CN112559973B - Adaptive multi-component linear frequency modulation signal parameter estimation method - Google Patents

Adaptive multi-component linear frequency modulation signal parameter estimation method Download PDF

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CN112559973B
CN112559973B CN202110209363.6A CN202110209363A CN112559973B CN 112559973 B CN112559973 B CN 112559973B CN 202110209363 A CN202110209363 A CN 202110209363A CN 112559973 B CN112559973 B CN 112559973B
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陈一畅
王万田
汤子跃
孙永健
朱勇
赵园青
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Air Force Early Warning Academy
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Abstract

The invention provides an adaptive multi-component linear frequency modulation signal parameter estimation method based on STFrFT, which comprises the steps of firstly realizing the optimal transformation order estimation of a linear frequency modulation component signal by a global search method based on a minimum entropy criterion, and estimating the signal amplitude of the linear frequency modulation component; secondly, estimating the instantaneous frequency of the linear frequency modulation component signal through short-time fractional Fourier transform; according to the estimated instantaneous frequency, a polynomial regression method based on linear least square estimation is adopted to realize the rough estimation of the initial frequency and the frequency modulation slope of the linear frequency modulation component signal; and then, obtaining a fine estimation result of the initial frequency and the frequency modulation slope of the linear frequency modulation component signal through frequency modulation removal, low-pass filtering and phase regression processing.

Description

Adaptive multi-component linear frequency modulation signal parameter estimation method
Technical Field
The invention relates to the field of communication, in particular to an adaptive multi-component linear frequency modulation signal parameter estimation method based on STFrFT.
Background
Linear Frequency Modulation (LFM) signals are taken as typical non-stationary signals, have the characteristic of large time-bandwidth product, are taken as transmitting signals, adopt the pulse compression technology, can effectively solve the contradiction between the detection distance and the distance resolution, and are widely applied to the fields of radar, communication, seismic exploration and the like. Therefore, how to accurately perform parameter estimation of LFM signals, especially multi-component LFM signals, has been a major issue in the field of signal processing.
At present, most of the LFM signal parameter estimation methods are linear Time-frequency analysis methods represented by Short Time Fourier Transform (STFT) and bilinear Time-frequency analysis methods represented by Wigner-Ville Distribution (WVD). Although the STFT solves the time-dependent variation of the frequency that cannot describe the signal local in the fourier transform, it is difficult to satisfy both high time domain resolution and high frequency domain resolution, i.e., its time-frequency resolution is not high, and thus the application is limited. The WVD transforms the LFM signal to a time-frequency domain through a quadratic function, has good time-frequency resolution for the LFM signal of a single component, but has inevitable cross-term interference for the LFM signal of a plurality of components, and seriously influences the parameter estimation of the LFM signal of the plurality of components. Although many scholars improve the WVD method by adding a smoothing window function, an adaptive kernel function, and the like in order to suppress the interference of the cross terms, the time-frequency resolution is reduced.
In recent years, many researchers have been dedicated to studying methods for estimating parameters of polynomial phase signals, including the Quasi-maximum likelihood estimation (QML) method based on STFT, which is proposed in 2014 by Igor Djurovi ć et al in IET Signal Processing international journal, volume 8, phase 4, and is proposed in Quasi-maximum-likelihood estimator of polymial phase signals. Compared with the traditional methods such as a high-order fuzzy function and a product high-order fuzzy function, the method has a lower signal-to-noise ratio threshold and reaches the lower boundary of Cramer-Lo when the signal-to-noise ratio is higher, in addition, the LFM signal is used as a simple polynomial phase signal of a second order, parameter estimation can be quickly and accurately realized by the method, but the STFT has lower time-frequency resolution, the parameter estimation performance and the signal-to-noise ratio threshold are influenced, and the method can only be used for realizing the polynomial phase signal of a single component. In a paper "Review of the square-maximum likelihood estimation for a poly phase Signal" published in Digital Signal Processing international journal by Igor Djurovi ć et al in 2017, it is proposed to realize multi-component polynomial phase Signal parameter estimation by combining a STFT-based quasi-maximum likelihood estimation method with sequential elimination of estimated signals, but the method is only limited to the case that the Signal amplitudes of the components are greatly different. When the signal amplitudes of different components are not greatly different, and the instantaneous frequencies of the component signals are estimated by using the STFT, the components with the signal amplitudes which are not greatly different mutually influence, so that the estimation performance of the parameters is seriously deteriorated. In addition, the method defaults that the number of component signals is known, and a decision condition for terminating the signal parameter estimation is not given.
Disclosure of Invention
The present invention aims to provide a parameter estimation method for adaptive multi-component chirp signal based on STFrFT, which can adaptively determine whether parameter estimation is terminated and determine the number of LFM component signals in the signal.
In order to achieve the purpose, the invention adopts the following technical scheme:
the invention provides an adaptive multi-component linear frequency modulation signal parameter estimation method based on STFrFT, which comprises the following steps:
s1, obtaining a multi-component chirp signal
Figure 555886DEST_PATH_IMAGE001
(ii) a Assuming a multi-component chirp signal consisting of
Figure 197083DEST_PATH_IMAGE002
The chirp component signal and the background noise component, namely:
Figure 820963DEST_PATH_IMAGE003
Figure 231215DEST_PATH_IMAGE004
wherein the content of the first and second substances,
Figure 547927DEST_PATH_IMAGE005
is shown as
Figure 676420DEST_PATH_IMAGE006
The frequency of the individual chirp component signals,
Figure 103991DEST_PATH_IMAGE007
which is representative of the background noise signal,
Figure 103171DEST_PATH_IMAGE008
、
Figure 590784DEST_PATH_IMAGE009
and
Figure 206573DEST_PATH_IMAGE010
respectively represent
Figure 440764DEST_PATH_IMAGE011
The signal amplitude, the start frequency and the chirp rate,
Figure 91188DEST_PATH_IMAGE012
represents the index of the sample point of the discrete signal, and has,
Figure 484123DEST_PATH_IMAGE013
，
Figure 587208DEST_PATH_IMAGE014
represents the total number of sample points of the discrete signal,
Figure 356581DEST_PATH_IMAGE015
which is indicative of the time of observation of the signal,
Figure 861512DEST_PATH_IMAGE016
which represents the time interval between the sampling of the samples,
Figure 425348DEST_PATH_IMAGE017
the number of the imaginary numbers is represented,
Figure 281309DEST_PATH_IMAGE018
expressed as natural constants
Figure 854373DEST_PATH_IMAGE019
An exponential operation of a base number;
s2, initializing the linear frequency modulation component signal number index
Figure 948231DEST_PATH_IMAGE020
Let the residual signal
Figure 948548DEST_PATH_IMAGE021
；
S3, the calculation corresponds to
Figure 26225DEST_PATH_IMAGE022
A signal of said chirp component
Figure 134471DEST_PATH_IMAGE023
Of the optimal transformation order
Figure 348415DEST_PATH_IMAGE024
(ii) a Firstly, judging whether the residual signal still has the linear frequency modulation component signal, if so, adopting a global search method based on a minimum entropy criterion to realize the optimal transformation order
Figure 785212DEST_PATH_IMAGE025
Otherwise, ending parameter estimation;
s4, estimating the
Figure 84606DEST_PATH_IMAGE026
A signal of said chirp component
Figure 999473DEST_PATH_IMAGE027
Signal amplitude of
Figure 67923DEST_PATH_IMAGE028
(ii) a In fractional Fourier transform, when the order is changed
Figure 410043DEST_PATH_IMAGE029
Equal to said chirp component signal
Figure 196733DEST_PATH_IMAGE030
Of the optimal transformation order
Figure 649711DEST_PATH_IMAGE031
Then, the signal amplitude of the component is estimated according to the maximum amplitude value in the fractional Fourier transform result and the total number of sampling points of the signal
Figure 838247DEST_PATH_IMAGE032
Namely:
Figure 616847DEST_PATH_IMAGE033
wherein the content of the first and second substances,
Figure 374324DEST_PATH_IMAGE034
is shown as
Figure 630993DEST_PATH_IMAGE035
A signal of a linear frequency-modulated component
Figure 674035DEST_PATH_IMAGE036
Is determined by the signal amplitude estimate of (a),
Figure 357957DEST_PATH_IMAGE037
representing the order of the transformation equal to
Figure 119240DEST_PATH_IMAGE038
Time signal
Figure 179600DEST_PATH_IMAGE039
The fractional order fourier transform result vector of (a);
s5, estimating the
Figure 77149DEST_PATH_IMAGE040
A signal of said chirp component
Figure 931972DEST_PATH_IMAGE041
The instantaneous frequency of (d); obtained according to said S3
Figure 180551DEST_PATH_IMAGE042
A signal of said chirp component
Figure 44602DEST_PATH_IMAGE043
Of the optimal transformation order
Figure 528148DEST_PATH_IMAGE044
Estimating the instantaneous frequency of the signal through short-time fractional Fourier transform, wherein the calculation formula of the short-time fractional Fourier transform is as follows:
Figure 819452DEST_PATH_IMAGE045
wherein the content of the first and second substances,
Figure 555327DEST_PATH_IMAGE046
representing a transformation order of
Figure 957489DEST_PATH_IMAGE047
Time, residual signal
Figure 564051DEST_PATH_IMAGE048
The result of the short-time fractional fourier transform,
Figure 291836DEST_PATH_IMAGE049
representing the window function length of
Figure 249428DEST_PATH_IMAGE050
And has a proper Gaussian window function
Figure 189702DEST_PATH_IMAGE051
When the temperature of the water is higher than the set temperature,
Figure 916349DEST_PATH_IMAGE052
；
Figure 549456DEST_PATH_IMAGE053
representing a discrete time sequence;
Figure 728765DEST_PATH_IMAGE054
a kernel function representing a fractional Fourier transform;
the instantaneous frequency of the signal is estimated by searching for the maximum of the short-time fractional order fourier transform results at different sampling points, namely:
Figure 738309DEST_PATH_IMAGE055
Figure 587972DEST_PATH_IMAGE056
wherein the content of the first and second substances,
Figure 126401DEST_PATH_IMAGE057
is shown as
Figure 58585DEST_PATH_IMAGE058
A signal of a linear frequency-modulated component
Figure 340661DEST_PATH_IMAGE059
The instantaneous frequency of the received signal,
Figure 41901DEST_PATH_IMAGE060
representing a modulo operation;
Figure 751231DEST_PATH_IMAGE061
is a frequency variable;
s6, rough estimation
Figure 170711DEST_PATH_IMAGE062
A signal of said chirp component
Figure 256479DEST_PATH_IMAGE063
Starting frequency of
Figure 77804DEST_PATH_IMAGE064
And chirp rate
Figure 223615DEST_PATH_IMAGE065
；
Estimated signal according to the S5
Figure 864812DEST_PATH_IMAGE066
And roughly estimating the starting frequency by a polynomial regression method based on linear least squares estimation
Figure 954603DEST_PATH_IMAGE067
And chirp rate
Figure 896014DEST_PATH_IMAGE068
The solving formula is as follows:
Figure 947147DEST_PATH_IMAGE069
wherein the content of the first and second substances,
Figure 75640DEST_PATH_IMAGE070
representing the frequency of the start
Figure 503210DEST_PATH_IMAGE071
And chirp rate
Figure 767969DEST_PATH_IMAGE072
The vector of coarse estimates of (a), i.e.:
Figure 255582DEST_PATH_IMAGE073
；
Figure 871372DEST_PATH_IMAGE074
and
Figure 102633DEST_PATH_IMAGE075
respectively represent
Figure 487478DEST_PATH_IMAGE076
A signal of a linear frequency-modulated component
Figure 137203DEST_PATH_IMAGE077
Starting frequency of
Figure 709130DEST_PATH_IMAGE078
And chirp rate
Figure 478503DEST_PATH_IMAGE079
Is determined by the coarse estimation value of (c),
Figure 717854DEST_PATH_IMAGE080
which represents the operation of transposition by means of a transposition operation,
Figure 547270DEST_PATH_IMAGE081
representing an inversion operation, a matrix
Figure 403230DEST_PATH_IMAGE082
Sum vector
Figure 710715DEST_PATH_IMAGE083
Respectively as follows:
Figure 70152DEST_PATH_IMAGE084
Figure 70469DEST_PATH_IMAGE085
Figure 616988DEST_PATH_IMAGE086
Figure 990813DEST_PATH_IMAGE087
s7, fine estimation
Figure 204757DEST_PATH_IMAGE088
A signal of said chirp component
Figure 375975DEST_PATH_IMAGE089
Starting frequency of
Figure 940949DEST_PATH_IMAGE090
And chirp rate
Figure 59078DEST_PATH_IMAGE091
(ii) a Obtaining rough estimated values of the initial frequency and the frequency modulation slope according to the S6
Figure 127528DEST_PATH_IMAGE092
And
Figure 469647DEST_PATH_IMAGE093
the signals are processed by frequency-modulation removal, low-pass filtering and phase regression
Figure 256338DEST_PATH_IMAGE094
Carrying out fine estimation on the initial frequency and the frequency modulation slope;
s8, the first obtained according to the S4
Figure 709316DEST_PATH_IMAGE095
A signal amplitude estimation value of the chirp component signal and the second obtained at said S7
Figure 163431DEST_PATH_IMAGE096
The initial frequency and the fine frequency modulation slope value of the linear frequency modulation component signal are reconstructed
Figure 413802DEST_PATH_IMAGE097
A signal of a linear frequency-modulated component
Figure 953368DEST_PATH_IMAGE098
；
S9, stepping the linear frequency modulation component signal number index
Figure 944458DEST_PATH_IMAGE099
And updates the residual signal to:
Figure 987500DEST_PATH_IMAGE100
Figure 671422DEST_PATH_IMAGE101
represents the reconstructed second
Figure 167126DEST_PATH_IMAGE102
A chirp component signal;
returning to the step S3 to continue execution.
Further, the step S3 includes the following steps:
s3.1, in a half period interval
Figure 227486DEST_PATH_IMAGE103
Inter-search step size
Figure 125035DEST_PATH_IMAGE104
To discretize the transformation order to obtain
Figure 979858DEST_PATH_IMAGE105
A discrete value, i.e.
Figure 228437DEST_PATH_IMAGE106
Wherein
Figure 558399DEST_PATH_IMAGE107
Represents a vector of transform order candidate values,
Figure 310455DEST_PATH_IMAGE108
is referred to as the first
Figure 867338DEST_PATH_IMAGE109
The candidate values of the individual transformation orders are,
Figure 337634DEST_PATH_IMAGE110
which represents a rounding-down operation, the rounding-down operation,
Figure 474217DEST_PATH_IMAGE111
representing a transpose operation; initialization iteration number
Figure 346358DEST_PATH_IMAGE112
；
S3.2, calculating the order of transformation equal to
Figure 808563DEST_PATH_IMAGE113
Time residual signal
Figure 500576DEST_PATH_IMAGE114
Fractional order fourier transform of (a); the calculation formula of the fractional Fourier transform is as follows:
Figure 706429DEST_PATH_IMAGE115
wherein the content of the first and second substances,
Figure 698656DEST_PATH_IMAGE116
representing a transformation order of
Figure 66183DEST_PATH_IMAGE117
Time residual signal
Figure 511071DEST_PATH_IMAGE118
The result of the fractional order fourier transform of (a),
Figure 316606DEST_PATH_IMAGE119
the kernel function of fractional Fourier transform is represented by the following mathematical expression:
Figure 897760DEST_PATH_IMAGE120
Figure 701768DEST_PATH_IMAGE121
Figure 368372DEST_PATH_IMAGE122
which represents an integer number of times,
Figure 916028DEST_PATH_IMAGE123
indicates a rotation angle, and is provided with
Figure 882847DEST_PATH_IMAGE124
Representing a square-on operation;
s3.3, calculating the order of transformation equal to
Figure 592177DEST_PATH_IMAGE125
Time residual signal
Figure 11657DEST_PATH_IMAGE126
Entropy of the fractional fourier transform result of (a); assuming a residual signal
Figure 97425DEST_PATH_IMAGE127
The vector form of (a) is:
Figure 653171DEST_PATH_IMAGE128
wherein the content of the first and second substances,
Figure 798982DEST_PATH_IMAGE129
representing a discretized signal vector; when the transformation order candidate is
Figure 437249DEST_PATH_IMAGE130
Then, the vector form of the signal fractional order fourier transform result is:
Figure 326708DEST_PATH_IMAGE131
Figure 736960DEST_PATH_IMAGE132
the entropy of the fractional fourier transform result can be calculated by:
Figure 788093DEST_PATH_IMAGE133
wherein the content of the first and second substances,
Figure 651007DEST_PATH_IMAGE134
expressed as natural constants
Figure 78577DEST_PATH_IMAGE135
A logarithmic operation of a base number;
s3.4, step iteration number
Figure 608916DEST_PATH_IMAGE136
Judgment of
Figure 96529DEST_PATH_IMAGE137
If the result is true, entering S3.5 if the result is true, otherwise entering S3.2 to continue execution;
s3.5 each transformation order candidate may calculate an entropy value from said S3.2 and said S3.3, from which a vector of candidate values corresponding to the transformation order may be obtained
Figure 712318DEST_PATH_IMAGE138
Vector of entropy values
Figure 943579DEST_PATH_IMAGE139
Comprises the following steps:
Figure 331354DEST_PATH_IMAGE140
s3.6, normalization entropy value vector
Figure 989868DEST_PATH_IMAGE141
The mathematical formula is as follows:
Figure 827374DEST_PATH_IMAGE142
wherein the content of the first and second substances,
Figure 596747DEST_PATH_IMAGE143
represents a vector of normalized entropy values that is,
Figure 836098DEST_PATH_IMAGE144
expressing the operation of solving the maximum value; computing an entropy vector
Figure 931093DEST_PATH_IMAGE145
Variance of (2)
Figure 255895DEST_PATH_IMAGE146
The mathematical expression is as follows:
Figure 828959DEST_PATH_IMAGE147
wherein the content of the first and second substances,
Figure 922817DEST_PATH_IMAGE148
representing vectors of entropy values
Figure 923134DEST_PATH_IMAGE149
And has a mean value of
Figure 732303DEST_PATH_IMAGE150
；
Judging whether the following formula is satisfied:
Figure 109057DEST_PATH_IMAGE151
wherein the content of the first and second substances,
Figure 57422DEST_PATH_IMAGE152
a threshold, here set to 0.02, representing a determination of the presence or absence of a chirp component signal;
if the above formula is true, the residual signal is determined
Figure 228640DEST_PATH_IMAGE153
If there is a chirp component signal, continuing to execute the step S3.7; otherwise, the residual signal is determined
Figure 528035DEST_PATH_IMAGE153
If only a noise signal exists in the signal, terminating the circulation and finishing the parameter estimation;
s3.7 by vector from entropy values
Figure 442901DEST_PATH_IMAGE154
To estimate the entropy corresponding to the second
Figure 511351DEST_PATH_IMAGE155
A signal of a linear frequency-modulated component
Figure 587891DEST_PATH_IMAGE156
The optimal transformation order of (a), namely:
Figure 109003DEST_PATH_IMAGE157
。
further, the pair signal in S7
Figure 827560DEST_PATH_IMAGE158
The initial frequency and the chirp rate are precisely estimated, specifically as follows:
s7.1, frequency modulation removing: first, based on the initial frequency and the rough estimation value of the chirp rate
Figure 7307DEST_PATH_IMAGE159
And
Figure 520328DEST_PATH_IMAGE160
to reconstruct the
Figure 263156DEST_PATH_IMAGE161
A signal of a linear frequency-modulated component
Figure 519825DEST_PATH_IMAGE162
The phase conjugate term of (a), namely:
Figure 828446DEST_PATH_IMAGE163
Figure 512369DEST_PATH_IMAGE164
wherein the content of the first and second substances,
Figure 273651DEST_PATH_IMAGE165
to be reconstructed
Figure 68432DEST_PATH_IMAGE166
A signal of a linear frequency-modulated component
Figure 231560DEST_PATH_IMAGE167
The phase conjugate term of (a);
secondly, according to the reconstructed second
Figure 351963DEST_PATH_IMAGE168
A signal of a linear frequency-modulated component
Figure 334962DEST_PATH_IMAGE169
Phase conjugate term of to the residual signal
Figure 930504DEST_PATH_IMAGE170
And (3) performing frequency modulation removal treatment, wherein the specific formula is as follows:
Figure 948139DEST_PATH_IMAGE171
Figure 973863DEST_PATH_IMAGE172
wherein the content of the first and second substances,
Figure 178580DEST_PATH_IMAGE173
the residual signal after frequency modulation processing is removed;
s7.2, low-pass filtering: performing low-pass filtering processing on the signal subjected to frequency modulation removal processing by adopting a moving average filter; the specific formula is as follows:
Figure 315163DEST_PATH_IMAGE174
Figure 187304DEST_PATH_IMAGE175
wherein the content of the first and second substances,
Figure 383930DEST_PATH_IMAGE176
，
Figure 341522DEST_PATH_IMAGE177
represents the length of the moving average filter;
Figure 547375DEST_PATH_IMAGE178
represents the result of the low-pass filtering process;
s7.3, phase regression treatment:
firstly, extracting the signal phase after the low-pass filtering processing, wherein the specific formula is as follows:
Figure 539602DEST_PATH_IMAGE179
Figure 907130DEST_PATH_IMAGE180
wherein the content of the first and second substances,
Figure 89368DEST_PATH_IMAGE181
which represents the operation of the arctan function,
Figure 98912DEST_PATH_IMAGE182
the imaginary part operation of the signal is expressed,
Figure 680066DEST_PATH_IMAGE183
representing the operation of taking the real part of the signal;
Figure 484074DEST_PATH_IMAGE184
representing the extracted signal phase;
the extracted signal phases are written in vector form
Figure 150679DEST_PATH_IMAGE185
Comprises the following steps:
Figure 432756DEST_PATH_IMAGE186
secondly, the polynomial regression method based on linear least square estimation is adopted to carry out phase matching on the extracted signal
Figure 868416DEST_PATH_IMAGE187
Fitting to estimate the initial frequency of the de-modulated signal
Figure 843325DEST_PATH_IMAGE188
And chirp rate
Figure 262806DEST_PATH_IMAGE189
The solution formula is as follows:
Figure 82994DEST_PATH_IMAGE190
wherein the content of the first and second substances,
Figure 104652DEST_PATH_IMAGE191
representing the estimated phase constant of the signal,
Figure 516042DEST_PATH_IMAGE192
and
Figure 157239DEST_PATH_IMAGE193
respectively representing estimated signal start frequencies
Figure 46697DEST_PATH_IMAGE194
And chirp rate
Figure 191371DEST_PATH_IMAGE195
And rest amount of
Figure 508083DEST_PATH_IMAGE196
，
Figure 636576DEST_PATH_IMAGE197
Matrix of
Figure 64146DEST_PATH_IMAGE198
Expressed as:
Figure 860064DEST_PATH_IMAGE199
wherein the content of the first and second substances,
Figure 550939DEST_PATH_IMAGE200
represents a time variable, and the mathematical expression thereof is as follows:
Figure 181377DEST_PATH_IMAGE201
Figure 412638DEST_PATH_IMAGE202
s7.4, obtaining a precise estimation value: according to the starting frequency
Figure 63062DEST_PATH_IMAGE203
And chirp rate
Figure 721577DEST_PATH_IMAGE204
The coarse estimation value and the estimated residual quantity can obtain a fine estimation value of the two parameters, and the solution formula is as follows:
Figure 559083DEST_PATH_IMAGE205
wherein the content of the first and second substances,
Figure 328455DEST_PATH_IMAGE206
and
Figure 567807DEST_PATH_IMAGE207
respectively represent
Figure 397223DEST_PATH_IMAGE208
Multiple multi-component chirp component signal
Figure 722025DEST_PATH_IMAGE209
Starting frequency of
Figure 560668DEST_PATH_IMAGE210
And chirp rate
Figure 920105DEST_PATH_IMAGE211
And (6) fine estimation value.
Further, the step S4
Figure 917492DEST_PATH_IMAGE212
A signal of a linear frequency-modulated component
Figure 729590DEST_PATH_IMAGE213
Signal amplitude estimate of
Figure 840766DEST_PATH_IMAGE214
And the second obtained in said S7.4
Figure 54710DEST_PATH_IMAGE215
A signal of a linear frequency-modulated component
Figure 960349DEST_PATH_IMAGE216
Fine estimation of the starting frequency of
Figure 400688DEST_PATH_IMAGE217
Sum chirp slope fine estimate
Figure 315555DEST_PATH_IMAGE218
To reconstruct the first
Figure 121355DEST_PATH_IMAGE219
A signal of a linear frequency-modulated component
Figure 463475DEST_PATH_IMAGE220
The concrete formula is as follows:
Figure 250166DEST_PATH_IMAGE221
Figure 968723DEST_PATH_IMAGE222
wherein the content of the first and second substances,
Figure 891679DEST_PATH_IMAGE223
represents the reconstructed second
Figure 670280DEST_PATH_IMAGE224
A chirp component signal.
The invention has the beneficial effects that: firstly, realizing the optimal transformation order estimation of a Linear Frequency Modulation (LFM) component signal by a global search method based on a minimum entropy criterion, and estimating the signal amplitude of the component; secondly, estimating the instantaneous frequency of the LFM component signal through short-time fractional Fourier transform; according to the estimated instantaneous frequency, a polynomial regression method based on linear least square estimation is adopted to realize the rough estimation of the initial frequency and the frequency modulation slope of the LFM component signal; and then, obtaining a fine estimation result of the initial frequency and the frequency modulation slope of the LFM component signal through frequency modulation removal, low-pass filtering and phase regression processing. And finally, reconstructing the LFM component signal according to the estimated signal amplitude, initial frequency and fine frequency modulation slope estimation value, and sequentially realizing parameter estimation of each LFM component signal by removing the LFM component signal from the multi-component LFM signal and circularly executing the operation. In addition, when the optimal transformation order of the LFM component signal is estimated, whether parameter estimation is terminated or not can be judged in a self-adaptive mode by calculating the variance of the entropy vector and comparing the variance with a fixed threshold value, and the number of the LFM component signals in the signal can be determined. The method is a quasi-maximum likelihood estimation, not only improves the instantaneous frequency estimation of component signals through short-time fractional Fourier transform with higher time-frequency resolution when a time-frequency spectrogram is obtained, but also reduces the influence of noise on parameter estimation through low-pass filtering processing based on a moving average filter, and has a lower signal-to-noise ratio threshold.
Drawings
Fig. 1 is a flow chart of the adaptive multi-component chirp signal parameter estimation method based on STFrFT of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Referring to fig. 1, the adaptive multi-component chirp signal parameter estimation method based on STFrFT includes the following steps:
s1, obtaining a multi-component chirp signal
Figure 944266DEST_PATH_IMAGE001
(ii) a Assuming a multi-component chirp signal consisting of
Figure 200935DEST_PATH_IMAGE002
The chirp component signal and the background noise component, namely:
Figure 243977DEST_PATH_IMAGE003
Figure 193479DEST_PATH_IMAGE225
wherein the content of the first and second substances,
Figure 954762DEST_PATH_IMAGE005
is shown as
Figure 15121DEST_PATH_IMAGE006
The frequency of the individual chirp component signals,
Figure 909741DEST_PATH_IMAGE007
which is representative of the background noise signal,
Figure 30143DEST_PATH_IMAGE008
、
Figure 278722DEST_PATH_IMAGE009
and
Figure 877194DEST_PATH_IMAGE010
respectively represent
Figure 894828DEST_PATH_IMAGE011
The signal amplitude, the start frequency and the chirp rate,
Figure 186132DEST_PATH_IMAGE012
represents the index of the sample point of the discrete signal, and has,
Figure 656428DEST_PATH_IMAGE013
，
Figure 324170DEST_PATH_IMAGE014
represents the total number of sample points of the discrete signal,
Figure 196311DEST_PATH_IMAGE015
which is indicative of the time of observation of the signal,
Figure 392937DEST_PATH_IMAGE016
which represents the time interval between the sampling of the samples,
Figure 616108DEST_PATH_IMAGE017
the number of the imaginary numbers is represented,
Figure 556382DEST_PATH_IMAGE018
expressed as natural constants
Figure 274240DEST_PATH_IMAGE019
An exponential operation of a base number;
s2, initializing the linear frequency modulation component signal number index
Figure 172926DEST_PATH_IMAGE020
Let the residual signal
Figure 617814DEST_PATH_IMAGE021
；
S3, the calculation corresponds to
Figure 361779DEST_PATH_IMAGE022
A signal of said chirp component
Figure 208512DEST_PATH_IMAGE023
Of the optimal transformation order
Figure 12520DEST_PATH_IMAGE024
(ii) a Firstly, judging whether the residual signal still has the linear frequency modulation component signal, if so, adopting a global search method based on a minimum entropy criterion to realize the optimal transformation order
Figure 944704DEST_PATH_IMAGE025
Otherwise, ending parameter estimation;
s3.1, in a half period interval
Figure 492360DEST_PATH_IMAGE103
Inter-search step size
Figure 193600DEST_PATH_IMAGE104
To discretize the transformation order to obtain
Figure 168509DEST_PATH_IMAGE105
A discrete value, i.e.
Figure 587989DEST_PATH_IMAGE106
Wherein
Figure 673757DEST_PATH_IMAGE107
Represents a vector of transform order candidate values,
Figure 495083DEST_PATH_IMAGE108
is referred to as the first
Figure 637963DEST_PATH_IMAGE109
The candidate values of the individual transformation orders are,
Figure 279160DEST_PATH_IMAGE110
which represents a rounding-down operation, the rounding-down operation,
Figure 168619DEST_PATH_IMAGE111
representing a transpose operation; initialization iteration number
Figure 844451DEST_PATH_IMAGE112
；
S3.2, calculating the order of transformation equal to
Figure 161163DEST_PATH_IMAGE113
Time residual signal
Figure 555235DEST_PATH_IMAGE114
Fractional order fourier transform of (a); the calculation formula of the fractional Fourier transform is as follows:
Figure 982805DEST_PATH_IMAGE115
wherein the content of the first and second substances,
Figure 247564DEST_PATH_IMAGE116
representing a transformation order of
Figure 757DEST_PATH_IMAGE117
Time residual signal
Figure 616546DEST_PATH_IMAGE226
The result of the fractional order fourier transform of (a),
Figure 847807DEST_PATH_IMAGE119
the kernel function of fractional Fourier transform is represented by the following mathematical expression:
Figure 235582DEST_PATH_IMAGE120
Figure 628517DEST_PATH_IMAGE121
Figure 731602DEST_PATH_IMAGE122
which represents an integer number of times,
Figure 500975DEST_PATH_IMAGE123
indicates a rotation angle, and is provided with
Figure 5906DEST_PATH_IMAGE124
Representing a square-on operation;
s3.3, calculating the order of transformation equal to
Figure 569742DEST_PATH_IMAGE125
Time residual signal
Figure 160124DEST_PATH_IMAGE126
Entropy of the fractional fourier transform result of (a); assuming a residual signal
Figure 998767DEST_PATH_IMAGE127
The vector form of (a) is:
Figure 92625DEST_PATH_IMAGE227
wherein the content of the first and second substances,
Figure 92942DEST_PATH_IMAGE129
representing a discretized signal vector; when the transformation order candidate is
Figure 170619DEST_PATH_IMAGE130
Then, the vector form of the signal fractional order fourier transform result is:
Figure 547374DEST_PATH_IMAGE131
Figure 758388DEST_PATH_IMAGE228
the entropy of the fractional fourier transform result can be calculated by:
Figure 929606DEST_PATH_IMAGE229
wherein the content of the first and second substances,
Figure 963421DEST_PATH_IMAGE134
expressed as natural constants
Figure 143867DEST_PATH_IMAGE135
A logarithmic operation of a base number;
s3.4, step iteration number
Figure 212317DEST_PATH_IMAGE136
Judgment of
Figure 554437DEST_PATH_IMAGE137
If the result is true, entering S3.5 if the result is true, otherwise entering S3.2 to continue execution;
s3.5 each transformation order candidate may calculate an entropy value from said S3.2 and said S3.3, from which a vector of candidate values corresponding to the transformation order may be obtained
Figure 606706DEST_PATH_IMAGE138
Vector of entropy values
Figure 59684DEST_PATH_IMAGE139
Comprises the following steps:
Figure 248220DEST_PATH_IMAGE140
s3.6, normalization entropy value vector
Figure 26820DEST_PATH_IMAGE141
The mathematical formula is as follows:
Figure 300807DEST_PATH_IMAGE230
wherein the content of the first and second substances,
Figure 837703DEST_PATH_IMAGE143
represents a vector of normalized entropy values that is,
Figure 880746DEST_PATH_IMAGE144
expressing the operation of solving the maximum value; computing an entropy vector
Figure 564668DEST_PATH_IMAGE145
Variance of (2)
Figure 325951DEST_PATH_IMAGE146
The mathematical expression is as follows:
Figure 386311DEST_PATH_IMAGE147
wherein the content of the first and second substances,
Figure 549439DEST_PATH_IMAGE231
representing vectors of entropy values
Figure 404262DEST_PATH_IMAGE232
And has a mean value of
Figure 652841DEST_PATH_IMAGE233
；
Judging whether the following formula is satisfied:
Figure 251312DEST_PATH_IMAGE151
wherein the content of the first and second substances,
Figure 3368DEST_PATH_IMAGE152
a threshold, here set to 0.02, representing a determination of the presence or absence of a chirp component signal;
if the above formula is true, the residual signal is determined
Figure 560251DEST_PATH_IMAGE153
If there is a chirp component signal, continuing to execute the step S3.7; otherwise, the residual signal is determined
Figure 30547DEST_PATH_IMAGE153
If only a noise signal exists in the signal, terminating the circulation and finishing the parameter estimation;
s3.7 by vector from entropy values
Figure 695359DEST_PATH_IMAGE154
To estimate the entropy corresponding to the second
Figure 301921DEST_PATH_IMAGE155
A signal of a linear frequency-modulated component
Figure 764126DEST_PATH_IMAGE156
The optimal transformation order of (a), namely:
Figure 987297DEST_PATH_IMAGE157
。
s4, estimating the
Figure 193150DEST_PATH_IMAGE026
A signal of said chirp component
Figure 919798DEST_PATH_IMAGE027
Signal amplitude of
Figure 552904DEST_PATH_IMAGE028
(ii) a In fractional Fourier transform, when the order is changed
Figure 997792DEST_PATH_IMAGE029
Equal to said chirp component signal
Figure 741757DEST_PATH_IMAGE030
Of the optimal transformation order
Figure 588491DEST_PATH_IMAGE031
Then, the signal amplitude of the component is estimated according to the maximum amplitude value in the fractional Fourier transform result and the total number of sampling points of the signal
Figure 392499DEST_PATH_IMAGE032
Namely:
Figure 324683DEST_PATH_IMAGE033
wherein the content of the first and second substances,
Figure 872339DEST_PATH_IMAGE034
is shown as
Figure 576508DEST_PATH_IMAGE035
A signal of a linear frequency-modulated component
Figure 285838DEST_PATH_IMAGE036
Is determined by the signal amplitude estimate of (a),
Figure 970897DEST_PATH_IMAGE037
representing the order of the transformation equal to
Figure 56665DEST_PATH_IMAGE038
Time signal
Figure 877990DEST_PATH_IMAGE039
The fractional order fourier transform result vector of (a);
s5, estimating the
Figure 23801DEST_PATH_IMAGE040
A signal of said chirp component
Figure 664998DEST_PATH_IMAGE041
The instantaneous frequency of (d); obtained according to said S3
Figure 554456DEST_PATH_IMAGE042
A signal of said chirp component
Figure 230288DEST_PATH_IMAGE043
Of the optimal transformation order
Figure 547000DEST_PATH_IMAGE044
Estimating the instantaneous frequency of the signal through short-time fractional Fourier transform, wherein the calculation formula of the short-time fractional Fourier transform is as follows:
Figure 675493DEST_PATH_IMAGE234
wherein the content of the first and second substances,
Figure 103064DEST_PATH_IMAGE046
representing a transformation order of
Figure 896052DEST_PATH_IMAGE047
Time, residual signal
Figure 383665DEST_PATH_IMAGE048
The result of the short-time fractional fourier transform,
Figure 265033DEST_PATH_IMAGE049
representing the window function length of
Figure 230715DEST_PATH_IMAGE050
And has a proper Gaussian window function
Figure 615560DEST_PATH_IMAGE051
When the temperature of the water is higher than the set temperature,
Figure 539654DEST_PATH_IMAGE052
；
Figure 642739DEST_PATH_IMAGE053
representing a discrete time sequence;
Figure 412112DEST_PATH_IMAGE054
a kernel function representing a fractional Fourier transform;
the instantaneous frequency of the signal is estimated by searching for the maximum of the short-time fractional order fourier transform results at different sampling points, namely:
Figure 917042DEST_PATH_IMAGE055
Figure 746458DEST_PATH_IMAGE235
wherein the content of the first and second substances,
Figure 336839DEST_PATH_IMAGE057
is shown as
Figure 909903DEST_PATH_IMAGE058
A signal of a linear frequency-modulated component
Figure 260551DEST_PATH_IMAGE059
The instantaneous frequency of the received signal,
Figure 260868DEST_PATH_IMAGE060
representing a modulo operation;
Figure 338546DEST_PATH_IMAGE061
is a frequency variable;
s6, rough estimation
Figure 715301DEST_PATH_IMAGE062
A signal of said chirp component
Figure 929244DEST_PATH_IMAGE063
Starting frequency of
Figure 366042DEST_PATH_IMAGE064
And chirp rate
Figure 665436DEST_PATH_IMAGE065
；
Estimated signal according to the S5
Figure 580302DEST_PATH_IMAGE066
And roughly estimating the starting frequency by a polynomial regression method based on linear least squares estimation
Figure 914332DEST_PATH_IMAGE067
And chirp rate
Figure 256451DEST_PATH_IMAGE068
The solving formula is as follows:
Figure 308721DEST_PATH_IMAGE069
wherein the content of the first and second substances,
Figure 761699DEST_PATH_IMAGE070
representing the frequency of the start
Figure 212885DEST_PATH_IMAGE071
And chirp rate
Figure 725906DEST_PATH_IMAGE072
The vector of coarse estimates of (a), i.e.:
Figure 999892DEST_PATH_IMAGE073
；
Figure 256561DEST_PATH_IMAGE074
and
Figure 565183DEST_PATH_IMAGE075
respectively represent
Figure 249105DEST_PATH_IMAGE076
A signal of a linear frequency-modulated component
Figure 10387DEST_PATH_IMAGE077
Starting frequency of
Figure 70747DEST_PATH_IMAGE078
And chirp rate
Figure 233875DEST_PATH_IMAGE079
Is determined by the coarse estimation value of (c),
Figure 88699DEST_PATH_IMAGE236
which represents the operation of transposition by means of a transposition operation,
Figure 337278DEST_PATH_IMAGE081
representing an inversion operation, a matrix
Figure 201329DEST_PATH_IMAGE082
Sum vector
Figure 956314DEST_PATH_IMAGE083
Respectively as follows:
Figure 247618DEST_PATH_IMAGE084
Figure 983492DEST_PATH_IMAGE237
Figure 120076DEST_PATH_IMAGE238
Figure 992217DEST_PATH_IMAGE239
s7, fine estimation
Figure 720001DEST_PATH_IMAGE088
A signal of said chirp component
Figure 412014DEST_PATH_IMAGE089
Starting frequency of
Figure 883446DEST_PATH_IMAGE090
And chirp rate
Figure 610094DEST_PATH_IMAGE091
(ii) a Obtaining rough estimated values of the initial frequency and the frequency modulation slope according to the S6
Figure 243201DEST_PATH_IMAGE092
And
Figure 688088DEST_PATH_IMAGE093
the signals are processed by frequency-modulation removal, low-pass filtering and phase regression
Figure 697633DEST_PATH_IMAGE094
Carrying out fine estimation on the initial frequency and the frequency modulation slope;
the pair signal in S7
Figure 278787DEST_PATH_IMAGE158
The initial frequency and the chirp rate are precisely estimated, specifically as follows:
s7.1, frequency modulation removing: first, based on the initial frequency and the rough estimation value of the chirp rate
Figure 79865DEST_PATH_IMAGE240
And
Figure 746470DEST_PATH_IMAGE241
to reconstruct the
Figure 559705DEST_PATH_IMAGE242
A signal of a linear frequency-modulated component
Figure 260945DEST_PATH_IMAGE162
The phase conjugate term of (a), namely:
Figure 970275DEST_PATH_IMAGE243
Figure 124176DEST_PATH_IMAGE164
wherein the content of the first and second substances,
Figure 475523DEST_PATH_IMAGE165
to be reconstructed
Figure 31269DEST_PATH_IMAGE166
A signal of a linear frequency-modulated component
Figure 911500DEST_PATH_IMAGE167
The phase conjugate term of (a);
secondly, according to the reconstructed second
Figure 818276DEST_PATH_IMAGE168
A signal of a linear frequency-modulated component
Figure 707735DEST_PATH_IMAGE169
Phase conjugate term of to the residual signal
Figure 132636DEST_PATH_IMAGE170
And (3) performing frequency modulation removal treatment, wherein the specific formula is as follows:
Figure 714927DEST_PATH_IMAGE171
Figure 843420DEST_PATH_IMAGE172
wherein the content of the first and second substances,
Figure 536570DEST_PATH_IMAGE173
the residual signal after frequency modulation processing is removed;
s7.2, low-pass filtering: performing low-pass filtering processing on the signal subjected to frequency modulation removal processing by adopting a moving average filter; the specific formula is as follows:
Figure 66908DEST_PATH_IMAGE174
Figure 554521DEST_PATH_IMAGE244
wherein the content of the first and second substances,
Figure 639152DEST_PATH_IMAGE176
，
Figure 135992DEST_PATH_IMAGE177
represents the length of the moving average filter;
Figure 520837DEST_PATH_IMAGE178
represents the result of the low-pass filtering process;
s7.3, phase regression treatment:
firstly, extracting the signal phase after the low-pass filtering processing, wherein the specific formula is as follows:
Figure 444931DEST_PATH_IMAGE179
Figure 548016DEST_PATH_IMAGE180
wherein the content of the first and second substances,
Figure 582968DEST_PATH_IMAGE181
which represents the operation of the arctan function,
Figure 819390DEST_PATH_IMAGE182
the imaginary part operation of the signal is expressed,
Figure 648806DEST_PATH_IMAGE183
representing the operation of taking the real part of the signal;
Figure 239187DEST_PATH_IMAGE184
representing the extracted signal phase;
the extracted signal phases are written in vector form
Figure 812251DEST_PATH_IMAGE185
Comprises the following steps:
Figure 171688DEST_PATH_IMAGE186
secondly, the polynomial regression method based on linear least square estimation is adopted to carry out phase matching on the extracted signal
Figure 437584DEST_PATH_IMAGE187
Fitting to estimate the initial frequency of the de-modulated signal
Figure 249682DEST_PATH_IMAGE188
And chirp rate
Figure 626437DEST_PATH_IMAGE189
The solution formula is as follows:
Figure 840381DEST_PATH_IMAGE190
wherein the content of the first and second substances,
Figure 277178DEST_PATH_IMAGE191
representing the estimated phase constant of the signal,
Figure 576573DEST_PATH_IMAGE192
and
Figure 757018DEST_PATH_IMAGE193
respectively representing estimated signal start frequencies
Figure 93977DEST_PATH_IMAGE194
And chirp rate
Figure 436097DEST_PATH_IMAGE195
And rest amount of
Figure 222787DEST_PATH_IMAGE245
，
Figure 941345DEST_PATH_IMAGE197
Matrix of
Figure 129881DEST_PATH_IMAGE198
Expressed as:
Figure 908481DEST_PATH_IMAGE199
wherein the content of the first and second substances,
Figure 916888DEST_PATH_IMAGE200
represents a time variable, and the mathematical expression thereof is as follows:
Figure 173557DEST_PATH_IMAGE201
Figure 216599DEST_PATH_IMAGE246
s7.4, obtaining a precise estimation value: according to the starting frequency
Figure 634942DEST_PATH_IMAGE203
And chirp rate
Figure 661804DEST_PATH_IMAGE204
The coarse estimation value and the estimated residual quantity can obtain a fine estimation value of the two parameters, and the solution formula is as follows:
Figure 456585DEST_PATH_IMAGE205
wherein the content of the first and second substances,
Figure 616783DEST_PATH_IMAGE206
and
Figure 737186DEST_PATH_IMAGE207
respectively represent
Figure 720186DEST_PATH_IMAGE208
Multiple multi-component chirp component signal
Figure 318657DEST_PATH_IMAGE209
Starting frequency of
Figure 883762DEST_PATH_IMAGE210
And chirp rate
Figure 838381DEST_PATH_IMAGE211
And (6) fine estimation value.
S8, the first obtained according to the S4
Figure 43097DEST_PATH_IMAGE095
A signal amplitude estimation value of the chirp component signal and the second obtained at said S7
Figure 445260DEST_PATH_IMAGE096
The initial frequency and the fine frequency modulation slope value of the linear frequency modulation component signal are reconstructed
Figure 317401DEST_PATH_IMAGE097
A signal of a linear frequency-modulated component
Figure 514027DEST_PATH_IMAGE098
；
Obtained according to said S4
Figure 737198DEST_PATH_IMAGE212
A chirp component signalNumber (C)
Figure 943051DEST_PATH_IMAGE213
Signal amplitude estimate of
Figure 935278DEST_PATH_IMAGE214
And the second obtained in said S7.4
Figure 302805DEST_PATH_IMAGE215
A signal of a linear frequency-modulated component
Figure 747693DEST_PATH_IMAGE216
Fine estimation of the starting frequency of
Figure 426412DEST_PATH_IMAGE217
Sum chirp slope fine estimate
Figure 7566DEST_PATH_IMAGE218
To reconstruct the first
Figure 811574DEST_PATH_IMAGE219
A signal of a linear frequency-modulated component
Figure 478178DEST_PATH_IMAGE220
The concrete formula is as follows:
Figure 291414DEST_PATH_IMAGE247
Figure 461495DEST_PATH_IMAGE222
wherein the content of the first and second substances,
Figure 436404DEST_PATH_IMAGE223
represents the reconstructed second
Figure 855884DEST_PATH_IMAGE224
A chirp component signal.
S9, stepping the linear frequency modulation component signal number index
Figure 941652DEST_PATH_IMAGE099
And updates the residual signal to:
Figure 497398DEST_PATH_IMAGE100
Figure 908788DEST_PATH_IMAGE101
represents the reconstructed second
Figure 552914DEST_PATH_IMAGE102
A chirp component signal;
returning to the step S3 to continue execution.
1) The quasi-maximum likelihood estimation method based on short-time fractional Fourier transform is provided, the estimation precision of each component instantaneous frequency is improved through higher signal time-frequency resolution, the estimation progress of initial frequency and frequency modulation slope is further improved, and the threshold value of signal-to-noise ratio is lower;
2) because the optimal transformation orders of all LFM component signals are different, only one LFM component signal has higher time-frequency resolution in single cycle estimation by adopting short-time fractional Fourier transform, the mutual influence among all LFM component signals in instantaneous frequency estimation is removed, and a foundation is laid for realizing the parameter estimation of the multi-component LFM signal;
3) when the optimal transformation order of the LFM component signals is estimated, whether parameter estimation is terminated or not can be judged in a self-adaptive mode by calculating the variance of the entropy vector and comparing the variance with a threshold value, and the number of the LFM component signals in the signals is determined;
4) according to the estimated signal amplitude, initial frequency and frequency modulation slope, reconstructing an LFM component signal, subtracting the LFM component signal from a multi-component LFM signal, removing the influence of the LFM component signal with the parameter estimation on the LFM component signal without the parameter estimation, and sequentially realizing the parameter estimation of each LFM component signal.
The above-mentioned embodiments only express the embodiments of the present invention, and the description thereof is more specific and detailed, but not construed as limiting the scope of the present invention. It should be noted that, for a person skilled in the art, several variations and modifications can be made without departing from the inventive concept, which falls within the scope of the present invention. Therefore, the protection scope of the present patent shall be subject to the appended claims.

Claims (4)

1. The adaptive multi-component chirp signal parameter estimation method based on the STFrFT is characterized by comprising the following steps of:
s1, obtaining a multi-component chirp signal
Figure 31063DEST_PATH_IMAGE001
(ii) a Assuming a multi-component chirp signal consisting of
Figure 885887DEST_PATH_IMAGE002
The chirp component signal and the background noise component, namely:
Figure 665624DEST_PATH_IMAGE003
Figure 998516DEST_PATH_IMAGE004
wherein the content of the first and second substances,
Figure 812889DEST_PATH_IMAGE005
is shown as
Figure 635351DEST_PATH_IMAGE006
The frequency of the individual chirp component signals,
Figure 840067DEST_PATH_IMAGE007
which is representative of the background noise signal,
Figure 38968DEST_PATH_IMAGE008
、
Figure 645529DEST_PATH_IMAGE009
and
Figure 638893DEST_PATH_IMAGE010
respectively represent
Figure 393223DEST_PATH_IMAGE011
The signal amplitude, the start frequency and the chirp rate,
Figure 67918DEST_PATH_IMAGE012
represents the index of the sample point of the discrete signal, and has,
Figure 591303DEST_PATH_IMAGE013
，
Figure 21147DEST_PATH_IMAGE014
represents the total number of sample points of the discrete signal,
Figure 200456DEST_PATH_IMAGE015
which is indicative of the time of observation of the signal,
Figure 475579DEST_PATH_IMAGE016
which represents the time interval between the sampling of the samples,
Figure 119050DEST_PATH_IMAGE017
representing imaginary numbers by natural constants
Figure 120821DEST_PATH_IMAGE019
An exponential operation of a base number;
s2, initializing the linear frequency modulation component signal number index
Figure 199636DEST_PATH_IMAGE020
Let the residual signal
Figure 635296DEST_PATH_IMAGE021
；
S3, the calculation corresponds to
Figure 406943DEST_PATH_IMAGE022
A signal of said chirp component
Figure 357582DEST_PATH_IMAGE023
Of the optimal transformation order
Figure 177770DEST_PATH_IMAGE024
(ii) a Firstly, judging whether the residual signal still has the linear frequency modulation component signal, if so, adopting a global search method based on a minimum entropy criterion to realize the optimal transformation order
Figure 530254DEST_PATH_IMAGE025
Otherwise, ending parameter estimation;
s4, estimating the
Figure 472802DEST_PATH_IMAGE026
A signal of said chirp component
Figure 848420DEST_PATH_IMAGE027
Signal amplitude of
Figure 269037DEST_PATH_IMAGE028
(ii) a In fractional Fourier transform, when the order is changed
Figure 741607DEST_PATH_IMAGE029
Equal to said chirp component signal
Figure 792739DEST_PATH_IMAGE030
Of the optimal transformation order
Figure 717970DEST_PATH_IMAGE031
Then, the signal amplitude of the component is estimated according to the maximum amplitude value in the fractional Fourier transform result and the total number of sampling points of the signal
Figure 942278DEST_PATH_IMAGE032
Namely:
Figure 941458DEST_PATH_IMAGE033
wherein the content of the first and second substances,
Figure 225809DEST_PATH_IMAGE034
is shown as
Figure 638336DEST_PATH_IMAGE035
A signal of a linear frequency-modulated component
Figure 604017DEST_PATH_IMAGE036
Is determined by the signal amplitude estimate of (a),
Figure 785600DEST_PATH_IMAGE037
representing the order of the transformation equal to
Figure 912956DEST_PATH_IMAGE038
Time signal
Figure 812779DEST_PATH_IMAGE039
The fractional order fourier transform result vector of (a);
s5, estimating the
Figure 378890DEST_PATH_IMAGE040
A signal of said chirp component
Figure 606522DEST_PATH_IMAGE041
In the momentA time frequency; obtained according to said S3
Figure 967096DEST_PATH_IMAGE042
A signal of said chirp component
Figure 354215DEST_PATH_IMAGE043
Of the optimal transformation order
Figure 724017DEST_PATH_IMAGE044
Estimating the instantaneous frequency of the signal through short-time fractional Fourier transform, wherein the calculation formula of the short-time fractional Fourier transform is as follows:
Figure 552296DEST_PATH_IMAGE045
wherein the content of the first and second substances,
Figure 349350DEST_PATH_IMAGE046
representing a transformation order of
Figure 958186DEST_PATH_IMAGE047
Time, residual signal
Figure 131679DEST_PATH_IMAGE048
The result of the short-time fractional fourier transform,
Figure 80043DEST_PATH_IMAGE049
representing the window function length of
Figure 47999DEST_PATH_IMAGE050
And has a proper Gaussian window function
Figure 144131DEST_PATH_IMAGE051
When the temperature of the water is higher than the set temperature,
Figure 793418DEST_PATH_IMAGE052
；
Figure 658606DEST_PATH_IMAGE053
representing a discrete time sequence;
Figure 797463DEST_PATH_IMAGE054
a kernel function representing a fractional Fourier transform;
the instantaneous frequency of the signal is estimated by searching for the maximum of the short-time fractional order fourier transform results at different sampling points, namely:
Figure 380891DEST_PATH_IMAGE055
Figure 568290DEST_PATH_IMAGE056
wherein the content of the first and second substances,
Figure 553564DEST_PATH_IMAGE057
is shown as
Figure 863322DEST_PATH_IMAGE058
A signal of a linear frequency-modulated component
Figure 934047DEST_PATH_IMAGE059
The instantaneous frequency of the received signal,
Figure 925136DEST_PATH_IMAGE060
representing a modulo operation;
Figure 764916DEST_PATH_IMAGE061
is a frequency variable;
s6, rough estimation
Figure 245576DEST_PATH_IMAGE062
An instituteSaid chirp component signal
Figure 538017DEST_PATH_IMAGE063
Starting frequency of
Figure 332798DEST_PATH_IMAGE064
And chirp rate
Figure 27085DEST_PATH_IMAGE065
；
Estimated signal according to the S5
Figure 678646DEST_PATH_IMAGE066
And roughly estimating the starting frequency by a polynomial regression method based on linear least squares estimation
Figure 661645DEST_PATH_IMAGE067
And chirp rate
Figure 56854DEST_PATH_IMAGE068
The solving formula is as follows:
Figure 605648DEST_PATH_IMAGE069
wherein the content of the first and second substances,
Figure 693689DEST_PATH_IMAGE070
representing the frequency of the start
Figure 898406DEST_PATH_IMAGE071
And chirp rate
Figure 97306DEST_PATH_IMAGE072
The vector of coarse estimates of (a), i.e.:
Figure 500605DEST_PATH_IMAGE073
；
Figure 759548DEST_PATH_IMAGE074
and
Figure 451561DEST_PATH_IMAGE075
respectively represent
Figure 188573DEST_PATH_IMAGE076
A signal of a linear frequency-modulated component
Figure 711958DEST_PATH_IMAGE077
Starting frequency of
Figure 141802DEST_PATH_IMAGE078
And chirp rate
Figure 993215DEST_PATH_IMAGE079
The coarse estimation value of (1) represents a transposition operation, represents an inversion operation, and represents a matrix
Figure 450238DEST_PATH_IMAGE082
Sum vector
Figure 179159DEST_PATH_IMAGE083
Respectively as follows:
Figure 257974DEST_PATH_IMAGE084
Figure 755951DEST_PATH_IMAGE085
Figure 199702DEST_PATH_IMAGE086
Figure 415920DEST_PATH_IMAGE087
s7, fine estimation
Figure 298425DEST_PATH_IMAGE088
A signal of said chirp component
Figure 650909DEST_PATH_IMAGE089
Starting frequency of
Figure 531140DEST_PATH_IMAGE090
And chirp rate
Figure 969075DEST_PATH_IMAGE091
(ii) a Obtaining rough estimated values of the initial frequency and the frequency modulation slope according to the S6
Figure 389692DEST_PATH_IMAGE092
And
Figure 862262DEST_PATH_IMAGE093
the signals are processed by frequency-modulation removal, low-pass filtering and phase regression
Figure 647815DEST_PATH_IMAGE094
Carrying out fine estimation on the initial frequency and the frequency modulation slope;
s8, the first obtained according to the S4
Figure 573046DEST_PATH_IMAGE095
A signal amplitude estimation value of the chirp component signal and the second obtained at said S7
Figure 797354DEST_PATH_IMAGE096
The initial frequency and the fine frequency modulation slope value of the linear frequency modulation component signal are reconstructed
Figure 62113DEST_PATH_IMAGE097
A signal of a linear frequency-modulated component
Figure 346464DEST_PATH_IMAGE098
；
S9, stepping the linear frequency modulation component signal number index
Figure 758991DEST_PATH_IMAGE099
And updates the residual signal to:
Figure 521410DEST_PATH_IMAGE100
Figure 640676DEST_PATH_IMAGE101
represents the reconstructed second
Figure 95928DEST_PATH_IMAGE102
A chirp component signal;
returning to the step S3 to continue execution.
2. The STFrFT-based adaptive multi-component chirp signal parameter estimation method of claim 1, wherein the S3 comprises the following implementation steps:
s3.1, in a half period interval
Figure 995751DEST_PATH_IMAGE103
Inter-search step size
Figure 561862DEST_PATH_IMAGE104
To discretize the transformation order to obtain
Figure 535634DEST_PATH_IMAGE105
A discrete value, i.e.
Figure 161787DEST_PATH_IMAGE106
Wherein
Figure 548906DEST_PATH_IMAGE107
Represents a vector of transform order candidate values,
Figure 590811DEST_PATH_IMAGE108
is referred to as the first
Figure 746986DEST_PATH_IMAGE109
A transform order candidate representing a round-down operation and a transposition operation; initialization iteration number
Figure 264052DEST_PATH_IMAGE112
；
S3.2, calculating the order of transformation equal to
Figure 274734DEST_PATH_IMAGE113
Time residual signal
Figure 242690DEST_PATH_IMAGE114
Fractional order fourier transform of (a); the calculation formula of the fractional Fourier transform is as follows:
Figure 338822DEST_PATH_IMAGE115
wherein the content of the first and second substances,
Figure 722530DEST_PATH_IMAGE116
representing a transformation order of
Figure 853297DEST_PATH_IMAGE117
Time residual signal
Figure 992154DEST_PATH_IMAGE118
The result of the fractional order fourier transform of (a),
Figure 575582DEST_PATH_IMAGE119
kernel function being fractional Fourier transformA number, whose mathematical expression is:
Figure 762981DEST_PATH_IMAGE120
Figure 748254DEST_PATH_IMAGE121
Figure 58013DEST_PATH_IMAGE122
which represents an integer number of times,
Figure 66420DEST_PATH_IMAGE123
indicates a rotation angle, and is provided with
Figure 119827DEST_PATH_IMAGE124
Representing a square-on operation;
s3.3, calculating the order of transformation equal to
Figure 959607DEST_PATH_IMAGE125
Time residual signal
Figure 440267DEST_PATH_IMAGE126
Entropy of the fractional fourier transform result of (a); assuming a residual signal
Figure 670391DEST_PATH_IMAGE127
The vector form of (a) is:
Figure 527489DEST_PATH_IMAGE128
wherein the content of the first and second substances,
Figure 221775DEST_PATH_IMAGE129
representing a discretized signal vector; when the transformation order candidate is
Figure 873336DEST_PATH_IMAGE130
Then, the vector form of the signal fractional order fourier transform result is:
Figure 856336DEST_PATH_IMAGE131
Figure 251545DEST_PATH_IMAGE132
the entropy of the fractional fourier transform result can be calculated by:
Figure 800338DEST_PATH_IMAGE133
wherein, natural constants are expressed
Figure 93096DEST_PATH_IMAGE135
A logarithmic operation of a base number;
s3.4, step iteration number
Figure 26417DEST_PATH_IMAGE136
Judgment of
Figure 695296DEST_PATH_IMAGE137
If the result is true, entering S3.5 if the result is true, otherwise entering S3.2 to continue execution;
s3.5 each transformation order candidate may calculate an entropy value from said S3.2 and said S3.3, from which a vector of candidate values corresponding to the transformation order may be obtained
Figure 891922DEST_PATH_IMAGE138
Vector of entropy values
Figure 380672DEST_PATH_IMAGE139
Comprises the following steps:
Figure 383263DEST_PATH_IMAGE140
s3.6, normalization entropy value vector
Figure 906649DEST_PATH_IMAGE141
The mathematical formula is as follows:
Figure 8597DEST_PATH_IMAGE142
wherein the content of the first and second substances,
Figure 250222DEST_PATH_IMAGE143
expressing a normalized entropy vector and expressing a maximum value calculation; computing an entropy vector
Figure 168817DEST_PATH_IMAGE145
Variance of (2)
Figure 707245DEST_PATH_IMAGE146
The mathematical expression is as follows:
Figure 170588DEST_PATH_IMAGE147
wherein the content of the first and second substances,
Figure 514982DEST_PATH_IMAGE148
representing vectors of entropy values
Figure 12959DEST_PATH_IMAGE149
And has a mean value of
Figure 456710DEST_PATH_IMAGE150
；
Judging whether the following formula is satisfied:
Figure 672927DEST_PATH_IMAGE151
wherein the content of the first and second substances,
Figure 555433DEST_PATH_IMAGE152
a threshold, here set to 0.02, representing a determination of the presence or absence of a chirp component signal;
if the above formula is true, the residual signal is determined
Figure 845600DEST_PATH_IMAGE153
If there is a chirp component signal, continuing to execute the step S3.7; otherwise, the residual signal is determined
Figure 522569DEST_PATH_IMAGE153
If only a noise signal exists in the signal, terminating the circulation and finishing the parameter estimation;
s3.7 by vector from entropy values
Figure 960503DEST_PATH_IMAGE154
To estimate the entropy corresponding to the second
Figure 646700DEST_PATH_IMAGE155
A signal of a linear frequency-modulated component
Figure 803092DEST_PATH_IMAGE156
The optimal transformation order of (a), namely:
Figure 916541DEST_PATH_IMAGE157
。
3. the STFrFT-based adaptive multi-component chirp signal parameter estimation method of claim 2, wherein the S7 is for a signal
Figure 841772DEST_PATH_IMAGE158
The initial frequency and the chirp rate are precisely estimated, specifically as follows:
s7.1, frequency modulation removing: first, based on the initial frequency and the rough estimation value of the chirp rate
Figure 66080DEST_PATH_IMAGE159
And
Figure 330839DEST_PATH_IMAGE160
to reconstruct the
Figure 615190DEST_PATH_IMAGE161
A signal of a linear frequency-modulated component
Figure 762138DEST_PATH_IMAGE162
The phase conjugate term of (a), namely:
Figure 727820DEST_PATH_IMAGE163
Figure 909402DEST_PATH_IMAGE164
wherein the content of the first and second substances,
Figure 364654DEST_PATH_IMAGE165
to be reconstructed
Figure 264477DEST_PATH_IMAGE166
A signal of a linear frequency-modulated component
Figure 768271DEST_PATH_IMAGE167
The phase conjugate term of (a);
secondly, according to the reconstructed second
Figure 804360DEST_PATH_IMAGE168
A signal of a linear frequency-modulated component
Figure 430514DEST_PATH_IMAGE169
Phase conjugate term of to the residual signal
Figure 552053DEST_PATH_IMAGE170
And (3) performing frequency modulation removal treatment, wherein the specific formula is as follows:
Figure 859538DEST_PATH_IMAGE171
Figure 15713DEST_PATH_IMAGE172
wherein the content of the first and second substances,
Figure 812767DEST_PATH_IMAGE173
the residual signal after frequency modulation processing is removed;
s7.2, low-pass filtering: performing low-pass filtering processing on the signal subjected to frequency modulation removal processing by adopting a moving average filter; the specific formula is as follows:
Figure 296969DEST_PATH_IMAGE174
Figure 142566DEST_PATH_IMAGE175
wherein the content of the first and second substances,
Figure 153247DEST_PATH_IMAGE176
，
Figure 121203DEST_PATH_IMAGE177
represents the length of the moving average filter;
Figure 155018DEST_PATH_IMAGE178
represents the result of the low-pass filtering process;
s7.3, phase regression treatment:
firstly, extracting the signal phase after the low-pass filtering processing, wherein the specific formula is as follows:
Figure 866622DEST_PATH_IMAGE179
Figure 997389DEST_PATH_IMAGE180
wherein, the method represents the operation of an arc tangent function, the operation of taking the imaginary part of a signal and the operation of taking the real part of the signal;
Figure 892347DEST_PATH_IMAGE184
representing the extracted signal phase;
the extracted signal phases are written in vector form
Figure 202106DEST_PATH_IMAGE185
Comprises the following steps:
Figure 944934DEST_PATH_IMAGE186
secondly, the polynomial regression method based on linear least square estimation is adopted to carry out phase matching on the extracted signal
Figure 998340DEST_PATH_IMAGE187
Fitting to estimate the initial frequency of the de-modulated signal
Figure 838120DEST_PATH_IMAGE188
And chirp rate
Figure 318780DEST_PATH_IMAGE189
The solution formula is as follows:
Figure 814484DEST_PATH_IMAGE190
wherein the content of the first and second substances,
Figure 406002DEST_PATH_IMAGE191
representing the estimated phase constant of the signal,
Figure 365868DEST_PATH_IMAGE192
and
Figure 955112DEST_PATH_IMAGE193
respectively representing estimated signal start frequencies
Figure 734849DEST_PATH_IMAGE194
And chirp rate
Figure 130059DEST_PATH_IMAGE195
And rest amount of
Figure 944431DEST_PATH_IMAGE196
，
Figure 704576DEST_PATH_IMAGE197
Matrix of
Figure 971610DEST_PATH_IMAGE198
Expressed as:
Figure 170510DEST_PATH_IMAGE199
wherein the content of the first and second substances,
Figure 839389DEST_PATH_IMAGE200
represents a time variable, and the mathematical expression thereof is as follows:
Figure 770435DEST_PATH_IMAGE201
Figure 524765DEST_PATH_IMAGE202
s7.4, obtaining a precise estimation value: according to the starting frequency
Figure 527356DEST_PATH_IMAGE203
And chirp rate
Figure 50741DEST_PATH_IMAGE204
The coarse estimation value and the estimated residual quantity can obtain a fine estimation value of the two parameters, and the solution formula is as follows:
Figure 152689DEST_PATH_IMAGE205
wherein the content of the first and second substances,
Figure 394315DEST_PATH_IMAGE206
and
Figure 935018DEST_PATH_IMAGE207
respectively represent
Figure 250592DEST_PATH_IMAGE208
Multiple multi-component chirp component signal
Figure 851338DEST_PATH_IMAGE209
Starting frequency of
Figure 314680DEST_PATH_IMAGE210
And chirp rate
Figure 659074DEST_PATH_IMAGE211
And (6) fine estimation value.
4. The STFrFT-based adaptive multi-factoring of claim 3The method for estimating the parameters of the volume linear frequency modulation signals is characterized in that: obtained according to said S4
Figure 829155DEST_PATH_IMAGE212
A signal of a linear frequency-modulated component
Figure 600802DEST_PATH_IMAGE213
Signal amplitude estimate of
Figure 551441DEST_PATH_IMAGE214
And the second obtained in said S7.4
Figure 433946DEST_PATH_IMAGE215
A signal of a linear frequency-modulated component
Figure 724113DEST_PATH_IMAGE216
Fine estimation of the starting frequency of
Figure 666661DEST_PATH_IMAGE217
Sum chirp slope fine estimate
Figure 104596DEST_PATH_IMAGE218
To reconstruct the first
Figure 728475DEST_PATH_IMAGE219
A signal of a linear frequency-modulated component
Figure 935466DEST_PATH_IMAGE220
The concrete formula is as follows:
Figure 48915DEST_PATH_IMAGE221
Figure 974146DEST_PATH_IMAGE222
wherein the content of the first and second substances,
Figure 136137DEST_PATH_IMAGE223
represents the reconstructed second
Figure 463213DEST_PATH_IMAGE224
A chirp component signal.
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