CN113075611B - Method for constructing fourth-order differential array based on expansion and displacement nested array - Google Patents
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Abstract
The invention discloses a method for constructing a fourth-order differential array based on an extended and shifted nested array, which is called as EAS-NA for short LS The method comprises the steps of carrying out a first treatment on the surface of the According to the invention, in the EAS-NA-NA, the freedom degree of estimating the direction of arrival can be remarkably increased and the performance of estimating the direction of arrival can be improved by further expanding the array element spacing of the nested array. The invention provides the EAS-NA LS Compared with other sparse arrays, the method has larger DOA (direction of arrival) estimation freedom, and the simulation experiment proves that the method has the advantage that the EAS-NA LS Is a performance of the (c). The invention is easy to realize, has good effect, can realize a large number of degrees of freedom, and has important theoretical and engineering values for improving the accuracy of the intelligent antenna.
Description
Technical Field
The invention relates to the technical field of array signal processing, in particular to a method for constructing a fourth-order differential array based on an expansion and displacement nested array.
Background
In the aspects of radar systems, wireless communication, voice processing and the like, the estimation of the direction of arrival has been greatly focused, global mobile internet service has been rapidly increased since 21 st century, limited wireless spectrum resources are more and more precious, a smart antenna is an array antenna structure consisting of a plurality of uniform and isotropic antenna array elements, a space division multiple access multiplexing mode is introduced in the smart antenna technology, different users can be distinguished according to different propagation paths of space under the condition that time slots, frequencies or address codes are the same, therefore, the capacity of a mobile communication system is doubled, the defect of small capacity of the traditional multiple access multiplexing mode is overcome, spectrum resources are efficiently utilized, and the direction of arrival estimation algorithm is one of key technologies in the field of array signal processing, in order to solve the problems, a nested array formed by two or more uniform arrays has appeared in recent years, and compared with the uniform arrays, the nested array can remarkably increase the degree of freedom, can solve the problem of more information sources than the actual number of physical array elements, and can improve the estimation accuracy under the condition of the same number of the array elements, but even if the nested array has various ingenious sparse array structures, the added degree of freedom is limited, and cannot meet the estimation of the high-precision direction of arrival, so the invention provides a method for constructing a fourth-order differential array based on the expanded and shifted nested arrays to solve the problems in the prior art.
Disclosure of Invention
Aiming at the problems, the invention aims to provide a method for constructing a fourth-order differential array based on an extended and shifted nested array, which can obviously increase the degree of freedom of the estimation of the direction of arrival by properly expanding the array element spacing of the nested array, and can obtain more degrees of freedom for estimation compared with other sparse arrays.
In order to achieve the purpose of the invention, the invention is realized by the following technical scheme: the method for constructing the fourth-order differential array based on the expansion and displacement nested array has the following related signal model and characteristics:
order the and />Represents the array element position and lists +.> and />Is set by the set operation of:
where c is a constant and Z represents the integer set { n: m }, i.e., a continuous set from n to m, p and q are each and />Any element in the collection;
the T array element positions of the array are represented as follows:
P＝{p 1 d,p 2 d,…,p t d,…,p τ d,t＝1,2,…,T} (2)
in the formula pT d represents the position of the array element, and d=λ/2 where λ represents the wavelength, and the array output of the L far-field narrowband source is represented by the following formula:
in which s (i) = [ s ] 1 (i),…,s L (i)] T Comprising L sources, ε (i) represents Gaussian noise,for theta l The steering vector of l=1, 2, …, L, I is the snapshot number, and the matrix form of the fourth order cumulative amount of X (I) is:
superscript in middle [ ·] H and [·]* Respectively represent the conjugate transpose and the conjugate,is an operator of the kronecker product,and Cur [. Cndot.]Is an addition operator; on the basis of the signal model formula (4), the fourth-order differential array is denoted as F and is expressed as:
F＝(P-P)+(P-P) (5)
definition of array EAS-NA LS For the spreading factor C E Greater than 2N 2 (N 1 +1) -1 EAS-NA-NA, wherein C E ＞2C s +1, array EAS-NA LS The array element number of the standard nested array is N1 and N2 respectively, the array element number of the extended nested array is M1 and M2, namely the array element number is T=M in total 2 +M 1 +N 2 +N 1 -1The EAS-NA array element positions are denoted as set P ENN ＝P 1 ∪P 2, wherein ,
in the formula CE Is an integer representing the spreading factor, C S Is a shift factor equal to N 2 (N 1 +1) -1, P for EAS-NA-NA 1 Represents the standard nested array element position, P 2 Representing the positions of array elements of a nested array which are expanded, in the EAS-NA-NA method, the maximum value of the expansion factor is proved to be C E ＝2N 2 (N 1 +1) -1, whereby the number of consecutive array elements of the fourth order differential array in the EAS-NA-NA is equal to (2M 2 M 1 +2M 2 -1)(2N 2 N 1 +2N 2 -1)+2(M 2 M 1 +M 2 -1) due to EAS-NA LS The position of the array element is expressed asThe second order differential array consists of four subsets +.>The concrete representation is as follows:
similarly, the fourth order differential array is expressed as wherein ,
the array element positions of the subset in the formula are as follows:
in the formula F12 As a continuous set, when C E ＝2C S At +1, F 12 Ranging from (-M) for a continuous set 2 M 1 -M 2 +1)C E -C S To (M) 2 M 1 +M 2 -1)C E +C S . And subset F 12 and F33 Is represented as two adjacent consecutive segments:
in the formula kCE -C S ＜kC E +C S -N 1 ＜kC E +C S According to EAS-NA LS Definition of (C) to obtain C E >2C S +1, when 2C S +1<C E ≤3C S At +1, F 12' and F33 ' are discontinuous and F 12' and F33 The' joint set comprises a continuum { -C s :(M 2 M 1 +M 2 -1)C E +2C s }， and />And aggregate contains from 0 to (M 2 M 1 +M 2 -1)C E +2C S According to the symmetry of FODC, +.>Also comprises a slave- (M) 2 M 1 +M 2 -1)C E -2C S To a non-positive subset of 0, to the extent that 2C S +1<C E ≤3C S +1，EAS-NA-NA LS The number of consecutive array elements of the fourth order differential array of (a) ranges from- (M) 2 M 1 +M 2 -1)C E -2C S To (M) 2 M 1 +M 2 -1)C E +2C S Thus, EAS-NA LS The number of consecutive elements of the fourth order differential array is greater than EAS-NA.
The further improvement is that: the sources are uncorrelated, and a non-Gaussian source assumption is used for the existing fourth-order accumulation of the source signal.
The further improvement is that: based on the fourth order cumulant matrix formula, the fourth order differential array corresponds to P ENN ＝P 1 ∪P 2 Denoted as F, expressed as:
F＝(P-P)+(P-P) (11)
the further improvement is that: maximum value C in the EAS-NA-NA method E ＝2N 2 (N 1 The number of consecutive array elements of the fourth order differential array in +1) -1, EAS-NA is (2M 2 M 1 +2M 2 -1)(2N 2 N 1 +2N 2 -1)+2(M 2 M 1 +M 2 -1)。
The further improvement is that: according to EAS-NA LS Definition, EAS-NA LS A sparse array structure similar to EAS-NA sharing with larger array element spacing in an extended nested array.
The further improvement is that: in the formula (9), if the condition (k+1) C E -C S -(kC E +2C S )≤1,i.e.,C E ≤3C S +1 is satisfied, for a particular k, the union set F 12' and F33 ' form a continuous set { kC } E -C S :(k+1)C E +2C s }。
The beneficial effects of the invention are as follows: the invention provides that in the EAS-NA, the freedom degree of the estimation of the direction of arrival can be obviously increased by further expanding the array element spacing of the nested array, the performance of the estimation of the direction of arrival is improved, and the EAS-NA-NA LS Compared with other sparse arrays, the method has larger DOA (direction of arrival) estimation freedom, and the simulation experiment proves that the method has the advantage that the EAS-NA LS The invention is easy to realize, has good effect, can realize a large number of degrees of freedom, and has important reason for improving the precision of the intelligent antennaTheory and engineering value.
Drawings
In order to more clearly illustrate the embodiments of the invention or the technical solutions of the prior art, the drawings which are used in the description of the embodiments or the prior art will be briefly described, it being obvious that the drawings in the description below are only some embodiments of the invention, and that other drawings can be obtained according to these drawings without inventive faculty for a person skilled in the art.
FIG. 1 is a graph comparing the number of consecutive array elements of a fourth order differential array of the present invention with other sparse arrays;
FIG. 2 is a graph comparing DOA estimation spectra of the invention with other sparse array MUSIC algorithms;
FIG. 3 is a graph comparing the RMSE versus signal to noise ratio for the present invention with other sparse arrays;
FIG. 4 is a graph comparing the RMSE of the present invention with other sparse arrays as a function of snapshot.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
In the description of the present invention, it should be noted that the directions or positional relationships indicated by the terms "center", "upper", "lower", "left", "right", "vertical", "horizontal", "inner", "outer", etc. are based on the directions or positional relationships shown in the drawings, are merely for convenience of describing the present invention and simplifying the description, and do not indicate or imply that the devices or elements referred to must have a specific orientation, be configured and operated in a specific orientation, and thus should not be construed as limiting the present invention. Furthermore, the terms "first," "second," "third," "fourth," and the like are used for descriptive purposes only and are not to be construed as indicating or implying relative importance.
In the description of the present invention, it should be noted that, unless explicitly specified and limited otherwise, the terms "mounted," "connected," and "connected" are to be construed broadly, and may be either fixedly connected, detachably connected, or integrally connected, for example; can be mechanically or electrically connected; can be directly connected or indirectly connected through an intermediate medium, and can be communication between two elements. The specific meaning of the above terms in the present invention will be understood in specific cases by those of ordinary skill in the art.
Referring to FIGS. 1,2, 3, and 4, the present embodiment provides a method for constructing a fourth order differential array based on an extended and shifted nested array, as shown in Table one, P 1 Representation of EAS-NA-NA and EAS-NA-NA LS Array element positions, P, of a standard sub-array of (a) 2 Representation of EAS-NA-NA and EAS-NA-NA LS Array element positions of the extended subarray, F 12 Representing sum arrays of differential arrays of the same subarray in fourth-order differential array FODC, F 33 Representing the differential array of the same sub-array and array in a fourth order differential array FODC. () +* The symbols represent the non-negative part of the set.
List one
Defined by: EAS-NA LS Is the spreading factor C E Greater than 2N 2 (N 1 +1) -1 EAS-NA, wherein C E >2C S +1, when 2C S +1<C E ≤3C S At +1, the fourth order differential array has a continuous array element ranging from- (M) 2 M 1 +M 2 -1)(3N 2 N 1 +3N 2 -2)d-2(N 2 N 1 +N 2 -1) d to (M 2 M 1 +M 2 -1)(3N 2 N 1 +3N 2 -2)d+2(N 2 N 1 +N 2 -1) d. The number of the fourth-order differential array continuous array elements is 2 (M 2 M 1 +M 2 -1)(3N 2 N 1 +3N 2 -2)+4(N 2 N 1 +N 2 -1) +1. For better understanding, the EAS-NA with 5 array elements is demonstrated LS For example, if EAS-NA LS C of (2) E ＝3C S +1. And compared to the EAS-NA of 5 array elements in table one. In this case N 1 ＝1,N 2 ＝2,M 1＝1 and M2 =2. Accordingly, C in EAS-NA-NA S＝3 and CE =7, while in EAS-NA LS Middle C S＝3 and CE ＝10。
The EAS-NA will be described below LS In contrast to other sparse arrays:
watch II
Here, the EAS-NA LS Comparison was made with several representative sparse arrays, including SAFO-NA, EAS-NA-NA and E-FL-NA. Their expressions concerning the number of successive array elements of the fourth order differential array are listed in Table two and are denoted as u, respectively SN ,u ENN ,u EFN Andfor fair comparison we have chosen a parameter that generates the maximum number of FODC consecutive elements per sparse array. The corresponding parameter settings are shown in Table III. Under the given parameters, u SN ,u ENN ,u EFN and />In figure one there is a comparison.
It is apparent that, no matter how many sensors are used,always larger than its corresponding array element, and u EFN Always greater than u SN ,u ENN Except when t=4. For example, when t=20, u SN ＝2223,u ENN ＝4259,u EFN＝4537 and
watch III
In the simulation, table four lists a sparse array structure with 5 array elements. According to the expression about the number of the continuous array elements of the fourth-order differential array shown in Table II, u SN ＝53,u ENN ＝55,u EFN＝61 and The estimation is performed by using the SS-MUSIC method, and the simulation result is shown in a figure II.
Table four
In the first simulation, the MUSIC spectrum of each sparse array is shown in a second graph, the simulation selection parameters are 25 signal sources, the signal to noise ratio is 0dB, and the snapshot number is 30 ten thousand. The results show that only the EAS-NA LS 25 sources can be handled accurately and E-FL-NA, EAS-NA and SAFO-NA fail to achieve this goal.
In the second simulation, the DOA estimation Root Mean Square Error (RMSE) for each sparse array was compared by 300 Monte Carlo experiments. 10000 samples are used for the simulation parameters, here 13 sources are used. As shown in FIG. three, EAS-NA LS The method is always superior to other sparse arrays in estimation accuracy. In the third simulation, 13 sources with a signal to noise ratio of 0dB were used. The simulation parameters used an initial number of samples of 5000. As shown in FIG. four, in a trend similar to that of FIG. three, EAS-NA LS Optimum performance can still be achieved.
The invention discloses a method for constructing a fourth-order differential array based on an expansion and displacement nested array, which is called E for shortAS-NA-NA LS It is proposed that in the EAS-NA, the degree of freedom of the direction of arrival estimation can be significantly increased by further expanding the array element spacing of the nested array, improving the performance of the direction of arrival estimation, EAS-NA LS Compared with other sparse arrays, the method has larger DOA (direction of arrival) estimation freedom, and the simulation experiment proves that the method has the advantage that the EAS-NA LS Is a performance of the (c). The invention is easy to realize, has good effect, can realize a large number of degrees of freedom, and has important theoretical and engineering values for improving the accuracy of the intelligent antenna.
The foregoing description of the preferred embodiments of the invention is not intended to be limiting, but rather is intended to cover all modifications, equivalents, alternatives, and improvements that fall within the spirit and scope of the invention.
Claims (6)
1. The method for constructing the fourth-order differential array based on the expansion and displacement nested array has the following related signal model and characteristics:
order the and />Represents the array element position and lists +.> and />Is set by the set operation of:
where c is a constant and Z represents the integer set { n: m }, i.e., a continuous set from n to m, p and q are each and />Any element in the collection;
the T array element positions of the array are represented as follows:
P＝{p 1 d,p 2 d,…,p t d,…,p τ d,t＝1,2,…,T} (2)
in the formula pT d represents the position of the array element, and d=λ/2 where λ represents the wavelength, and the array output of the L far-field narrowband source is represented by the following formula:
in which s (i) = [ s ] 1 (i),…,s L (i)] T Comprises L sources, epsilon (i) represents Gaussian noise, A= [ a (theta) 1 ) j …a(θ L )]，For theta l The steering vector of l=1, 2, …, L, I is the snapshot number, and the matrix form of the fourth order cumulative amount of X (I) is:
superscript in middle [ ·] H and [·]* Respectively represent the conjugate transpose and the conjugate,is an operator of the kronecker product,and Cur [. Cndot.]Is an addition operator; on the basis of the signal model formula (4), the fourth-order differential array is denoted as F and is expressed as:
F＝(P-P)+(P-P) (5)
definition of array EAS-NA LS For the spreading factor C E Greater than 2N 2 (N 1 +1) -1 EAS-NA-NA, wherein C E ＞2C s +1, array EAS-NA LS The array element number of the standard nested array is N1 and N2 respectively, the array element number of the extended nested array is M1 and M2, namely the array element number is T=M in total 2 +M 1 +N 2 +N 1 -1 EAS-NA array element position is represented as set P ENN ＝P 1 ∪P 2, wherein ,
in the formula CE Is an integer representing the spreading factor, C S Is a shift factor equal to N 2 (N 1 +1) -1, P for EAS-NA-NA 1 Represents the standard nested array element position, P 2 Representing the positions of array elements of a nested array which are expanded, in the EAS-NA-NA method, the maximum value of the expansion factor is proved to be C E ＝2N 2 (N 1 +1) -1, whereby the number of consecutive array elements of the fourth order differential array in the EAS-NA-NA is equal to (2M 2 M 1 +2M 2 -1)(2N 2 N 1 +2N 2 -1)+2(M 2 M 1 +M 2 -1) due to EAS-NA LS The position of the array element is expressed asThe second order differential array consists of four subsets +.>The concrete representation is as follows:
similarly, the fourth order differential array is expressed as wherein ,
the array element positions of the subset in the formula are as follows:
in the formula F12 As a continuous set, when C E ＝2C S At +1, F 12 Ranging from (-M) for a continuous set 2 M 1 -M 2 +1)C E -C S To (M) 2 M 1 +M 2 -1)C E +C S The method comprises the steps of carrying out a first treatment on the surface of the And subset F 12 and F33 Is represented as two adjacent consecutive segments:
in the formula kCE -C S ＜kC E +C S -N 1 ＜kC E +C S According to EAS-NA LS Definition of (C) to obtain C E >2C S +1, when 2C S +1<C E ≤3C S At +1, F 12' and F33 ' are discontinuous and F 12' and F33 The' joint set comprises a continuum { -C s :(M 2 M 1 +M 2 -1)C E +2C s }， and />And aggregate contains from 0 to (M 2 M 1 +M 2 -1)C E +2C S According to the symmetry of FODC,also comprises a slave- (M) 2 M 1 +M 2 -1)C E -2C S To a non-positive subset of 0, to the extent that 2C S +1<C E ≤3C S +1，EAS-NA-NA LS The number of consecutive array elements of the fourth order differential array of (a) ranges from- (M) 2 M 1 +M 2 -1)C E -2C S To (M) 2 M 1 +M 2 -1)C E +2C S Thus, EAS-NA LS The number of consecutive elements of the fourth order differential array is greater than EAS-NA.
2. The method of constructing a fourth order differential array based on an extended and shifted nested array of claim 1, wherein: the sources are uncorrelated, and a non-Gaussian source assumption is used for the existing fourth-order accumulation of the source signal.
3. The method of constructing a fourth order differential array based on an extended and shifted nested array of claim 1, wherein: based on the fourth order cumulant matrix formula, the fourth order differential array corresponds to P ENN ＝P 1 ∪P 2 Denoted as F, expressed as:
F＝(P-P)+(P-P) (11)。
4. the method of constructing a fourth order differential array based on an extended and shifted nested array of claim 1, wherein: maximum value C in the EAS-NA-NA method E ＝2N 2 (N 1 The number of consecutive array elements of the fourth order differential array in +1) -1, EAS-NA is (2M 2 M 1 +2M 2 -1)(2N 2 N 1 +2N 2 -1)+2(M 2 M 1 +M 2 -1)。
5. The method of constructing a fourth order differential array based on an extended and shifted nested array of claim 1, wherein: according to EAS-NA LS Definition, EAS-NA LS A sparse array structure similar to EAS-NA sharing with larger array element spacing in an extended nested array.
6. The method of constructing a fourth order differential array based on an extended and shifted nested array of claim 1, wherein: in the formula (9), if the condition (k+1) C E -C S -(kC E +2C S )≤1,i.e.,C E ≤3C S +1 is satisfied, for a particular k, the union set F 12' and F33 ' form a continuous set { kC } E -C S :(k+1)C E +2C s }。
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| CN112115595A (en) * | 2020-09-07 | 2020-12-22 | 北京理工大学 | Three-level expansion transformation nested array design method for estimating direction of arrival of non-circular signal |
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