CN108920418B - Self-adaptive window length-variable time-frequency conversion method based on skewness - Google Patents

Self-adaptive window length-variable time-frequency conversion method based on skewness Download PDF

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CN108920418B
CN108920418B CN201810432818.9A CN201810432818A CN108920418B CN 108920418 B CN108920418 B CN 108920418B CN 201810432818 A CN201810432818 A CN 201810432818A CN 108920418 B CN108920418 B CN 108920418B
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罗钐
徐起
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Abstract

The invention belongs to the field of time-frequency analysis in signal processing, and particularly relates to a skewness-based self-adaptive window-variable length-time frequency-time conversion technology, which aims at nonlinear frequency-modulated signals. The invention uses skewness to control the adaptive process of window function length: firstly, taking the starting point of the window length as the signal starting point, taking the window length as the maximum, gradually reducing the window length based on skewness control until the self-adaptive result of the window length is obtained, and then moving the starting point of the added window forward (N)1/4) windowing at points; get the starting point of this windowing backward (3N)1And 4) taking the point position as the starting point of the next windowing; and after finishing all windowing, collecting results of each time into a time-frequency diagram. The invention solves the problem that the signal aggregation and resolution ratio are not high enough when the existing self-adaptive time frequency transformation is used for analyzing the multi-component frequency modulation signal.

Description

Self-adaptive window length-variable time-frequency conversion method based on skewness
Technical Field
The invention belongs to the field of time-frequency analysis in signal processing, and particularly relates to a self-adaptive window-variable length time-frequency conversion method based on skewness, which is mainly used for nonlinear frequency modulation signals.
Background
Frequency modulated signals refer to signals that vary continuously in frequency for a duration, and are widely used in a variety of information systems including radar, sonar, and communications. Depending on the form of the frequency variation of the signal, the frequency modulated signal may be divided into a chirp signal and a non-chirp signal.
The chirp signal may be expressed using the following general formula:
Figure GDA0002986012140000011
where s denotes the input signal, t denotes the time variable, K denotes the number of components of the signal, AkRepresenting the amplitude of the k-th component of the signal, e representing the natural logarithm, fkRepresenting the centre frequency, gamma, of the k-th component of the signalkIs the tuning frequency of the k component of the signal.
The non-chirp signal may be expressed by the following general formula:
Figure GDA0002986012140000012
wherein theta isk(t) is a phase function of the k-th component of the signal. It can be seen that formula (1) is a special case of formula (2).
The frequency modulated signal is typically analyzed by Short-Time Fourier Transform (STFT) and Wigner-willi Transform (WVD). Both of these methods have some drawbacks. The signal energy concentration of STFT is low, while WVD has strong cross terms. To improve the above-mentioned defects, a Lv Distribution (LVD) has been proposed. The LVD not only has high signal energy concentration, but also can eliminate cross terms. The inverse transform ilvd (inverse LVD) of LVD is a time-frequency analysis method based on time-frequency representation, and the processing result is a time-frequency graph of a signal. However, ILVD cannot be directly applied to a non-chirp signal, so that a Short-Time Lv Transform (STLVT) has been proposed by using the concept of windowed segmentation to solve this problem by taking advantage of the concept of STFT. The effect of the STLVT processing of the non-chirped signal is better than the STFT.
Both STFT and STLVT use fixed window function lengths, and for some signals with large frequency variation difference in different time periods, the processing effect of both STFT and STLVT is not ideal. Adaptive concepts have been introduced to implement a variable window length STFT, which adaptively adjusts the window length according to signal frequency variations. Since the performance of the STLVT processing of non-chirped signals is better than the stlft, the performance of the STLVT is better than the stlft if it is possible to implement an adaptive variable window length STLVT. However, the existing adaptive window length-changing method cannot be directly applied to the STLVT, so the STLVT still has the defect that the window length cannot be adjusted according to the change of the signal frequency.
Disclosure of Invention
Aiming at the problems or the defects, in order to solve the problem that the STLVT cannot adjust the Window length according to the signal characteristics, the invention provides an offset-based Adaptive Window length-based time-frequency conversion method, which is called Adaptive Window Lv Transform (AWLT).
The specific technical scheme of the self-adaptive window length-variable time-frequency conversion method is shown in fig. 1 and comprises the following steps.
Step 1: inputting the nonlinear frequency modulation signal shown in the formula (2), and selecting the starting point of the input signal as the starting point of the windowing. The ratio Q (0< Q <1) is set for each reduction of the window length.
Step 2: let N0Equal to the distance from the start of the windowing to the end of the signal.
And step 3: with N0Windowing is started from the starting point of windowing as the length of the window function, and the following equations (3), (4), (5), (6), (9), (10), (11) and (12) are sequentially executed on the windowed signal, and the result of equation (12) is denoted as P0。
The parametric autocorrelation function of a signal is expressed as:
Figure GDA0002986012140000021
wherein C isrFor cross terms between different signal components, RzFor the autocorrelation terms of the components of the signal, the expression is as follows:
Figure GDA0002986012140000022
the main idea of LVD is to perform a scaling operation on the parametric autocorrelation function of the signal as shown in the following equation:
Figure GDA0002986012140000023
tsfor the amount of time after the scaling transformation, called the scale time, tsT (τ + 1). Post-scaling parametric autocorrelation function RsThe following steps are changed:
Figure GDA0002986012140000024
Figure GDA0002986012140000025
called the scale parameter autocorrelation function, is the scale time tsAnd the delay τ.
The scale parameter autocorrelation function of equation (6)
Figure GDA0002986012140000026
Successive edges τ dimension, tsThe LVD is obtained by performing two Fourier transforms on the dimension as shown in the following formula:
Figure GDA0002986012140000027
wherein Fτ{·}、Fts{. denotes the dimension along τ, along t, respectivelysFourier transform of dimension, the first term of equation (7) represents that the energy of each component of the signal is gathered on a frequency-frequency modulation plane in the form of delta functionk,γk) At these points, the second term is the result of the operation of the cross term.
ILVD is obtained by performing inverse operation on the signal portion of LVD and then performing fourier transform, as shown in equations (8) and (9):
Figure GDA0002986012140000031
Is(t,f)＝|Fτ{Γ-1[Ff -1{Fγ -1{M[Ls(f,γ)]}}]}| (9)
in the formula (8), S represents a region where an input signal exists in a frequency-modulation frequency region, i.e., (f)k,γk) A collection of (a). In the formula (9) Ff -1{·}、Fγ -1{. denotes the inverse Fourier transform along the f-dimension, respectively along the y-dimension, Γ-1{. is the inverse of the stretching operation Γ in equation (5).
In time-frequency analysis, the signal energy concentration isAn important performance indicator is generally the higher the demand concentration the better. Skewness is a concept that can measure whether distribution is concentrated or not in probability theory, so that skewness can be applied to measuring signal energy aggregation in time-frequency analysis, and the higher skewness indicates the higher signal energy aggregation. First calculate LsMean and standard deviation of (f, γ) modulus values:
μ＝E[|Ls(f,γ)|] (10)
Figure GDA0002986012140000032
the skewness of the LVD is then calculated as follows:
Figure GDA0002986012140000033
and 4, step 4: the starting point of the windowing is not changed, order
Figure GDA0002986012140000034
With N1The signal is re-windowed as a window function length.
And 5: the windowed signals are sequentially subjected to the expressions (3), (4), (5), (6), (9), (10), (11) and (12), and the result of the expression (12) is denoted as P1. If P0≤1.2P1Then step 6 is entered. Otherwise, go to step 7.
Step 6: will N1Is given by N0The windowing starting point is unchanged. Then step 3 is entered.
And 7: moving the windowing starting point forward by N1Point 4 as new window starting point (at most, only moving to the starting point of the input signal, not exceeding the range of the input signal), and counting by N1Windowing is performed for the window function length from the new windowing starting point. N is set to record the number of times step 8 is performed, and n is made 0.
And 8: the value of n is increased by 1, and the windowed signals of step 7 are sequentially subjected to the operations (3), (4), (5), (6), (7), (8) and (9). The result of the formula (9) is recorded as
Figure GDA0002986012140000041
And step 9: according to the new window starting point and window function length N in the step 71And determining a windowing end point.
If the windowing end point is not at the input signal end point, taking the new windowing starting point of the step 7 and then 3N1The point of/4 is the starting point of the next windowing, and the step 2 is entered.
If the end of the windowing is at the end of the input signal, then all windowing is complete and step 10 is entered.
Step 10: the results of formula (9) obtained by windowing each time
Figure GDA0002986012140000042
And assembling a time-frequency graph in sequence, wherein the time-frequency graph comprises the processing results of all time sampling points.
The method uses the skewness to control the self-adaptive process of the window function length, and the higher the skewness is, the higher the signal energy concentration is, and the better the subsequent processing performance is. P0Representing the degree of deviation, P, of windowing by the current window length1Indicating the skewness of the window being windowed with the window length after being reduced once (i.e., K times the current window length). In step 5, if P0＞1.2P1If the window length is continuously reduced, the skewness is obviously reduced, which means that the window length is not suitable to be continuously reduced at this time, so the current window length is selected as the result of the window length adaptive process. If P0≤1.2P1The window length should be reduced continuously, so the window length after one reduction becomes a new window length, and the process of comparing skewness enters the next time.
In step 9, the starting point of the current windowing is taken (3/4N)1) The starting point of the next windowing is at the starting point of the next windowing, so that the starting point of the next windowing is exactly positioned at the end point of the time-frequency diagram obtained by the current windowing. And in step 7, the windowing starting point is moved forward (1/4N)1) The point (at most, only moving to the starting point of the input signal, and not exceeding the range of the input signal) is to make the time-frequency diagram obtained by this windowing end-to-end with the time-frequency diagram obtained by the last windowing.
In summary, the present invention can adaptively use window functions of different lengths according to the waveform change of the signal frequency function, and the window function used at a position where the waveform of the signal frequency function is closer to linearity is longer. The invention is primarily directed to non-chirped signals but is equally applicable to chirped signals and single frequency signals. The invention solves the problem that the STLVT can not adjust the window length according to the signal characteristics, and realizes the self-adaptive window Lv transformation, namely AWLT. The performance of the STLVT processing nonlinear frequency modulation signal is superior to that of the STFT, so that the method has more excellent performance than the existing adaptive window length-variable time-frequency conversion method.
Drawings
FIG. 1 is a flow chart of a specific embodiment of the present invention;
fig. 2(a), (b) are the raw time-frequency diagram and the time-frequency diagram of AWLT, respectively, of the input signal in example 1.
FIGS. 3(a), (b), (c) are the raw, AWLT and STLVT plots, respectively, of the input signal of example 2.
Detailed Description
The invention is further described with reference to the following drawings and detailed description.
Example 1: in a computer MATLAB environment, a two-component simulation signal is generated according to the following equation: each parameter of frequency modulation is f1＝-30Hz、 f2＝20Hz、γ1＝0.6Hz/s、γ220 Hz/s; sampling frequency fs256Hz, the number of signal sampling points Ns＝8192。
Figure GDA0002986012140000051
In this example, the ratio Q of each window length reduction is set to 0.5.
Fig. 2(a) is an original time-frequency diagram of an input signal. Fig. 2(b) is a time-frequency diagram of the use of the present invention, AWLT. Comparing fig. 2(a) and fig. 2(b) shows that the original time-frequency diagram of the input signal and the time-frequency diagram of the AWLT are highly overlapped, which indicates that the present invention can process such signals well.
To illustrate the advantages of the present invention over the STLVT, another example is given below.
Example 2: in a computer MATLAB environment, a one-component simulation signal is generated according to the following equation: each parameter of frequency modulation is f1＝-5Hz、γ16 Hz/s; sampling frequency fs256Hz, the number of signal sampling points Ns＝8192。
Figure GDA0002986012140000052
In this example, the ratio Q of each window length reduction is set to 0.8.
Fig. 3(a) is an original time-frequency diagram of an input signal. Fig. 3(b) is a time-frequency diagram of the processing of an input signal with the present invention, AWLT. Fig. 3(c) is a time-frequency diagram of STLVT, using a window length of 1536 points. It can be seen from the figure that the curve of the time-frequency diagram of the AWLT is smooth, and the waveform is very close to the original waveform. The time-frequency graph curve of the STLVT is not smooth enough, and the waveform is not close to the original waveform. This is due to the fact that the fixed window length used by the STLVT cannot be made everywhere appropriate.
We can also try to let STLVT use a shorter window length for comparison with AWLT. We calculated the average window length used by the AWLT in fig. 3(b), resulting in 906. Let STLVT select 906 process the same signal as a fixed window length and compare the processing result to the AWLT. The comparison method comprises the following steps: firstly, a mask plane is manufactured by using a time-frequency diagram of an input signal, and the following formula is shown:
Figure GDA0002986012140000053
wherein S(t,f)Representing the signal domain in the original time-frequency diagram of the input signal.
By M of formula (15)(t,f)Multiplying the results of AWLT and STLVT respectively to obtain two time-frequency graphs (in Matlab, in matrix form, respectively marked as H)AWLTAnd HSTLVT) The elements not equal to 0 in the matrix are the overlapped parts of the input signal and the time-frequency diagram of the processing result, and then the signal energy of the overlapped parts is accumulatedShown below as follows:
Figure GDA0002986012140000061
Figure GDA0002986012140000062
wherein h isAWLT(i,j)Represents HAWLTElement of row i and column j, hSTLVT(i,j)Representation matrix HSTLVTRow i and column j.
A higher total energy indicates a higher degree of coincidence of the processing result with the input signal, and the processing result is more excellent. Calculated total energy E of the STLVT using 906 as the fixed window lengthSTLVT42, and total energy E of AWLTAWLTIs 141. Therefore, even if a relatively appropriate average window length is calculated and applied to the STLVT, the energy concentration of the processing result is still far less than that of AWLT.
In summary, from the simulation results of the two examples, the present invention, AWLT, can effectively process non-chirped signals. The invention solves the problem that the STLVT can not adjust the window length according to the signal characteristics. Because the STLVT performance is better than the STFT performance, the performance of the method is superior to that of the existing self-adaptive window length-variable time-frequency conversion method.

Claims (1)

1. A self-adaptive window-changing long, short and time frequency conversion method based on skewness comprises the following specific steps:
step 1: inputting a non-linear frequency modulation signal shown as a formula (2), selecting a starting point of the input signal as a windowing starting point, and setting a ratio Q (0< Q <1) of each time of window length reduction;
Figure FDA0002986012130000011
where s denotes the input signal, t denotes the time variable, K denotes the number of components of the signal, AkRepresents a signal ofAmplitude of the k component, e denotes the natural logarithm, fkRepresenting the centre frequency, gamma, of the k-th component of the signalkFrequency modulation of the k-th component of the signal, θk(t) is a phase function of the kth component of the signal;
step 2: let N0Equal to the distance from the start of the windowing to the end of the signal;
and step 3: with N0Windowing is started from the starting point of windowing as the length of the window function, and the following equations (3), (4), (5), (6), (9), (10), (11) and (12) are sequentially executed on the windowed signal, and the result of equation (12) is denoted as P0；
The parametric autocorrelation function of a signal is expressed as:
Figure FDA0002986012130000012
wherein C isrFor cross terms between different signal components, RzFor the autocorrelation terms of the components of the signal, the expression is as follows:
Figure FDA0002986012130000013
LVD is a scaling operation performed on the parametric autocorrelation function of the signal as shown in the following equation:
Figure FDA0002986012130000014
tsfor the amount of time after the scaling transformation, called the scale time, ts(τ +1) t; post-scaling parametric autocorrelation function RsThe following steps are changed:
Figure FDA0002986012130000015
Figure FDA0002986012130000016
called the scale parameter autocorrelation function, is the scale time tsAnd the delay τ;
the scale parameter autocorrelation function of equation (6)
Figure FDA0002986012130000017
Successive edges τ dimension, tsThe dimension is Fourier transformed twice to obtain LVD as shown in the following formula:
Figure FDA0002986012130000018
wherein Fτ{·}、
Figure FDA0002986012130000021
Respectively representing the dimension along tau, the edge tsFourier transform of dimension, the first term of equation (7) represents that the energy of each component of the signal is gathered on a frequency-frequency modulation plane in the form of delta functionk,γk) At these points, the second term is the operation result of the cross term;
ILVD is obtained by performing inverse operation on the signal portion of the LVD and then performing fourier transform, as shown in equations (8) and (9):
Figure FDA0002986012130000022
Figure FDA0002986012130000023
in the formula (8), S represents a region having an input signal in the frequency-modulation frequency domain, i.e., (f)k,γk) A set of (a); in the formula (9)
Figure FDA0002986012130000024
Figure FDA0002986012130000025
Denotes the inverse Fourier transform along the f-dimension, respectively along the y-dimension, Γ-1{. is the inverse of the stretching operation Γ in equation (5);
first calculate LsMean and standard deviation of (f, γ) modulus values:
μ＝E[|Ls(f,γ)|] (10)
Figure FDA0002986012130000026
the skewness of the LVD is then calculated as follows:
Figure FDA0002986012130000027
and 4, step 4: the starting point of the windowing is not changed, order
Figure FDA0002986012130000028
With N1Re-windowing the signal as a window function length;
and 5: the windowed signals are sequentially subjected to the expressions (3), (4), (5), (6), (9), (10), (11) and (12), and the result of the expression (12) is denoted as P1(ii) a If P0≤1.2P1If not, directly entering step 7;
step 6: will N1Is given by N0The windowing starting point is unchanged, and then the step 3 is carried out;
and 7: moving the windowing starting point forward by N1A point of/4 is used as a new windowing starting point, and only moves to the starting point of the input signal at most without exceeding the range of the input signal; then with N1Windowing from the new windowing starting point for the window function length; taking n to record the times of executing the step 8, and enabling n to be 0;
and 8: increasing the value of n by 1, and sequentially executing the formulas (3), (4), (5), (6), (7), (8) and (9) on the signal subjected to windowing in the step 7; the result of the formula (9) is recorded as
Figure FDA0002986012130000029
And step 9: starting point of new windowing and window function length N according to step 71Determining a windowing end point;
if the windowing end point is not at the input signal end point, taking the new windowing starting point of the step 7 and then 3N1The point of/4 is used as the starting point of the next windowing, and the step 2 is carried out;
if the windowing end point is at the input signal end point, all windowing is finished, and the step 10 is entered;
step 10: the results of formula (9) obtained by windowing each time
Figure FDA0002986012130000031
And assembling a time-frequency graph in sequence, wherein the time-frequency graph comprises the processing results of all time sampling points.
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