CN113422663B - Mixed signal separation method and device based on signal decomposition and sparse reconstruction - Google Patents

Mixed signal separation method and device based on signal decomposition and sparse reconstruction Download PDF

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CN113422663B
CN113422663B CN202110599578.3A CN202110599578A CN113422663B CN 113422663 B CN113422663 B CN 113422663B CN 202110599578 A CN202110599578 A CN 202110599578A CN 113422663 B CN113422663 B CN 113422663B
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CN113422663A (en
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晋本周
柴恒
周子烨
高墨昀
李建峰
张小飞
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention provides a mixed signal separation method and a mixed signal separation device based on signal decomposition and sparse reconstruction, wherein the method comprises the following steps: carrying out Fourier decomposition on each component signal under the condition of knowing the number of information sources and the instantaneous frequency of each component signal in the mixed signal to obtain the decomposition coefficient of each component signal; establishing an observation model of the received signal based on the decomposition coefficients of the constituent signals and the received signal; converting a signal separation problem into a sparse optimization problem based on an observation model of a received signal, and establishing a sparse optimization objective function; and solving the sparse optimization problem based on a sparse reconstruction algorithm, realizing reconstruction of the decomposition coefficient of each component signal, and obtaining the time domain signal of each separated component signal on the basis. The method can realize effective separation of signals such as radar and communication under the condition of aliasing in three domains of space, time and frequency.

Description

Mixed signal separation method and device based on signal decomposition and sparse reconstruction
Technical Field
The invention relates to the field of radar and communication signal processing, in particular to a mixed signal separation method and device based on signal decomposition and sparse reconstruction.
Background
With the increasing complexity of electromagnetic environments, signals received by electronic reconnaissance systems often show the superposition of multiple signals in space, time and frequency domains, and overlap exists among the multiple signals. How to separate the signal of interest from other signals is the key for the parameter estimation and signal type identification of the reconnaissance system. In the case of a single channel (or multiple signals enter from the main lobe and cannot be distinguished in the spatial domain), the constituent signals overlap in the time domain, the frequency domain, and the time-frequency domain, and the signal separation task is also generally called single-channel signal separation. Since the problem of single-channel signal separation is quite specific, the separation algorithm will generally rely on some characteristics of the source signal itself. The current single-channel signal separation method mainly comprises a method for converting a single channel into multiple channels, a multi-parameter joint estimation method, an estimation method based on finite symbol set characteristics, a transform domain filtering method and the like. However, the above method mainly has the following problems: (1) In an actual scene, the prior information of a signal to be separated is insufficient, and a plurality of methods are difficult to directly apply; (2) The method has less research aiming at the separation method under the radar and communication signal overlapping scene; (3) Many algorithms are only suitable for scenes with 2 overlapped signals, and the separation performance is poor for scenes with 3 or more overlapped signals; and (4) an algorithm which has both performance and complexity is lacked.
At present, a pervasive algorithm is lacked to solve the problem of single-channel signal separation.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the application requirements and challenges of the separation of the space, time and frequency mixed signals, the invention provides a mixed signal separation method based on signal decomposition and sparse reconstruction, which can be applied to electronic reconnaissance tasks in complex electromagnetic environments.
Another object of the present invention is to provide a mixed signal separating apparatus and a computer device based on signal decomposition and sparse reconstruction.
The technical scheme is as follows: in a first aspect, a mixed signal separation method based on signal decomposition and sparse reconstruction includes the following steps:
1) Carrying out Fourier decomposition on each component signal under the condition that the number of mixed signal information sources and the instantaneous frequency of each component signal are known, and obtaining a decomposition coefficient vector p of each component signal;
2) Establishing an observation model y = Gp + n of the received signal based on the decomposition coefficient vector p of each component signal and the received signal, wherein y is the received signal, G is an observation matrix, and n is noise;
3) Converting the signal separation problem into a sparse optimization problem based on an observation model of the received signal, and establishing a target function of the sparse optimization problem;
4) Solving the sparse optimization problem based on a sparse reconstruction algorithm, realizing sparse reconstruction of the decomposition coefficients of each component signal, and obtaining sparse reconstruction of each component signalOf the decomposition coefficient vector
Figure BDA0003092291650000021
5) Decomposition coefficient vector based on sparse reconstruction of constituent signals
Figure BDA0003092291650000022
And observing the matrix G to obtain each separated component signal.
In a second aspect, a mixed signal separation apparatus based on signal decomposition and sparse reconstruction is provided, the apparatus comprising:
the mixed signal decomposition module is used for carrying out Fourier decomposition on each component signal under the condition that the number of mixed signal information sources and the instantaneous frequency of each component signal are known, so as to obtain a decomposition coefficient vector p of each component signal;
the signal observation model establishing module is used for establishing an observation model y = Gp + n of the received signal based on the decomposition coefficient vector p and the received signal of each component signal, wherein y is the received signal, G is an observation matrix, and n is noise;
the optimization problem establishing module is used for converting the signal separation problem into a sparse optimization problem based on the observation model of the received signal and establishing a target function of the sparse optimization problem;
a sparse reconstruction module for solving the established sparse optimization problem, realizing the sparse reconstruction of the decomposition coefficient of each component signal, and obtaining the decomposition coefficient vector of the sparse reconstruction of each component signal
Figure BDA0003092291650000023
A separate signal calculation module for calculating a decomposition coefficient vector based on sparse reconstruction of each constituent signal
Figure BDA0003092291650000024
And observing the matrix G to obtain each separated component signal.
In a third aspect, a computer device is provided, the device comprising:
a memory having one or more programs stored therein, which when executed by the one or more processors, cause the one or more processors to perform a method for mixed signal separation based on signal decomposition and sparse reconstruction as described in the first aspect of the present invention.
Has the advantages that: according to the method, under the condition of known information source number and instantaneous frequency, fourier decomposition is carried out on all component signals, a sparse observation model of the received signals is established, the signal separation problem is converted into the sparse reconstruction problem of multiple signals, separation of signals such as radar and communication in a complex electromagnetic environment is achieved, and a new technical approach is provided for mixed multiple-signal separation under a single-channel condition. While the separation performance is ensured, the algorithm complexity is controlled to an acceptable level by adopting a sparse reconstruction method based on generalized message transmission. Simulation results show that under the condition that signals are overlapped in time domain, frequency domain and time-frequency domain, a plurality of signals can be effectively separated, and the correlation coefficient of the separated signals and the source signals is more than 0.9 on average.
Drawings
FIG. 1 is a flow chart of a mixed signal separation method according to the present invention;
FIG. 2 is a time-frequency diagram of a mixed signal in a typical scenario according to the present invention;
FIG. 3 is a graph comparing the time domain (real part) of each separated signal with the source signal provided by the present invention;
fig. 4 is a schematic diagram of the correlation coefficient between each separated signal and the source signal provided by the present invention.
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings.
The method is suitable for separating single-channel time-frequency overlapped signals, wherein the overlapped signals are called mixed signals and are also called aliasing signals. The overall scheme is shown in figure 1 and comprises the following steps:
step 1) carrying out Fourier decomposition on each signal under the condition that the number of mixed signal sources and the instantaneous frequency of each signal are known, and obtaining a coefficient vector p of each signal decomposition.
The invention is suitable for signal separation of a plurality of intrinsic signal mixtures.
For a continuous signal s, an intrinsic signal is defined if the following equation is satisfied:
Figure BDA0003092291650000031
wherein A (t) and f (t) satisfy:
Figure BDA0003092291650000032
Figure BDA0003092291650000033
Figure BDA0003092291650000034
Figure BDA0003092291650000035
a (t) represents the amplitude of the signal, f (t) represents the instantaneous frequency of the signal, θ 0 ∈[0，2π]Representing the initial phase of the signal, gamma > 0 is a parameter controlling the chirp rate.
The received mixed signal can be regarded as a linear superposition of multi-component unsteady signals, and the signal model of the received mixed signal can be represented as:
Figure BDA0003092291650000036
wherein M is the number of constituent signals in the mixed signal,
Figure BDA0003092291650000037
is the instantaneous complex envelope of the m-th component, phi m (t) is the instantaneous phase of the mth component,
Figure BDA0003092291650000038
IF m (t) is the instantaneous frequency of the mth component, and n (t) is white Gaussian noise.
Number of sources M and instantaneous frequency IF of each component signal in a mixed signal m (t) can be obtained by an existing method, and is assumed to be known in the present invention. Known methods for obtaining the number of sources and the instantaneous frequency of each component signal are referred to as "W.Lu, J.Xie, H.Wang and C.Sheng", "Parameterized time-frequency analysis to separate multi-radio signals", "Journal of Systems Engineering and Electronics, vol.28, no.3, pp.493-502".
Based on the signal model, the number of mixed signal sources and the instantaneous frequency of each signal can be sent to a signal decomposition module, and the signal envelope is subjected to Fourier expansion, namely
Figure BDA0003092291650000041
Can be unfolded as follows:
Figure BDA0003092291650000042
wherein the content of the first and second substances,
Figure BDA0003092291650000043
is a complex Fourier coefficient;
Figure BDA0003092291650000044
is a fundamental frequency, wherein F s The method is characterized in that the method is used for controlling the thickness of frequency interval division, wherein the signal sampling frequency is N, the number of sampling points is N, Q is a positive integer, and Q =2 can be generally selected;
Figure BDA00030922916500000410
is Fourier order, B m Is the bandwidth of the signal.
The decomposition coefficients of the constituent signals are represented by a vector p, and equation (7) is rewritten as: a is m ＝G′ m p m Wherein a is m ＝[a m (t 0 )，…，a m (t N-1 )] T ；
Figure BDA0003092291650000045
Wherein the content of the first and second substances,
Figure BDA0003092291650000046
Figure BDA0003092291650000047
then a coefficient vector p may be constructed,
Figure BDA0003092291650000048
G′ m is N × (2K) m + 1) of the elements (G' m ) cd Expressed as:
Figure BDA0003092291650000049
wherein, t c-1 Denotes a matrix (G' m ) cd The time corresponding to row c.
And 2) establishing an observation model of the received signal based on each signal decomposition result and the received signal.
Will be provided with
Figure BDA00030922916500000411
Bringing the received signal into the expansion of (1) can result in the following equation:
y＝Gp+n. (9)
wherein the content of the first and second substances,
y＝[y(t 0 )，y(t 1 )，…，y(t N-1 )] T . (10)
n＝[n(t 0 )，n(t 1 )，…，n(t N-1 )] T . (11)
Figure BDA0003092291650000051
Figure BDA0003092291650000052
G＝[G′ 1 ，…，G′ m ，…，G′ M ]. (14)
the dimension of the matrix G is N (2K) m +1)M。
In constructing the vector p m Then the fourier order K needs to be known m . The fourier order may be determined by the degree of oscillation of the instantaneous amplitude of each component signal, i.e. the baseband bandwidth of the complex envelope signal. Thus, it is natural to obtain
Figure BDA0003092291650000053
Wherein, K m In order of the fourier transform,
Figure BDA0003092291650000054
is the previously defined fundamental frequency.
In practice, the bandwidth of each component signal is usually unknown and difficult to measure, and taking a multi-term code radar waveform as an example, the bandwidth is very large at a phase jump point and very small at a non-jump point, and the bandwidth is difficult to measure in practice. For such signals, the conventional decomposition-based separation method has poor reconstruction and separation effects due to many decomposed coefficients close to zero when a large bandwidth value is selected, and has poor signal detail information reconstruction and separation effects due to partial decomposition coefficients not included when a small bandwidth value is selected. In embodiments of the invention, a larger bandwidth value B is given, e.g., B > F s /2, making the bandwidth of each signal satisfy B m < B. The decomposition order of all signals
Figure BDA0003092291650000055
The dimension of matrix G is N × (2K + 1) M. At this time, the vector p m There are many elements of zero, then p is generally sparse.
And 3) converting the signal separation problem into a sparse optimization problem to solve based on the signal observation model, and establishing a sparse optimization objective function.
Based on equation (9), solving the coefficient vector p can be converted into the following sparse optimization problem vector:
Figure BDA0003092291650000056
alpha is a regularization coefficient, generally, the value of alpha is equivalent to the noise power, a good alpha value can be calculated by a plurality of methods, but simulation results show that the parameter has little influence on the problem in the invention and can be 0.5.
Step 4) solving the sparse optimization problem based on a sparse reconstruction algorithm, realizing sparse reconstruction of decomposition coefficients of each component signal, and obtaining a sparse reconstructed coefficient vector
Figure BDA0003092291650000057
The sparse reconstruction method may adopt a Generalized Applied Message Passing (GAMP) algorithm or other sparse reconstruction algorithms to solve the formula (16) to obtain a sparsely reconstructed coefficient vector
Figure BDA0003092291650000064
The GAMP algorithm has a complexity of
Figure BDA0003092291650000065
Lower than matching pursuit, etc. Since the implementation process of the GAMP algorithm is the prior art, those skilled in the art should understand how to solve the sparse optimization problem based on the GAMP algorithm, and the solving process is not the core of the present invention, and therefore, it is not described herein again.
Step 5) based on sparse reconstruction results
Figure BDA0003092291650000061
And observing the matrix G, and calculating to obtain each separated component signal.
After obtaining the sparse reconstruction coefficient vector, each component signal can be reconstructed by the following formula:
Figure BDA0003092291650000062
and calculating to obtain the separated time domain signal.
Signal separation algorithms recover individual source signals from a received composite signal and typically evaluate the performance of the algorithm by comparing source signal estimates to true values. The invention adopts the similarity coefficient as a performance evaluation method.
Suppose a separated signal y m ＝[y m (1)，y m (2)，…，y m (T)] T Corresponding to the estimate of the mth source signal, T =1,2, …, T denotes the sampling instant. The m-th source signal is s m ＝[s m (1)，s m (2)，…，s m (T)] T . The correlation coefficient of the estimated signal with the actual signal can be used as a measure of separation performance, which is defined as:
Figure BDA0003092291650000063
where M is the number of sources, i.e., the number of constituent signals in the mixed signal.
The steps for carrying out the separation method of the present invention are described above, and in order to verify the effectiveness of the method, the following description is made by simulation experiments. Hereinafter, the method of the present invention is abbreviated as CSICCD (Compressive Sensing-inductive chip component decomposition). Fig. 2 shows a time-frequency image (TFI) of the mixed signal before separation according to the present invention. Simulation parameters as shown in table 1, the received aliased signal is an alias of three signals: the 3 signals are respectively a Linear Frequency Modulation (LFM) signal, a phase coding signal and a quadrature amplitude modulation (16-QAM) signal, which respectively correspond to (a), (b) and (c) in FIG. 2, the SNR is respectively 15dB, 15dB and 13dB, the duration is respectively 54 mu s, 72 mu s and 744 mu s, and the starting time of the 3 signals is the same. The LFM signal bandwidth is 30MHz, the center frequency is 0MHz, the code type of the phase coding signal is 13-bit Barker code, the center frequency is-6 MHz, the number of 16-QAM signal code elements is 1500, the code element rate is 2MHz, and the center frequency is 6MHz.
TABLE 1 simulation parameters
Figure BDA0003092291650000071
Fig. 3 shows a time domain (real part) contrast diagram and a local enlarged diagram of the reconstructed separation signal and the source signal in the case of primary simulation, and respectively shows, from top to bottom, (a) a time domain contrast diagram and a local enlarged diagram of a chirp-like modulation (LFM) signal, (b) a time domain contrast diagram and a local enlarged diagram of a phase-coded signal, and (c) a time domain contrast diagram and a local enlarged diagram of a quadrature amplitude modulation (16-QAM) signal. The 3 signal separation performances are shown in table 2.
Table 2 signal separation performance of 3 in one simulation
Signal LFM Phase encoding QAM Remarks to note
Correlation coefficient 0.89 0.96 0.91 First simulation result
Fig. 4 shows the separation performance (correlation coefficient) of 3 signals at a time under 100 simulation conditions, the simulation parameters are still as shown in table 1, and the separation performance (average value) of 3 signals in 100 simulations is shown in table 3.
TABLE 3 Signal separation Performance (average)
Figure BDA0003092291650000072
The feasible idea of solving the single-channel signal separation is to convert the signal separation problem into the reconstruction problem of each component signal. According to the invention, under the condition of known information source number and signal instantaneous frequency, a sparse observation model of a received signal is established by performing signal decomposition on each component signal, so that the problem of signal separation is converted into the problem of sparse reconstruction of the signal, and signal separation under the condition of complex signal overlapping is realized. The method has wide application range and is suitable for effectively separating signals of 3 or more radars, communication and the like.
According to another embodiment of the present invention, there is provided a mixed signal separation apparatus based on signal decomposition and sparse reconstruction, including:
the mixed signal decomposition module is used for carrying out Fourier decomposition on each component signal under the condition that the number of mixed signal information sources and the instantaneous frequency of each component signal are known, so as to obtain a decomposition coefficient vector p of each component signal;
the signal observation model establishing module is used for establishing an observation model y = Gp + n of the received signal based on the decomposition coefficient vector p and the received signal of each component signal, wherein y is the received signal, G is an observation matrix, and n is noise;
the optimization problem establishing module is used for converting the signal separation problem into a sparse optimization problem based on the observation model of the received signal and establishing a target function of the sparse optimization problem;
the sparse reconstruction module is used for solving the established sparse optimization problem, realizing sparse reconstruction of the decomposition coefficient of each component signal and obtaining a decomposition coefficient vector p of the sparse reconstruction of each component signal;
and the separation signal calculation module is used for obtaining each separated component signal based on the sparse reconstructed decomposition coefficient vector p and the observation matrix G of each component signal.
Specifically, the mixed signal decomposition module includes:
a signal representation unit for treating the received mixed signal as a linear superposition of the multi-component non-stationary signal, the signal model of which is represented as:
Figure BDA0003092291650000081
wherein M is the number of each constituent signal in the mixed signal,
Figure BDA0003092291650000082
for the instantaneous complex envelope, phi, of the m-th component signal at time t m (t) is the instantaneous phase of the mth component signal at time t,
Figure BDA0003092291650000083
IF m (t) is the instantaneous frequency of the mth component signal at the time t, and n (t) is white Gaussian noise at the time t;
a Fourier decomposition unit for dividing the envelope
Figure BDA0003092291650000084
Performing Fourier decomposition:
Figure BDA0003092291650000085
wherein the content of the first and second substances,
Figure BDA0003092291650000091
is a complex Fourier coefficient;
Figure BDA0003092291650000092
is a fundamental frequency of, wherein F s The method comprises the following steps of (1) sampling frequency of a signal, N being the number of sampling points, and Q being a positive integer control parameter, wherein the positive integer control parameter is used for controlling the thickness of frequency interval division;
Figure BDA0003092291650000093
is Fourier order, B m Is the bandwidth of the signal;
a coefficient vector construction unit for expressing the decomposition coefficients of the respective constituent signals by a vector p, rewriting the formula (20) as: a is m ＝G′ m p m Wherein a is m ＝[a m (t 0 )，…，a m (t N-1 )] T ；
Figure BDA0003092291650000094
Wherein the content of the first and second substances,
Figure BDA0003092291650000095
a coefficient vector p is constructed which is,
Figure BDA0003092291650000096
G′ m is N × (2K) m + 1) of the elements (G' m ) cd Expressed as:
Figure BDA0003092291650000097
wherein, t c-1 Represents a matrix (G' m ) cd The time corresponding to row c.
Further, the signal observation model establishing module comprises:
the model building unit is used for building an observation model y = Gp + n of the received signal;
a mathematical representation unit, configured to mathematically represent each part according to an observation model of the received signal as follows:
y＝[y(t 0 )，y(t 1 )，…，y(t N-1 )] T . (22)
n＝[n(t 0 )，n(t 1 )，…，n(t N-1 )] T . (23)
Figure BDA0003092291650000098
Figure BDA0003092291650000099
G＝[G′ 1 ，…，G′ m ，…，G′ M ]. (26)
wherein the dimension of the matrix G is N (2K) m + 1) M, N being the number of sampling points, M being the number of each constituent signal in the mixed signal, K m For the order of the fourier, the superscript T represents the matrix transpose,
Figure BDA00030922916500000910
is a complex Fourier coefficient;
a sparse constraint unit for satisfying B by the bandwidth of each signal m < B to keep p a sparse vector, where B is a pre-given bandwidth value;
the objective function of the sparse optimization problem established by the optimization problem establishing module is expressed as:
Figure BDA0003092291650000101
wherein
Figure BDA0003092291650000102
And expressing the decomposed sparse vector after sparse reconstruction, wherein alpha is a regularization coefficient.
The sparse reconstruction module solves the established sparse optimization problem by using a generalized message transfer algorithm or other sparse reconstruction algorithms.
It should be understood that the mixed signal separation apparatus based on signal decomposition and sparse reconstruction provided in this embodiment may implement all technical solutions in the foregoing method embodiments, functions of each functional module may be specifically implemented according to the method in the foregoing method embodiments, and a specific implementation process thereof may refer to relevant descriptions in the foregoing embodiments, which is not described herein again.
Based on the same technical concept as the method embodiment, according to another embodiment of the present invention, there is provided a computer apparatus including: one or more processors; a memory; and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, which when executed by the processors implement the steps in the method embodiments.
As will be appreciated by one skilled in the art, embodiments of the present invention may be provided as a method, system, or computer program product. Accordingly, the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present invention may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present invention is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each flow and/or block of the flowchart illustrations and/or block diagrams, and combinations of flows and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solutions of the present invention and not for limiting the same, and although the present invention is described in detail with reference to the above embodiments, those of ordinary skill in the art should understand that: modifications and equivalents may be made to the embodiments of the invention without departing from the spirit and scope of the invention, which is to be covered by the claims.

Claims (5)

1. A mixed signal separation method based on signal decomposition and sparse reconstruction is characterized by comprising the following steps:
1) Carrying out Fourier decomposition on each component signal under the condition that the number of mixed signal information sources and the instantaneous frequency of each component signal are known, and obtaining a decomposition coefficient vector p of each component signal;
2) Establishing an observation model y = Gp + n of the received signal based on the decomposition coefficient vector p of each component signal and the received signal, wherein y is the received signal, G is an observation matrix, and n is noise;
3) Converting the signal separation problem into a sparse optimization problem based on an observation model of the received signal, and establishing a target function of the sparse optimization problem;
4) Solving the sparse optimization problem, realizing the sparse reconstruction of the decomposition coefficient of each component signal, and obtaining the decomposition coefficient vector of the sparse reconstruction of each component signal
Figure FDA0003722945210000011
5) Base ofSparse reconstructed decomposition coefficient vector for each component signal
Figure FDA0003722945210000012
And observing the matrix G to obtain each separated component signal;
wherein, the step 1) specifically comprises:
regarding the received mixed signal as a linear superposition of multi-component unsteady signals, the signal model is expressed as:
Figure FDA0003722945210000013
wherein M is the number of each constituent signal in the mixed signal,
Figure FDA0003722945210000014
for the instantaneous complex envelope, phi, of the m-th component signal at time t m (t) is the instantaneous phase of the mth component signal at time t,
Figure FDA0003722945210000015
IF m (t) is the instantaneous frequency of the mth component signal at the time t, and n (t) is white Gaussian noise at the time t;
for envelope
Figure FDA0003722945210000016
Performing Fourier decomposition:
Figure FDA0003722945210000017
wherein the content of the first and second substances,
Figure FDA0003722945210000018
is a complex Fourier coefficient;
Figure FDA0003722945210000019
is a fundamental frequency, wherein F s The method comprises the following steps of (1) sampling frequency of a signal, N being the number of sampling points, and Q being a positive integer control parameter, wherein the positive integer control parameter is used for controlling the thickness of frequency interval division;
Figure FDA00037229452100000110
is Fourier order, B m Is the bandwidth of the signal;
the decomposition coefficients of the constituent signals are represented by a vector p, and equation (2) is rewritten as: a is m ＝G′ m p m Wherein a is m ＝[a m (t 0 )，...，a m (t N-1 )] T ；
Figure FDA00037229452100000111
Wherein the content of the first and second substances,
Figure FDA00037229452100000112
Figure FDA0003722945210000021
a coefficient vector p is constructed which is,
Figure FDA0003722945210000022
G′ m is N × (2K) m + 1) of the elements (G' m ) cd Expressed as:
Figure FDA0003722945210000023
wherein, t c-1 Represents a matrix (G' m ) cd The time corresponding to the c-th line;
the step 2) further comprises the following steps: from the observation model y = Gp + n of the received signal, the mathematical representation of the parts is as follows:
y＝[y(t 0 )，y(t 1 )，...，y(t N-1 )] T . (5)
n＝[n(t 0 )，n(t 1 )，...，n(t N-1 )] T . (6)
Figure FDA0003722945210000024
Figure FDA0003722945210000025
G＝[G′ 1 ，...，G′ m ，...，G′ M ]. (9)
wherein the dimension of the matrix G is N (2K) m + 1) M, N being the number of sampling points, M being the number of each constituent signal in the mixed signal, K m For the order of the fourier, the superscript T represents the matrix transpose,
Figure FDA0003722945210000026
the bandwidth of each signal is made to satisfy B for complex Fourier coefficient m < B, where B is a pre-given bandwidth value, in which case p is typically a sparse vector;
the objective function of the sparse optimization problem in the step 3) is expressed as:
Figure FDA0003722945210000027
wherein
Figure FDA0003722945210000028
And expressing the decomposed sparse vector after sparse reconstruction, wherein alpha is a regularization coefficient.
2. The method for separating mixed signals based on signal decomposition and sparse reconstruction as claimed in claim 1, wherein said step 4) adopts generalized message passing algorithm or other sparse reconstruction algorithm to solve the sparse optimization problem.
3. The method for separating a mixed signal based on signal decomposition and sparse reconstruction as claimed in claim 1, wherein the separated component signals in step 5) are obtained by:
Figure FDA0003722945210000031
wherein
Figure FDA0003722945210000032
Is a sparsely reconstructed decomposition coefficient vector.
4. A mixed signal separation apparatus based on signal decomposition and sparse reconstruction, comprising:
a mixed signal decomposition module for performing Fourier decomposition on each component signal under the condition of known mixed signal source number and instantaneous frequency of each component signal to obtain decomposition coefficient vector of each component signal
Figure FDA0003722945210000033
The signal observation model establishing module is used for establishing an observation model y = Gp + n of the received signal based on the decomposition coefficient vector p and the received signal of each component signal, wherein y is the received signal, G is an observation matrix, and n is noise;
the optimization problem establishing module is used for converting the signal separation problem into a sparse optimization problem based on the observation model of the received signal and establishing a target function of the sparse optimization problem;
a sparse reconstruction module for solving the established sparse optimization problem, realizing the sparse reconstruction of the decomposition coefficient of each component signal, and obtaining the decomposition coefficient vector of the sparse reconstruction of each component signal
Figure FDA0003722945210000034
A separate signal calculation module for calculating a decomposition coefficient vector based on sparse reconstruction of each constituent signal
Figure FDA0003722945210000035
And observing the matrix G to obtain each separated component signal;
wherein the mixed signal decomposition module comprises:
a signal representation unit for treating the received mixed signal as a linear superposition of the multi-component non-stationary signal, the signal model of which is represented as:
Figure FDA0003722945210000036
wherein M is the number of each constituent signal in the mixed signal,
Figure FDA0003722945210000037
for the instantaneous complex envelope, phi, of the m-th component signal at time t m (t) is the instantaneous phase of the mth component signal at time t,
Figure FDA0003722945210000038
IF m (t) is the instantaneous frequency of the mth component signal at the time t, and n (t) is white Gaussian noise at the time t;
a Fourier decomposition unit for decomposing the envelope
Figure FDA0003722945210000039
Performing Fourier decomposition:
Figure FDA00037229452100000310
wherein the content of the first and second substances,
Figure FDA00037229452100000311
is a complex Fourier coefficient;
Figure FDA00037229452100000312
is a fundamental frequency, wherein F s The method comprises the following steps of (1) sampling frequency of a signal, N being the number of sampling points, and Q being a positive integer control parameter, wherein the positive integer control parameter is used for controlling the thickness of frequency interval division;
Figure FDA0003722945210000041
is Fourier order, B m Is the bandwidth of the signal;
a coefficient vector construction unit for expressing the decomposition coefficients of the respective constituent signals by a vector p, rewriting formula (2) as: a is m ＝G′ m p m Wherein a is m ＝[a m (t 0 )，...，a m (t N-1 )] T ；
Figure FDA0003722945210000042
Wherein the content of the first and second substances,
Figure FDA0003722945210000043
a coefficient vector p is constructed which is,
Figure FDA0003722945210000044
G′ m is N × (2K) m + 1) of the elements (G' m ) cd Expressed as:
Figure FDA0003722945210000045
wherein, t c-1 Represents a matrix (G' m ) cd The time corresponding to the c-th line;
the signal observation model establishing module comprises:
the model building unit is used for building an observation model y = Gp + n of the received signal;
a mathematical representation unit, configured to mathematically represent each part according to an observation model of the received signal as follows:
y＝[y(t 0 )，y(t 1 )，...，y(t N-1 )] T . (5)
n＝[n(t 0 )，n(t 1 )，...，n(t N-1 )] T . (6)
Figure FDA0003722945210000046
Figure FDA0003722945210000047
G＝[G′ 1 ，...，G′ m ，...，G′ M ]. (9)
wherein the dimension of the matrix G is N (2K) m + 1) M, N being the number of sampling points, M being the number of each constituent signal in the mixed signal, K m For the order of the fourier, the superscript T represents the matrix transpose,
Figure FDA0003722945210000048
is a complex Fourier coefficient;
a sparse constraint unit for satisfying B by the bandwidth of each signal m < B to keep p a sparse vector, where B is a pre-given bandwidth value;
the objective function of the sparse optimization problem established by the optimization problem establishing module is expressed as:
Figure FDA0003722945210000051
wherein
Figure FDA0003722945210000052
And expressing the decomposed sparse vector after sparse reconstruction, wherein alpha is a regularization coefficient.
5. A computer device, comprising:
memory having stored therein one or more computer programs that, when executed by one or more processors, cause the one or more processors to perform the method of mixed signal separation based on signal decomposition and sparse reconstruction as claimed in any one of claims 1-3.
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