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CN108594165B - A Method for Estimating Direction of Arrival for Narrowband Signals Based on Expectation-Maximization Algorithm - Google Patents

A Method for Estimating Direction of Arrival for Narrowband Signals Based on Expectation-Maximization Algorithm Download PDF

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CN108594165B
CN108594165B CN201810347711.4A CN201810347711A CN108594165B CN 108594165 B CN108594165 B CN 108594165B CN 201810347711 A CN201810347711 A CN 201810347711A CN 108594165 B CN108594165 B CN 108594165B
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段慧芳
赵宣植
刘增力
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Kunming University of Science and Technology
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Abstract

本发明涉及一种波达方向估计方法，特别是一种基于期望最大化算法的窄带信号波达方向估计方法，属于信号处理技术领域。本发明提出将期望最大化算法运用于的窄带信号的波达方向估计中，引入隐含变量进行更新迭代，并利用隐含变量构造关于待估波达方向的极大似然函数从而求解出信号的波达方向角的算法。本发明与现有技术相比，主要解决了MUSIC等子空间类算法在低信噪比时估计性能下降；最大似然估计方法多维搜索容易出现误差、估计分辨率低等问题。本方法利用期望最大化迭代分别计算，尽可能地避免了误差，保证了在信噪比低和快拍数少的情况下，波达方向估计的优良性能，且其具有很好的稳定性和分辨率。

Figure 201810347711

The invention relates to a direction of arrival estimation method, in particular to a narrowband signal direction of arrival estimation method based on an expectation maximization algorithm, and belongs to the technical field of signal processing. The present invention proposes to apply the expectation maximization algorithm to the estimation of the direction of arrival of the narrow-band signal, introduce hidden variables to update and iterate, and use the hidden variables to construct a maximum likelihood function about the direction of arrival to be estimated to solve the signal algorithm for the direction of arrival angle. Compared with the prior art, the invention mainly solves the problem that the estimation performance of the subspace algorithms such as MUSIC is reduced when the signal-to-noise ratio is low; the multi-dimensional search of the maximum likelihood estimation method is prone to errors and the estimation resolution is low. This method uses expectation-maximization iteration to calculate separately, avoids errors as much as possible, ensures the excellent performance of DOA estimation under the condition of low signal-to-noise ratio and small number of snapshots, and has good stability and resolution.

Figure 201810347711

Description

Narrow-band signal direction-of-arrival estimation method based on expectation maximization algorithm
Technical Field
The invention relates to a direction of arrival estimation method, in particular to a narrowband signal direction of arrival estimation method based on an expectation maximization algorithm, and belongs to the technical field of signal processing.
Background
The direction of arrival estimation based on the antenna array is an important research direction in array signal processing, the main purpose of the method is to estimate and extract parameters such as the direction of incoming waves, the number of signals and the like of a target signal of a space to be detected by utilizing measurement data received by the sensor array in a certain arrangement mode in the space, and the method has wide application prospects in military and civil fields such as radar, passive sonar, biomedicine, radio astronomy and seismic exploration.
A subspace classification algorithm such as a multiple signal classification algorithm (MUSIC) is a space spectrum estimation method utilizing orthogonality of a signal subspace and a noise subspace, the estimation performance of the method is influenced by characteristic value decomposition, and the estimation performance is estimated under the conditions of low signal-to-noise ratio and few fast beatsIs unstable; the maximum likelihood algorithm (ML) is an important high-resolution spatial spectrum estimation method, has excellent estimation performance and has the advantages of good robustness and stability. However, the direction estimation likelihood function of the ML algorithm is non-linear and is performed at theta1，θ2，…θPThe resolution of the multi-dimensional nonlinear maximum is reduced due to the easy generation of errors in the global search.
Disclosure of Invention
The invention provides a narrow-band signal direction-of-arrival estimation method based on an expectation maximization algorithm, which mainly solves the problem that the estimation performance of subspace algorithms such as MUSIC (multiple signal to noise ratio) is reduced when the signal to noise ratio is low; the multi-dimensional search of the maximum likelihood estimation method is easy to have the problems of error, low estimation resolution and the like.
The technical scheme of the invention is as follows: a narrow-band signal direction-of-arrival estimation method based on an expectation-maximization algorithm comprises the following specific steps:
1) supposing that the receiving antenna array consists of M array element uniform linear arrays, the spacing of the array elements is d, P independent target signal sources from a space far-field narrow band are incident on the equal-spacing uniform linear arrays, and the incident direction of the signals is theta1,θ2,…θPWherein M is more than or equal to P,
Figure GDA0003117496540000011
λ is the wavelength of the incident signal;
2) sampling the space signal by an array antenna receiver to obtain a receiving signal Y (t);
3) constructing hidden variables
Figure GDA0003117496540000012
Which represents the output of the ith signal produced on the array, the output of the P signals on the array can be represented as
Figure GDA0003117496540000021
4) Construction of an implicit variable yi(t) a log-likelihood function for a parameter θ to be estimated;
5) deducing a narrow-band signal direction of arrival estimation objective function by utilizing an expectation maximization algorithm;
6) searching the target function respectively to obtain the direction of arrival of the information source;
further, an implicit variable y is constructedi(t) a log-likelihood function for a parameter θ to be estimated, said construction method being as follows:
the output produced on the array by the ith signal is:
Figure GDA0003117496540000022
in the formula, am(θi)＝exp[(m-1)dsinθi]Representing the phase delay, theta, of the ith source at the mth element relative to the reference elementiIndicating the incident direction of the ith signal, si(t) denotes the envelope of the i-th signal, ni(t) represents the noise of the ith signal on the array, and the noise is assumed to be white gaussian noise with zero mean;
Figure GDA0003117496540000023
Figure GDA0003117496540000024
in the formula, Θ represents signal parameters including parameters such as amplitude and direction of arrival of a signal; p (y)i(t) | Θ) represents the probability density function of the array output under the condition of the signal parameter;
construction of an implicit variable yi(t) log-likelihood function for the parameter θ to be estimated:
∵InL(Θ|yi(t))＝Inp(yi(t)|Θ)
Figure GDA0003117496540000025
Figure GDA0003117496540000026
Figure GDA0003117496540000027
wherein In represents logarithmic operation with e as the base, oc represents proportional proportion, and | | represents Frobenius norm of matrix;
further, the direction of arrival of each source is calculated by using an EM algorithm, and the calculation steps are as follows:
E-step:
Q(Θk+1|Θk)＝E[InL(Θk+1|yi)|X，Θk]
Figure GDA0003117496540000031
wherein, E [. C]Expressing the mathematical expectation, and the superscript k expressing the estimated values of the variables, Q (theta), obtained during the k-th iterationk+1|Θk) The condition expectation that is indicative of the signal parameter,
Figure GDA0003117496540000032
representing the estimated value.
M-step:
Obtaining the output generated by the signal i in k steps of iteration by using E-step solution
Figure GDA0003117496540000033
Combining the log-likelihood functions of the signal parameters to obtain an iterative formula of signal envelope and direction of arrival:
envelope of the signal:
Figure GDA0003117496540000034
direction of arrival:
Figure GDA0003117496540000035
where H represents the conjugate transpose of the array.
The invention has the beneficial effects that: the invention introduces the expectation maximization algorithm into the estimation of the direction of arrival of the narrow-band signal, avoids errors as much as possible, ensures the excellent performance of the estimation of the direction of arrival under the conditions of low signal-to-noise ratio and few fast beats, and has good stability and resolution.
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FIG. 1 is a general flow chart of the present invention;
FIG. 2 is a comparison graph of mean square error under different SNR conditions of the present invention and the maximum likelihood estimation method and the MUSIC method.
Detailed Description
The invention is further described with reference to the following figures and specific examples.
Example 1: as shown in fig. 1, a method for estimating the direction of arrival of a narrowband signal based on expectation maximization specifically includes the following steps:
step 1: and forming an equidistant and uniform linear array by using the antenna receiver.
1 antenna receiver is placed at each interval distance d, M antenna receivers are placed together to form a uniform linear array, each antenna receiver becomes an array element, P far-field narrow-band signals are supposed to be incident on the uniform linear array, zero-mean Gaussian white noise is added in the signals in the transmission process, wherein M is more than or equal to P,
Figure GDA0003117496540000041
λ is the wavelength of the incident signal.
Step 2: the spatial signal is sampled to obtain a received signal y (t).
And step 3: constructing hidden variables
Figure GDA0003117496540000042
Which represents the output of the ith signal produced on the array, the output of the P signals on the array can be represented as
Figure GDA0003117496540000043
And 4, step 4: implicit variable y in the constructioni(t) a log-likelihood function with respect to the parameter θ to be estimated.
1) The output produced on the array by the ith signal is:
Figure GDA0003117496540000044
in the formula, am(θi)＝exp[(m-1)d sinθi]Indicating the phase delay, theta, of the ith signal at the mth array element relative to the reference array elementiIndicating the incident direction of the ith signal, si(t) denotes the envelope of the i-th signal, ni(t) represents the noise of the ith signal on the array, and the noise is assumed to be white gaussian noise with zero mean;
2) the output of the P sources on the array can be represented as
Figure GDA0003117496540000045
3)
Figure GDA0003117496540000046
Figure GDA0003117496540000047
In the formula, Θ represents a signal parameter and includes parameters p (y) such as the amplitude and the direction of arrival of a signali(t) | Θ) represents the probability density function of the array output under the condition of the signal parameter;
4) construction of an implicit variable yi(t) log-likelihood function for the parameter θ to be estimated:
∵InL(Θ|yi(t))＝Inp(yi(t)|Θ)
Figure GDA0003117496540000048
Figure GDA0003117496540000049
Figure GDA00031174965400000410
wherein In represents logarithmic operation with e as the base, oc represents proportional proportion, and | | represents Frobenius norm of matrix;
and 5: and (3) deriving a narrow-band signal direction-of-arrival estimation objective function by using an expectation-maximization algorithm.
1)E-step:
Q(Θk+1|Θk)＝E[InL(Θk+1|yi)|X，Θk]
Figure GDA0003117496540000051
Wherein, E [. C]Expressing the mathematical expectation, and the superscript k expressing the estimated values of the variables, Q (theta), obtained during the k-th iterationk+1|Θk) The condition expectation that is indicative of the signal parameter,
Figure GDA0003117496540000052
representing the estimated value.
2)M-step:
Obtaining the output generated by the signal i in k steps of iteration by using E-step solution
Figure GDA0003117496540000053
Combining the log-likelihood functions of the signal parameters to obtain an iterative formula of signal envelope and direction of arrival:
envelope of the signal:
Figure GDA0003117496540000054
direction of arrival:
Figure GDA0003117496540000055
where H represents the conjugate transpose of the array.
Step 6: and searching the target function respectively to obtain the direction of arrival of the information source.
The method utilizes expectation-maximization algorithm (EM algorithm) to simplify calculation and construct hidden variable yi(t) and constructing an implicit variable yi(t) a log-likelihood function InL of a parameter theta to be estimated, wherein the InL is a nonlinear function of the parameter theta, and the optimal solution is solved only by one-dimensional search of the theta. The algorithm effectively avoids errors generated by multi-dimensional search of the ML algorithm on the space spectrum of the received signal, and simultaneously can ensure the performance and stability of estimation of the direction of arrival at low signal-to-noise ratio and few fast beats.
Example 2: the calculation was performed according to the method in example 1, wherein a uniform linear array consisting of 8 omnidirectional array elements was considered, the array element spacing was 0.5, the number of sampling points was 100, and the spatial angle search range was [ -90 ° 90 ° ].
The mean square error calculation equation is:
Figure GDA0003117496540000056
wherein I represents the number of Monte Carlo experiments,
Figure GDA0003117496540000057
denotes the angle of arrival, θ, of the ith testpRepresenting the true direction of arrival angle of the signal.
Assuming that three independent information sources respectively enter an equidistant uniform linear array consisting of 8 omnidirectional array elements at an angle of-30 degrees to 60 degrees, the signal-to-noise ratio of the experiment is increased from-10 dB to 10dB, 50 independent direction-of-arrival estimation tests are carried out under each signal-to-noise ratio, the method disclosed by the invention is respectively compared with the direction-of-arrival estimation values obtained by the existing maximum likelihood estimation method and the MUSIC estimation method, the mean square errors of the three methods under different signal-to-noise ratios are respectively calculated, and the test result is shown in figure 2, wherein: in fig. 2, the abscissa represents the signal-to-noise ratio and the ordinate represents the mean square error.
It can be seen from fig. 2 that the estimation errors of the three methods are all reduced along with the increase of the signal-to-noise ratio, but under the conditions of low signal-to-noise ratio and few snapshot numbers, the mean square error of the method is obviously smaller than that of the other two algorithms, thereby illustrating the effectiveness of the method.
In conclusion, the method has excellent estimation performance, good stability and resolution ratio under the conditions of low signal-to-noise ratio and few fast beats.
While the present invention has been described in detail with reference to the embodiments shown in the drawings, the present invention is not limited to the embodiments, and various changes can be made without departing from the spirit of the present invention within the knowledge of those skilled in the art.

Claims (2)

1.一种基于期望最大化算法的窄带信号波达方向估计方法，其特征在于：具体步骤如下：1. a narrowband signal direction of arrival estimation method based on expectation maximization algorithm, is characterized in that: concrete steps are as follows: (1)利用天线接收机形成等距均匀线阵(1) Using the antenna receiver to form an equidistant uniform linear array 假设接收天线阵由M个阵元的均匀线阵组成，阵元间距为d，P个来自空间远场窄带的独立目标信号入射到等距均匀线阵上，信号的入射方向为θ1,θ2,…θP，其中，M≥P，
Figure FDA0003233867410000011
λ为入射信号的波长；Assuming that the receiving antenna array is composed of a uniform linear array of M array elements, the distance between the array elements is d, and P independent target signals from the space far-field narrowband are incident on the equidistant uniform linear array, and the incident directions of the signals are θ 1 , θ 2 ,…θ P , where M≥P,
Figure FDA0003233867410000011
λ is the wavelength of the incident signal; (2)由阵列天线接收机对空间信号采样得到接收信号Y(t)；(2) The space signal is sampled by the array antenna receiver to obtain the received signal Y(t); (3)构造隐含变量yi(t)，其表示第i个信号在阵列上产生的输出，则P个信号在阵列上的输出可表示为
Figure FDA0003233867410000012
(3) Construct the implicit variable y i (t), which represents the output generated by the i-th signal on the array, then the output of the P signals on the array can be expressed as
Figure FDA0003233867410000012
 (4)构造隐含变量yi(t)关于待估参数θ的对数似然函数；(4) Construct the log-likelihood function of the latent variable y i (t) about the parameter to be estimated θ; 第i个信号在阵列上产生的输出为：
Figure FDA0003233867410000013
式中，am(θi)＝exp[(m-1)dsinθi]表示第i个信号在第m个阵元处相对于参考阵元的相位延迟，θi表示第i个信号的入射方向，si(t)表示第i个信号的包络，ni(t)表示第i个信号在阵列上的噪声，且假设噪声为零均值的高斯白噪声；The output produced by the ith signal on the array is:
Figure FDA0003233867410000013
In the formula, a m (θ i )=exp[(m-1)dsinθ i ] represents the phase delay of the i-th signal relative to the reference array element at the m-th array element, and θ i represents the incidence of the i-th signal direction, s i (t) represents the envelope of the ith signal, ni (t) represents the noise of the ith signal on the array, and assumes that the noise is Gaussian white noise with zero mean;
Figure FDA0003233867410000014
Figure FDA0003233867410000014
Figure FDA0003233867410000015
Figure FDA0003233867410000015
 式中，Θ表示信号参数，包含信号的振幅、波达方向，σi表示第i个信号在阵列上产生的噪声方差，p(yi(t)|Θ)表示在信号参数的条件下阵列输出的概率密度函数；In the formula, Θ represents the signal parameters, including the amplitude and direction of arrival of the signal, σ i represents the noise variance generated by the ith signal on the array, and p(y i (t)|Θ) represents the array under the condition of signal parameters. The output probability density function; 构造隐含变量yi(t)关于待估参数θ的对数似然函数：Construct the log-likelihood function of the latent variable y i (t) with respect to the parameter θ to be estimated: InL(Θ|yi(t))＝Inp(yi(t)|Θ)InL(Θ|y i (t))=Inp(y i (t)|Θ)
Figure FDA0003233867410000021
Figure FDA0003233867410000021
Figure FDA0003233867410000022
Figure FDA0003233867410000022
 其中，In表示以e为底的对数运算，∝表示成正比例，||||表示矩阵的Frobenius范数；Among them, In represents the logarithmic operation with the base e, ∝ represents the proportionality, and |||| represents the Frobenius norm of the matrix; (5)利用期望最大化算法推导窄带信号波达方向估计目标函数；(5) Deriving the objective function of narrowband signal DOA estimation by using expectation maximization algorithm; (6)分别对目标函数进行搜索，得到该信号的波达方向。(6) Search the objective function respectively to obtain the direction of arrival of the signal. 2.根据权利要求1所述的基于期望最大化算法的窄带信号波达方向估计方法，其特征在于：所述步骤(5)的具体过程如下：2. the method for estimating direction of arrival of narrowband signals based on expectation maximization algorithm according to claim 1, is characterized in that: the concrete process of described step (5) is as follows: E-stepE-step Q(Θk+1|Θk)＝E[InL(Θk+1|yi(t))|X，Θk]Q(Θ k+1 |Θ k )=E[InL(Θ k+1 |y i (t))|X,Θ k ]
Figure FDA0003233867410000023
Figure FDA0003233867410000023
 其中，E[·]表示求数学期望，上标k表示第k步迭代过程中所得到的各变量的估计值，Q(Θk+1|Θk)表示信号参数的条件期望，
Figure FDA0003233867410000024
表示估计值；Among them, E[ ] represents the mathematical expectation, the superscript k represents the estimated value of each variable obtained in the k-th iteration process, Q(Θ k+1 |Θ k ) represents the conditional expectation of the signal parameter,
Figure FDA0003233867410000024
represents an estimated value; M-stepM-step 利用E-step求解得到k步迭代时信号i产生的输出
Figure FDA0003233867410000025
联合信号参数的对数似然函数即可得到信号包络和波达方向的迭代式：Use E-step to solve to get the output generated by signal i during k-step iteration
Figure FDA0003233867410000025
The log-likelihood function of the joint signal parameters gives the iterative formula for the signal envelope and direction of arrival: 信号包络：Signal envelope:
Figure FDA0003233867410000026
Figure FDA0003233867410000026
 波达方向：Direction of arrival:
Figure FDA0003233867410000027
Figure FDA0003233867410000027
 其中，H表示阵列的共轭转置。where H represents the conjugate transpose of the array.
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