Joint computation of doa and doppler frequency under unequal time domain parameter space-time coprime sampling

Abstract

The application discloses a DOA and Doppler frequency joint calculation method under unequal time domain parameter space-time coprime sampling, belongs to the field of signal processing, and is particularly aimed at the joint estimation of a direction of arrival and a Doppler frequency under the condition that physical resources are limited, and is suitable for joint coprime sampling and optimization of a space array and a time sequence. In the scheme, a coprime array is used in the space domain, coprime sampling is performed in the time domain, and different time domain coprime parameters are allowed to be selected at each antenna. Through the scheme, a larger virtual aperture is generated under the condition that the number of antennas and the number of sampling points are limited, so that the degree of freedom and the estimation resolution are improved.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of signal processing, and is particularly directed to the problem of joint estimation of direction of arrival and Doppler frequency under the condition of limited physical resources, and is suitable for joint coprime sampling and optimization of spatial array and time sequence. BACKGROUND

[0002] In array signal processing, joint estimation of multiple parameters is an important problem that has been widely studied. For joint estimation of parameters, early methods include linear prediction methods, but such methods have low estimation accuracy and poor noise resistance. Then, there are methods that combine maximum likelihood algorithms with Fourier transforms, but maximum likelihood algorithms require multiple iterations and multiple searches, resulting in long computation time and complexity. In the late 1970s, MUSIC algorithms and ESPRIT algorithms were proposed, achieving estimation accuracy beyond the original aperture, and not requiring parameter matching processes when jointly estimating parameters. In addition, there are some improved algorithms, such as FSF-MUSIC algorithms that apply MUSIC algorithms in the frequency domain and spatial domain in sequence.

[0003] In the problem of parameter joint estimation, the uniform linear array is usually used, which directly limits the spatial resolution. Therefore, sparse array has attracted much attention in recent years. Given the number of antennas, the array aperture of sparse array is larger than that of uniform linear array, thus increasing the estimation accuracy. Given the array aperture, the number of antennas required by sparse array is less than that of uniform linear array, thus reducing physical consumption. In sparse array, nested array and co-prime array are the most common two. Compared with nested array, co-prime array is more sparse, so the antenna coupling effect is smaller, and the influence on the estimation performance is also smaller. In addition to the co-prime array in the spatial domain, the co-prime sampling structure is also applied in the time domain (see the literature: P. P. Vaidyanathan and P. Pal, "Sparse Sensing With Co-Prime Samplers and Arrays," IEEE Transactions on Signal Processing, vol. 59, no. 2, pp. 573-586, Feb. 2011.). In addition, there are also studies considering the case of applying co-prime sampling structure in space and time domain at the same time, but for each antenna, the time domain co-prime sampling parameters are the same (see the literature: C. Liu and P. P. Vaidyanathan, "Coprime arrays and samplers for space-time adaptive processing," in Proceedings of 2015 IEEE International Conference on Acoustics, Speech and Signal Processing, Apr. 2015, pp. 2364-2368.).

[0004] At present, the related research on sparse-based parameter joint estimation is mostly to estimate separately and then match parameters, and rarely to estimate directly. In addition, the time domain is directly sampled according to the antenna arrangement mode in the spatial domain, and the time domain sampling parameters at each antenna are the same, so the optimal design of the space-time domain co-prime sampling structure is missed. Therefore, the present application considers the problem of DOA and Doppler frequency joint estimation under the space-time domain co-prime sampling structure when the time domain sampling parameters of different antennas are different. SUMMARY

[0005] The application provides a scheme for joint estimation of DOA and Doppler frequency based on a space-time coprime sampling structure of unequal time domain parameters.

[0006] The application discloses a joint calculation method for DOA and Doppler frequency based on a space-time coprime sampling structure of unequal time domain parameters.

[0007] Step 1: constructing a coprime antenna array;

[0008] The coprime array is composed of two uniform linear antenna arrays, and the total number of antennas L = 2M1+N1-1, wherein M1 and N1 are the number of antennas of the two uniform linear arrays, and M1 and N1 are coprime; the distribution sets of the two uniform linear arrays are and wherein the set and represent the relative positions of the antennas in the two uniform linear arrays, n and m represent the antenna serial numbers, and d represents the antenna spacing; the sets and are combined, and then the elements are sorted to obtain the antenna distribution set of the coprime antenna array wherein represents the relative position of the antenna in the coprime array;

[0009] Step 2: performing coprime sampling on the received signals of each antenna in the time domain with different parameters; at the lth antenna, the coprime sampling parameters P l and Q l are taken to sample the received signals in two ways; the first sampling point set is the second sampling point set is wherein the sets and represent the relative positions of the sampling points in the two sampling ways, n' and m' represent the sampling point serial numbers, and T represents the sampling interval. The two sampling point sets are combined and then sorted, so that the sampling point set at the lth antenna is obtained wherein represents the relative position of the sampling point in the coprime time sampling sequence;

[0010] Step 3: calculating the space-time autocorrelation matrix of the received signal

[0011]

[0012] ​where b = 1, …, B, B denotes the number of sources, σ 2 denotes the variance of the source amplitude, denotes the space-time steering vector, and denotes the normalized DOA and Doppler frequency, "s" and "f" represent the spatial and temporal domains, respectively, denotes the variance of the noise, denotes the identity matrix, (·) H denotes the conjugate transpose, denotes the expectation;

[0013] Let where vec(·) denotes the vectorization of a matrix, the b-th column of

[0014]

[0015] where (·) * denotes the conjugate, denotes the element of the matrix in the l-th row and l'-th column, denotes the spatial steering matrix, constitutes the temporal steering matrix respectively,

[0016]

[0017] and

[0018]

[0019] where and denote the spatial and temporal steering vectors, respectively, denotes the element of the spatial steering vector corresponding to the l-th antenna, denotes the temporal steering vector of the l-th antenna, is the temporal steering vector of the l-th antenna;

[0020] Step 4: According to the spatial coprime array and the temporal coprime sequence, the difference sets corresponding to the spatial and temporal domains are obtained as:

[0021]

[0022] and

[0023]

[0024] where l, l' = 1, …, L, u1 and u2 are sets any two elements in u′ 1,l and u′ 2,l is a set any two elements in u′; the continuous sets of spatial and time domain are the largest continuous central parts of the respective differential sets, respectively, and and are sorted, merged and de-duplicated, obtaining a set and where i = 1, …, L0, represents the number of virtual antennas, |·| represents the number of elements in the set; based on the differential set and the continuous set, two kinds of space-time coprime sampling structures based on unequal time domain parameters are considered, the first one is STCD_D based on the differential set, and the second one is STCD_C based on the continuous set; next, the subsequent steps are introduced taking STCD_D as an example, and the operation steps of STCD_C are the same;

[0025] Step 5: Constructing virtual time domain and spatial domain steering vectors, obtaining an equivalent received signal model;

[0026] In and pick out virtual antenna points and time sample points, respectively, obtaining virtual spatial and time domain steering vectors

[0027]

[0028] and

[0029]

[0030] wherein, represents the number of virtual samples at the i-th virtual antenna; the b-th column of the virtual space-time steering matrix W1is represented as

[0031]

[0032] wherein, represents the element in the i-th row and i'-th column of , W1is represented as represents a virtual spatial domain steering matrix, constitutes a virtual time domain steering matrix respectively,

[0033]

[0034] and

[0035]

[0036] wherein, Therefore, an equivalent signal model is constructed as follows:

[0037]

[0038] in, It is a vector in which only the (V+1) / 2th element is 1. Indicates degrees of freedom;

[0039] Step 6: Jointly estimate DOA and Doppler frequency using compressed sensing technology;

[0040] Based on compressed sensing theory, an overcomplete dictionary matrix is ​​constructed. Obtain the sparse signal model

[0041]

[0042] in, It is a sparse vector containing B non-zero elements. Next, we establish the following optimization problem.

[0043]

[0044] Where ε is a set threshold, obtained by solving the above optimization problem using the CVX toolbox. Then, find The final estimate is the DOA and frequency of the B largest elements in the dictionary matrix at their corresponding positions.

[0045] Step 7: If the number of antennas L and the total number of time-domain sampling points R are limited, the optimization problem for selecting the spatial-temporal coprime parameter is established as follows:

[0046]

[0047] in, V represents the degrees of freedom. Let STCD be based on difference sets; the above optimization problem is difficult to solve, so it is transformed into the following optimization problem.

[0048]

[0049] in, and Given the coprime parameters in the time domain when the same coprime sampling is performed at each antenna, gcd(a,b)=1 indicates that the greatest common divisor of a and b is returned; solve the transformed optimization problem to obtain the DOA and frequency.

[0050] This invention employs coprime arrays in both the spatial and temporal domains and allows for different temporal coprime parameters for each antenna, thereby generating a larger virtual aperture in both the spatial and temporal domains. This results in higher degrees of freedom and estimation resolution even when the number of antennas and sampling points is limited. Attached Figure Description

[0051] Figure 1 This is a diagram of the STCD (Spatial-Time Coprime Sampling) structure with unequal time-domain parameters.

[0052] Figure 2 A comparison of Doppler frequency estimation resolution under STCD_D, STCD_C, STCI_D, and STCI_C sampling structures;

[0053] Figure 3 A comparison chart of DOA estimation resolution under STCD_D, STCD_C, STCI_D, and STCI_C sampling structures;

[0054] Figure 4 Performance curves for DOA estimation under the STCD_D, STCD_C, STCI_D, and STCI_C sampling structures;

[0055] Figure 5 The graphs show the performance curves of Doppler frequency estimation under the STCD_D, STCD_C, STCI_D, and STCI_C sampling structures. Detailed Implementation

[0056] For ease of description, we first define the following:

[0057] Bold uppercase letters represent matrices, bold lowercase letters represent vectors, and (·) * Indicates conjugation, (·) T Indicates transpose, (·) H This indicates the conjugate transpose. The Kronecker product is represented by `vec(·)`, which vectorizes the matrix. `gcd(a,b) = 1` returns the greatest common divisor of `a` and `b`. `|·|` represents the number of elements in the set. Represents the identity matrix.

[0058] Consider a coprime array containing L antennas. Assume there are B independent far-field narrowband sources, each with a velocity v. b The incident light is incident on a coprime antenna array at an angle of θ. b The Doppler frequency is f b =v b Given λ / 2, b = 1, ..., B, the received signal of the l-th antenna is:

[0059]

[0060] In the above formula, A b It is the amplitude of the information source, satisfying and ω l (t) is a variable with mean 0 and variance . Gaussian white noise. Assume the incident angle of the source remains constant throughout the observation period.

[0061] At the l-th antenna, according to the coprime parameter P l and Q l The received signal is sampled twice in the time domain, resulting in a total R. l The r-th sample, where the r-th sample... l r l =1,…,R l Each sampled signal is represented as

[0062]

[0063] In the above formula, Represents the r-th position at the l-th antenna. l The noise at each sampling point, different antennas, or different time samples is statistically independent, i.e. Normalized DOA and Doppler frequency and Defined as

[0064]

[0065] Then, the sample vector at the l-th antenna is represented as

[0066]

[0067] in, Collect all samples from L antennas to obtain the received signal.

[0068]

[0069] in,

[0070]

[0071]

[0072]

[0073] According to the received signal The expression yields the space-time autocorrelation matrix of the received signal as follows:

[0074]

[0075] Vectorizing the above equation yields

[0076]

[0077] where the b-th column of matrix is denoted as

[0078]

[0079] where,

[0080]

[0081] and

[0082]

[0083] The present application considers two kinds of space-time co-prime sampling structures (STCD) based on unequal time domain parameters, the first one is STCD based on differential set (STCD_D), and the second one is STCD based on continuous set (STCD_C). Next, the following operations are taken as an example of selecting STCD_D, and STCD_C has the same operation steps.

[0084] In and pick out virtual antenna points and time sample points, respectively get virtual space and time domain steering vector

[0085]

[0086] and

[0087]

[0088] Then the b-th column of virtual space-time steering matrix is denoted as

[0089]

[0090] In the above formula,

[0091]

[0092] and

[0093]

[0094] Therefore, the equivalent signal model can be constructed as

[0095]

[0096] The joint estimation of DOA and Doppler frequency is processed by using the theory of compressed sensing. The ranges of interest in space and time are discretized into X and Y grid points, respectively, satisfying XY>>B, and the DOA and Doppler frequency corresponding to each grid point are Based on the theory of compressed sensing, an overcomplete dictionary matrix is constructed with the dimension of V*XY, and each column of the dictionary contains a DOA and Doppler frequency parameter. Thus, the sparse signal based on the dictionary matrix is obtained

[0097]

[0098] wherein, is the sparse signal vector to be solved. The parameter values corresponding to the indexes of the non-zero elements in are the DOA and Doppler frequency of the source to be estimated. Therefore, the following optimization problem is established

[0099]

[0100] The above problem is a convex optimization problem, which is solved by using the CVX toolbox. Through sparse recovery, is obtained Then, the B largest elements are found, and the DOA and frequency at the corresponding positions in the dictionary matrix are the final estimated values.

[0101] The present application considers the case where the maximum available space physical antenna number L and the time domain sampling point number R are limited. The following optimization problem of space-time domain coprime parameter selection is established:

[0102]

[0103] wherein, 2M1+N1-1=L ensures the number of space physical antennas, ensures the total number of time domain samples, gcd(P l ,Q l )=1 and gcd(M1,N1)=1 ensure the coprimality. The above optimization problem is an integer programming problem, and there are many solutions that satisfy the constraint conditions. It is very complex to find the solution that maximizes the objective function, and there is no good algorithm to solve the above optimization problem with low complexity. In the existing research, the time domain at each antenna is sampled using the same coprime parameters, so we consider converting the optimization problem to optimize the space-time coprime parameters so that the degree of freedom is greater than or equal to the degree of freedom of the time domain coprime parameters. For the case of equal time domain coprime parameters, when the number of antennas is L and the total number of time domain sampling points is R, the space domain coprime parameters are and the time domain coprime parameters are and Thus, the following optimization problem is established

[0104]

[0105] The converted optimization problem can be solved quickly by a greedy algorithm or the like.

[0106] Working principle of the application

[0107] For the spatially-orthogonal array and the temporally-orthogonal sampling sequence, their difference sets are

[0108]

[0109] And

[0110]

[0111] From the expressions of And , it can be seen that the difference sets are obtained by subtracting the elements in the sets And two by two, so the difference sets And contain many new elements and many repeated elements. This also means that And contain many repeated elements, and these repeated elements indicate that there is a lot of information from the same time sample of the same antenna in the second-order statistics of the received signal, and these repeated information is called redundant information. In order to reduce the complexity of subsequent processing, it is necessary to remove these redundant information. Therefore, the elements in And are sorted, merged and de-duplicated to obtain the sets And

[0112] Based on the second-order statistics of the received signal after removing the redundant information, a new virtual equivalent model is considered. According to And , virtual spatial and temporal steering vectors are established to obtain an equivalent spatial array and a temporal sequence with a virtual aperture. The virtual aperture is greatly improved compared to the aperture and degrees of freedom of the original orthogonal array and the orthogonal sequence, thereby improving the estimation performance. In addition, the elements in the virtual aperture are not completely continuous, in order to avoid spatial aliasing and reduce complexity, the largest continuous part in the virtual aperture is selected to establish the equivalent signal model, but this method may cause loss of useful information.

[0113] For DOA and Doppler frequency, they are respectively regarded as discrete sparse quantities in space and time, so the compressed sensing theory is used to jointly recover DOA and Doppler frequency to realize the estimation of the parameters, thereby avoiding the waste of the virtual aperture and obtaining high-performance estimation results in the case of high degrees of freedom.

[0114] Two examples are given to verify the theoretical results and illustrate the advantage of the STCD structure with unequal time-domain parameters in terms of estimation resolution and overall estimation performance. In both examples, the STCD structure with unequal time-domain parameters is compared with the STCI structure with equal time-domain parameters. Since the virtual aperture is constructed based on the differential set and the continuous set respectively, four sampling structures are considered in the simulation: (1) STCD sampling based on the differential set (STCD_D); (2) STCD sampling based on the continuous set (STCD_C); (3) STCI sampling based on the differential set (STCI_D); (4) STCI sampling based on the continuous set (STCI_C).

[0115] Example 1 considers the comparison of the resolution under different sampling structures. Assume that the number of antennas L = 10, the total number of time-domain sampling points R = 100, the signal-to-noise ratio SNR = 0 dB, and the coprime parameters under the STCI structure are and The coprime parameters under the STCD structure are obtained by establishing the optimization problem of selecting the space-time-domain coprime parameters. In the first case, the Doppler frequencies of the two sources are close to each other, 200 Hz and 220 Hz respectively, and the DOAs are both 30°, and 105 Hz ~ 305 Hz is discretized into 120 uniform grid points. In the second case, the DOAs of the two sources are close to each other, 30° and 32° respectively, and the Doppler frequencies are both 200 Hz, and the observation angle range 0° ~ 60° is discretized into 121 uniform grid points. Example 2 considers the relationship between the RMSE of the estimation variable and the SNR. Assume that there are two sources, their DOAs are [θ1, θ2] = [-10°, 10°], their Doppler frequencies are [f1, f2] = [200 Hz, 300 Hz], the number of antennas L = 6, the total number of time-domain sampling points R = 54, and the coprime parameters under the STCI structure are: The coprime parameters under the STCD structure are obtained by establishing the optimization problem of selecting the space-time-domain coprime parameters.

[0116] For example 1, in Figure 2 , STCI_C cannot identify two close sources, and other methods can effectively identify, and the peak height of the curve corresponding to the STCD method is higher than that of the STCI method, because under the same physical resources and parameter conditions, the aperture of the virtual array and sequence obtained by the STCD sampling structure is larger than that of the STCI sampling structure. Figure 3In the simulation, all the methods can effectively identify two sources, and STCD_C and STCI_C have similar performance, and STCD_D and STCI_D have similar performance, because the aperture of the virtual array and sequence corresponding to the differential set is superior to that of the continuous set. For example 2, it can be seen that Figure 4 and Figure 5 For the STCI sampling structure, STCI_D has better performance than STCI_C, because the virtual aperture is increased and the degree of freedom is improved. For the STCD sampling structure, STCD_D has better performance than STCD_C at a lower signal-to-noise ratio, and STCD_C has better performance than STCD_D when the signal-to-noise ratio is-5 to-10 dB, because there is a discontinuous point in the virtual aperture under the STCD sampling structure, resulting in a non-continuous aperture and causing space-time aliasing. With the increase of the signal-to-noise ratio, the performance of the two sampling structures is consistent. In addition, the performance of STCD_C and STCD_D is better than that of STCI_C and STCI_D, respectively, because the degree of freedom of STCD_D is greater, which indicates that the performance of the STCD sampling structure is superior to that of the STCI sampling structure under the same physical resources.

Claims

1. A method for joint calculation of DOA and Doppler frequency under spatiotemporally coprime sampling with unequal time-domain parameters, the method comprising: Step 1: Construct a coprime antenna array; The co-prime array is composed of two uniform linear antenna arrays, and the total number of antennas \(L = 2M_1+N_1 - 1\), where \(M_1\) and \(N_1\) are the numbers of antennas of the two uniform linear arrays, and \(M_1\) and \(N_1\) are co-prime; the distribution sets of the antennas of the two uniform linear arrays are respectively and where the sets and represent the relative positions of the antennas in the two uniform linear arrays, \(n\) and \(m\) represent the antenna numbers, and \(d\) represents the antenna spacing; the sets and are merged and then the elements are sorted to obtain the antenna distribution set of the co-prime antenna array where represents the relative position of the antennas in the co-prime array; Step 2: Perform coprime sampling of the received signals from each antenna in the time domain with different parameters; at the l-th antenna, take the coprime sampling parameter P. l and Q l The received signal is sampled in two ways; the first set of sampling points is... The second set of sampling points is Among them, set and The relative positions of the sampling points in the two sampling paths are represented by n' and m', where n' and m' represent the sampling point numbers and T represents the sampling interval. Merging and sorting the two sampling point sets yields the sampling point set at the l-th antenna. in, Indicates the relative position of the sampling points in a coprime time sampling sequence; Step 3: Calculate the space-time autocorrelation matrix R of the received signal x. xx ; Where b = 1, ..., B, B represents the number of information sources, σ 2 The variance of the source amplitude is represented by... Represents the spacetime steering vector. and The "s" and "f" represent the normalized DOA and Doppler frequency, respectively, and represent the spatial and time domains. Let I represent the variance of the noise, and let (·) represent the identity matrix. H This indicates the conjugate transpose. This indicates a demand for expectation; make Where p = σ 2 I, vec(·) represents vectorizing the matrix, where the b-th column of W is: in,(·) * U represents conjugate. s (l,l'), where l=1,…,L represents matrix U. s The element in the l-th row and l'-th column, U s Represents the spatial guidance matrix. Constructing the time-domain steering matrix U f They are respectively: and in, and These represent the spatial and temporal steering vectors, respectively. Represents the spatial guidance vector The element corresponding to the l-th antenna. This represents the time-domain steering vector of the l-th antenna. Let be the time-domain steering vector of the l-th antenna; Step 4: Based on the spatial coprime array and the temporal coprime sequence, the corresponding difference sets in the spatial and temporal domains are obtained as follows: and Where l, l' = 1, ..., L, u1 and u2 are sets For any two elements, u1' ,l and u' 2,l For set Any two elements in the form; the continuous sets in the spatial and temporal domains are the maximal continuous center parts of their respective difference sets, respectively. and The elements in the dataset are sorted, merged, and deduplicated to obtain a set representing the relative positions of the virtual antenna and virtual sampling points. and Where i = 1, ..., L0, The number of virtual antennas is represented by |·|, and the number of elements in the set is represented by |·|. Based on differential sets and continuous sets, two spatiotemporal coprime sampling structures based on unequal time-domain parameters are considered. The first is STCD_D based on differential sets, and the second is STCD_C based on continuous sets. The following describes the subsequent steps using STCD_D as an example. The operation steps for STCD_C are the same. Step 5: Construct virtual time-domain and spatial-domain steering vectors to obtain an equivalent received signal model; exist and Virtual antenna points and time sample points are selected from the data to obtain virtual spatial and temporal steering vectors, respectively. and in, The number of virtual samples at the i-th virtual antenna is represented by the b-th column of the virtual space-time steering matrix W1. Among them, V s (i,i'),i,i'=1,...,L0 represents V s The element in the i-th row and i'-th column, V s This represents a virtual spatial guidance matrix. Constructing a virtual time-domain steering matrix V f , respectively and in, Therefore, an equivalent signal model is constructed as follows: in, It is a vector in which only the (V+1) / 2th element is 1. Indicates degrees of freedom; Step 6: Jointly estimate DOA and Doppler frequency using compressed sensing technology; Based on compressed sensing theory, an overcomplete dictionary matrix is ​​constructed. Obtain the sparse signal model in, It is a sparse vector containing B non-zero elements. Next, we establish the following optimization problem. Where ε is a set threshold, obtained by solving the above optimization problem using the CVX toolbox. Then, find The final estimate is the DOA and frequency of the B largest elements in the dictionary matrix at their corresponding positions. Step 7: If the number of antennas L and the total number of time-domain sampling points R are limited, the optimization problem for selecting the spatial-temporal coprime parameter is established as follows: Where P = [P1,...,P] L ] T Q = [Q1,...,Q] L ] T V represents the degrees of freedom. Let STCD be based on difference sets; the above optimization problem is difficult to solve, so it is transformed into the following optimization problem. in, and Given the coprime parameters in the time domain when the same coprime sampling is performed at each antenna, gcd(a,b)=1 indicates that the greatest common divisor of a and b is returned; solve the transformed optimization problem to obtain the DOA and frequency.

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