CN109490819B - Sparse Bayesian learning-based method for estimating direction of arrival of wave in a lattice - Google Patents

Sparse Bayesian learning-based method for estimating direction of arrival of wave in a lattice Download PDF

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CN109490819B
CN109490819B CN201811365309.5A CN201811365309A CN109490819B CN 109490819 B CN109490819 B CN 109490819B CN 201811365309 A CN201811365309 A CN 201811365309A CN 109490819 B CN109490819 B CN 109490819B
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CN109490819A (en
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吴晓欢
张泽云
朱卫平
颜俊
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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
    • G01S3/04—Details
    • G01S3/12—Means for determining sense of direction, e.g. by combining signals from directional antenna or goniometer search coil with those from non-directional antenna

Abstract

The invention discloses a sparse Bayesian learning-based method for estimating a direction of arrival of a lattice, which comprises the following steps: constructing a sparse array consisting of M array elements at a receiving end and establishing an array receiving signal model; establishing an ultra-complete dictionary based on array steering vectors according to a compressive sensing theory, and expanding an array received signal model into a sparse signal reconstruction model X; constructing a sparse signal reconstruction model Y corresponding to the virtual signal by the sparse signal reconstruction model X; initializing the value of a designated parameter, and determining the noise power of the airspace signal in the transmission process; calculating a virtual array output signal; checking and calculating the mean value and the variance of the probability density function after the virtual signal is output; calculating the power spectrum, the noise power and the quantization error of the virtual signal output signal by using Bayes learning iteration; setting a termination criterion; drawing a waveform of the power spectrum, searching a peak value on the power spectrum, and obtaining an estimation result of the estimated direction of arrival based on the peak value; the invention can estimate the number of signals more than the number of array elements and improves the estimation precision.

Description

Sparse Bayesian learning-based off-grid direction of arrival estimation method
Technical Field
The invention belongs to the technical field of signal processing, particularly relates to direction of arrival estimation of radar signals, acoustic signals and electromagnetic signals, and particularly relates to a sparse direction of arrival estimation method based on sparse Bayesian learning, which is used for passive positioning and target detection.
Background
DOA (Direction-of-Arrival) estimation is an important branch of the field of array signal processing, and refers to receiving spatial domain signals by using an array antenna, and processing the received signals by using a statistical signal processing technology and various optimization methods to recover incoming information of incident signals, and the DOA estimation has wide application in the fields of radar, sonar, voice, wireless communication and the like.
Uniform linear arrays are one of the most common array structures in existing DOA estimation methods because they satisfy the nyquist sampling theorem and enable efficient DOA estimation. However, the DOA estimation method using a uniform linear array has its degree of freedom limited by the number of actual antenna elements. The sparse array can break through the bottleneck that the degree of freedom of the traditional uniform linear array is limited, and the degree of freedom of DOA estimation number is increased on the premise that the number of antenna array elements is certain.
The direction-of-arrival estimation method based on compressed sensing is a typical method in the direction-finding field, and makes coherent signal direction-finding possible without a space smoothing method. This type of method can be applied not only to uniform arrays but also to sparse arrays, however, the property of aperture expansion that sparse arrays have is not effectively utilized. Some methods obtain more degrees of freedom by performing vectorization operation on a covariance matrix output by an array, but the methods all assume that an incident signal is an incoherent signal and the number of snapshots is not too small, so that application scenarios of the methods in direction finding are limited.
More importantly, the direction of arrival estimation method based on compressed sensing divides the grid in advance and assumes that the signal falls on the grid without error, which is contradictory to the fact that the incoming direction of the incident signal is uniform and continuous in the real environment. Therefore, a new signal model needs to be designed to accurately depict the physical signal model and accurately describe the existing quantization error, so as to achieve a more accurate estimation result.
Disclosure of Invention
The invention mainly aims to provide a sparse Bayesian learning-based off-grid DOA direction estimation method aiming at the problem of low accuracy of DOA estimation in the prior art, which utilizes the fact that a virtual array corresponding to a sparse array has more degrees of freedom, recovers signals received by the virtual array, utilizes a sparse Bayesian learning theory to realize joint solution of incident angles and quantization errors, and fully utilizes all information contained in the signals received by the sparse array, and the specific technical scheme is as follows:
a sparse Bayesian learning-based method for estimating a direction of arrival (DOA) of a discrete lattice is applied to receiving a space domain signal by an array antenna, wherein the space domain signal is a narrow-band incident signal, and the method comprises the following steps:
s1, constructing a sparse array consisting of M array elements at a receiving end and establishing an array receiving signal model: firstly, N array elements are utilized to construct an incident signal with narrow-band interval between adjacent array elementsA uniform linear array of half a wavelength; keeping the head and the tail of the uniform linear array unchanged, and removing N-M array elements in the middle of the uniform linear array to form the sparse array; and constructing an array receiving signal model X which is formed by K narrow-band incident signals as A Ω S+E Ω Wherein X ═ X (1),.., X (l)]For array received signals, L is the number of fast beats received by the array, S is the incident signal waveform, A Ω ＝[a Ω (θ 1 ),...,a Ω (θ K )]In the form of an array manifold matrix,
Figure BDA0001868362120000031
a steering vector corresponding to the angle of incidence of the narrowband incident signal,
Figure BDA0001868362120000032
Ω m represents the m-th element in the set omega, [ ·] T Denotes a transpose operation, E Ω The noise matrix is a noise matrix, and the noises received by different array elements are mutually independent;
s2, uniformly dividing the angle domain space to be observed to establish a grid set
Figure BDA0001868362120000033
Based on the grid set
Figure BDA0001868362120000034
And an expanded array manifold matrix corresponding to the grid set
Figure BDA0001868362120000035
Constructing a sparse signal reconstruction model of the array received signal model
Figure BDA0001868362120000036
Wherein the content of the first and second substances,
Figure BDA0001868362120000037
represents a virtual signal, an
Figure BDA0001868362120000038
Satisfy a Gaussian distribution with a mean of zero and a variance of Γ, Γ ═ diag (η) denotes that the vector of diagonal elements of Γ is η, and η denotes that the virtual signal is represented by
Figure BDA0001868362120000039
The power spectrum of (a) is,
Figure BDA00018683621200000310
is composed of
Figure BDA00018683621200000311
About
Figure BDA00018683621200000312
Is a diagonal matrix, δ represents the quantization error of the incident angle of the K signals compared to the nearest grid;
s3, setting a uniform virtual array composed of N array elements, and reconstructing a model based on the sparse signal
Figure BDA00018683621200000313
Constructing a sparse signal reconstruction model of the uniform virtual array received signal
Figure BDA00018683621200000314
Wherein, Y represents a virtual array reception signal,
Figure BDA00018683621200000315
an array manifold matrix representing a correspondence of the virtual array,
Figure BDA00018683621200000316
is composed of
Figure BDA00018683621200000317
About
Figure BDA00018683621200000318
E represents the noise matrix received by the virtual array;
s4, LiReconstructing the model for the sparse signal using a sparse Bayesian learning concept
Figure BDA00018683621200000319
Obtaining the sparse signal reconstruction model by adopting expectation maximization solution
Figure BDA00018683621200000320
The output signal Y of (2);
and S5, drawing the waveform of the power spectrum eta of the output signal Y, searching peak values on the power spectrum according to a one-dimensional spectrum peak searching method, arranging the peak values from large to small, taking the angle direction phi corresponding to the first K peak values as a preliminary estimation result of the direction of arrival, and taking theta (phi + delta) as a final estimation result of the direction of arrival.
Further, in step S1, the uniform linear array is located at Ω ═ { 1., N }, and the sparse array is located at Ω ═ Ω · Ω ·, N } 1 ,...,Ω M The incidence angle of K incidence narrow-band signals is theta ═ theta 1 ,...,θ K }。
In a further aspect of the present invention,
Figure BDA0001868362120000041
is the grid set
Figure BDA0001868362120000042
The steering vector of (1).
Further, the virtual signal
Figure BDA0001868362120000043
Is a row sparse matrix with sparsity K, and the virtual signal
Figure BDA0001868362120000044
Each column contains only K non-zero values; and K of the non-zero values and the set of grids
Figure BDA0001868362120000045
And the two are arranged in a one-to-one correspondence manner.
Further, step S4 includes:
s41, initializing the values of the designated parameters: order to
Figure BDA0001868362120000046
And determining the noise power of the space domain signal in the transmission process
Figure BDA0001868362120000047
S42, calculating a virtual array output signal: by the formula
Figure BDA0001868362120000048
Calculating to obtain the output signal Y, wherein
Figure BDA0001868362120000049
Wherein P is a selection matrix [ ·] H Representing a conjugate transpose operation, wherein only the omega-th element of the mth row of P is 1, and the rest elements are all 0;
s43, checking virtual signal
Figure BDA00018683621200000410
Mean and variance after output: by the formula
Figure BDA00018683621200000411
Calculating said mean value from the formula
Figure BDA00018683621200000412
Calculating the variance, wherein,
Figure BDA00018683621200000413
s44, calculating the iterative formula of the power spectrum eta by using the Bayes learning idea
Figure BDA00018683621200000414
And iterative formula of noise power sigma
Figure BDA00018683621200000415
S45、δ＝U -1 G calculates the quantization error, δ, where,
Figure BDA0001868362120000051
Figure BDA0001868362120000052
s46, setting a loop termination criterion
Figure BDA0001868362120000053
Wherein eta (i) The output power of the ith iteration is represented, whether the termination criterion is satisfied or not is judged, and if not, the step S41 is returned; if so, the iteration terminates and proceeds to step S5.
The invention relates to a sparse Bayesian learning-based off-grid direction of arrival estimation method, which comprises the steps of firstly constructing a sparse array and establishing an array signal model; then establishing a sparse signal reconstruction model
Figure BDA0001868362120000054
Carrying out iterative solution by using sparse Bayesian learning; in iteration, setting a termination criterion, and after the termination criterion is met, estimating the direction of arrival by using the recovered sparse signal; compared with the prior art, the invention has the following beneficial effects: the covariance matrix output by the vectorization array is avoided, the incident signal is not required to be assumed to be an incoherent signal, and the method can be suitable for relevant and coherent signal scenes; the high-degree-of-freedom property of the sparse array is fully utilized, and the number of signals more than the number of array elements can be estimated; the condition that the signal deviates from the grid is fully considered, the fitting error between the constructed model and the real physical scene is reduced, and the estimation accuracy of the direction of arrival is improved.
Drawings
FIG. 1 is a flowchart illustrating a method for estimating a direction of arrival of a outlier based on Bayesian learning according to an embodiment of the present invention;
FIG. 2 is a schematic structural diagram of a sparse array in an embodiment of the present invention;
fig. 3 is a graph illustrating performance comparison between the sparse bayesian learning-based direction of arrival estimation method and other existing methods according to an embodiment of the present invention.
Fig. 4 is a schematic diagram of a spatial power spectrum effect of the sparse bayesian learning-based direction of arrival estimation method in the embodiment of the present invention.
Detailed Description
In order to make those skilled in the art better understand the technical solution of the present invention, the technical solution in the embodiment of the present invention will be clearly and completely described below with reference to the drawings in the embodiment of the present invention.
Referring to fig. 1 and fig. 2, in an embodiment of the present invention, a sparse bayesian learning-based direction of arrival estimation method for receiving a spatial domain signal by an array antenna is provided, where the spatial domain signal is a narrow-band incident signal, and specifically, the method includes the steps of:
s1, constructing a sparse array consisting of M array elements at a receiving end and establishing an array receiving signal model;
firstly, constructing a uniform linear array with the distance between adjacent array elements being half of the wavelength of a narrow-band incident signal by using N array elements, wherein the position of the uniform linear array is omega { 1.., N }; then, keeping the head and the tail of the uniform linear array unchanged, removing N-M array elements in the middle of the uniform linear array to form a sparse array, wherein the position of the sparse array is omega { omega ═ omega 1 ,...,Ω M }; finally, a set of K incidence angles theta ═ theta is constructed 1 ,...,θ K An array receiving signal model X formed by narrow-band incident signals is A Ω S+E Ω Wherein X ═ X (1),.., X (l)]For array received signals, L is the number of fast beats received by the array, S is the incident signal waveform, A Ω ＝[a Ω (θ 1 ),...,a Ω (θ K )]In the form of an array manifold matrix,
Figure BDA0001868362120000061
a steering vector corresponding to the angle of incidence of the narrowband incident signal,
Figure BDA0001868362120000062
Ω m represents the m-th element in the set omega, [ ·] T Denotes a transpose operation, E Ω The noise matrix is a noise matrix, and the noises received by different array elements are independent.
S2, establishing an array-steering-vector-based overcomplete dictionary according to a compressive sensing theory, and expanding an array receiving signal model into a sparse signal reconstruction model; specifically, the angle domain space with observation is uniformly divided, and the established grid set
Figure BDA0001868362120000071
And an expanded array manifold matrix corresponding to the grid set
Figure BDA0001868362120000072
Constructing a sparse signal reconstruction model of the array receiving signal model on the basis
Figure BDA0001868362120000073
Model reconstruction in sparse signals
Figure BDA0001868362120000074
In (1),
Figure BDA0001868362120000075
representing a virtual signal in which, among others,
Figure BDA0001868362120000076
it can be seen that the lines of S are extended according to a spatial grid and the virtual signal
Figure BDA0001868362120000077
Each column contains only K non-zero values, and the K non-zero values are combined with the grid set
Figure BDA0001868362120000078
The components are arranged in a one-to-one correspondence manner; due to the fact that
Figure BDA0001868362120000079
Thus, the device
Figure BDA00018683621200000710
Is a matrix with sparse rows, the sparsity is K,
Figure BDA00018683621200000711
a gaussian distribution having a mean value of zero and a variance of Γ, where Γ ═ diag (η) denotes η that is a vector composed of diagonal elements of Γ, and η denotes a virtual signal
Figure BDA00018683621200000712
The power spectrum of (a) is,
Figure BDA00018683621200000713
is composed of
Figure BDA00018683621200000714
About
Figure BDA00018683621200000715
Is a diagonal matrix, δ represents the quantization error of the incident angle of the K signals compared to the nearest grid.
S3, setting a uniform virtual array composed of N array elements, and reconstructing the model based on the sparse signal
Figure BDA00018683621200000716
Constructing a sparse signal reconstruction model of a uniform virtual array receiving signal, wherein the assumption is that
Figure BDA00018683621200000717
Wherein, Y represents a virtual array reception signal,
Figure BDA00018683621200000718
an array manifold matrix representing the correspondence of the virtual array,
Figure BDA00018683621200000719
is composed of
Figure BDA00018683621200000720
About
Figure BDA00018683621200000721
E represents the noise matrix received by the virtual array;
s4, reconstructing the model for the sparse signal by using the sparse Bayesian learning idea
Figure BDA00018683621200000722
Obtaining a sparse signal reconstruction model by adopting expectation maximization solution
Figure BDA00018683621200000723
The output signal Y of (1);
firstly, initializing a quantization error delta and a power spectrum eta of specified parameters, making delta equal to 0,
Figure BDA0001868362120000081
and determining the noise power of the space domain signal in the transmission process
Figure BDA0001868362120000082
And reconstructing the model based on the sparse signal obtained in step S3
Figure BDA0001868362120000083
Calculation formula for obtaining output signal of virtual array
Figure BDA0001868362120000084
Calculating to obtain the output signal Y of the virtual array, wherein
Figure BDA0001868362120000085
Then, by the formula
Figure BDA0001868362120000086
Computing and validating virtual signals
Figure BDA0001868362120000087
The mean value after output is calculated by formula
Figure BDA0001868362120000088
Computing and validating virtual signals
Figure BDA0001868362120000089
The output variance of the measured data is calculated by the following formula,
Figure BDA00018683621200000810
then, by iterative formula
Figure BDA00018683621200000811
Calculating the value of the power spectrum eta, likewise, by an iterative formula
Figure BDA00018683621200000812
Calculating the numerical value of the noise power sigma; at the same time, by the formula δ ═ U -1 G calculates the quantization error, delta, where,
Figure BDA00018683621200000813
finally, a cycle termination criterion is set
Figure BDA00018683621200000814
Wherein eta (i) Indicating the output power of the ith iteration according to a set cycle termination criterion
Figure BDA00018683621200000815
Judging whether the process using Bayesian learning thought is terminated or not, and if so, judging whether the process using Bayesian learning thought is terminated or not
Figure BDA00018683621200000816
The iteration is successful and the process goes to step S5, otherwise, the process goes back to step S41 to re-execute the iteration operation until the iteration is successful.
S5, drawing the termination criterion satisfied by the Bayesian learning idea through the virtual array after iteration
Figure BDA00018683621200000817
The waveform of the power spectrum eta of the output signals Y in the last series is searched by adopting a one-dimensional spectral peak searching methodAnd arranging the peak values on the waveform of the power spectrum eta from large to small, taking the angle direction phi corresponding to the first K peak values as a preliminary estimation result of the direction of arrival, and finally calculating according to a formula theta to phi + delta to obtain a final estimation result of the direction of arrival theta.
Example two
The effect of the present invention will be further described with reference to the simulation example.
Simulation example 1: specifically, the incident signal is received by a 4-element sparse array with the element number Ω {1,2,5, 7 }. Assume that the number of incident narrow-band coherent signals is 2, and the incident direction is θ [ -5 °,5 ° ]](ii) a The signal-to-noise ratio is set to 5 dB; the termination criterion parameter oa is set to 10 -4 . The relation between the root mean square error and the fast beat number of the sparse Bayesian learning-based direction-of-arrival estimation method is shown in FIG. 3, and it can be seen that the estimation error of the method is the minimum.
Simulation example 2: specifically, the incident signal is received by using a 4-element sparse array with an element number of Ω ═ {1,2,5, 7 }. Assuming that the number of incident narrow-band coherent signals is 4 and the incident direction is theta [ -32 °, -10 °,5 °,25 ° ]](ii) a The receiving fast beat number is 100; the signal-to-noise ratio is set to 10 dB; the termination criterion parameter oa is set to 10 -4 . The relation between the root mean square error and the fast beat number of the sparse Bayesian learning-based direction of arrival estimation method is shown in FIG. 4, and it can be known that the method provided by the invention can realize the increase of the degree of freedom in the coherent signal scene.
The invention relates to a sparse Bayesian learning-based off-grid direction of arrival estimation method, which comprises the steps of firstly constructing a sparse array and establishing an array signal model; then, establishing a discrete sparse reconstruction model and carrying out iterative solution by using sparse Bayesian learning; in iteration, setting a termination criterion, and after the termination criterion is met, estimating the direction of arrival by using the recovered sparse signal; compared with the prior art, the invention has the beneficial effects that: the covariance matrix output by the vectorization array is avoided, the incident signal is not required to be assumed to be an incoherent signal, and the method can be suitable for relevant and coherent signal scenes; the high-degree-of-freedom property of the sparse array is fully utilized, and the number of signals more than the number of array elements can be estimated; the condition that the signal deviates from the grid is fully considered, the fitting error between the constructed model and the real physical scene is reduced, and the estimation accuracy of the direction of arrival is improved.
Although the present invention has been described in detail with reference to the foregoing embodiments, it will be apparent to those skilled in the art that modifications may be made to the embodiments described in the foregoing detailed description, or equivalent changes may be made in some of the features of the embodiments described above. All equivalent structures made by using the contents of the specification and the attached drawings of the invention can be directly or indirectly applied to other related technical fields, and all the equivalent structures are within the protection scope of the invention.

Claims (5)

1. A sparse Bayesian learning-based off-grid direction-of-arrival estimation method is applied to receiving space domain signals by an array antenna, wherein the space domain signals are narrow-band incident signals, and is characterized by comprising the following steps:
s1, constructing a sparse array consisting of M array elements at a receiving end and establishing an array receiving signal model: firstly, N array elements are utilized to construct a uniform linear array with the adjacent array element spacing being half of the wavelength of a narrow-band incident signal; keeping the head and the tail of the uniform linear array unchanged, and removing N-M array elements in the middle of the uniform linear array to form the sparse array; and constructing an array receiving signal model X (A) consisting of K narrow-band incident signals Ω S+E Ω Wherein X ═ X (1),.., X (l)]For array received signals, L is the number of fast beats received by the array, S is the incident signal waveform, A Ω ＝[a Ω (θ 1 )，...，a Ω (θ K )]Is a matrix of the array manifold,
Figure FDA0003691310400000011
a steering vector corresponding to the angle of incidence of the narrowband incident signal,
Figure FDA0003691310400000012
Ω M represents the Mth element in the set Ω, [ ·] T Denotes a transpose operation, E Ω The noise matrix is a noise matrix, and the noises received by different array elements are mutually independent;
s2, uniformly dividing the angle domain space to be observed to establish a grid set
Figure FDA0003691310400000013
Based on the grid set
Figure FDA0003691310400000014
And an expanded array manifold matrix corresponding to the grid set
Figure FDA0003691310400000015
Constructing a sparse signal reconstruction model of the array received signal model
Figure FDA0003691310400000016
Wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003691310400000017
represents a virtual signal, an
Figure FDA0003691310400000018
Satisfy a Gaussian distribution with a mean of zero and a variance of Γ, Γ ═ diag (η) denotes that the vector of diagonal elements of Γ is η, and η denotes that the virtual signal is represented by
Figure FDA0003691310400000019
The power spectrum of (a) is,
Figure FDA00036913104000000110
is composed of
Figure FDA00036913104000000111
About
Figure FDA00036913104000000112
Is a diagonal matrix, δ represents the quantization error of the incident angle of the K signals compared to the nearest grid;
s3, setting a uniform virtual array composed of N array elements, and reconstructing a model based on the sparse signal
Figure FDA0003691310400000021
Constructing a sparse signal reconstruction model of the uniform virtual array received signal
Figure FDA0003691310400000022
Wherein, Y represents a virtual array reception signal,
Figure FDA0003691310400000023
an array manifold matrix representing a correspondence of the virtual array,
Figure FDA0003691310400000024
is composed of
Figure FDA0003691310400000025
About
Figure FDA0003691310400000026
E represents the noise matrix received by the virtual array;
s4, reconstructing the model for the sparse signal by using the sparse Bayesian learning idea
Figure FDA0003691310400000027
Obtaining the sparse signal reconstruction model by adopting expectation maximization solution
Figure FDA0003691310400000028
The output signal Y of (1);
and S5, drawing the waveform of the power spectrum eta of the output signal Y, searching peak values on the power spectrum according to a one-dimensional spectrum peak searching method, arranging the peak values from large to small, taking the angle direction phi corresponding to the first K peak values as a preliminary estimation result of the direction of arrival, and taking theta (phi + delta) as a final estimation result of the direction of arrival.
2. The sparse bayesian learning-based direction-of-arrival estimation method according to claim 1, wherein in step S1, the position of the uniform linear array is Ω ═ { 1., N }, and the position of the sparse array is Ω ═ { Ω ·, N } 1 ,...,Ω M K narrow band incident signals are incident at an angle θ ═ θ 1 ,...,θ K }。
3. The sparse Bayesian learning based direction of arrival dispersion method according to claim 1,
Figure FDA0003691310400000029
is set for the grid
Figure FDA00036913104000000210
The steering vector of (1).
4. The sparse Bayesian learning-based direction of arrival estimation method of the de-lattice, as recited in claim 1, wherein the virtual signal is derived from a set of signals
Figure FDA00036913104000000211
Is a row sparse matrix with sparsity K, and the virtual signal
Figure FDA00036913104000000212
Each column contains only K non-zero values; and K of the non-zero values and the set of grids
Figure FDA0003691310400000031
And the two are arranged in a one-to-one correspondence manner.
5. The sparse bayesian learning based direction of arrival estimation method according to claim 1, wherein step S4 comprises:
s41, initializing the values of the designated parameters: let δ be equal to 0 (0),
Figure FDA0003691310400000032
and determining the noise power of the space domain signal in the transmission process
Figure FDA0003691310400000033
S42, calculating a virtual array output signal: by the formula
Figure FDA0003691310400000034
Calculating to obtain the output signal Y, wherein
Figure FDA0003691310400000035
Wherein P is a selection matrix [ ·] H Representing a conjugate transpose operation, wherein only the omega-th element of the mth row of P is 1, and the rest elements are all 0;
s43, checking virtual signal
Figure FDA0003691310400000036
Mean and variance after output: by the formula
Figure FDA0003691310400000037
Calculating said mean value from the formula
Figure FDA0003691310400000038
Calculating the variance, wherein,
Figure FDA0003691310400000039
s44, calculating the iterative formula of the power spectrum eta by using the Bayes learning idea
Figure FDA00036913104000000310
And iterative formula of noise power sigma
Figure FDA00036913104000000311
S45、δ＝U -1 G calculates the quantization error, delta, where,
Figure FDA00036913104000000312
Figure FDA00036913104000000313
s46, setting a loop termination criterion
Figure FDA00036913104000000314
Wherein eta (i) The output power of the ith iteration is represented, whether the termination criterion is satisfied or not is judged, and if not, the step S41 is returned; if true, the iteration terminates and proceeds to step S5.
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