CN109344754A - A kind of improvement type shortest path is deficient to determine source signal restoration methods - Google Patents

Abstract

The invention belongs to radar-reconnaissance technical field, a kind of disclosed improvement type shortest path is deficient to determine source signal restoration methods.Under the conditions of being known to the observation signal, using the hybrid matrix of estimation as condition, it is converted into according to Sparse Component Analysis recovery resource signal problem and solves following optimization problems, it is greater than 2 or the source signal recovery situation equal to 2 in observation signal number, 1st and m-th of observation signal are formed a two-dimensional observation signal combination by the step of by step 1 to 10；For each combination, corresponding source signal is recovered using classical critical path method (CPM), then always there are m two-dimensional observation signal combination；A new matrix, the final estimation for obtaining n source signal are formed by isolated signal is combined by m two-dimensional observation signalSource signal as to be restored.The present invention can be handled radar signal, signal of communication, biomedicine signals etc., realize the deficient recovery for determining blind source separating source signal in the case where hybrid matrix estimated completion.

Description

Improved shortest path underdetermined source signal recovery method

Technical Field

The invention belongs to the technical field of radar reconnaissance, and further relates to an improved shortest path underdetermined source signal recovery method in the technical field of radar signal sorting. The technical scheme of the invention can process radar signals, communication signals, biomedical signals and the like, and realizes the recovery of underdetermined blind source separation source signals under the condition that the mixed matrix is estimated.

Background

Underdetermined blind source separation is a signal processing technique that estimates source signals only by using observation signals under the condition that prior information of radiation source signals and propagation channel parameters are unknown and the number of observation signals is less than that of the source signals. In view of its unique capabilities, underdetermined blind source separation has become a hot research topic in the international signal processing community in recent years. At present, a two-step method idea is generally adopted to solve the underdetermined blind source separation problem, namely, a mixed matrix is estimated by using an observation signal, and then a source signal is recovered by using the estimated mixed matrix and the observation signal. Since the mixing matrix is underdetermined, the matrix cannot be directly inverted to recover the source signal, which also involves a series of complex algorithms. The source signal recovery effect is directly related to success or failure of signal blind separation processing, so that the research of a source signal recovery algorithm has important theoretical value and practical significance.

The literature assumes that the source signals have strict orthogonality or quasi-orthogonality in the time-frequency domain, i.e. completely or almost not coinciding in each time-frequency point, and the source signal separation is achieved by a time-frequency mask. The degradation separation estimation technique (DUET) proposed by Yilmaz et al is a typical time-frequency masking method, and the improvement of the DUET by Cobos et al improves the accuracy of signal separation, but both methods are only applicable to the situation that 2 observed signals exist. The literature relaxes the assumption condition of orthogonality, and only requires that the number of source signals existing at the time-frequency point at the same time is less than that of observation signals. Araki et al extend the time-frequency masking approach to situations where there are more than 2 observed signals. The time-frequency masking method has high requirements on the sparsity of signals. If the sparsity of the source signals is not enough, the frequency points overlap each other at certain time, and the separation effect is seriously reduced. The literature further relaxes the assumed conditions, and the number of the source signals allowing the simultaneous existence of the time-frequency points is not more than the number of the sensors.

Bofill et al propose a shortest path source signal recovery algorithm based on source signal sparsity^[1]The method has good effect, but the method is only suitable for the situation that the observation signal is 2-dimensional. Zibulevsky et al propose an L1 norm source signal recovery algorithm based on Maximum a posteriori Probability (MPA) for sufficiently sparse signals. Under the condition that source signals are sufficiently sparse, the L1 norm method can obtain a good separation effect, and Theis et al proves that the minimum L1 norm method is equivalent to the shortest path method. The invention patent of the national invention 'an underdetermined blind source separation source signal recovery method (acceptance number: 2017102051088) based on an improved shortest path method', which is declared by a national emphasis laboratory of complex electromagnetic environment effect of an electronic information system, improves a classical shortest path source signal recovery method, so that an algorithm can be suitable for the source signal recovery problem under the condition that the number of observed signals is more than 2, but the classical shortest path method still has an improved space to improve the effect of the underdetermined source signal recovery.

Xiao et al propose a statistical sparse Decomposition-based source signal recovery algorithm (SSDP) that estimates a source signal by minimizing the correlation coefficient of the source signal over a fixed time interval, but this algorithm requires that no more than 2 non-zero source signals are present in each fixed time interval, and is not suitable for the insufficiently sparse underdetermined source signal recovery problem. The SSDP algorithm is extended by the Zhao-Min, etc., and a Non-Sparse decomposition principle (SNSDP) for 2 observed signals is obtained.

Compressed sensing theory is also used to achieve underdetermined source signal recovery. Under the condition that the mixed matrix completes estimation, the underdetermined blind separation source signal recovery problem is similar to a compressed sensing reconstruction model, and the difference is that compressed sensing has high requirements on sparsity and mainly aims at the problem of large data volume. The complementary matching tracking algorithm is improved strictly, and the time complexity of the algorithm is reduced. The effects of a greedy algorithm, an L1 norm algorithm and a smooth L0 norm algorithm in the aspect of underdetermined source signal recovery are compared by W.H.Fu and the like, an SCMP algorithm is provided, the signal recovery effect is improved, and the calculation time is shortened. The main problems existing when the compressive sensing theory is applied to the recovery of an underdetermined blind source separation source signal are that the requirement on the sparsity of the source signal is extremely high, the requirement on the number of data sampling points is high, and the calculation time is long. Reference documents:

[1]P.Bofill,M.Zibulevsky.Underdeterminedblind source separation usingsparse representations[J].Signal Process.,2001(81):2353-2362。

[2] chengehu, research on compressed sensing-based underdetermined blind separation source signal recovery algorithm [ D ]. Saian, university of Saian electronic technology, 2015.

[3] Duweihong, nong and Chengehu, et al. underdetermined blind separation source signal recovery based on RBF networks [ J ]. proceedings of Beijing post and telecommunications university, 2017,40(1): 94-98.

Disclosure of Invention

Aiming at a source signal recovery method based on a shortest path method, the invention aims to overcome the defects of the prior art and provides an improved shortest path underdetermined source signal recovery method so as to realize the accurate recovery of source signals under the condition that the number of observed signals is more than 2.

In order to achieve the purpose, the invention adopts the following technical scheme:

an improved shortest path underdetermined source signal recovery method is characterized in that under the condition that an observed signal is known, an estimated mixed matrix is taken as a condition, and according to a sparse component analysis theory, a source signal recovery problem is converted into the following optimization problem:

where x (t) is observed signal, the number of observed signals is m, A is estimated mixing matrix, the number of source signals is n, a_iIs the ith column, s of the mixing matrix_i(t) is the ith source signal, which is minimizedThe method is to perform linear decomposition on the observation signals along the direction of two columns of the mixed matrix to find out the shortest path from the origin to the observation signals. For the case of 2 observed signals, the solution idea of the above problem is shown as 1, in which case the minimum is requiredAs can be seen from the figure, the shortest path from the origin to the observed signal x is the two vectors a and b that are closest in angle to x;

when the number of observed signals is greater than 2 or equal to 2 in the case of recovering the source signals, for m observed signals, two adjacent observed signals are taken each time, which are not represented as the ith and jth observed signals, i is 1,2, …, m-1, j is i +1, that is, the observed signal processed each time is the combination of two adjacent observed signals, so as to obtain the target signalSeed combination; the method comprises the following specific steps:

step 1: of m observation signals x (t) obtained in one measurement, x (t) ═ x₁(t),x₂(t),…,x_m(t)]^T(ii) a In the formula, the superscript T is positive and indicates transposition, and the sampling time T is 1,2, …, T, and two adjacent observation signals x are selected each time_i(t) and x_j(t)，i＝1,2,…,m-1，j＝i+1；

Step 2: combining x for each observed signal in step 1_k(t), k is 1,2, …, m is preprocessed, column vectors with all zero observation signals are removed, and then the directions are unified;

and step 3: calculating the angle of each base vector of the mixing matrix A: the angle of the basis vector is defined asA_jRepresents the jth column vector of the mixing matrix, j is 1,2, …, n is the number of source signals, and superscripts 2 and 1 respectively represent the 2 nd row and the 1 st row of the column vector;

and 4, step 4: at each observation time, aiming at m combinations with only 2 observation signals, respectively calculating observation signal vectors x in each combination by applying a classical shortest path method^tThe angle of (d);

and 5: finding out the vector angle theta closest to the observed signal at the moment^tAnd recording two column vectors a of the corresponding mixing matrixⁱAnd bⁱWherein a isⁱ,bⁱE is A, i represents the serial number of the ith combination in m combinations, i is 1,2, …, m;

step 6: suppose A_r＝[aⁱbⁱ]，A_rIs a of the mixing matrix AⁱAnd bⁱTwo columns forming a 2 x 2 sub-matrix, aⁱAnd bⁱIs closest to x at time t^tTwo vectors of (1), order

And 7: the source signal at time t recovers as follows:

wherein,is x along the vector aⁱAnd bⁱTwo directional components;

in the case of source signal recovery where the number of observed signals is greater than 2 or equal to 2, after the observed signal processed each time is a combination of two adjacent observed signals;

combining the 1 st observation signal and the m-th observation signal into a two-dimensional observation signal combination, thereby obtaining m two-dimensional observation signal combinations in total; for each combination, recovering corresponding source signals by using a classical shortest path method, wherein the recovered source signals have m groups, if the number of the source signals is n, n separated signals can be respectively obtained after the two-dimensional combination of m observation signals obtained by combining the original observation signals is used for signal recovery, and the signal obtained by separating each combination is represented asWherein, i is 1,2, …, m represents the serial number of each two-dimensional observation signal combination, k is 1,2, …, n represents the serial number of the signal obtained by separating each two-dimensional combination, i represents the number of sampling points; the m groups of separated signals are combined into a new matrix which is expressed as

ThenIs a vector combination matrix with m multiplied by n dimensions; wherein, the upper label^TRepresenting a transpose;

for matrixSolving the vector included angle between the lines to obtain a square matrix Q with the dimension of mn multiplied by mn; for the first n rows × mn columns of the square matrix, this indicates the matrix by detecting whether the matrix element is greater than 0 and less than 20 °, which indicates that the matrix is a non-uniform matrixThe included angle of the medium signal vector is less than 20 degrees, the signals with the included angle less than 20 degrees are respectively averaged to be used as an estimation of the source signal, and finally the estimation of n source signals is obtainedNamely the source signal to be recovered;

wherein, step 1, two adjacent observation signals x are selected each time_i(t) and x_j(t), i is 1,2, …, m-1, j is i + 1; combining the 1 st and the m-th observed signals into a two-dimensional observed signal combination results in m signal combinations of only 2 observed signals, denoted x_k(t)＝[x_i(t),x_j(t)]^T，k＝1,2,…,m；

Step 8 adopted: for each combination, recovering corresponding source signals by using a classical shortest path method, wherein the recovered source signals have m groups, the number of the source signals is n, the n signals are respectively obtained after the signal recovery is carried out on the m observation signal two-dimensional combinations, and the signal obtained by separating each combination is represented asWherein i is 1,2, …, m represents each two-dimensional observation signal combination, k is 1,2, …, n represents the signal separated by each two-dimensional combination, and represents the number of sampling points;

step 9 adopted: combining and separating the signals obtained by combining and separating the m two-dimensional observation signals to form a new matrix expressed as The method comprises the following steps of (1) forming a vector combination matrix of m multiplied by n dimensions, wherein each vector represents a signal, and the number of sampling points is T;

step 10: for matrixSolving the vector included angle between the lines to obtain a square matrix Q with the dimension of mn multiplied by mn; for the first n rows × mn columns of the square matrix, this indicates the matrix by detecting whether the matrix element is greater than 0 and less than 20 °, which indicates that the matrix is a non-uniform matrixThe included angle of the medium signal vector is less than 20 degrees, the signals with the included angle less than 20 degrees are respectively averaged to be used as an estimation of the source signal, and finally the estimation of n source signals is obtainedI.e. the source signal to be recovered.

The technical scheme of the invention brings the following advantages:

firstly, the invention overcomes the problem that the existing shortest path method underdetermined source signal recovery algorithm cannot be applied to the recovery of underdetermined blind source separation signals with the number of observation signals more than 2.

Secondly, the method has the advantages of clearer principle, simpler algorithm steps, better signal recovery effect and lower calculation time cost.

Drawings

FIG. 1 is a schematic diagram of a shortest path method when 2 observation signals are provided;

fig. 2-4 are graphs of simulation results of the snr of the present invention with m 2, n 5, … … and the snr of the prior art method.

Detailed description of the preferred embodiments

The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this application, illustrate embodiment(s) of the invention and together with the description serve to explain the invention without limiting the invention.

As shown in fig. 1,2, 3, and 4, an improved method for recovering an underdetermined source signal with a shortest path is that under the condition that an observed signal is known, a problem of recovering a source signal can be converted into a solution of the following optimization problem according to a sparse component analysis theory under the condition of an estimated mixing matrix:

where x (t) is observed signal, the number of observed signals is m, A is estimated mixing matrix, the number of source signals is n, a_iIs the ith column, s of the mixing matrix_i(t) is the ith source signal, which is minimizedThe method is to perform linear decomposition on the observation signals along the direction of two columns of the mixed matrix to find out the shortest path from the origin to the observation signals. For the case of 2 observed signals, the solution idea of the above problem is shown as 1, in which case the minimum is requiredAs can be seen from the figure, the shortest path from the origin to the observed signal x is the two vectors a and b that are closest in angle to x.

The method aims at the problem of source signal recovery of which the number of observed signals is more than 2 or equal to 2. The improved idea is to take two adjacent observation signals at a time for m observation signals, which are not represented as the ith and jth observation signals, i is 1,2, …, m-1, j is i +1, that is, the observation signal processed at each time is a combination of two adjacent observation signals, and the improvement can be thatTo obtainAnd (4) combining the 1 st observation signal and the m-th observation signal into a two-dimensional observation signal combination, so that m two-dimensional observation signal combinations can be obtained in total. For each combination, recovering corresponding source signals by using a classical shortest path method, wherein the recovered source signals have m groups, if the number of the source signals is n, n separated signals can be respectively obtained after the two-dimensional combination of m observation signals obtained by combining the original observation signals is used for signal recovery, and the signal obtained by separating each combination is represented asWherein i is 1,2, …, m represents the serial number of each two-dimensional observation signal combination, k is 1,2, …, n represents the serial number of the signal separated by each two-dimensional combination, and represents the number of sampling points. The m groups of separated signals are combined into a new matrix which can be expressed as

ThenIs a vector combination matrix with dimension of m x n. Wherein, the upper label^TIndicating transposition.

For matrixThe vector angle between the lines is calculated to obtain a mn × mn dimensional square matrix Q. For the first n rows × mn columns of the square matrix, this indicates the matrix by detecting whether the matrix element is greater than 0 and less than 20 °, which indicates that the matrix is a non-uniform matrixThe included angle of the medium signal vector is less than 20 degrees, the signals with the included angle less than 20 degrees are respectively averaged to be used as an estimation of the source signal, and finally the estimation of n source signals can be obtainedMeterI.e. the source signal to be recovered.

The specific steps for realizing the algorithm are as follows:

step 1: of m observation signals x (t) obtained in one measurement, x (t) ═ x₁(t),x₂(t),…,x_m(t)]^T(in the formula, the superscript T is positive and indicates transposition, the same applies below), the sampling time T is 1,2, …, T, and two adjacent observation signals x are selected each time_i(t) and x_j(t), i-1, 2, …, m-1, j-i +1, and combining the 1 st and m-th observed signals into a two-dimensional observed signal combination results in m signal combinations of only 2 observed signals, denoted as x_k(t)＝[x_i(t),x_j(t)]^T，k＝1,2,…,m；

Step 2: combining x for each observed signal in step 1_k(t), k is 1,2, …, m is preprocessed, column vectors with all zero observation signals are removed, and then the directions are unified;

and step 3: calculating the angle of each base vector of the mixing matrix A: the angle of the basis vector is defined asA_jRepresents the jth column vector of the mixing matrix, j is 1,2, …, n is the number of source signals, and superscripts 2 and 1 respectively represent the 2 nd row and the 1 st row of the column vector;

and 4, step 4: at each observation time, aiming at m combinations with only 2 observation signals, respectively calculating observation signal vectors x in each combination by applying a classical shortest path method^tThe angle of (d);

and 5: finding out the vector angle theta closest to the observed signal at the moment^tAnd recording two column vectors a of the corresponding mixing matrixⁱAnd bⁱWherein a isⁱ,bⁱE.g. A, i represents the th of m combinationsi combination numbers, i ═ 1,2, …, m;

step 6: suppose A_r＝[aⁱbⁱ]，A_rIs a of the mixing matrix AⁱAnd bⁱTwo columns forming a 2 x 2 sub-matrix, aⁱAnd bⁱIs closest to x at time t^tTwo vectors of (1), order

And 7: the source signal at time t recovers as follows:

wherein,is x along the vector aⁱAnd bⁱTwo directional components;

and 8: for each combination, the corresponding source signal is restored by using a classical shortest path method, the restored source signals have m groups, for example, the number of the source signals is n, n signals can be respectively obtained after the signal restoration is carried out by the two-dimensional combination of m observation signals, and the signal obtained by separating each combination is represented asWherein i is 1,2, …, m represents each two-dimensional observation signal combination, k is 1,2, …, n represents the signal separated by each two-dimensional combination, and represents the number of sampling points;

and step 9: the signals obtained by combining and separating the m two-dimensional observation signals form a new matrix which can be expressed as The method comprises the following steps of (1) forming a vector combination matrix of m multiplied by n dimensions, wherein each vector represents a signal, and the number of sampling points is T;

step 10: for matrixThe vector angle between the lines is calculated to obtain a mn × mn dimensional square matrix Q. For the first n rows × mn columns of the square matrix, this indicates the matrix by detecting whether the matrix element is greater than 0 and less than 20 °, which indicates that the matrix is a non-uniform matrixThe included angle of the medium signal vector is less than 20 degrees, the signals with the included angle less than 20 degrees are respectively averaged to be used as an estimation of the source signal, and finally the estimation of n source signals can be obtainedI.e. the source signal to be recovered.

The method comprises the following specific implementation steps:

step 1: of m observation signals x (t) obtained in one measurement, x (t) ═ x₁(t),x₂(t),…,x_m(t)]^T(in the formula, the superscript T is positive and indicates transposition, the same applies below), the sampling time T is 1,2, …, T, and two adjacent observation signals x are selected each time_i(t) and x_j(t), i-1, 2, …, m-1, j-i +1, and combining the 1 st and m-th observed signals into a two-dimensional observed signal combination results in m signal combinations of only 2 observed signals, denoted as x_k(t)＝[x_i(t),x_j(t)]^T，k＝1,2,…,m；

Step 2: combining x for each observed signal in step 1_k(t), k is 1,2, …, m is preprocessed, column vectors with all zero observation signals are removed, and then the directions are unified;

and step 3: calculating the angle of each base vector of the mixing matrix A: the angle of the basis vector is defined asA_jRepresents the jth column vector of the mixing matrix, j is 1,2, …, n is the number of source signals, and superscripts 2 and 1 respectively represent the 2 nd row and the 1 st row of the column vector;

and 4, step 4: at each observation time, aiming at m combinations with only 2 observation signals, respectively calculating observation signal vectors x in each combination by applying a classical shortest path method^tThe angle of (d);

and 5: finding out the vector angle theta closest to the observed signal at the moment^tAnd recording two column vectors a of the corresponding mixing matrixⁱAnd bⁱWherein a isⁱ,bⁱE is A, i represents the serial number of the ith combination in m combinations, i is 1,2, …, m;

step 6: suppose A_r＝[aⁱbⁱ]，A_rIs a of the mixing matrix AⁱAnd bⁱTwo columns forming a 2 x 2 sub-matrix, aⁱAnd bⁱIs closest to x at time t^tTwo vectors of (1), order

And 7: the source signal at time t recovers as follows:

wherein,is x along the vector aⁱAnd bⁱTwo directional components;

and 8: for each combination, the corresponding source signals are recovered by using a classical shortest path method, and the recovered source signals have m groups, for example, if the number of the source signals is n, m observation signalsAfter signal recovery is performed on the two-dimensional combinations of the signals, n signals can be obtained respectively, and the signal obtained by separating each combination is represented asWherein i is 1,2, …, m represents each two-dimensional observation signal combination, k is 1,2, …, n represents the signal separated by each two-dimensional combination, and represents the number of sampling points;

and step 9: the signals obtained by combining and separating the m two-dimensional observation signals form a new matrix which can be expressed as The method comprises the following steps of (1) forming a vector combination matrix of m multiplied by n dimensions, wherein each vector represents a signal, and the number of sampling points is T;

step 10: for matrixThe vector angle between the lines is calculated to obtain a mn × mn dimensional square matrix Q. For the first n rows × mn columns of the square matrix, this indicates the matrix by detecting whether the matrix element is greater than 0 and less than 20 °, which indicates that the matrix is a non-uniform matrixThe included angle of the medium signal vector is less than 20 degrees, the signals with the included angle less than 20 degrees are respectively averaged to be used as an estimation of the source signal, and finally the estimation of n source signals can be obtainedI.e. the source signal to be recovered.

1. Simulation conditions are as follows:

the experimental verification of the invention is carried out under the simulation condition of a DELL9020MT personal computer, an Intel (R) core (TM) i7-4770CPU @3.40GHz and a 64-bit Windows operating system, and M is adopted as simulation softwareATLABR2010 a. The recovery effect of the source signal adopts a separation signal-to-interference ratio and a similarity coefficient index, and the calculation formulas of the two indexes are respectively shown as formulas (4) and (5). Wherein,andrespectively representing the i and j signals, s, separated_i(t) denotes the ith source signal.

In the following simulation experiments, the signal-to-noise ratio range of the mixed signal is 8-20dB, the simulation step size is 2dB, and Monte Carlo simulation is performed 100 times at each signal-to-noise ratio. In order to fully verify the effectiveness of the improved algorithm, three sets of simulation experiments were carried out for different situations.

Setting simulation parameters:

experiment one: the number of the radiation source signals is 5, the 5 signals are sufficiently sparse in a time domain, and the signal patterns and parameters are as follows:

s₁for a conventional pulse signal, carrier frequency f_c15MHz, pulse width t_r110 mus, pulse repetition period T_r1100 mus, pulse start time t₀₁＝0；

s₂For a conventional pulse signal, carrier frequency f_c25MHz, pulse width t_r27 mus, pulse repetition period T_r2100 mus, pulse start time t₀₂＝10μs；

s₃For a linear frequency-modulated signal, carrier frequency f_c35MHz, pulse width t_r3＝10μs，Pulse repetition period T_r3100 mus, pulse start time t₀₃20 mus, intra-pulse bandwidth of B₃＝10MHz；

s₄For a linear frequency-modulated signal, carrier frequency f_c45MHz, pulse width t_r48 mus, pulse repetition period T_r4100 mus, pulse start time t₀₄30 mus, intra-pulse width B₄＝15MHz；

s₅Being sine-phase modulated signals, carrier frequency f_c55MHz, pulse width t_r58 mus, pulse repetition period T_r5100 mus, pulse start time t₀₅40 mus, modulation signal frequency f_a5100kHz, modulation index a₅＝5。

The sampling frequency of the receiver is 50MHz, the number of signal sampling points is 10000, the number of observation signals is 2, a mixing matrix A is generated by using a rand function,when m is 2 and n is 5, i.e. the observed signals are 2, the method of the present invention is essentially a classical shortest path source signal recovery method, and the complementary matching pursuit method (CMP) of the present invention and the literature is used^[2]Complementary matching pursuit method based on L1 norm (L1CMP)^[2]And radial basis network method (RBF network)^[3]The source signal is recovered and the results are shown in fig. 2(a) - (c).

Experiment two: the signal and the number of the radiation source are the same as those of the first experiment, the sampling frequency of the receiver is 50MHz, the number of signal sampling points is 10000, the number of observation signals is 3, a mixing matrix A is generated by using a rand function,when m is 3 and n is 5, that is, when there are 3 observed signals, the source signal is recovered by using the CMP method, the L1CMP method and the RBF network method in the present document and the literature, and the results are shown in fig. 3(a) - (c).

Experiment three: the number of the radiation source signals is 7, the 7 signals are sufficiently sparse in a time domain, and the signal patterns and parameters are as follows:

s₁for non-linearly frequency-modulated signals, carrier frequency f_c110MHz, pulse width t_r116 mus, pulse repetition period T_r1200 mus, intra-pulse bandwidth B₁10MHz, pulse start time t₀₁＝0；

s₂For a conventional pulse signal, carrier frequency f_c28MHz, pulse width t_r215 mus, pulse repetition period T_r2180 mus, pulse start time t₀₂＝20μs；

s₃For a linear frequency-modulated signal, carrier frequency f_c35MHz, pulse width t_r315 mus, pulse repetition period T_r3180 mus, pulse start time t₀₃40 mus, intra-pulse bandwidth B₃＝20MHz；

s₄For a linear frequency-modulated signal, carrier frequency f_c45MHz, pulse width t_r420 mus, pulse repetition period T_r4180 mus, pulse start time t₀₄60 mus, intra-pulse width B₄＝15MHz；

s₅Being sine-phase modulated signals, carrier frequency f_c55MHz, pulse width t_r520 mus, pulse repetition period T_r5200 mus, pulse start time t₀₅80 mus, modulation signal frequency f_a5200kHz, modulation index a₅＝5；

s₆Being sine-phase modulated signals, carrier frequency f_c65MHz, pulse width t_r615 mus, pulse repetition period T_r6200 mus, pulse start time t₀₅100 mus, modulation signal frequency f_a6200kHz, modulation index a₆＝2；

s₇For non-linearly frequency-modulated signals, carrier frequency f_c715MHz, pulse width t_r720 mus, pulse repetition period T_r7200 mus, intra-pulse bandwidth B₇5MHz, pulse StartTime t₀₇＝115μs。

The sampling frequency of the receiving signal of the receiver is 50MHz, and the number of sampling points is 10000. The mixing matrix is generated using an arbitrary function,when m is 4 and n is 7, that is, when 4 observed signals exist, the source signal is recovered by using the CMP method of the present invention and the CMP method of the document, the L1CMP method and the RBF network method, and the results are shown in fig. 4(a) - (c).

2. And (3) simulation result analysis:

when m is 2 and n is 5, that is, when there are 2 observed signals, the present invention is a classical shortest path method, and as the signal-to-noise ratio of the mixed signal increases, as can be seen from fig. 2(a) and (b), the signal-to-interference ratios and the similarity coefficients obtained by different signal recovery algorithms increase, which indicates that the signal separation effect tends to be good, and is consistent with general knowledge. However, in the aspect of signal effect comparison of different algorithms, under the simulation condition, the separation effect of the method provided by the invention is slightly inferior to that of a CMP method, and is superior to that of L1CMP and RBF network methods. In terms of computational efficiency, fig. 2(c) reflects that the computation time of the method proposed herein is much less than that of the other methods.

When m is 3 and n is 5, the conditions are set for the experiment, and it can be seen from fig. 3(a), 3(b) and 3(c) that the separation effect of the improved method is better than that of the other three methods, and the computational efficiency is higher than that of the CMP method and the RBF network method and is slightly inferior to that of the L1CMP method.

When m is 4 and n is 7, as can be seen from fig. 4(a) and (b), the separation effect of the method is slightly worse than that of the other three methods, but from the viewpoint of separating the signal-to-interference ratio and the similarity coefficient value, such a value indicates that the improved algorithm is enough to accurately separate the 7 source signals from the 4 received signals, and the calculation efficiency is basically the same as that of the two experimental results.

The simulation experiment shows that the method can be applied under the condition that the number of the observed signals is 2 or more than 2. The method can realize more ideal source signal recovery with higher calculation efficiency under the conditions of sufficient and insufficient sparsity of the source signal time domain. Under the condition that the source signal is not sufficiently sparse, the method can be applied to recover the source signal by performing sparse representation on the observation signal.

Claims (1)

1. An improved shortest path underdetermined source signal recovery method is characterized in that under the condition that an observed signal is known, an estimated mixed matrix is taken as a condition, and according to a sparse component analysis theory, a source signal recovery problem is converted into the following optimization problem:

where x (t) is observed signal, the number of observed signals is m, A is estimated mixing matrix, and the number of source signals isn，a_iIs the ith column, s of the mixing matrix_i(t) is the ith source signal, which is minimizedThe method comprises the steps of performing linear decomposition on observation signals along the direction of two columns of a mixed matrix, and finding out the shortest path from an origin to the observation signals; for the case of 2 observed signals, the minimum isThe shortest path from the origin to the observed signal x is the two vectors a and b closest to the angle of x;

when the number of observed signals is greater than 2 or equal to 2 in the case of recovering the source signals, for m observed signals, two adjacent observed signals are taken each time, which are not represented as the ith and jth observed signals, i is 1,2, …, m-1, j is i +1, that is, the observed signal processed each time is the combination of two adjacent observed signals, so as to obtain the target signalSeed combination; the method comprises the following specific steps:

step 1: of m observation signals x (t) obtained in one measurement, x (t) ═ x₁(t),x₂(t),…,x_m(t)]^T(ii) a In the formula, the superscript T is positive and indicates transposition, and the sampling time T is 1,2, …, T, and two adjacent observation signals x are selected each time_i(t) and x_j(t)，i＝1,2,…,m-1，j＝i+1；

Step 2: combining x for each observed signal in step 1_k(t), k is 1,2, …, m is preprocessed, column vectors with all zero observation signals are removed, and then the directions are unified;

and step 3: calculating the angle of each base vector of the mixing matrix A: the angle of the basis vector is defined asA_jRepresents the jth column vector of the mixing matrix, j being 1,2, …, nN is the number of source signals, and superscripts 2 and 1 respectively represent the 2 nd row and the 1 st row of the column vector;

and 4, step 4: at each observation time, aiming at m combinations with only 2 observation signals, respectively calculating observation signal vectors x in each combination by applying a classical shortest path method^tThe angle of (d);

and 5: finding out the vector angle theta closest to the observed signal at the moment^tAnd recording two column vectors a of the corresponding mixing matrixⁱAnd bⁱWherein a isⁱ,bⁱE is A, i represents the serial number of the ith combination in m combinations, i is 1,2, …, m;

step 6: suppose A_r＝[aⁱbⁱ]，A_rIs a of the mixing matrix AⁱAnd bⁱTwo columns forming a 2 x 2 sub-matrix, aⁱAnd bⁱIs closest to x at time t^tTwo vectors of (1), order

And 7: the source signal at time t recovers as follows:

wherein,is x along the vector aⁱAnd bⁱTwo directional components; the method is characterized in that:

in the case of source signal recovery where the number of observed signals is greater than 2 or equal to 2, after the observed signal processed each time is a combination of two adjacent observed signals;

combining the 1 st observation signal and the m-th observation signal into a two-dimensional observation signal combination, thereby obtaining m two-dimensional observation signal combinations in total; for each combination, recovering corresponding source signals by using a classical shortest path method, wherein the recovered source signals have m groups, and if the number of the source signals is n, m groups are obtained by combining original observation signalsAfter signal recovery is performed on two-dimensional combinations of the observed signals, n separated signals can be obtained respectively, and the signal obtained by separation of each combination is represented asWherein, i is 1,2, …, m represents the serial number of each two-dimensional observation signal combination, k is 1,2, …, n represents the serial number of the signal obtained by separating each two-dimensional combination, i represents the number of sampling points; the m groups of separated signals are combined into a new matrix which is expressed as

ThenIs a vector combination matrix with m multiplied by n dimensions; wherein, the upper label^TRepresenting a transpose;

for matrixSolving the vector included angle between the lines to obtain a square matrix Q with the dimension of mn multiplied by mn; for the first n rows × mn columns of the square matrix, this indicates the matrix by detecting whether the matrix element is greater than 0 and less than 20 °, which indicates that the matrix is a non-uniform matrixThe included angle of the medium signal vector is less than 20 degrees, the signals with the included angle less than 20 degrees are respectively averaged to be used as an estimation of the source signal, and finally the estimation of n source signals is obtainedNamely the source signal to be recovered;

wherein, step 1, two adjacent observation signals x are selected each time_i(t) and x_j(t), i is 1,2, …, m-1, j is i + 1; combining the 1 st and the m-th observed signals into a two-dimensional observed signal combination results in m signals with only 2 observed signalsCombination, denoted x_k(t)＝[x_i(t),x_j(t)]^T，k＝1,2,…,m；

Step 8 adopted: for each combination, recovering corresponding source signals by using a classical shortest path method, wherein the recovered source signals have m groups, the number of the source signals is n, the n signals are respectively obtained after the signal recovery is carried out on the m observation signal two-dimensional combinations, and the signal obtained by separating each combination is represented asWherein i is 1,2, …, m represents each two-dimensional observation signal combination, k is 1,2, …, n represents the signal separated by each two-dimensional combination, and represents the number of sampling points;

step 9 adopted: combining and separating the signals obtained by combining and separating the m two-dimensional observation signals to form a new matrix expressed as The method comprises the following steps of (1) forming a vector combination matrix of m multiplied by n dimensions, wherein each vector represents a signal, and the number of sampling points is T;

step 10: for matrixSolving the vector included angle between the lines to obtain a square matrix Q with the dimension of mn multiplied by mn; for the first n rows × mn columns of the square matrix, this indicates the matrix by detecting whether the matrix element is greater than 0 and less than 20 °, which indicates that the matrix is a non-uniform matrixThe included angle of the medium signal vector is less than 20 degrees, the signals with the included angle less than 20 degrees are respectively averaged to be used as an estimation of the source signal, and finally the estimation of n source signals is obtainedI.e. the source signal to be recovered.