CN112731273B - Low-complexity signal direction-of-arrival estimation method based on sparse Bayesian - Google Patents

Low-complexity signal direction-of-arrival estimation method based on sparse Bayesian Download PDF

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CN112731273B
CN112731273B CN202011426111.0A CN202011426111A CN112731273B CN 112731273 B CN112731273 B CN 112731273B CN 202011426111 A CN202011426111 A CN 202011426111A CN 112731273 B CN112731273 B CN 112731273B
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iteration
incident signal
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CN112731273A (en
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王一凡
王芳芳
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Nanjing University of Posts and Telecommunications
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
    • G01S3/14—Systems for determining direction or deviation from predetermined direction
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N7/00—Computing arrangements based on specific mathematical models
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Abstract

The invention discloses a sparse Bayesian-based low-complexity signal direction-of-arrival estimation method, and aims to solve the technical problems of high computational complexity and low computational efficiency of an MMV model in the prior art when time correlation problems are considered. It comprises the following steps: expanding an incident signal, and obtaining the prior of the expanded incident signal by using a Markov probability prior model; and (3) obtaining an input function and an output function of a GAMP algorithm according to the priori of the expanded incident signal, performing GAMP iteration, obtaining a recovered expanded incident signal, and obtaining a signal direction of arrival according to the recovered expanded incident signal. The method can reduce the calculation complexity of DOA estimation on the premise of considering time-related factors, improve the calculation efficiency and accurately estimate the signal arrival direction.

Description

Low-complexity signal direction-of-arrival estimation method based on sparse Bayesian
Technical Field
The invention relates to a sparse Bayesian-based low-complexity signal direction-of-arrival estimation method, and belongs to the technical field of signal processing.
Background
The signal arrival direction (Direction of Arrival, DOA) is the incoming wave direction of an incident signal estimated by using the received data of an antenna array in a noisy environment, the basic principle is to estimate the space angle of the signal arrival array by using the phase difference existing between the received data of different array elements of a space array, and the method can be applied to the fields of radio communication, radar, sonar, navigation, seismic detection, biomedicine and the like and has important significance.
The SBL algorithm developed in recent years was originally proposed as a machine learning algorithm by Tipping et al before and after 2001, and then was introduced into the sparse signal recovery/compressed sensing field, and an initial SBL was applied to a model of a single measurement vector (Signal Measurement Vector, SMV) and then gradually expanded to a multi-measurement vector (Multiple Measurement Vectors, MMV), which has the advantages of higher resolution and a small reduction in calculation amount, but the DOA model used in MMV is more ideal and does not consider more practical factors. Zhang Zhilin et al in 2011 applied SBL to a time-dependent direction finding scene, proposed a time-dependent sparse bayesian learning (Temporally Sparse Bayesian Learning, TSBL) algorithm, introduced a super parameter to control the time correlation between snapshots, which can solve the time-dependent problem, but the introduced super parameter can cause an increase in the calculated amount, and the efficiency becomes very low, if on a large scale of problems such as a large number of array elements, accurate direction finding cannot be realized, and further algorithm optimization is necessary.
Disclosure of Invention
In order to solve the problems of high computational complexity and low computational efficiency of an MMV model in the prior art when time correlation problems are considered, the invention provides a sparse Bayesian-based low-complexity signal direction-of-arrival estimation method, which optimizes a derivation process of posterior probability by using Markov prior probability distribution on the basis of considering time correlation factors, decouples by using a GAMP algorithm, replaces matrix inversion steps in the prior art, reduces the calculated amount and improves the computational efficiency.
In order to solve the technical problems, the invention adopts the following technical means:
the invention provides a sparse Bayesian-based low-complexity signal direction-of-arrival estimation method, which comprises the following steps:
expanding an incident signal based on spatial domain meshing;
obtaining the priori of the expanded incident signal by using a Markov probability priori model;
obtaining an input function and an output function of a GAMP algorithm according to the priori of the expanded incident signal;
performing GAMP iteration on the snapshot of the expanded incident signal by using an input function and an output function of a GAMP algorithm to obtain a restored expanded incident signal;
and obtaining the signal arrival direction according to the recovered expanded incident signal.
Further, the specific steps of the incident signal expansion are as follows:
let the incident signal be x, the receiving array is made up of M homogeneous linear array antennas, when the receiving array receives T snapshots of the incident signal, obtain the output model of the array:
y＝A(θ)x+e (1)
wherein y is a receiving signal of a receiving array, A (theta) is an array flow pattern matrix, and e is Gaussian white noise received by the receiving array;
grid dividing the space domain of the receiving array by using an exhaustion method to obtain an ultra-complete angle set
Figure BDA0002824903900000031
Wherein (1)>
Figure BDA0002824903900000032
Indicating the nth overcomplete angle, H is the overcomplete angle set +.>
Figure BDA0002824903900000033
N=1, 2, …, H;
based on
Figure BDA0002824903900000034
And A (theta) obtains an expanded array flow pattern matrix:
Figure BDA0002824903900000035
wherein A (θ) _spark represents the expanded array flow pattern matrix;
obtaining an ultra-complete array output model according to the expanded array flow pattern matrix:
Figure BDA0002824903900000036
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0002824903900000037
representing the expandedReceive signal,/->
Figure BDA0002824903900000038
Representing the spread incident signal.
Further, the spatial domain of the receiving array is [ -90 °,90 ° ].
Further, the prior acquisition of the expanded incident signal includes the following steps:
modeling the expanded incident signal by using a Markov probability prior model to obtain the relation between the nth snapshot and the (t-1) th snapshot of the nth expanded incident signal:
Figure BDA0002824903900000039
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA00028249039000000310
an nth expanded incident signal representing a nth snapshot, +.>
Figure BDA00028249039000000311
An n-th expanded incident signal representing the t-1 th snapshot, +.>
Figure BDA00028249039000000312
Representation->
Figure BDA00028249039000000313
And->
Figure BDA00028249039000000314
The relation between beta and beta is the time correlation coefficient, beta epsilon (-1, 1), gamma n Is the a priori error of the nth spread incident signal, t=1, 2, …, T;
obtaining the influence of the previous snapshot according to the relation between the front snapshot and the rear snapshot of the expanded incident signal
Figure BDA00028249039000000315
And the latterInfluence of snapshot->
Figure BDA00028249039000000316
Wherein (1)>
Figure BDA0002824903900000041
Representing the incident signal +.>
Figure BDA0002824903900000042
Is>
Figure BDA0002824903900000043
Represents the mean value of forward transmission of the t-1 snapshot in the nth expanded incident signal t snapshot,/->
Figure BDA0002824903900000044
Representing the variance of the forward transmission of the t-1 snapshot in the nth expanded incident signal t snapshot,/for>
Figure BDA0002824903900000045
Representing the incident signal +.>
Figure BDA0002824903900000046
Is>
Figure BDA0002824903900000047
Represents the mean value of backward transmission of t+1 snapshot in the time of t snapshot of the n-th expanded incident signal,/and>
Figure BDA0002824903900000048
representing the variance of backward transfer of the t+1 snapshot in the snapshot of the nth expanded incident signal t;
according to
Figure BDA0002824903900000049
And->
Figure BDA00028249039000000410
Obtain the expanded incident signal +.>
Figure BDA00028249039000000411
Is a priori of:
Figure BDA00028249039000000412
further, the nth expanded incident signal of the nth snapshot
Figure BDA00028249039000000413
The expression of (2) is as follows:
Figure BDA00028249039000000414
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA00028249039000000415
further, the expression of the input function of the GAMP algorithm is as follows:
Figure BDA00028249039000000416
wherein g s Represents the input function, q (t) An approximation of the received signal representing the t-th snapshot without noise effects,
Figure BDA00028249039000000417
represents q (t) Noise variance of y (t) Received signal, sigma, representing the t-th snapshot receiving matrix 2 Representing y (t) Is a noise variance of (1);
the expression of the output function of the GAMP algorithm is as follows:
Figure BDA0002824903900000051
wherein g x The output function is represented as a function of the output,
Figure BDA0002824903900000052
representing an approximation of the nth expanded incident signal of the nth snapshot,
Figure BDA0002824903900000053
representation->
Figure BDA0002824903900000054
Is a noise variance of (a).
Further, the specific operation of obtaining the recovered expanded incident signal is as follows:
initializing an expanded incident signal of a t-th snapshot
Figure BDA0002824903900000055
Input function and output function pairs using GAMP algorithm
Figure BDA0002824903900000056
Performing GAMP iteration, and updating the mean value and variance of the expanded incident signal;
updating iteration parameters by using an EM algorithm in each iteration process, and carrying out iteration convergence judgment by using iteration convergence conditions;
obtaining a recovered expanded incident signal according to the mean value and the variance of the expanded incident signal meeting the iterative convergence judgment;
wherein the iteration convergence condition comprises a GAMP iteration convergence condition and an EM iteration convergence condition.
Further, if the maximum iteration number of the GAMP algorithm is G, the iteration process of each iteration is specifically as follows:
according to the output of the ith-1 th iteration, obtaining the expanded incident signal of the t snapshot in the ith iteration
Figure BDA0002824903900000057
Variance of->
Figure BDA0002824903900000058
Mean value X i(t) Priori error gamma i And the noise variance (sigma) of the received signal 2 ) i Wherein i is [1, G ]]；
Let s= |a (θ) _spark| 2 By using
Figure BDA0002824903900000059
And X i(t) Calculating the approximation q of the received signal of the t snapshot without noise effect in the i-th iteration i(t) ：
Figure BDA0002824903900000061
Wherein g s (i-1)(t) An input function representing a t-snapshot of the i-1 th iteration;
using gamma i 、(σ 2 ) i And q i(t) Updating the input function g of the t snapshot of the ith iteration s i(t) ；
Using updated input function g s i(t) Respectively calculating approximate values r of the expanded incident signals of the t-th snapshot i(t) Noise variance of
Figure BDA0002824903900000062
Figure BDA0002824903900000063
Figure BDA0002824903900000064
Wherein A (θ) _spark T Represents a transpose of a (θ) _space,
Figure BDA0002824903900000065
Figure BDA0002824903900000066
for inputting damping coefficient，/>
Figure BDA0002824903900000067
s (i-1)(t) Representing the value of the i-1 th iteration, S T The transition of S is represented (g) s i(t) ) ' represents the input function g s i(t) Is a derivative of (2);
using gamma i 、r i(t) And
Figure BDA0002824903900000068
updating the output function g of the t snapshot of the ith iteration x i(t) ；
According to the updated output function g x i(t) Updating the variance and the mean value of the expanded incident signal:
Figure BDA0002824903900000069
Figure BDA00028249039000000610
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA00028249039000000611
representing the variance of the expanded incident signal output from the ith iteration, (g) x i(t) ) ' represents the output function g x i(t) Derivative of X (i+1)(t) Representing the mean value of the expanded incident signal output from the ith iteration, < >>
Figure BDA00028249039000000612
For outputting damping coefficient->
Figure BDA00028249039000000613
Pair X using GAMP iteration convergence conditions (i+1)(t) And
Figure BDA00028249039000000614
performing GAMP iteration convergence judgment, ending iteration when GAMP iteration convergence conditions are met, outputting the mean value and the variance of the incident signal after the expansion of the ith iteration, and entering the next step when GAMP iteration convergence conditions are not met;
x output by ith iteration based on EM algorithm (i+1)(t) And
Figure BDA0002824903900000071
calculating a priori error gamma in the (i+1) th iteration i+1 And noise variance (sigma) 2 ) i+1 The calculation formula is as follows:
Figure BDA0002824903900000072
(σ 2 ) i+1 ＝||y (t) -A(θ)_sparse·X (i+1)(t) || 2 (15)
pair X using EM iteration convergence conditions (i+1)(t) And
Figure BDA0002824903900000073
and (3) carrying out EM iteration convergence judgment, outputting the mean value and the variance of the incident signal after the expansion of the (i+1) th iteration when the EM iteration convergence condition is met, ending the iteration, and continuing the iteration when the EM iteration convergence condition is not met.
Further, the GAMP iteration convergence condition is:
Figure BDA0002824903900000074
or the iteration number i reaches the maximum iteration number G of the GAMP algorithm, wherein epsilon gamp Representing normalized tolerance parameters for the GAMP algorithm iterations.
Further, the EM iteration convergence condition is:
Figure BDA0002824903900000075
or the iteration number i reaches the maximum iteration number of the EM algorithmNumber K, where ε em Representing normalized tolerance parameters for EM algorithm iterations.
The following advantages can be obtained by adopting the technical means:
the invention provides a low-complexity signal direction-of-arrival estimation method based on sparse Bayes, which utilizes a Markov probability prior model to carry out modeling, considers factors related to the time when a plurality of beats are estimated by the direction-of-arrival estimation, improves the accuracy of a DOA model under a large-scale beat and time-related MMV model, is suitable for DOA model estimation in a non-ideal environment, and improves the robustness of the DOA estimation under the condition of low signal-to-noise ratio. After the priori of the incident signal is obtained, the invention combines with the GAMP idea, carries out GAMP iteration by introducing intermediate parameters, replaces the step of matrix inversion when solving the posterior probability density under the traditional Bayesian framework by iterative operation, greatly reduces the calculation complexity, improves the calculation efficiency and improves the resolution of signal arrival direction estimation.
In the signal direction of arrival estimation process, the method considers non-ideal environments such as time correlation and the like, balances good performances such as high resolution, low complexity and the like, breaks through the limitation of the existing SBL algorithm in the aspects of precision, complexity, robustness and the like, and has very important significance for the research of modern application scenes.
Drawings
Fig. 1 is a flowchart of steps of a sparse bayesian-based low-complexity signal direction-of-arrival estimation method according to the present invention.
FIG. 2 is a diagram illustrating messaging between multiple snapshot time frames in an embodiment of the present invention.
Fig. 3 is a normalized power spectrum diagram of the method and TMSBL algorithm under the condition of low signal to noise ratio in the embodiment of the present invention.
Fig. 4 is a normalized power spectrum diagram of the method and TMSBL algorithm under the condition of high signal to noise ratio in the embodiment of the present invention.
Fig. 5 is a graph comparing the RMSE of the method of the present invention and the TMSBL algorithm in an embodiment of the present invention.
FIG. 6 is a graph showing the comparison between the method of the present invention and TMSBL algorithm at CUP-time in the embodiment of the present invention.
Detailed Description
The technical scheme of the invention is further described below with reference to the accompanying drawings:
the invention provides a sparse Bayesian-based low-complexity signal direction-of-arrival estimation method, which is shown in fig. 1 and specifically comprises the following steps:
step 1, expanding an incident signal based on space domain grid division;
step 2, obtaining the priori of the expanded incident signal by using a Markov probability priori model;
step 3, obtaining an input function and an output function of a GAMP algorithm according to the priori of the expanded incident signal;
step 4, performing GAMP iteration on the snapshot of the expanded incident signal by utilizing an input function and an output function of a GAMP algorithm to obtain a restored expanded incident signal;
and step 5, obtaining the signal arrival direction according to the restored expanded incident signal.
In the embodiment of the invention, W narrow-band far-field signal sources with wavelength lambda are arranged, and theta is used for w Is injected into the receiving end, W is {1,2, and W, the incident signal is x, the receiving array of the receiving end consists of M uniform linear array antennas.
When the receiving array receives T snapshots of the incident signal, an array output model is obtained:
y＝A(θ)x+e (16)
wherein y is a receiving signal of a receiving array, A (theta) is an array flow pattern matrix, e is Gaussian white noise received by the receiving array, e meets normal distribution, and e-N (0, sigma) 2 I)，σ 2 Representing the noise variance of the received signal.
The expression of the array flow pattern matrix is as follows:
Figure BDA0002824903900000091
based on thinSparse representation theory, the incoming direction of the incident signal is sparse in the whole spatial domain, the resolvable spatial domain of the receiving array is [ -90 °,90 °]Grid division is carried out on the space domain of the receiving array by using an exhaustion method to obtain an ultra-complete angle set
Figure BDA0002824903900000101
Expanding the incident signal, wherein +_>
Figure BDA0002824903900000102
Represents the nth overcomplete angle, H is the overcomplete angle set +.>
Figure BDA0002824903900000103
N=1, 2, …, H.
Based on
Figure BDA0002824903900000104
And A (theta) obtains an expanded array flow pattern matrix A (theta) _spark:
Figure BDA0002824903900000105
because the incident signal falls onto the grid, an overcomplete array output model is obtained from the expanded array flow pattern matrix:
Figure BDA0002824903900000106
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0002824903900000107
representing the spread received signal, < >>
Figure BDA0002824903900000108
Representing the expanded incident signal, < >>
Figure BDA0002824903900000109
The invention recovers the expanded incident signal by using the received signal and A (theta) _spark
Figure BDA00028249039000001010
Then use the expanded incident signal +.>
Figure BDA00028249039000001013
The DOA direction of the incident signal x is obtained.
In step 2, the prior acquisition of the expanded incident signal is as follows:
step 201, under the bayesian framework, taking time-related factors, namely that an incident signal is affected by front and rear snapshots at the current moment, modeling a signal source by using a markov probability prior model, and combining the products of the front and rear snapshots into one message.
The invention introduces a time correlation coefficient beta, beta epsilon (-1, 1), beta can represent the relation between the previous snapshot and the next snapshot of a signal. Nth expanded incident signal of the nth snapshot
Figure RE-GDA00029702718600001013
Can be expressed as:
Figure BDA00028249039000001012
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0002824903900000111
an n-th expanded incident signal representing the t-1 th snapshot, +.>
Figure BDA0002824903900000112
γ n T=1, 2, …, T, a priori error of the nth spread incident signal.
Obtaining the relation between the nth snapshot and the (t-1) th snapshot of the nth expanded incident signal according to formula (19):
Figure BDA0002824903900000113
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0002824903900000114
representation->
Figure BDA0002824903900000115
And->
Figure BDA0002824903900000116
Relationship between them.
Under MMV model, let
Figure BDA0002824903900000117
The message passing between the multiple snapshots is shown in fig. 2 for the first pass of the nth expanded incident signal of the nth snapshot.
Assuming that the prior of forward message transmission is a Gaussian distribution, introducing parameters eta and psi as the mean value and variance of the parameters eta and psi, the influence of the previous snapshot is
Figure BDA0002824903900000118
Let the first pass of backward message be a gaussian distribution, introduce parameters v and phi as their mean and variance, then the effect of the latter snapshot is +.>
Figure BDA0002824903900000119
Wherein (1)>
Figure BDA00028249039000001110
Representing the incident signal +.>
Figure BDA00028249039000001111
Is>
Figure BDA00028249039000001112
Represents the mean value of forward transmission of the t-1 snapshot in the nth expanded incident signal t snapshot,/->
Figure BDA00028249039000001113
Representing the variance of the forward transmission of the t-1 snapshot in the nth expanded incident signal t snapshot,
Figure BDA00028249039000001114
representing the incident signal +.>
Figure BDA00028249039000001115
Is>
Figure BDA00028249039000001116
Represents the mean value of backward transmission of t+1 snapshot in the time of t snapshot of the n-th expanded incident signal,/and>
Figure BDA00028249039000001117
representing the variance of the backward transmission of the t+1 snapshot of the nth expanded incident signal t snapshot.
Merging
Figure BDA00028249039000001118
And->
Figure BDA00028249039000001119
Obtain the n expanded incident signal of the t snapshot +.>
Figure BDA00028249039000001120
Is a priori of:
Figure BDA0002824903900000121
after the priori of the incident signal is obtained, the invention combines with the GAMP idea, and performs subsequent calculation through intermediate parameters to replace the related step of matrix inversion, thereby reducing the calculation complexity. The invention introduces an intermediate parameter r as an approximation of the spread incident signal, giving the noise variance τ of r r Introducing an intermediate parameter q as an approximation of A (θ) x, i.e. joint without noise effectsApproximation of the received signal, given the noise variance τ of q q 。
According to Gaussian probability density function and convolution operation, substituting the formula (20) to obtain the average value of forward extinction transfer prior
Figure BDA0002824903900000122
Sum of variances->
Figure BDA0002824903900000123
Is represented by the expression:
Figure BDA0002824903900000124
so that:
Figure BDA0002824903900000125
Figure BDA0002824903900000126
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0002824903900000127
approximation of the n expanded incident signal representing the t-1 th snapshot,/>
Figure BDA00028249039000001212
Representation->
Figure BDA0002824903900000128
Is a noise variance of (a).
Similarly, a mean value of the prior of the backward message transmission can be obtained
Figure BDA0002824903900000129
Sum of variances->
Figure BDA00028249039000001210
Is represented by the expression:
Figure BDA00028249039000001211
Figure BDA0002824903900000131
the expression of the input function can be derived from the definition of the input function in the GAMP algorithm:
Figure BDA0002824903900000132
wherein g s Represents the input function, q (t) An approximation of the received signal representing the t-th snapshot without noise effects,
Figure BDA0002824903900000133
represents q (t) Is z=a (θ) x, y (t) Received signal, sigma, representing the t-th snapshot receiving matrix 2 Representing y (t) Is a noise variance of (a).
From the definition and prior of output functions in the GAMP algorithm
Figure RE-GDA0002970271860000134
The expression of the output function can be deduced:
Figure BDA0002824903900000141
wherein g x The output function is represented as a function of the output,
Figure BDA0002824903900000142
representing an approximation of the nth expanded incident signal of the nth snapshot,
Figure BDA0002824903900000143
representation->
Figure BDA0002824903900000144
Is a noise variance of (a).
In the embodiment of the present invention, the specific operation of step 4 is as follows:
initializing an expanded incident signal of a t-th snapshot
Figure BDA0002824903900000145
Input function and output function pairs using GAMP algorithm
Figure BDA0002824903900000146
Performing GAMP iteration, and updating the mean value and variance of the expanded incident signal;
updating iteration parameters by using an EM algorithm in each iteration process, and carrying out iteration convergence judgment by using iteration convergence conditions;
obtaining a recovered expanded incident signal according to the mean value and the variance of the expanded incident signal meeting the iterative convergence judgment;
the iteration convergence conditions comprise a GAMP iteration convergence condition and an EM iteration convergence condition, two convergence judgments are carried out according to 2 iteration convergence conditions in each iteration process, any one convergence judgments pass, and the iteration is terminated.
Setting the maximum iteration number of the GAMP algorithm as G, when the first iteration is carried out, initializing the values of the iteration number, the mean value, the variance, the priori error, the noise variance of the received signal and the like of the expanded incident signal, and outputting the recovered expanded incident signal when the iteration number is increased and the values are changed along with the increase of the iteration number until the iteration convergence condition is met.
Taking the ith iteration as an example, the iterative process is specifically as follows:
(1) According to the output of the ith-1 th iteration, obtaining the expanded incident signal of the t snapshot in the ith iteration
Figure BDA0002824903900000151
Variance of->
Figure BDA0002824903900000152
Mean value X i(t) Priori error gamma i And the noise variance (sigma) of the received signal 2 ) i Wherein i is [1, G ]]。
(2) Let s= |a (θ) _spark| 2 A calculation formula of variance of approximation of the received signal is obtained:
Figure BDA0002824903900000153
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0002824903900000154
representing the variance of the approximation of the received signal for the t snapshot without noise effects in the i-th iteration.
(3) By means of
Figure BDA0002824903900000155
And X i(t) Calculating the approximation q of the received signal of the t snapshot without noise effect in the i-th iteration i(t) ：
Figure BDA0002824903900000156
Wherein g s (i-1)(t) An input function representing a t-snapshot of the i-1 th iteration.
(4) Will gamma i 、(σ 2 ) i And q i(t) Equal substitution formula (28), updating the input function g of t snapshot of the ith iteration s i (t) 。
(5) For convergence of GAMP iterations, an input damping coefficient is introduced
Figure BDA0002824903900000161
And output damping coefficient +.>
Figure BDA0002824903900000162
Furthermore, an intermediate parameter s is introduced, the expression of s is:
Figure BDA0002824903900000163
wherein s is i(t) The value representing the ith iteration, which has no actual physical meaning, is simply a mathematical equation, s (i -1)(t) Representing the value of the i-1 th iteration.
(6) Using updated input function g s i(t) Respectively calculating approximate values r of the expanded incident signals of the t-th snapshot i(t) Noise variance of
Figure BDA0002824903900000164
Figure BDA00028249039000001610
Figure BDA0002824903900000165
Wherein A (θ) _spark T Represents the transpose of A (θ) _space, S T The transpose of S is represented (g) s i(t) ) ' represents the input function g s i(t) Is a derivative of (a).
(7) Will gamma i 、r i(t) And τ r i(t) Equal substitution formula (29), updating output function g of t snapshot of the ith iteration x i (t) 。
(8) According to the updated output function g x i(t) Updating the variance and the mean value of the expanded incident signal:
Figure BDA0002824903900000166
Figure BDA0002824903900000167
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0002824903900000168
representing the variance of the expanded incident signal output from the ith iteration, (g) x i(t) ) ' represents the output function g x i(t) Derivative of X (i+1)(t) Representing the mean value of the expanded incident signal output by the ith iteration.
(9) Pair X using GAMP iteration convergence conditions (i+1)(t) And
Figure BDA0002824903900000169
and (3) performing GAMP iteration convergence judgment, wherein the GAMP iteration convergence condition is as follows: />
Figure BDA0002824903900000171
Or the iteration number i reaches the maximum iteration number G of the GAMP algorithm, wherein epsilon gamp Representing normalized tolerance parameters for the GAMP algorithm iterations.
Ending iteration and outputting when GAMP iteration convergence condition is satisfied
Figure BDA0002824903900000172
And X (i+1)(t) And when the GAMP iteration convergence condition is not satisfied, entering the next step.
(10) Iteration parameters are updated based on the EM algorithm (expectation maximization algorithm), and the ith iteration output X is utilized (i+1)(t) And
Figure BDA0002824903900000173
calculating a priori error gamma in the (i+1) th iteration i+1 And noise variance (sigma) 2 ) i+1 The calculation formula is as follows:
Figure BDA0002824903900000174
(σ 2 ) i+1 ＝||y (t) -A(θ)_sparse·X (i+1)(t) || 2 (37)
(11) Pair X using EM iteration convergence conditions (i+1)(t) And
Figure BDA0002824903900000175
and (3) carrying out EM iteration convergence judgment, wherein the EM iteration convergence condition is as follows:
Figure BDA0002824903900000176
or the iteration number i reaches the maximum iteration number K of the EM algorithm, wherein ε em Representing normalized tolerance parameters for EM algorithm iterations.
When the EM iteration convergence condition is met, the (i+1) th iteration is carried out, only steps (1) - (8) are carried out in the (i+1) th iteration, the iteration is ended, and the average value X of the expanded incident signal of the (i+1) th iteration is output (i+2)(t) Sum of variances
Figure BDA0002824903900000177
And when the EM iteration convergence condition is not met, continuing iteration.
In step 5, each row of the recovered expanded incident signal contains only W non-zero values, the positions of which correspond to the DOA direction of the incident signal x.
Several groups of simulation experiments are provided for verifying the effect of the method:
simulation experiment 1:
the uniform linear array is adopted to receive the incident signals, the number M=12 of array elements of the receiving array is set, 4 signal sources are set, and the incoming wave angles are respectively as follows: -30.15, -10.23, 40.74, 28.39. In the MMV model, the sampling snapshot number T is 100, the signal direction of arrival estimation is carried out by using the method and the existing TMSBL algorithm respectively, normalized power spectrograms of the method and the existing TMSBL algorithm under different noise environments are shown in figures 3 and 4, and as can be seen from the figures, under the condition that the SNR=10dB of the low signal-to-noise ratio, the TMSBL algorithm taking time correlation factors into consideration can not be estimated, and the method can still accurately carry out the signal direction of arrival estimation; under the condition of high signal-to-noise ratio, namely SNR=40 dB, the peak of the power spectrum of the TMSBL algorithm is still inaccurate, and a false peak exists, so that the method is applicable to DOA model estimation of a non-ideal environment.
Simulation experiment 2:
let the number of array elements m=10 of the receiving array, there are 2 signal sources, the incoming wave direction is-11.21 and 0.74 respectively, the snapshot number T is 40. The predetermined grid point range is [ -90 °,90 ° ], and the resolution is 1 °. The performance comparison condition of the RMSE under the variation of signal-to-noise ratio (SNR) is researched by using the method and the existing TMSBL algorithm, and the root mean square error is obtained by averaging 500 Monte Carlo experiments aiming at each group of simulation parameter setting.
Fig. 5 is a graph comparing the RMES with the TMSBL algorithm, and it can be seen that, in consideration of the time correlation factor, the error of the method is far smaller than that of the TMSBL algorithm at the time of low signal to noise ratio, so that the accuracy of the method in a non-ideal environment is higher.
Simulation experiment 3:
under the condition that other simulation conditions are identical to the simulation experiment 2, the SNR=30dB is set, the performance comparison condition of CUP time along with the change of the snapshot number is researched by using the method and TMSBL algorithm, and the result is shown in figure 6, compared with TMSBL, the time required by the method is far smaller than TMSBL under the condition of large snapshot, the calculation complexity of the method under the condition of large snapshot number is lower, and the calculation efficiency is higher.
In summary, the method solves the problem of time correlation of the signal source in MMV direction of arrival estimation under a Bayesian framework, utilizes GAMP to replace a matrix inversion part in the traditional SBL algorithm, reduces the calculation complexity, and can estimate DOA more rapidly and with high precision. Under the non-ideal environment considering time correlation and the like, the method has the advantages of high resolution, low complexity, high accuracy, good robustness and the like.
The foregoing is merely a preferred embodiment of the present invention, and it should be noted that modifications and variations could be made by those skilled in the art without departing from the technical principles of the present invention, and such modifications and variations should also be regarded as being within the scope of the invention.

Claims (8)

1. A low-complexity signal direction-of-arrival estimation method based on sparse Bayes is characterized by comprising the following steps:
expanding an incident signal based on spatial domain meshing;
obtaining the priori of the expanded incident signal by using a Markov probability priori model;
obtaining an input function and an output function of a GAMP algorithm according to the priori of the expanded incident signal;
performing GAMP iteration on the snapshot of the expanded incident signal by using an input function and an output function of a GAMP algorithm to obtain a restored expanded incident signal;
obtaining a signal direction of arrival according to the recovered expanded incident signal;
the specific steps of the incident signal expansion are as follows:
let the incident signal be x, the receiving array is made up of M homogeneous linear array antennas, when the receiving array receives T snapshots of the incident signal, obtain the output model of the array:
y＝A(θ)x+e
wherein y is a receiving signal of a receiving array, A (theta) is an array flow pattern matrix, and e is Gaussian white noise received by the receiving array;
grid dividing the space domain of the receiving array by using an exhaustion method to obtain an ultra-complete angle set
Figure FDA0004231760040000011
Wherein (1)>
Figure FDA0004231760040000012
Represents the nth overcomplete angle, H is the overcomplete angle set +.>
Figure FDA0004231760040000013
N=1, 2, …, H;
based on
Figure FDA0004231760040000014
And A (theta) obtains an expanded array flow pattern matrix:
Figure FDA0004231760040000015
wherein A (θ) _spark represents the expanded array flow pattern matrix;
obtaining an ultra-complete array output model according to the expanded array flow pattern matrix:
Figure FDA0004231760040000021
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure FDA0004231760040000022
representing the spread received signal, < >>
Figure FDA0004231760040000023
Representing the expanded incident signal;
the a priori acquisition of the expanded incident signal is as follows:
modeling the expanded incident signal by using a Markov probability prior model to obtain the relation between the nth snapshot and the (t-1) th snapshot of the nth expanded incident signal:
Figure FDA0004231760040000024
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure FDA0004231760040000025
representing the t-th speedN-th expanded incident signal of beat, < >>
Figure FDA0004231760040000026
An n-th expanded incident signal representing the t-1 th snapshot, +.>
Figure FDA0004231760040000027
Representation->
Figure FDA0004231760040000028
And->
Figure FDA0004231760040000029
The relation between beta and beta is the time correlation coefficient, beta epsilon (-1, 1), gamma n Is the a priori error of the nth spread incident signal, t=1, 2, …, T;
obtaining the influence of the previous snapshot according to the relation between the front snapshot and the rear snapshot of the expanded incident signal
Figure FDA00042317600400000210
And the influence of the latter snapshot +.>
Figure FDA00042317600400000211
Wherein (1)>
Figure FDA00042317600400000212
Representing the incident signal +.>
Figure FDA00042317600400000213
Is>
Figure FDA00042317600400000214
Represents the mean value of forward transmission of the t-1 snapshot in the nth expanded incident signal t snapshot,/->
Figure FDA00042317600400000215
Representing the nth expanded incident signalVariance of forward transmission of t-1 snapshot at t-th snapshot, +.>
Figure FDA00042317600400000216
Representing the incident signal +.>
Figure FDA00042317600400000217
Is used to determine the prior probability of (c) for a given channel,
Figure FDA00042317600400000218
represents the mean value of backward transmission of t+1 snapshot in the time of t snapshot of the n-th expanded incident signal,/and>
Figure FDA00042317600400000219
representing the variance of backward transmission of the t+1 snapshot in the snapshot of the nth expanded incident signal t;
according to
Figure FDA00042317600400000220
And->
Figure FDA00042317600400000221
Obtaining an expanded incident signal
Figure FDA00042317600400000222
Is a priori of:
Figure FDA0004231760040000031
2. a sparse bayesian-based low complexity signal direction of arrival estimation method according to claim 1, wherein the spatial domain of the receive array is [ -90 °,90 ° ].
3. A sparse bayesian-based low complexity signal direction of arrival estimation method according to claim 1, wherein the first stepN-th expanded incident signal of t snapshots
Figure FDA0004231760040000032
The expression of (2) is as follows:
Figure FDA0004231760040000033
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure FDA0004231760040000034
4. the sparse bayesian-based low-complexity signal direction of arrival estimation method according to claim 1, wherein the expression of the input function of the GAMP algorithm is as follows:
Figure FDA0004231760040000035
wherein g s Represents the input function, q (t) An approximation of the received signal representing the t-th snapshot without noise effects,
Figure FDA0004231760040000036
represents q (t) Noise variance of y (t) Received signal, sigma, representing the t-th snapshot receiving matrix 2 Representing y (t) Is a noise variance of (1);
the expression of the output function of the GAMP algorithm is as follows:
Figure FDA0004231760040000037
wherein g x The output function is represented as a function of the output,
Figure FDA0004231760040000038
representing the t-th speedApproximation of the n-th expanded incident signal of the beat,/and>
Figure FDA0004231760040000041
representation->
Figure FDA0004231760040000042
Is a noise variance of (a).
5. The sparse bayesian-based low-complexity signal direction of arrival estimation method of claim 4, wherein the specific operation of obtaining the recovered expanded incident signal is as follows:
initializing an expanded incident signal of a t-th snapshot
Figure FDA0004231760040000043
Input function and output function pairs using GAMP algorithm
Figure FDA0004231760040000044
Performing GAMP iteration, and updating the mean value and variance of the expanded incident signal;
updating iteration parameters by using an EM algorithm in each iteration process, and carrying out iteration convergence judgment by using iteration convergence conditions;
obtaining a recovered expanded incident signal according to the mean value and the variance of the expanded incident signal meeting the iterative convergence judgment;
wherein the iteration convergence condition comprises a GAMP iteration convergence condition and an EM iteration convergence condition.
6. The sparse bayesian-based low-complexity signal direction of arrival estimation method according to claim 5, wherein if the maximum iteration number of the GAMP algorithm is G, the iteration process of each iteration is specifically as follows:
according to the output of the ith-1 th iteration, obtaining the expanded incident signal of the t snapshot in the ith iteration
Figure FDA0004231760040000045
Variance of (2)
Figure FDA0004231760040000046
Mean value X i(t) Priori error gamma i And the noise variance (sigma) of the received signal 2 ) i Wherein i is [1, G ]]；
Let s= |a (θ) _spark| 2 By using
Figure FDA0004231760040000047
And X i(t) Calculating the approximation q of the received signal of the t snapshot without noise effect in the i-th iteration i(t) ：
Figure FDA0004231760040000048
Wherein g s (i-1)(t) An input function representing a t-snapshot of the i-1 th iteration;
using gamma i 、(σ 2 ) i And q i(t) Updating the input function g of the t snapshot of the ith iteration s i(t) ；
Using updated input function g s i(t) Respectively calculating approximate values r of the expanded incident signals of the t-th snapshot i(t) Noise variance of
Figure FDA0004231760040000051
Figure FDA0004231760040000052
Figure FDA0004231760040000053
Wherein A (θ) _sparse T Represents a transpose of a (θ) _space,
Figure FDA0004231760040000054
Figure FDA0004231760040000055
for inputting damping coefficient->
Figure FDA0004231760040000056
s (i-1)(t) Representing the value of the i-1 th iteration, S T The transpose of S is represented (g) s i(t) ) ' represents the input function g s i(t) Is a derivative of (2);
using gamma i 、r i(t) And
Figure FDA0004231760040000057
updating the output function g of the t snapshot of the ith iteration x i(t) ；
According to the updated output function g x i(t) Updating the variance and the mean value of the expanded incident signal:
Figure FDA0004231760040000058
Figure FDA0004231760040000059
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure FDA00042317600400000510
representing the variance of the expanded incident signal output from the ith iteration, (g) x i(t) ) ' represents the output function g x i(t) Derivative of X (i+1)(t) Representing the mean value of the expanded incident signal output from the ith iteration, < >>
Figure FDA00042317600400000511
In order to output the damping coefficient,
Figure FDA00042317600400000512
pair X using GAMP iteration convergence conditions (i+1)(t) And
Figure FDA00042317600400000513
performing GAMP iteration convergence judgment, ending iteration when GAMP iteration convergence conditions are met, outputting the mean value and the variance of the incident signal after the expansion of the ith iteration, and entering the next step when GAMP iteration convergence conditions are not met;
x output by ith iteration based on EM algorithm (i+1)(t) And
Figure FDA00042317600400000514
calculating a priori error gamma in the (i+1) th iteration i+1 And noise variance (sigma) 2 ) i+1 The calculation formula is as follows:
Figure FDA0004231760040000061
(σ 2 ) i+1 ＝||y (t) -A(θ)_sparse·X (i+1)(t) || 2
pair X using EM iteration convergence conditions (i+1)(t) And
Figure FDA0004231760040000062
and (3) carrying out EM iteration convergence judgment, outputting the mean value and the variance of the incident signal after the expansion of the (i+1) th iteration when the EM iteration convergence condition is met, ending the iteration, and continuing the iteration when the EM iteration convergence condition is not met.
7. The sparse bayesian-based low-complexity signal direction of arrival estimation method of claim 6, wherein the GAMP iterative convergence condition is:
Figure FDA0004231760040000063
or the iteration number i reaches the maximum iteration number G of the GAMP algorithm, wherein epsilon gamp Representing normalized tolerance parameters for the GAMP algorithm iterations.
8. The sparse bayesian-based low-complexity signal direction of arrival estimation method of claim 6, wherein said EM iteration convergence condition is:
Figure FDA0004231760040000064
or the iteration number i reaches the maximum iteration number K of the EM algorithm, wherein epsilon em Representing normalized tolerance parameters for EM algorithm iterations.
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