CN111007488A - Information source number estimation method based on Gerr circle transformation and modified Rao score test - Google Patents

Information source number estimation method based on Gerr circle transformation and modified Rao score test Download PDF

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CN111007488A
CN111007488A CN201911150063.4A CN201911150063A CN111007488A CN 111007488 A CN111007488 A CN 111007488A CN 201911150063 A CN201911150063 A CN 201911150063A CN 111007488 A CN111007488 A CN 111007488A
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王川川
曾勇虎
董晓博
汪连栋
张静克
姜林
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Abstract

The information source number estimation method based on the Gerr transform and the modified Rao score test comprises the steps of firstly calculating a sample covariance matrix of an observed signal, then carrying out the Gerr transform on the sample covariance matrix, detecting the structural characteristics of a large-dimension covariance matrix on the basis of the modified Rao score test idea by utilizing an estimated value of a feature value of the transformed sample covariance matrix, then constructing an observation statistic for establishing an information theory criterion likelihood function by testing whether the covariance matrix of a noise part in the observed signal is in direct proportion to a unit matrix, wherein the statistic is also the statistic of the sample feature value, and then carrying out information source number estimation through a generalized Bayes information criterion on the basis. The method provided by the invention has wider applicability, is suitable for estimating the information source number under a classical asymptotic system, and is also suitable for estimating the information source number under a general asymptotic system; the method is suitable for estimating the number of the information sources in the Gaussian white noise environment and the number of the information sources in the color noise environment.

Description

Information source number estimation method based on Gerr circle transformation and modified Rao score test
Technical Field
The invention belongs to the technical field of radar and communication reconnaissance, and further relates to an information source number estimation method based on Gaier circle transformation and Rao score correction test in the technical field of radar and communication reconnaissance signal processing.
Background
The estimation of the number of radiation sources has important application in many fields, such as phased array radar, communication, brain imaging, neural networks, voice signal separation, signal direction of arrival estimation and the like.
The information source number estimation method is essentially based on the statistical analysis theory of the observed data and the moment function thereof, such as a hypothesis testing method and an information theory criterion method commonly used in the information source number estimation, which mainly utilize the statistical distribution of the observed data and the statistics of the sample characteristic value. At present, the source number estimation method is mainly based on a classical asymptotic system established, namely, a classical statistical signal analysis theory that the dimension of an observed data matrix is fixed and the number of samples tends to be infinite, and is suitable for small-scale array signals of which the number of samples is far greater than the number of array elements.
However, in large-scale sensor arrays such as phased array radar and Multiple Input Multiple Output (MIMO) systems, due to the limitation of data storage space and the requirement of signal processing real-time, in practice, it is often difficult for observation data to satisfy the condition that the number of signal samples is far greater than the number of array elements, and the observation data usually belongs to high-dimensional limited sample data or even small sample data, that is, the number of signal samples and the number of array elements are in the same order of magnitude, or even the number of signal samples is smaller than the number of array elements. The proportional relation between the signal sampling number and the array element number of the large-scale array observation data often does not meet the requirements of the classical statistical theory, so that the appearance of the large-scale array brings new challenges to the classical information source number estimation technology.
At the present stage, in the information source number estimation method under the classical asymptotic system, the hypothesis test method includes spherical test, characteristic value detection and the like, and observation statistics for hypothesis test are mainly constructed by using a statistical distribution rule of sample characteristic values, and a decision threshold is set. The Information theory Criterion method includes Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Minimum Description Length (MDL), expected Description Length (PDL), etc., and usually assumes the observed data as gaussian distribution, establishes a Criterion of Information source number estimation according to a likelihood function of observed data joint probability distribution, and an expression of the Information source number estimation is a function of a sample characteristic value. These methods are suitable for use in white gaussian noise environments. Under a classical asymptotic system, methods suitable for source number estimation under a color noise environment mainly include a Gerr circle method and an information theory criterion method based on diagonal loading, but both methods are not suitable for large-scale arrays.
The source number estimation under the general asymptotic system is mainly an estimation method based on a random matrix theory, and comprises an RMT-AIC method, a BN-AIC method, a BIC-variant method, an LS-MDL method and an estimation method based on a spike model which are suitable for array elements with the number less than the number of signal samples, an estimation method based on a spherical test and an estimation method based on a modified Rao score test, wherein the estimation method is suitable for array elements with the number more than, less than or equal to the number of signal samples. The methods are not only suitable for source number estimation under a general asymptotic system, but also suitable for source number estimation under a classical asymptotic system, but the methods are only suitable for white noise environment, and source number estimation fails under a color noise environment.
The comprehensive analysis of domestic and foreign documents shows that an information source number estimation method which is suitable for both a classical asymptotic system and a general asymptotic system and is in a white noise or color noise environment is not available at present. Considering that the proportional relation between the number of antenna arrays and the number of signal samples in the actual array receiving signal environment and whether the noise of the aliasing of the observed signal is white gaussian noise or color noise are unknown, in order to improve the reliability of signal source number estimation, a signal source number estimation method which is suitable for both a classical asymptotic system and a general asymptotic system and is suitable for both white gaussian noise and color noise environments needs to be developed.
Disclosure of Invention
Aiming at the actual situation that the proportional relation of the antenna array element number and the signal sampling number and the fact that the noise of the aliasing of the observed signal is Gaussian white noise or color noise are unknown when the array antenna is applied to receive signals in the actual environment, the invention provides the information source number estimation method based on the Gauer circle transformation and the modified Rao score test, the relation of the antenna array element number and the signal sampling number does not need to be judged in advance (the application condition of the invention needs to be met, the relation of the antenna array element number M and the information source number K and the signal sampling number N is that M-K is more than or equal to 1, K is less than N, M can be more than, equal to or less than N, and the aliasing noise of the observed signal is Gaussian white noise or color noise, the antenna array observed signal is directly processed, and the blind estimation can be carried out on the number of narrow-band signal sources such as radar and communication under the complex electromagnetic environment.
The mathematical model of the information source number estimation method provided by the invention is as follows:
suppose K far-field signals are from direction θ1,…,θKIncident on an array consisting of M sensors, and observing signals of the array at the time t are X (t) and are expressed as
Figure BDA0002283294390000021
Wherein X (t) ═ X1(t),X2(t),…,XM(t)]T(the superscript T denotes transposition) into the array observation signal vector, a (θ)k) Is an array direction vector, A (theta) ═ a (theta)1),a(θ2),…,a(θK)]Is a matrix of directional vectors, theta ═ theta1,…θK]TIs the incoming wave angle parameter vector of the signal, s (t) ═ s1(t),s2(t),…,sK(t)]TFor the incident signal vector, w (t) ═ w1(t),w2(t),…,wM(t)]TFor an additive noise vector, the sampling time t is 1,2, …, and N is the number of signal samples. Formula (1)) The basic assumptions of the array observed signal model shown are as follows:
(1) incident signals are mutually independent narrow-band steady signals, and satisfy the mean value E { s (t) } ═ covariance matrix
Figure BDA0002283294390000031
Wherein
Figure BDA0002283294390000032
Is the power of the kth signal;
(2) the superimposed noise in the array observation signal vector is additive noise (white Gaussian noise or color noise) and is independent from the signal;
(3) the number of incident signals is smaller than the number of array elements and the number of samples at the same time, namely K is less than min (M, N);
(4) the signals are propagated in an ideal space, and the array sensor has omnidirectional consistency.
In practice, the sample data received by the antenna array contains noise, and may not be ideal white gaussian noise, but rather complex spatial color noise. In a complex spatial color noise environment, the noise eigenvalue part of the covariance matrix of the antenna array received data becomes very divergent and does not vibrate around the noise power as the noise eigenvalue part under gaussian white noise. The color noise results can disable various algorithms for estimating the number of information sources by using hypothesis testing and information theory criteria, and the information source number estimation method based on the Gerr theorem and the information source number estimation method based on the eigenvalue diagonal loading combined information theory criteria can only be applied to the classical asymptotic system, namely, the relationship between the antenna array element number M and the signal sampling number N is as follows: m is fixed and M/N is less than 1, under the general asymptotic system, namely the relation between the antenna array element number M and the signal sampling number N is as follows: m and N tend to infinity at the same rate, M, N → ∞ and M/N → c ∈ (0, ∞), the above method usually fails for source number estimation, whether the noise is gaussian white noise or chromatic noise.
The existing source number estimation method based on the random matrix theory cannot solve the source number estimation problem under the condition of observing signal aliasing noise under a general asymptotic system.
By analyzing the eigenvalue of the covariance matrix of the observation signals of the antenna array, the noise eigenvalue is found to be very divergent in a color noise environment. The existing information source number estimation method based on the Gerr theorem can estimate the information source number under the condition of observing aliasing Gaussian white noise or color noise of a signal under a classical asymptotic system. When the method is applied, the covariance matrix of an observation signal sample needs to be specially transformed, and after the transformation, the radius of a signal Gerr circle and the radius of a noise Gerr circle can be more obviously distinguished. In order to solve the difficult problem of information source number estimation under the condition of mixed color and noise of an observed signal under a general asymptotic system, the invention calculates a sample covariance matrix of the observed signal by means of a Gerr circle transformation idea, then performs Gerr circle transformation on the sample covariance matrix, and distinguishes more obvious characteristics by means of the radius of the transformed Gerr circle and the radius of the noise Gerr circle, and utilizes the estimation value of the characteristic value of the transformed sample covariance matrix on the basis of a modified Rao score inspection idea, namely CRST is the modified Rao score inspection and is used for detecting the structural characteristics of a large-dimensional covariance matrix. The sphere test statistic in the CRST can test whether the covariance matrix of the noise part in the observation signal is in direct proportion to the unit matrix, the observation statistic for establishing the likelihood function of the information theory criterion can be constructed according to the principle, the statistic is also the statistic of the sample characteristic value, and on the basis, the information source number estimation is realized through the Generalized Bayesian Information Criterion (GBIC). The method is improved aiming at the existing source number estimation method based on the modified Rao score test, and can be used for the source number estimation under the conditions that M is less than N in a classical asymptotic system, and the signal number M and the signal sample number N are observed to be in the same order of magnitude, or M is more than or equal to N in a general asymptotic system, and the signal aliasing white Gaussian noise or color noise is observed.
The invention relates to a method for estimating the number of information sources based on Gerr circle transformation and modified Rao score test, which specifically comprises the following steps:
step 1: setting M array elements in the array, wherein M observation signals obtained by measuring at t moment are X (t), and X (t) is [ X [)1(t),X2(t),...,XM(t)]T(the superscript T denotes transposition), the sampling time T is 1,2, …, N, N is the number of signal samples, and the sample covariance matrix of the observed signal is calculated
Figure BDA0002283294390000041
Step 2: blocking the sample covariance matrix r (t):
Figure BDA0002283294390000042
the M-1 dimensional square matrix R '(t) is a covariance matrix of the observed signal X' (t) obtained by removing the last array element. For convenience of explanation, R (t) and R '(t) will be abbreviated as R and R', respectively, in the following. Taking a characteristic matrix V of R, and constructing a unitary transformation matrix T:
Figure BDA0002283294390000043
and performing unitary transformation on a sample covariance matrix of the observation signal by using the constructed unitary transformation matrix T:
Figure BDA0002283294390000044
and step 3: setting the spectral decomposition of the covariance matrix R of the M-dimensional observation signal and the covariance matrix R' of the M-1-dimensional observation signal as follows:
Figure BDA0002283294390000045
and 4, step 4: performing eigenvalue decomposition on the covariance matrix R of the M-dimensional observation signal and the covariance matrix R' of the M-1-dimensional observation signal, which are respectively expressed as:
R＝U∑λUHand R '═ V Σ'λVH
U, V and sigmaλPartitioning:
Figure BDA0002283294390000046
V＝[v1v2… vM-1]
Figure BDA0002283294390000047
wherein U 'is [ U'1u′2… u′M-1]，u′i＝[u1iu2i… u(M-1)i]H(i＝1,2,…,M)，e＝[uM1uM2… uM(M-1)]H，∑′λ＝diag(λ1,λ2,...,λM-1)；
And 5: applying unitary transformation matrix T and block matrix U, V pair sigma in equation (2)λPerforming a unitary transform yields:
Figure BDA0002283294390000051
step 6: taking lines 1 to M-1 and column M of formula (4) in step (5), denoted as ρ'i(i-1, 2, …, M-1) and taking the absolute value | rho |'i(i-1, 2, …, M-1), and let ri＝|ρ|′iTaking the covariance matrix as an estimated value of M-1 eigenvalues of the covariance matrix R';
and 7: for ri(i-1, 2, …, M-1), and let rM＝rM-1R is toiAnd rMIs represented by a sequence r'i＝ri(i ═ 1,2, …, M), and r 'is judged'iIs according to r'1≥r′2≥…≥r′MAre arranged in sequence, if yes, then r'iReserving the sequence and entering the next step; r'iIs r'1≤r′2≤…≤r′MIn the order of (1), r'iReverse order, i.e. the values are arranged in descending order, denoted ri′fThe serial number is 1,2, … and M; for convenience of explanation, r 'will be'iOr ri′fIs represented by ri′new＝r′iOr ri′f；
And 8: according to a sequence of characteristic values ri′newIntroducing a modified Rao score inspection method to estimate the number of information sources; definition of
Figure BDA0002283294390000052
Calculating T according to(k)：
Figure BDA0002283294390000053
Wherein the content of the first and second substances,
Figure BDA0002283294390000054
and step 9: defining an expression of a source number estimation algorithm based on the Gerr theorem and the modified Rao score test according to the following formula:
GDE-CRST-GBIC(k)＝(T(k))2+(k+1)logN
step 10: estimating the number of the information sources according to the following formula
Figure BDA0002283294390000055
Figure BDA0002283294390000056
The information source number estimation method based on the Gerr circle transformation and the modified Rao score test can estimate the number of narrow-band signal sources such as radar signals and communication signals in a complex electromagnetic environment, has wide applicability, and can obtain the following beneficial effects:
firstly, in terms of the relation between the array number and the signal sampling number, the method does not need to preset or assume the relation between the array element number of the array antenna and the signal sampling number, and is suitable for the information source number estimation under a classical asymptotic system (the array element number M is fixed and is far less than the signal sampling number N) and the information source number estimation under a general asymptotic system (the array element number M is close to, equal to or more than the signal sampling number N);
secondly, from the noise characteristics, the method can be suitable for estimating the number of information sources in a Gaussian white noise environment and in a color noise environment, particularly for estimating the number of information sources in the case of observing the mixed color noise of signals under a general asymptotic system, and an effective solution is provided for the problem of the lack of the number of information sources estimation technology under the condition;
thirdly, the method can provide important support for technologies requiring the number of the information sources as conditions, such as the number estimation of the radiation sources in the electromagnetic environment, the estimation of the direction of arrival of signals and the like.
Drawings
Fig. 1(a) to 1(d) are graphs comparing the GDE-CRST-GBIC method proposed by the present invention with the information theory criterion method and the information source number estimation result of the geuer circle method under the condition of white gaussian noise.
Fig. 2(a) to 2(d) are source number estimation results of GDE-CRST-GBIC method, information theory criterion method, and geuer circle method under color noise condition.
Fig. 3(a) to 3(d) are source number estimation results of the GDE-CRST-GBIC method proposed by the present invention and the source number estimation method based on the random matrix theory under the color noise condition.
Detailed Description
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.
The method comprises the following specific implementation steps:
step 1: setting an antenna array to have M array elements, wherein M observation signals obtained by one measurement are X (t), and X (t) is [ X [)1(t),X2(t),...,XM(t)]T(the superscript T denotes transposition), the sampling time T is 1,2, …, N, N is the number of signal samples, and the sample covariance matrix of the observed signal is calculated
Figure BDA0002283294390000061
Step 2: blocking the sample covariance matrix r (t):
Figure BDA0002283294390000062
the M-1 dimensional square matrix R '(t) is a covariance matrix of the observed signal X' (t) obtained by removing the last array element. For convenience of explanation, R (t) and R '(t) will be abbreviated as R and R', respectively, in the following. Taking a characteristic matrix V of R, and constructing a unitary transformation matrix T:
Figure BDA0002283294390000063
and performing unitary transformation on a sample covariance matrix of the observation signal by using the constructed unitary transformation matrix T:
Figure BDA0002283294390000071
and step 3: setting the spectral decomposition of the covariance matrix R of the M-dimensional observation signal and the covariance matrix R' of the M-1-dimensional observation signal as follows:
Figure BDA0002283294390000072
and 4, step 4: performing eigenvalue decomposition on the covariance matrix R of the M-dimensional observation signal and the covariance matrix R' of the M-1-dimensional observation data, which are respectively expressed as:
R＝U∑λUHand R '═ V Σ'λVH
U, V and sigmaλPartitioning:
Figure BDA0002283294390000073
V＝[v1v2… vM-1]
Figure BDA0002283294390000074
wherein U 'is [ U'1u′2… u′M-1]，u′i＝[u1iu2i… u(M-1)i]H(i＝1,2,...,M)，e＝[uM1uM2… uM(M-1)]H，∑′λ＝diag(λ1,λ2,…,λM-1)。
And 5: applying unitary transformation matrix T and block matrix U, V pair sigma in equation (2)λPerforming a unitary transform yields:
Figure BDA0002283294390000075
step 6: taking lines 1 to M-1 and column M of formula (4) in step (5), denoted as ρ'i(i-1, 2, …, M-1) and taking the absolute value | rho |'i(i-1, 2, …, M-1), and let ri＝|ρ|′iThis is considered as an estimate of the M-1 eigenvalues of the covariance matrix R'.
And 7: for ri(i-1, 2, …, M-1), and let rM＝rM-1R is toiAnd rMIs represented by a sequence r'i＝ri(i＝1,2,…,M)。
And 8: and introducing a modified Rao score test method for source number estimation. Definition of
Figure BDA0002283294390000076
Calculating T according to(k)：
Figure BDA0002283294390000081
Wherein the content of the first and second substances,
Figure BDA0002283294390000082
and step 9: defining an expression of a source number estimation algorithm based on the Gerr theorem and the modified Rao score test according to the following formula:
GDE-CRST-GBIC(k)＝(T(k))2+(k+1)logN
step 10: estimating the number of the information sources according to the following formula
Figure BDA0002283294390000083
Figure BDA0002283294390000084
The invention is further described below in connection with experimental testing.
1. Setting experimental conditions:
the experimental verification of the invention is carried out under the simulation condition of a DELL9020MT personal computer, Intel (R) core (TM) i7-4770CPU @3.40GHz and 64-bit Windows operating system, and MATLAB R2010a is adopted as simulation software. In order to fully verify the effectiveness of the method (named as GDE-CRST-GBIC method), the technical scheme of the invention is compared with the technical scheme recorded in the existing literature, and three groups of experimental tests are carried out.
Experiment one: the GDE-CRST-GBIC method provided by the invention is compared with information theory criterion methods (BIC method, AIC method, MDL method, KIC method) and Goll circle method (MGDE method) disclosed in the monograph document research on information source number estimation method under the environment of Gaussian white noise. The experimental conditions were set as follows:
1)s1is BPSK signal, the sampling frequency is 120MHz, the sub-pulse width is 3X 10-7s, carrier frequency is 10 MHz;
2)s2is a CW signal, the sampling frequency is 120MHz, and the sub-pulse width is 1.5X 10-5s, carrier frequency is 10 MHz;
3)s3the carrier frequency is 10MHz and the pulse repetition frequency is 0.1MHz for LFM signals;
4)s4is FSK signal, has a sampling frequency of 120MHz and a sub-pulse width of 10-7s, the carrier frequency changes between two frequency points of 25MHz and 50MHz along with the binary baseband signal;
5)s5for MPSK signal, the sampling frequency is 120MHz, the sub-pulse width is 4 × 10-7s, carrier frequency 50 MHz.
If the source number K is set to 4 in the simulation, the source signal is divided by s1、s2、s3、s5And (4) forming. Different array antenna array element numbers M are set, a mixing matrix A is generated by a random function randn, the sampling frequency is 120MHz, the number of signal sampling points is N, Gaussian white noise is superposed on a mixed signal, the variation range of the signal-to-noise ratio is-10 dB-30 dB, the step length is 2dB, 1000 times of Monte Carlo simulation is carried out on each signal-to-noise ratio, and the experimental result is shown in figure 1.
Experiment two: the GDE-CRST-GBIC method provided by the invention is compared with information theory criterion methods (BIC method, AIC method, MDL method, KIC method) and Geller circle method (MGDE method) based on characteristic value to angle loading under the color noise environment. The experimental conditions were set as follows:
1)s1is BPSK signal, the sampling frequency is 120MHz, the sub-pulse width is 3X 10-7s, carrier frequency is 10 MHz;
2)s2is a CW signal, the sampling frequency is 120MHz, and the sub-pulse width is 1.5X 10-5s, carrier frequency is 10 MHz;
3)s3the carrier frequency is 10MHz and the pulse repetition frequency is 0.1MHz for LFM signals;
4)s4is FSK signal, has a sampling frequency of 120MHz and a sub-pulse width of 10-7s, the carrier frequency changes between two frequency points of 25MHz and 50MHz along with the binary baseband signal;
5)s5for MPSK signal, the sampling frequency is 120MHz, the sub-pulse width is 4 × 10-7s, carrier frequency 50 MHz.
If the source number K is set to 4 in the simulation, the source signal is divided by s1、s2、s3、s5And (4) forming. Setting different array antenna array element numbers M, generating a mixing matrix A by a random function randn, sampling frequency of 120MHz, the number of signal sampling points of N, and observing signal superposition spatial color noise, wherein the elements of a covariance matrix are given by the following formula:
Figure BDA0002283294390000091
i, k is 1,2, …, M, where σnIs an adjustable parameter, is used for setting the signal-to-noise ratio of the mixed signal, the change range of the signal-to-noise ratio is-10 dB to 30dB, the step length is 2dB, and 1000 times of Monte Carlo simulation is carried out on each signal-to-noise ratio, and the experimental result is shown in figure 2.
Experiment three: the invention provides a GDE-CRST-GBIC method and a source number estimation method (BN-AIC, RMT-AIC, BIC-variant, LS-MDL and CRST-GBIC) based on a random matrix theory, which is disclosed by a academic paper application of a large-dimension random matrix theory in array signal parameter estimation, and the source number estimation performance under the condition of color noise. The radiation source signals used in the experiment were the same as in experiment two, and the number of source signals was 5. Setting different array antenna array element numbers M, generating a mixing matrix A by a random function randn, sampling frequency of 120MHz, the number of signal sampling points of N, and observing signal superposition spatial color noise, wherein the elements of a covariance matrix are given by the following formula:
Figure BDA0002283294390000092
i, k is 1,2, …, M, where σnIs an adjustable parameter, is used for setting the signal-to-noise ratio of the mixed signal, the change range of the signal-to-noise ratio is-10 dB to 30dB, the step length is 2dB, and 1000 times of Monte Carlo simulation is carried out on each signal-to-noise ratio, and the experimental result is shown in figure 3.
2. And (3) simulation result analysis:
FIG. 1 is a comparison of the GDE-CRST-GBIC method proposed by the present invention with the information theoretic criterion method (BIC, AIC, MDL, KIC) and the Goll circle Method (MGDE) under the white Gaussian noise environment. As can be seen from FIG. 1(a), at this time, M/N is less than 1, the relationship between the antenna array element number and the sample number meets the requirement of the classical asymptotic system, and under the condition of Gaussian white noise, when the signal-to-noise ratio is above about 3dB, the GDE-CRST-GBIC method, the MDL method and the BIC method can accurately realize the information source number estimation with 100% probability, but the Gerr method needs to be above about 26dB to realize the information source number estimation with 100% probability; in FIGS. 1(b), 1(c) and 1(d),
Figure BDA0002283294390000093
the relation between the array element number of the antenna array and the sample number meets the requirement of a general asymptotic system, under the condition of Gaussian white noise, the technical scheme of the invention (GDE-CRST-GBIC method) can accurately realize the information source number estimation with 100% probability when the signal to noise ratio is about more than 14dB, more than 8dB and more than 7dB, and the estimation of other information theory criterion methods and the Gerr method fails.
FIG. 2 is a comparison of the GDE-CRST-GBIC method proposed by the present invention with the information theory rule method (BIC, AIC, MDL, KIC) and the Geller circle Method (MGDE) based on the characteristic value to the angle loading under the color noise environment. In fig. 2(a), at this time, M/N is less than 1, the relationship between the number of antenna array elements and the number of samples meets the requirement of the classical asymptotic system, and under the condition of color noise, when the signal-to-noise ratio is about 5dB or more, the GDE-CRST-GBIC method can accurately realize the estimation of the number of information sources with 100% probability, and the signal-to-noise ratio required by other methods is higher; in FIGS. 2(b), 2(c) and 2(d),
Figure BDA0002283294390000101
the relation between the array element number of the antenna array and the sample number meets the requirement of a general asymptotic system, and under the condition of color noise, when the signal-to-noise ratio is about more than 9dB, more than 15dB and more than 19dB, the information source number estimation can be accurately realized with 100 percent probability, and the estimation effect of other methods is poor or the estimation fails.
FIG. 3 is a comparison between the GDE-CRST-GBIC method and the source number estimation method based on the random matrix theory (BN-AIC, RMT-AIC, BIC-variant, LS-MDL, CRST-GBIC) in the color noise environment. It can be seen from fig. 3(a) that at this time, M/N is less than 1, the relationship between the antenna array element number and the sample number meets the requirement of the classical asymptotic system, and under the condition of color noise, the GDE-CRST-GBIC method can accurately realize the source number estimation with 100% probability when the signal-to-noise ratio is about 23dB or more compared with the multiple source number estimation methods based on the random matrix theory, and the experimental result is slightly inferior to that of other methods in this case; in FIGS. 3(b), 3(c) and 3(d),
Figure BDA0002283294390000102
the relationship between the array element number of the antenna array and the sample number belongs to a classical asymptotic system, under the condition of color noise, the method can realize the information source number estimation with the probability of 100% when the signal-to-noise ratio is respectively about 10dB, 15dB and more than 13dB, and the estimation of other methods fails.

Claims (2)

1. The information source number estimation method based on the Gerr transform and the modified Rao score test is characterized in that a sample covariance matrix of an observed signal is calculated, then the Gerr transform is carried out on the sample covariance matrix, the estimation value of the characteristic value of the transformed sample covariance matrix is utilized, the structural characteristic of a large-dimensional covariance matrix is detected on the basis of the modified Rao score test idea, then the observation statistic used for establishing an information theory criterion likelihood function is constructed by checking whether the covariance matrix of a noise part in the observed signal is in direct proportion to a unit matrix, the statistic is also the statistic of the sample characteristic value, and the information source number estimation is carried out on the basis of the generalized Bayesian information criterion.
2. The method of claim 1, wherein the source number estimation method comprises the following steps:
step 1: setting M array elements in the array, wherein M observation signals obtained by measuring at t moment are X (t), and X (t) is [ X [)1(t),X2(t),...,XM(t)]T(the superscript T denotes transposition), the sampling time T is 1,2, …, N, N is the number of signal samples, and the sample covariance matrix of the observed signal is calculated
Figure FDA0002283294380000011
Step 2: blocking the sample covariance matrix r (t):
Figure FDA0002283294380000012
the M-1 dimensional square matrix R '(t) is a covariance matrix of an observed signal X' (t) obtained by removing the last array element; for convenience of description, R (t) and R '(t) will be abbreviated as R and R', respectively, later; taking a characteristic matrix V of R, and constructing a unitary transformation matrix T:
Figure FDA0002283294380000013
and performing unitary transformation on a sample covariance matrix of the observation signal by using the constructed unitary transformation matrix T:
Figure FDA0002283294380000014
and step 3: setting the spectral decomposition of the covariance matrix R of the M-dimensional observation signal and the covariance matrix R' of the M-1-dimensional observation signal as follows:
Figure FDA0002283294380000015
and 4, step 4: performing eigenvalue decomposition on the covariance matrix R of the M-dimensional observation signal and the covariance matrix R' of the M-1-dimensional observation signal, which are respectively expressed as:
R＝U∑λUHand R '═ V Σ'λVH
U, V and sigmaλPartitioning:
Figure FDA0002283294380000021
V＝[v1v2… vM-1]
Figure FDA0002283294380000022
wherein U 'is [ U'1u′2… u′M-1]，u′i＝[u1iu2i… u(M-1)i]H(i＝1,2,...,M)，e＝[uM1uM2… uM(M-1)]H，∑′λ＝diag(λ1,λ2,…,λM-1)；
And 5: applying unitary transformation matrix T and block matrix U, V pair sigma in equation (2)λPerforming a unitary transform yields:
Figure FDA0002283294380000023
step 6: lines 1 to M-1 and column M of formula (4) in step 5 are each represented by ρ'i(i-1, 2, …, M-1) and taking the absolute value | rho |'i(i-1, 2, …, M-1), and let ri＝|ρ|′iTaking the covariance matrix as an estimated value of M-1 eigenvalues of the covariance matrix R';
and 7: for ri(i-1, 2, …, M-1), and let rM＝rM-1R is toiAnd rMIs represented by a sequence r'i＝ri(i ═ 1,2, …, M), and r 'is judged'iIs according to r'1≥r′2≥…≥r′MAre arranged in sequence, if yes, then r'iReserving the sequence and entering the next step; r'iIs r'1≤r′2≤…≤r′MIn the order of (1), r'iReverse order, i.e. the values are arranged in descending order, denoted as
Figure FDA00022832943800000213
The serial number is 1,2, … and M; for convenience of explanation, will be ri' or
Figure FDA00022832943800000211
Is shown as
Figure FDA0002283294380000024
Or
Figure FDA00022832943800000212
；
And 8: according to a sequence of characteristic values
Figure FDA0002283294380000025
Introducing a modified Rao score inspection method to estimate the number of information sources; definition of
Figure FDA0002283294380000026
Calculating T according to(k)：
Figure FDA0002283294380000027
Wherein the content of the first and second substances,
Figure FDA0002283294380000028
and step 9: defining an expression of a source number estimation algorithm based on the Gerr theorem and the modified Rao score test according to the following formula:
GDE-CRST-GBIC(k)＝(T(k))2+(k+1)logN
step 10: estimating the number of the information sources according to the following formula
Figure FDA0002283294380000029
Figure FDA00022832943800000210
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