CN111007488B - Information source number estimation method based on Gell circle transformation and modified Rao score test - Google Patents
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Abstract
The method comprises the steps of firstly calculating a sample covariance matrix of an observed signal, then carrying out the Gerr transformation on the sample covariance matrix, detecting structural features of a large-dimensional covariance matrix by utilizing an estimated value of a characteristic value of the transformed sample covariance matrix on the basis of a correction Rao score test thought, and then constructing an observed statistic for establishing an information theory criterion likelihood function by checking whether the covariance matrix of a noise part in the observed signal is in direct proportion to a unit matrix or not, wherein the statistic is also a statistic of the sample characteristic value, and carrying out information source number estimation by a generalized Bayesian information criterion on the basis. The method provided by the invention has wider applicability, and is suitable for estimating the information source number under a classical asymptotic system and also suitable for estimating the information source number under a general asymptotic system; the method is suitable for estimating the source number in the Gaussian white noise environment and also suitable for estimating the source number in the color noise environment.
Description
Technical Field
The invention belongs to the technical field of radar and communication reconnaissance, and further relates to a signal source number estimation method based on the Gerr circle transformation and the correction Rao score test in the technical field of radar and communication reconnaissance signal processing.
Background
The estimation of the number of radiation sources has important applications in many fields, such as phased array radar, communication, brain imaging, neural networks, speech signal separation, and signal direction of arrival estimation.
The information source number estimation method is essentially based on the statistical analysis theory of observed data and moment functions thereof, such as a hypothesis test type method and an information theory criterion type method which are commonly used in information source number estimation mainly utilize the statistical distribution of the observed data and the statistics of sample characteristic values. At present, the information source number estimation method is mainly based on a classical statistical signal analysis theory which is established in a classical asymptotic system, namely, the dimension of an observation data matrix is fixed, and the number of samples tends to infinity, and is suitable for small-scale array signals with the number of samples far larger than the number of array elements.
However, in large-scale sensor arrays such as phased array radar, multiple input multiple output (Multiple Input Multiple Output, MIMO) systems, due to limitations of data storage space and requirements of signal processing instantaneity, in practice, it is often difficult for observed data to satisfy the condition that the number of signal samples is far greater than the number of array elements, and the observed data usually belongs to high-dimensional finite sample data or even small sample data, that is, the number of signal samples is in the same order of magnitude as the number of array elements or even the number of signal samples is less 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 requirement of the classical statistical theory, so the appearance of the large-scale array brings new challenges to the classical information source number estimation technology.
In the prior art, in the information source number estimation method under the classical asymptotic system, the hypothesis test method comprises sphere test, eigenvalue detection and the like, and the observation statistics used for the hypothesis test are mainly constructed by utilizing the statistical distribution rule of the sample eigenvalue, and a decision threshold is set. The information theory criterion class method comprises an Akaike information criterion (Akaike Information Criterion, AIC), a bayesian information criterion (Bayesian Information Criterion, BIC), a minimization description length (Minimum Description Length, MDL), an expected description length (Predictive Description Length, PDL) and the like, the observed data is generally assumed to be gaussian distribution, and a criterion of estimating the number of sources is established according to likelihood functions of the joint probability distribution of the observed data, wherein an expression of the number of sources is a function of the characteristic values of the samples. These methods are applicable in gaussian white noise environments. Under a classical asymptotic system, the methods suitable for estimating the source number in the color noise environment mainly comprise a Gerr round method and an information theory criterion type method based on diagonal loading, but both methods are not suitable for large-scale arrays.
The estimation method based on random matrix theory is mainly used for estimating the information source number under the general asymptotic system, 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 estimating the information source number with more than, less than or equal to the signal sampling number based on a sphere test and an estimation method based on a corrected Rao score test. These methods are not only suitable for estimating the number of the sources in a general asymptotic system, but also suitable for estimating the number of the sources in a classical asymptotic system, but are only suitable for white noise environments, and fail in estimating the number of the sources in a color noise environment.
The comprehensive analysis of the literature at home and abroad shows that the method for estimating the information source number in the white noise or the color noise environment is not suitable for a classical asymptotic system and a general asymptotic system at the present stage. Considering that the proportional relation between the antenna array number and the signal sampling number in the actual array receiving signal environment and whether the noise of the observation signal aliasing is Gaussian white noise or color noise are unknown, in order to improve the reliability of signal source number estimation, the signal source number estimation method applicable to a classical asymptotic system and a general asymptotic system must be developed, and the signal source number estimation method applicable to the Gaussian white noise or the color noise environment is an effective technical scheme provided by the invention to meet the requirement.
Disclosure of Invention
Aiming at the practical situation that the proportional relation between the array antenna array element number and the signal sampling number and whether the noise of the aliasing of the observed signal is Gaussian white noise or color noise are unknown when an array antenna is applied to signal receiving in an actual environment, the invention provides an information source number estimation method based on the Gerr circle transformation and the correction Rao score test, and the relation between the antenna array element number and the signal sampling number is not required to be judged in advance (the applicable condition of the invention is required to be met, the relation between the antenna array element number M and the information source number K and the relation between the antenna array element number N are that M-K is more than or equal to 1, K is less than N, M can be more than or equal to N, and M can be more than or equal to N), and whether the aliasing noise of the observed signal is Gaussian white noise or color noise can be directly processed, so that the blind estimation of the number of radar, communication and other narrow-band signal sources can be carried out 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 θ 1 ,…,θ K Incident on an array of M sensors, the array observed signal at time t is X (t), expressed as
Wherein X (t) = [ X ] 1 (t),X 2 (t),…,X M (t)] T (superscript T denotes transpose) is the array observation signal vector, a (θ k ) For the array direction vector, a (θ) = [ a (θ) 1 ),a(θ 2 ),…,a(θ K )]For a matrix of direction vectors, θ= [ θ ] 1 ,…θ K ] T As the angle parameter vector of the incoming wave of the signal, s (t) = [ s ] 1 (t),s 2 (t),…,s K (t)] T For the incident signal vector, w (t) = [ w 1 (t),w 2 (t),…,w M (t)] T For the additive noise vector, the sampling time t=1, 2, …, N is the number of signal samples. The basic assumption condition of the array observation signal model shown in the formula (1) is as follows:
(1) The incident signal is a mutually independent narrow-band stable signal, and meets the mean value E { s (t) } =0 and covariance matrixWherein p is sk Power for the kth signal;
(2) The superimposed noise in the array observation signal vector is additive noise (Gaussian white noise or color noise) and is independent of the signal;
(3) The number of the incident signals is smaller than the number of array elements and the number of samples at the same time, namely K is smaller than min (M, N);
(4) The signals propagate 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 be non-ideal gaussian white noise, but complex spatially colored noise. In a complex space-color noise environment, however, the noise eigenvalue portion of the covariance matrix of the data received by the antenna array will become very divergent, not vibrate around noise power as the noise eigenvalue portion under gaussian white noise. The result caused by the color noise can cause various algorithms for estimating the number of the signal sources by using the hypothesis test and the information theory criterion to fail, and the method for estimating the number of the signal sources by using the information source based on the Galois circle theorem and the method for estimating the number of the signal sources by combining the information theory criterion based on the characteristic value diagonal loading can only be applied to the classical asymptotic system, namely the relation between the number M of antenna array elements and the number N of signal samples is as follows: m is fixed and M/N is less than 1, under a general asymptotic system, namely, the relation between the number M of antenna array elements and the number N of signal samples is as follows: m and N tend to infinity at the same rate, M, N → infinity and M/N → c e (0, +), the above method typically fails to estimate the source number, whether the noise is Gaussian white or colored.
The existing information source number estimation method based on the random matrix theory can not solve the information source number estimation problem under the condition of observing signal aliasing color noise in a general asymptotic system.
By analyzing the eigenvalues of the covariance matrix of the observed signals of the antenna array, the noise eigenvalues are found to be very divergent in the color noise environment. The existing information source number estimation method based on the Galois circle theorem can estimate the information source number under the condition of observing signal aliasing Gaussian white noise or color noise in a classical asymptotic system. When the method is applied, the covariance matrix of the observed signal sample is required to be subjected to special transformation, and after transformation, the signal guerre radius and the noise guerre radius are more obviously distinguished. In order to solve the problem of estimating the number of information sources under the condition of aliasing color noise of an observation signal under a general asymptotic system, the invention calculates a sample covariance matrix of the observation signal by means of a Gerr circle transformation idea, then carries out Gerr circle transformation on the sample covariance matrix, and uses the characteristic that the difference between the Gerr circle radius of the transformed signal and the Gerr circle radius of the noise is more obvious to utilize the estimated value of the characteristic value of the transformed sample covariance matrix to detect the structural characteristic of a large-dimensional covariance matrix on the basis of correcting a Rao score checking idea, namely CRST is the corrected Rao score checking. The spherical test statistic in CRST can test whether the covariance matrix of noise part in the observed signal is proportional to identity matrix, and can construct the observed statistic for establishing information theory criterion likelihood function according to the principle, and the statistic is also the statistic of sample characteristic value, and can implement source number estimation by using generalized bayesian information rule (GBIC) on the basis. The method is improved aiming at the prior information source number estimation method based on correction Rao score test, and can be used for not only under a classical asymptotic system of M < N, but also under a general asymptotic system of which the number of observed signals M and the number of signal samples N are in the same order of magnitude or M is more than or equal to N, and the information source number estimation problem under the condition of the aliasing Gaussian white noise or the chromatic noise of the observed signals.
The invention discloses a source number estimation method based on a guerre circle transformation and a modified Rao score test, which specifically comprises the following steps of:
step 1: let M array elements in antenna array, M observation signals obtained by measuring at time t be X (t), X (t) = [ X ] 1 (t),X 2 (t),…,X M (t)] T (superscript T indicates transpose), sampling time t=1, 2, …, N is the number of signal samples, and the sample covariance matrix of the observed signal is calculated
Step 2: partitioning a sample covariance matrix R (t):
the M-1 dimensional square matrix R '(t) is the covariance matrix of the observed signal X' (t) obtained by removing the last element. For convenience of explanation, R (t) and R '(t) will be abbreviated as R and R', respectively, hereinafter. Taking the characteristic matrix V of R, and constructing a unitary transformation matrix T:
unitary transformation is performed on a sample covariance matrix of an observation signal by using the constructed unitary transformation matrix T:
step 3: let the covariance matrix R of the M-dimensional observation signal and the covariance matrix R' of the M-1-dimensional observation signal be respectively:
step 4: performing eigenvalue decomposition on a covariance matrix R of the M-dimensional observation signal and a covariance matrix R' of the M-1-dimensional observation signal, wherein the eigenvalue decomposition is respectively expressed as follows:
R＝U∑ λ U H and R '=v Σ' λ V H
U, V sum sigma λ And (3) performing blocking:
V＝[v 1 v 2 … v M-1 ]
wherein U '= [ U ]' 1 u′ 2 …u′ M-1 ]，u′ i ＝[u 1i u 2i … u (M-1)i ] H (i＝1,2,…,M)，e＝[u M1 u M2 … u M(M-1) ] H ，∑′ λ ＝diag(λ 1 ,λ 2 ,…,λ M-1 )；
Step 5: applying unitary transformation matrix T and blocking matrix U, V pair Sigma in (2) λ Performing unitary transformation, one can obtain:
step 6: taking the 1 st to M-1 st rows, M th columns, expressed as ρ 'of the formula (4) in the step (5) similarly to the formula (3) in the step (2)' i (i=1, 2, …, M-1) and taking its absolute value |ρ|' i (i=1, 2, …, M-1), let r i ＝|ρ|′ i Taking the covariance matrix R 'as an estimated value of M-1 eigenvalues of the covariance matrix R';
step 7: for r i (i=1, 2, …, M-1), let r M ＝r M-1 Will r i And r M Represented as a sequence r i ′＝r i (i=1, 2, …, M), r is determined i Whether or not the value of 'is in accordance with r' 1 ≥r′ 2 ≥…≥r′ M Is arranged in sequence, if so, r i ' sequence reservation, go to the next step; if r i The value of' is according to r 1 ′≤r 2 ′≤…≤r′ M Is arranged in sequence, r i ' reverse order, i.e. the values are arranged in order from big to small, denoted r i ′ f The serial numbers are i=1, 2, …, M; for convenience of explanation, r will be i ' or r i ′ f Denoted as r i ′ new ＝r i ' or r i ′ f ；
Step 8: according to the characteristic value sequence r i ′ new Introducing a correction Rao score test method to estimate the information source number; definition of the definitionCalculate T according to (k) ：
Wherein,,
step 9: the source number estimation algorithm expression based on the guerre theorem and the modified Rao score test is defined according to the following:
GDE-CRST-GBIC(k)＝(T (k) ) 2 +(k+1)logN
step 10: estimating the number of sources according to the following
The information source number estimation method based on the Gerr circle transformation and the modified Rao score test can estimate the number of the radar signals, the communication signals and other narrow-band signal sources in the complex electromagnetic environment, has wider applicability, and can obtain the following beneficial effects:
firstly, in terms of the relation between the array number and the signal sampling number, the relation between the array antenna array element number and the signal sampling number is not required to be preset or assumed, namely the method is applicable to the information source number estimation under a classical asymptotic system (the array element number M is fixed and is far smaller 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 larger than the signal sampling number N);
secondly, from the noise characteristic, the invention can be suitable for the information source number estimation under the Gaussian white noise environment, also can be suitable for the information source number estimation under the color noise environment, especially under the general asymptotic system, the information source number estimation under the condition of observing the signal aliasing color noise, and provides an effective solution for the problem of lack of the information source number estimation technology under the condition;
thirdly, the invention can provide important support for techniques requiring the number of the information sources as conditions, such as the number estimation of the radiation sources, the signal arrival direction estimation and the like in the electromagnetic environment.
Drawings
Fig. 1 (a) to 1 (d) are comparison of the estimation results of the source number under the gaussian white noise condition by the GDE-CRST-GBIC method and the information theory criterion type method and the guerre circle method proposed by the present invention.
Fig. 2 (a) to 2 (d) are comparison of the source number estimation results of the GDE-CRST-GBIC method and the information theory criterion type method and the guerre circle method under the condition of color noise.
Fig. 3 (a) to 3 (d) are comparison of the GDE-CRST-GBIC method proposed by the present invention with the source number estimation result of the source number estimation method based on the random matrix theory under the condition of color noise.
Detailed Description
Embodiments of the present invention will now be described in detail with reference to the accompanying drawings and examples, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this application, and to illustrate and explain the invention by way of example and not limitation.
The specific implementation steps of the invention are as follows:
step 1: let M array elements in antenna array, M observation signals obtained by one measurement be X (t), X (t) = [ X ] 1 (t),X 2 (t),…,X M (t)] T (superscript T indicates transpose), sampling time t=1, 2, …, N is the number of signal samples, and the sample covariance matrix of the observed signal is calculated
Step 2: partitioning a sample covariance matrix R (t):
the M-1 dimensional square matrix R '(t) is the covariance matrix of the observed signal X' (t) obtained by removing the last element. For convenience of explanation, R (t) and R '(t) will be abbreviated as R and R', respectively, hereinafter. Taking the characteristic matrix V of R, and constructing a unitary transformation matrix T:
unitary transformation is performed on a sample covariance matrix of an observation signal by using the constructed unitary transformation matrix T:
step 3: let the covariance matrix R of the M-dimensional observation signal and the covariance matrix R' of the M-1-dimensional observation signal be respectively:
step 4: performing eigenvalue decomposition on a covariance matrix R of the M-dimensional observation signal and a covariance matrix R' of M-1-dimensional observation data, wherein the eigenvalue decomposition is expressed as follows:
R＝U∑ λ U H and R '=v Σ' λ V H
U, V sum sigma λ And (3) performing blocking:
V＝[v 1 v 2 … v M-1 ]
wherein U '= [ U ]' 1 u′ 2 … u′ M-1 ]，u′ i ＝[u 1i u 2i … u (M-1)i ] H (i＝1,2,…,M)，e＝[u M1 u M2 … u M(M-1) ] H ，∑′ λ ＝diag(λ 1 ,λ 2 ,…,λ M-1 )。
Step 5: application ofPair Σ of unitary transformation matrix T and blocking matrix U, V in equation (2) λ Performing unitary transformation, one can obtain:
step 6: taking the 1 st to M-1 st rows, M th columns, expressed as ρ 'of the formula (4) in the step (5) similarly to the formula (3) in the step (2)' i (i=1, 2, …, M-1) and taking its absolute value |ρ|' i (i=1, 2, …, M-1), let r i ＝|ρ|′ i This is considered as an estimate of the M-1 eigenvalues of the covariance matrix R'.
Step 7: for r i (i=1, 2, …, M-1), let r M ＝r M-1 Will r i And r M Represented as a sequence r i ′＝r i (i＝1,2,…,M)。
Step 8: and (4) introducing a correction Rao score checking method to estimate the information source number. Definition of the definitionCalculate T according to (k) ：
Wherein,,
step 9: the source number estimation algorithm expression based on the guerre theorem and the modified Rao score test is defined according to the following:
GDE-CRST-GBIC(k)＝(T (k) ) 2 +(k+1)logN
step 10: estimating the number of sources according to the following
The invention is further described in connection with experimental tests.
1. Experimental condition setting:
the experimental verification of the invention is carried out under the simulation condition of a DELL9020MT personal computer and an Intel (R) Core (TM) i7-4770 CPU@3.40GHz,64 bit Windows operating system, and MATLAB R2010a is adopted as simulation software. In order to fully verify the effectiveness of the invention (called GDE-CRST-GBIC method), the technical scheme of the invention is compared with the technical scheme recorded in the prior literature, and three groups of experimental tests are carried out.
Experiment one: the GDE-CRST-GBIC method provided by the invention is compared with the information theory criterion method (BIC method, AIC method, MDL method, KIC method) and the Gerr circle method (MGDE method) disclosed in the study of the information source number estimation method of the monograph literature. The experimental conditions were set as follows:
1)s 1 for BPSK signals, the sampling frequency is 120MHz, and the sub-pulse width is 3×10 -7 s, the carrier frequency is 10MHz;
2)s 2 for CW signals, the sampling frequency is 120MHz and the sub-pulse width is 1.5X10 -5 s, the carrier frequency is 10MHz;
3)s 3 for LFM signals, the carrier frequency is 10MHz, and the pulse repetition frequency is 0.1MHz;
4)s 4 for FSK signal, the sampling frequency is 120MHz, the sub-pulse width is 10 -7 s, the carrier frequency changes between two frequency points of 25MHz and 50MHz along with the binary baseband signal;
5)s 5 for MPSK signals, the sampling frequency is 120MHz, and the sub-pulse width is 4×10 -7 s, the carrier frequency is 50MHz.
If the source number k=4 is set in the simulation, the source signal is composed of s 1 、s 2 、s 3 、s 5 Composition is prepared. Setting different array antenna array element numbers M, generating a mixed matrix A by a random function randn, wherein the sampling frequency is 120MHz, the number of signal sampling points is N, and mixing signals are overlappedThe variation range of the signal to noise ratio is-10 dB to 30dB, the step length is 2dB, 1000 Monte Carlo simulations are carried out on each signal to noise ratio, and the experimental result is shown in figure 1.
Experiment II: the GDE-CRST-GBIC method provided by the invention is compared with an information theory criterion method (BIC method, AIC method, MDL method, KIC method) and a Gerr circle method (MGDE method) based on characteristic value diagonal loading under a color noise environment. The experimental conditions were set as follows:
1)s 1 for BPSK signals, the sampling frequency is 120MHz, and the sub-pulse width is 3×10 -7 s, the carrier frequency is 10MHz;
2)s 2 for CW signals, the sampling frequency is 120MHz and the sub-pulse width is 1.5X10 -5 s, the carrier frequency is 10MHz;
3)s 3 for LFM signals, the carrier frequency is 10MHz, and the pulse repetition frequency is 0.1MHz;
4)s 4 for FSK signal, the sampling frequency is 120MHz, the sub-pulse width is 10 -7 s, the carrier frequency changes between two frequency points of 25MHz and 50MHz along with the binary baseband signal;
5)s 5 for MPSK signals, the sampling frequency is 120MHz, and the sub-pulse width is 4×10 -7 s, the carrier frequency is 50MHz.
If the source number k=4 is set in the simulation, the source signal is composed of s 1 、s 2 、s 3 、s 5 Composition is prepared. Setting different array antenna array element numbers M, wherein a mixed matrix A is generated by a random function randn, the sampling frequency is 120MHz, the number of signal sampling points is N, the observed signals are superimposed with space color noise, and the elements of a covariance matrix are given by the following formula:i, k=1, 2, …, M, wherein σ n Is an adjustable parameter used for setting the signal-to-noise ratio of the mixed signal, the variation range of the signal-to-noise ratio is-10 dB to 30dB, the step length is 2dB, 1000 Monte Carlo simulations are carried out on each signal-to-noise ratio, and the experimental result is shown in figure 2.
Experiment III: the GDE-CRST-GBIC method and the academic position provided by the inventionThe paper application of the large-dimension random matrix theory in array signal parameter estimation discloses the information source number estimation performance of an information source number estimation method (BN-AIC, RMT-AIC, BIC-variant, LS-MDL and CRST-GBIC) based on the random matrix theory under the condition of color noise. The radiation source signals used in the experiment are the same as those used in the experiment II, and the number of the source signals is 5. Setting different array antenna array element numbers M, wherein a mixed matrix A is generated by a random function randn, the sampling frequency is 120MHz, the number of signal sampling points is N, the observed signals are superimposed with space color noise, and the elements of a covariance matrix are given by the following formula:i, k=1, 2, …, M, wherein σ n Is an adjustable parameter used for setting the signal-to-noise ratio of the mixed signal, the variation range of the signal-to-noise ratio is-10 dB to 30dB, the step length is 2dB, 1000 Monte Carlo simulations are carried out on each signal-to-noise ratio, and the experimental result is shown in figure 3.
2. Simulation result analysis:
FIG. 1 shows the comparison of the GDE-CRST-GBIC method and the information theory criterion method (BIC, AIC, MDL, KIC) and the Gerr round Method (MGDE) in Gaussian white noise environment. From FIG. 1 (a), it can be seen that, at this time, M/N < 1, the relation 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 Gaussian white noise, when the signal-to-noise ratio is more than about 3dB, the GDE-CRST-GBIC method, the MDL method and the BIC method can accurately realize the estimation of the number of sources with 100% probability, but the Gerr method needs to realize the estimation of the number of sources with more than about 26dB with 100% probability; in figures 1 (b), 1 (c) and 1 (d),the relation between the array element number and the sample number of the antenna array meets the general asymptotic system requirement, and under the Gaussian white noise condition, the signal-to-noise ratio (GDE-CRST-GBIC) technical scheme can accurately realize the estimation of the information source number with 100% probability when the signal-to-noise ratio is more than 14dB, more than 8dB and more than 7dB respectively, and other information theory criterion methods and the Gal circle method fail to estimate.
FIG. 2 is a schematic view of the present inventionThe GDE-CRST-GBIC method is compared with an information theory criterion method (BIC, AIC, MDL, KIC) and a Gerr circle Method (MGDE) which are loaded diagonally based on characteristic values in a color noise environment. In fig. 2 (a), at this time, M/N < 1, the relation 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 more than about 5dB, the GDE-CRST-GBIC method can accurately estimate the number of signal sources with 100% probability, and other methods need higher signal-to-noise ratio; in figures 2 (b), 2 (c) and 2 (d),the relation between the array element number and the sample number of the antenna array meets the requirements of a general asymptotic system, and under the condition of color noise, when the signal to noise ratio is more than 9dB, more than 15dB and more than 19dB, the estimation of the information source number can be accurately realized with 100% probability, and the estimation effect of other methods is poor or the estimation fails.
FIG. 3 is a comparison of the GDE-CRST-GBIC method according to the present invention and the source number estimation method (BN-AIC, RMT-AIC, BIC-variable, LS-MDL, CRST-GBIC) based on the random matrix theory under the color noise environment. As can be seen from FIG. 3 (a), at this time, M/N < 1, the relation between the array element number and the sample number of the antenna array meets the requirement of the classical asymptotic system, and under the condition of color noise, compared with various signal source number estimation methods based on random matrix theory, the GDE-CRST-GBIC method can accurately realize signal source number estimation with 100% probability when the signal to noise ratio is more than about 23dB, and the experimental result is slightly inferior to other methods under the condition of the invention; in figures 3 (b), 3 (c) and 3 (d),the relation between the array element number and the sample number of the antenna array belongs to a classical asymptotic system, and under the condition of color noise, the method can realize the estimation of the information source number with 100% probability when the signal to noise ratio is respectively about 10dB, 15dB and more than 13dB, and the estimation of other methods fails.
Claims (1)
1. The information source number estimation method based on the Gerr circle transformation and the modified Rao score test is characterized by comprising the following steps of:
step 1: let M array elements in antenna array and M observation signals obtained by measuring at time t be X (t, X (t) = [ X) 1 (t), 2 (t),…, M (t)] T (superscript T indicates transpose), sampling time t=1, 2, …, N is the number of signal samples, and the sample covariance matrix of the observed signal is calculated
Step 2: partitioning a sample covariance matrix R (t):
m-1 dimensional matrix R ’ (t) is the observed signal X obtained by removing the last array element ‘ (covariance matrix of t; for convenience of explanation, R (t) and R will be described later ’ (t) is abbreviated as R and R, respectively ’ The method comprises the steps of carrying out a first treatment on the surface of the Taking the characteristic matrix V of R, and constructing a unitary transformation matrix T:
unitary transformation is performed on a sample covariance matrix of an observation signal by using the constructed unitary transformation matrix T:
step 3: setting covariance matrix R of M-dimensional observation signals and covariance matrix R of M-1-dimensional observation signals ‘ The spectral decomposition of (a) is:
step 4: synergism of M-dimensional observation signalsCovariance matrix R of variance matrix R and M-1 dimension observation signal ’ The eigenvalue decomposition is performed as follows:
R＝U∑ λ U H and R is ’ ＝V∑ ′ λ V H
U, V sum sigma λ And (3) performing blocking:
V＝[v 12 …v M-1 ]
wherein U '= [ U ]' 1 u′ 2 … u′ M-1 ]，u′ i ＝[u 1i u 2i … u (M-1)i ] H (i＝1,2,…,M)，e＝[u M1 u M2 … u M(M-1) ] H ，∑′ λ ＝diag(λ 1 ,λ 2 ,…,λ M-1 )；
Step 5: applying unitary transformation matrix T and blocking matrix U, V pair Sigma in (2) λ Performing unitary transformation, one can obtain:
step 6: taking the 1 st to M-1 st rows and M columns of the formula (4) in the step 5, denoted as ρ' i (i=1, 2, …, M-1) and taking its absolute value |ρ|' i (i=1, 2,., M-1), let r i ＝|ρ|′ i Taking the covariance matrix R 'as an estimated value of M-1 eigenvalues of the covariance matrix R';
step 7: for r i (i=1, 2, …, M-1), let r M ＝r M-1 Will r i And r M Expressed as a sequence r' i ＝r i (i＝1,2,…,M) Judging r' i Whether or not the value of (C) is in accordance with r' 1 ≥r′ 2 ≥...≥r′ M Is the order of (c), if so, r' i Sequence reservation, and entering the next step; if r' i The value of (2) is in accordance with r' 1 ≤r′ 2 ≤…≤r′ M Is arranged in sequence, r' i The reverse order, i.e. the values are arranged in order from the largest to the smallest, is expressed asThe serial numbers are i=1, 2, … and M; for convenience of explanation, r 'will be' i Or->Represented asOr->
Step 8: according to the characteristic value sequenceIntroducing a correction Rao score checking method to estimate the information source number; definition of the definitionCalculate T according to (k) ：
Wherein,,
step 9: the source number estimation algorithm expression based on the guerre theorem and the modified Rao score test is defined according to the following:
GDE-CRST-GBIC(k)＝(T (k) ) 2 +(k+1)logN
step 10: estimating the number of sources according to the following
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