

本发明公开了一种基于可调窗函数的S变换时频分析方法，对输入信号进行时域采样，获得离散序列；对离散序列进行FFT变换，获得信号频谱；根据信号频谱和分辨率要求，确定窗长控制函数以及窗函数；对窗函数进行FFT得到窗函数频谱；对信号频谱进行周期延拓，并将扩维后的信号频谱跟窗函数频谱相乘；对相乘的结果进行傅里叶逆变换，得到单个频率点的时间分布结果；完成所有频率点的计算，最终获得二维矩阵时频谱。本发明能够实现高分辨率时频分析，方法计算量适中，便于工程实现，具有良好的实时性。

Figure 201911320328

The invention discloses an S-transform time-frequency analysis method based on an adjustable window function. The input signal is sampled in the time domain to obtain a discrete sequence; the discrete sequence is subjected to FFT transformation to obtain a signal spectrum; according to the signal spectrum and resolution requirements, Determine the window length control function and the window function; perform FFT on the window function to obtain the window function spectrum; perform periodic extension on the signal spectrum, and multiply the expanded signal spectrum with the window function spectrum; perform Fourier on the multiplied result. Inverse leaf transform, the time distribution result of a single frequency point is obtained; the calculation of all frequency points is completed, and the time spectrum of the two-dimensional matrix is finally obtained. The invention can realize high-resolution time-frequency analysis, and the method has a moderate calculation amount, is convenient for engineering realization, and has good real-time performance.

Figure 201911320328

Description

S-transform time-frequency analysis method based on adjustable window function
Technical Field
The invention relates to a radar signal processing technology, in particular to an S-transform time-frequency analysis method based on an adjustable window function.
Background
The radar echo signals are processed, and information such as the number, the types and the states of the targets can be extracted. However, most of radar echo signals are non-stationary signals and have characteristics such as time-varying property, so that a time-frequency analysis method needs to be researched to determine the change characteristic of frequency along with time. Currently, common time-frequency analysis methods include short-time fourier transform (STFT), Wavelet Transform (WT), wigner-wiler distribution (WVD), S-transform (ST), and the like. The S-transform is proposed by Stockwell on the basis of short-time fourier transform (STFT) and Wavelet Transform (WT). The method adopts a Gaussian window, and sets the window length of the Gaussian window to be inversely proportional to the frequency of a signal to be analyzed, so that the window length is adaptively changed along with the frequency of the signal, and the Gaussian window has better time-frequency analysis performance.
The Chinese patent with the patent application number of CN201710397687.0 and the name of 'synchronous extrusion generalized S transform signal time-frequency decomposition and reconstruction method' integrates four-parameter generalized S transform and synchronous extrusion transform, deduces an inverse transformation formula of the synchronous extrusion generalized S transform, combines the inverse transformation formula with different orders of fractional order Fourier transform, carries out order search by using an angle constraint method, and integrates to obtain an optimal result. The method can obtain a high-precision time-frequency analysis result, but the calculation comprises more variable parameters, the conversion performance is seriously deteriorated when the parameter setting is improper, and meanwhile, the calculation amount is large and the real-time performance is poor.
Disclosure of Invention
The invention aims to provide an S-transform time-frequency analysis method based on an adjustable window function.
The technical solution for realizing the purpose of the invention is as follows: an S-transform time-frequency analysis method based on an adjustable window function comprises the following steps:
step 1: performing time domain sampling on an input signal to obtain a discrete sequence;
step 2: performing FFT (fast Fourier transform) on the discrete sequence to obtain a signal frequency spectrum;
and step 3: determining a window length control function and a window function according to the signal frequency spectrum and the resolution requirement;
and 4, step 4: performing FFT on the window function to obtain a window function frequency spectrum;
and 5: carrying out periodic extension on the signal frequency spectrum, and multiplying the signal frequency spectrum after dimension expansion by a window function frequency spectrum;
step 6: performing inverse Fourier transform on the multiplied result of the step 5 to obtain a time distribution result of a single frequency point;
and 7: and (4) repeating the steps 4-6 until the calculation of all the frequency points is completed, and finally obtaining the time spectrum of the two-dimensional matrix.
Compared with the prior art, the invention has the following remarkable advantages: 1) the window function is adaptively adjusted according to the signal frequency, high-resolution time-frequency analysis can be realized, the calculated amount of the method is moderate, engineering realization is facilitated, and good real-time performance is achieved; 2) the Sigmoid function is introduced into the window function to limit the window length variation range, so that the time-frequency analysis performance at low frequency and high frequency is improved.
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FIG. 1 is a flow chart of the S-transform time-frequency analysis method based on the improved window function of the present invention.
FIG. 2 is a flowchart of an algorithm of the S-transform time-frequency analysis method based on the improved window function.
Fig. 3 is a time-frequency analysis result diagram of the S transform embodiment 1.
FIG. 4 is a time-frequency analysis result diagram in embodiment 1 of the present invention.
Fig. 5 is a time-frequency analysis result diagram of the S transform embodiment 2.
FIG. 6 is a time-frequency analysis result diagram in embodiment 2 of the present invention.
Fig. 7 is a time-frequency analysis result diagram of the S transform embodiment 3.
FIG. 8 is a time-frequency analysis result diagram in embodiment 3 of the present invention.
Detailed Description
The invention will be further described with reference to the accompanying drawings and specific embodiments.
Referring to fig. 1 and 2, the S-transform time-frequency analysis method based on the improved window function includes the following steps:
step 1, performing time domain sampling on an input signal s (t), wherein the sampling frequency is fsThe sampling time interval is
Figure BDA0002326974230000021
The number of sampling points is
Figure BDA0002326974230000022
Wherein t is the signal duration, obtaining a discrete sequence s [ kT ]]，k＝0,1,2,…,N-1；
Step 2, for the discrete sequence s [ kT ]]FFT to obtain signal spectrum
Figure BDA0002326974230000023
n＝0,1,2…,N-1；
Step 3, determining the values of the parameters a, b and c according to the frequency spectrum characteristics, and determining a window length control function and a window function, wherein the specific steps are as follows:
step 3-1, the signal sampling rate is fsGet it
Figure BDA0002326974230000024
Due to the fact that
Figure BDA0002326974230000025
Thus can get
Figure BDA0002326974230000026
Determining the maximum frequency f of a signalmaxMinimum frequency fminAnd the maximum and minimum values Δ f of the width of the frequency domain of the window function allowed by the practical analysismaxAnd Δ fminDetermining the value ranges of a and c by the following inequalities:
Figure BDA0002326974230000031
and 3-2, determining values of a and c in a value range, substituting each parameter value into a window length control function:
Figure BDA0002326974230000032
and 3-3, substituting the window length control function into the Gaussian window function to obtain an improved window function expression:
Figure BDA0002326974230000033
step 4, FFT is carried out on the window function to obtain a window function frequency spectrum
Figure BDA0002326974230000034
The specific formula is as follows:
Figure BDA0002326974230000035
wherein n starts to take a value from 0;
step 5, multiplying the signal frequency spectrum after the dimension expansion by a window function frequency spectrum, and the specific steps are as follows:
step 5-1, making signal frequency spectrum
Figure BDA0002326974230000036
Dimension expansion to obtain signal spectrum
Figure BDA0002326974230000037
Wherein m is 0,1,2,3 … N-1;
step 5-2, the signal frequency spectrum after dimension expansion
Figure BDA0002326974230000038
Multiplying by a window function spectrum G (m, n);
step 6, carrying out inverse Fourier transform on the multiplication result in the step 5-2 to obtain signal time domain information of the frequency point
Figure BDA0002326974230000039
And 7, repeating the step 4, the step 5 and the step 6 until all the frequency points are completely calculated, and obtaining the high-resolution time spectrum. Judging whether all frequency points are calculated or not, wherein the specific mode is to judge whether N is more than or equal to N-1 or not, if not, repeating the steps 4, 5 and 6 after adding 1 to N; and if so, outputting a time-frequency spectrum result.
On one hand, the invention realizes the self-adaptive change of the window length of the window function along with the signal frequency; on the other hand, the frequency window width is controlled to be changed within a certain range by introducing a Sigmoid function, so that the spectrum has higher resolution in the whole time.
The following describes the time-frequency analysis method of the present invention with three signals as examples.
Example 1
The simulation signal is the superposition of four single-frequency sinusoidal signals, the signal frequency is respectively 100Hz, 200Hz, 300Hz and 400Hz, and the analytic formula is as follows:
s1(t)＝ej200πt+ej400πt+ej600πt+ej800πtt∈[0,1]
signal sampling frequency fs1024Hz, so b is 50. The signal has four fixed frequency components, and for a single-frequency signal, only the frequency resolution is considered, and the width of a frequency window is controlled within a small range by taking a to 5 and c to 5. Fig. 4 is a time-frequency spectrum obtained by using an improved S-transform time-frequency analysis method in which a Sigmoid function is used as a window function control function. As can be seen from fig. 4, this method can achieve very good frequency resolution.
Example 2
The simulation signal is the superposition of two Linear Frequency Modulation (LFM) signals, and the analytic formula is as follows:
s(t)＝s1(t)+s2(t)
wherein
Figure BDA0002326974230000041
Signal sampling frequency fs1024Hz, so b is 50. The signal is when t is equal to [0,1 ]]Time fmax＝400，f min0, let the frequency resolution Δ f ∈ [5,6 ]]Therefore, the adjustment factors for improving the S-transform may be taken as a-60 and c-45, respectively. FIG. 6 shows the use of SigThe moid function is an analysis result of the LFM signal by an improved S-transform time-frequency analysis method of the window function control function, and the frequency change of the LFM signal is large. As can be seen from FIG. 6, the method solves the problems of signal divergence and poor energy aggregation at the original S-transform high frequency, and has good time-frequency performance.
Example 3
The simulation signal is a nonlinear frequency modulation signal with sine variation frequency, and the analytic formula is as follows:
s(t)＝ej2π[6cos(10πt)+260t]t∈[0,1]
signal sampling frequency fs1024Hz, so b is 50. The frequency of the signal varies sinusoidally with time, the signal fmax≈450，fminAbout 70, and let the frequency resolution delta f ∈ [10,12 ]]Therefore, the adjustment factors for improving the S-transform may be taken as a-60 and c-90, respectively. FIG. 8 is a time-frequency diagram obtained after the nonlinear frequency-modulated signal is processed by the method of the present invention, and at this time, the time-frequency result can clearly see the change track of the frequency along with the time, and the time-frequency diagram has good time-frequency analysis performance.

Claims (8)

1.一种基于可调窗函数的S变换时频分析方法，其特征在于，包括以下步骤：1. an S-transform time-frequency analysis method based on an adjustable window function, is characterized in that, comprises the following steps: 步骤1：对输入信号进行时域采样，获得离散序列；Step 1: Sampling the input signal in the time domain to obtain a discrete sequence; 步骤2：对离散序列进行FFT变换，获得信号频谱；Step 2: Perform FFT transformation on the discrete sequence to obtain the signal spectrum; 步骤3：根据信号频谱和分辨率要求，确定窗长控制函数以及窗函数；Step 3: Determine the window length control function and the window function according to the signal spectrum and resolution requirements; 步骤4：对窗函数进行FFT得到窗函数频谱；Step 4: perform FFT on the window function to obtain the window function spectrum; 步骤5：对信号频谱进行周期延拓，并将扩维后的信号频谱跟窗函数频谱相乘；Step 5: Periodically extend the signal spectrum, and multiply the expanded signal spectrum by the window function spectrum; 步骤6：对步骤5相乘的结果进行傅里叶逆变换，得到单个频率点的时间分布结果；Step 6: Inverse Fourier transform is performed on the result of the multiplication in step 5 to obtain the time distribution result of a single frequency point; 步骤7：重复步骤4-6，直至完成所有频率点的计算，最终获得二维矩阵时频谱。Step 7: Repeat steps 4-6 until the calculation of all frequency points is completed, and finally the spectrum of the two-dimensional matrix is obtained. 2.根据权利要求1所述的基于可调窗函数的S变换时频分析方法，其特征在于，步骤1中，对输入信号s(t)进行时域采样，采样频率为fs，采样时间间隔为
Figure FDA0002326974220000011
采样点数为
Figure FDA0002326974220000012
其中t为信号时长，获得离散序列s[kT]，k＝0,1,2,…,N-1。2. the S transform time-frequency analysis method based on adjustable window function according to claim 1, is characterized in that, in step 1, time domain sampling is carried out to input signal s (t), sampling frequency is f s , sampling time interval is
Figure FDA0002326974220000011
The number of sampling points is
Figure FDA0002326974220000012
Among them, t is the signal duration, and a discrete sequence s[kT] is obtained, k=0, 1, 2, ..., N-1. 3.根据权利要求1所述的基于可调窗函数的S变换时频分析方法，其特征在于，步骤2中，对离散序列s[kT]进行FFT变换，获得信号频谱
Figure FDA0002326974220000013
3. the S transform time-frequency analysis method based on adjustable window function according to claim 1, is characterized in that, in step 2, carry out FFT transformation to discrete sequence s [kT], obtain signal spectrum
Figure FDA0002326974220000013
 4.根据权利要求1所述的基于可调窗函数的S变换时频分析方法，其特征在于，步骤3中，确定窗长控制函数以及窗函数的具体方法为：4. the S-transform time-frequency analysis method based on adjustable window function according to claim 1, is characterized in that, in step 3, the concrete method that determines window length control function and window function is: 步骤3-1、根据信号频谱和分辨率要求，确定参数a、b、c；Step 3-1. Determine parameters a, b, and c according to the signal spectrum and resolution requirements; 设信号采样率为fs，由于
Figure FDA0002326974220000014
因此取
Figure FDA0002326974220000015
设信号的最大频率和最小频率为fmax和fmin，实际分析所允许的窗函数频域宽度的最大和最小值Δfmax和Δfmin，则通过下列不等式确定a和c的取值范围：Assuming the signal sampling rate f s , since
Figure FDA0002326974220000014
Therefore take
Figure FDA0002326974220000015
Assuming that the maximum and minimum frequencies of the signal are f max and f min , and the maximum and minimum values Δf max and Δf min of the frequency domain width of the window function allowed in the actual analysis, the value ranges of a and c are determined by the following inequalities:
Figure FDA0002326974220000016
Figure FDA0002326974220000016
 步骤3-2、在取值范围内确定a和c的值，将各个参数值代入窗长控制函数中：Step 3-2. Determine the values of a and c within the value range, and substitute each parameter value into the window length control function:
Figure FDA0002326974220000017
Figure FDA0002326974220000017
 步骤3-3、将窗长控制函数带入到高斯窗函数中，得到改进后的窗函数表达式：Step 3-3, bring the window length control function into the Gaussian window function, and obtain the improved window function expression:
Figure FDA0002326974220000021
其中，
Figure FDA0002326974220000021
in, 其中，α(f)为窗函数尺度因子。Among them, α(f) is the window function scale factor. 5.根据权利要求1所述的基于可调窗函数的S变换时频分析方法，其特征在于，步骤4中，窗函数频谱是以
Figure FDA0002326974220000022
为起始频率点，具体公式为：5. the S transform time-frequency analysis method based on adjustable window function according to claim 1, is characterized in that, in step 4, the window function spectrum is based on
Figure FDA0002326974220000022
is the starting frequency point, the specific formula is:
Figure FDA0002326974220000023
Figure FDA0002326974220000023
 其中，n从0开始取值，N为频率点个数，T为采样周期。Among them, n starts from 0, N is the number of frequency points, and T is the sampling period. 6.根据权利要求1所述的基于可调窗函数的S变换时频分析方法，其特征在于，步骤5中，将扩维后的信号频谱跟窗函数频谱相乘，具体为：6. the S-transform time-frequency analysis method based on adjustable window function according to claim 1, is characterized in that, in step 5, the signal spectrum after dimension expansion is multiplied with window function spectrum, is specially: 步骤5-1、将信号频谱
Figure FDA0002326974220000024
扩维得到信号频谱
Figure FDA0002326974220000025
其中m＝0,1,2,3…N-1；Step 5-1, convert the signal spectrum
Figure FDA0002326974220000024
Expand the dimension to get the signal spectrum
Figure FDA0002326974220000025
Where m=0,1,2,3...N-1; 步骤5-2、将扩维后的信号频谱
Figure FDA0002326974220000026
与窗函数频谱G(m,n)相乘。Step 5-2, the expanded signal spectrum
Figure FDA0002326974220000026
Multiply by the window function spectrum G(m,n). 7.根据权利要求1所述的基于可调窗函数的S变换时频分析方法，其特征在于，步骤6中对相乘结果进行傅里叶逆变换，得到第n个频率点的时域信息
Figure FDA0002326974220000027
其中m＝0,1,2,3…N-1。7. the S-transform time-frequency analysis method based on adjustable window function according to claim 1, is characterized in that, in step 6, multiplication result is carried out inverse Fourier transform, obtains the time domain information of the nth frequency point
Figure FDA0002326974220000027
where m=0,1,2,3...N-1. 8.根据权利要求1所述的基于可调窗函数的改进S变换时频分析方法，其特征在于，步骤7中判断所有频率点是否都已计算具体判断方式是：判断n≥N-1是否成立，若不成立，n加1后重复步骤4、步骤5、步骤6；若成立，输出时频谱结果。8. the improved S-transform time-frequency analysis method based on adjustable window function according to claim 1, is characterized in that, in step 7, judge whether all frequency points have all calculated concrete judgment mode is: judge whether n≥N-1 If yes, if not, add 1 to n and repeat steps 4, 5, and 6; if yes, output the spectral result.
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Application publication date: 20200526