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The estimation of direction-of-arrival (DOA) of signals is a basic and important problem in sensor array signal processing. To solve this problem, many algorithms have been proposed, among which the Stochastic Maximum Likelihood (SML) is one of the most concerned algorithms because of its high accuracy of DOA. However, the estimation of SML generally involves the multidimensional nonlinear optimization problem. As a result, its computational complexity is rather high. This paper addresses the issue of reducing computational complexity of SML estimation of DOA based on the Alternating Minimization (AM) algorithm. We have the following two contributions. First using transformation of matrix and properties of spatial projection, we propose an efficient AM (EAM) algorithm by dividing the SML criterion into two components. One depends on a single variable parameter while the other does not. Second when the array is a uniform linear array, we get the irreducible form of the EAM criterion (IAM) using polynomial forms. Simulation results show that both EAM and IAM can reduce the computational complexity of SML estimation greatly, while IAM is the best. Another advantage of IAM is that this algorithm can avoid the numerical instability problem which may happen in AM and EAM algorithms when more than one parameter converges to an identical value.

The localization of multiple signal sources by a passive sensor array is of great importance in a wide variety of fields, such as radar, geophysics, radio-astronomy, biomedical engineering, communications, and underwater acoustics. The basic problem in this context is to estimate direction-of-arrival (DOA) of narrow-band signal sources located in the far field of the array.

A number of super-resolution techniques have been introduced, such as the Maximum Likelihood (ML) method [

Among these techniques, the ML technique is well known for its high accuracy of DOA. There are two famous ML criterions, that is, Deterministic or Conditional Maximum Likelihood (DML) [

To solve the multidimensional nonlinear optimization problem, the Alternating Minimization (AM) algorithm is one of the most classic techniques. It is an iterative technique and usually needs one-dimensional global search in the updating process. There are also some other efficient techniques, such as Alternating Projection (AP) [

Therefore, this paper addresses the issue of reducing computational complexity of SML estimation of DOA based on the conventional AM algorithm.

Firstly we show a brief description of SML estimation of DOA and the conventional solving method, AM algorithm. Then using transformation of matrix and properties of spatial projection, we propose an efficient AM (EAM) algorithm by dividing the AM criterion of SML into two components. One depends on a single variable parameter while the other does not. The computational complexity of EAM can be greatly reduced compared to AM algorithm. However, numerical instability may happen in calculation of EAM criterion when more than one parameter converges to an identical value. As a result, oscillation may happen in the convergence phase and extra calculation is needed. To solve this problem and reduce computational complexity further, based on the EAM criterion we get the irreducible form of the EAM criterion (IAM) using a uniform linear array. In this way, the EAM criterion can be written into polynomial forms. The common zeros can be easily canceled in numerator and denominator of the EAM criterion. Furthermore, the computational complexity is also reduced. Finally, simulation results are also shown to demonstrate the validity of the proposed EAM and IAM algorithms.

The rest of this paper is organized as follows. In Section

Without loss of generality, consider that there are

Note that the received signals may be coherent because of multipath propagation. In the case where there are signals coherent, the independent signal number is less than

Furthermore, the sensor configuration can be arbitrary and we assume that all the sensors are omnidirectional and not coupled [

Using complex envelope representation, the

In the matrix notation, (

Suppose that the received vector

The problem of DOA finding is to be stated as follows. Given the sampled data

In this section, brief descriptions of the exact SML criterion are shown [

To solve the problem of ML estimation of DOA, we make the following assumptions.

The array configuration is known and any

The noise samples

According to [

The SML criterion is shown as follows [

From (

Note that there are literatures [

The AM algorithm is the most classic estimation algorithm for multidimensional nonlinear optimization problem in DOA estimation. In this section, we will introduce the conventional AM algorithm firstly, and then we will introduce our proposed efficient AM algorithms.

The AM method is a popular iterative technique for solving a nonlinear multivariate minimization problem with a multimodal criterion [

Let

Although a global minimum is not guaranteed in the AM algorithm, global solutions can be obtained in most cases because of one-dimensional global searches performed in each update process.

In this section, we propose an efficient version of the AM algorithm. We call it the EAM algorithm. Using transformation of matrix and properties of spatial projection, the EAM algorithm divides the SML criterion into two components. One depends on a variable parameter and the other one is independent of the variable parameter. In order to simplify expressions, parameters to be estimated are represented without the accent hat, and the argument

In each updating process, let

Substituting (

Define

In (

The linear combination

On the other hand, when

Next, we consider calculation of the EAM criterion

In the case that the DOA can be solved, the numerical instability does not occur, since each parameter in the convergence phase of the EAM criterion comes apart from others. However, at the threshold region, when more than one signal approaches an identical value, the numerical instability becomes significant. In practice, when this case happens, the sequence of DOA obtained in the convergence phase of the EAM criterion shows oscillation that is because the estimated directions can not converge well due to the numerical instability and would oscillate around that identical value.

Let us give an example. In the simulation there are two sources located in 0 and 8 degrees (the true DOAs). When SNR = 0 dB and with specific noise samples, the solutions of SML criterion are 3.999 and 4.001 degrees (the solutions of SML). However, in this case, when calculating

To solve these problems, the key point is to cancel the common zeros in the numerator and denominator of the EAM criterion. Next we try to establish the irreducible form of the EAM criterion using a uniform linear array.

In this section, we derive the irreducible form of the EAM criterion using a uniform linear array. With a uniform linear array, the EAM criterion can be written into polynomial forms. Then we can easily cancel the common zero in both numerator and denominator. Thus numerical instability never happens in IAM criterion. Furthermore, we can find that the IAM algorithm can reduce the computational order of EAM criterion from square to one in each updating process.

The array configuration is uniform linear array composed of omnidirectional sensors, of which steering vector is represented as

Using the uniform linear array, we derive the irreducible form of (

First we derive the irreducible form of (

Using the uniform linear array defined above, the steering vector

Let

The irreducible form of the EAM criterion of

Using the coefficients of

Using the expressions

As for the irreducible form of

Define the following polynomials

Then we have the final form of

Therefore the irreducible form of efficient AM criterion (IAM) is shown like this.

Furthermore, we can find that

Here we should note that the IAM algorithm is only applicable to the ULA that is because the IAM algorithm is formulated based on (

In this section, we show some simulation results to demonstrate the validity of the EAM and IAM algorithms. In simulation, the array configuration is a uniform linear array as discussed above.

The SNR is defined as

The Root-Mean-Square-Error (RMSE) is defined as

In Figure

Numerical instability in EAM and stability in IAM criterion.

Due to the numerical instability, oscillation may happen as Figure

Oscillation in EAM algorithm while not in IAM.

The iteration does not stop until it reaches the maximum number, 800, using EAM criterion

The iteration stops at about 85 using IAM criterion

Oscillation rate of EAM and IAM algorithms

Figure

Comparison of RMSE between SML, MUSIC, and ESPRIT.

Figures

Comparison of computational complexity of AM, EAM, and IAM. The scenario is the same as Figure

Average iteration times | Main computational process | |
---|---|---|

AM | 127 | |

EAM | 127 | 127 times of calculation of |

IAM | 127 | 127 times of calculation of |

Average amount of operations for AM, EAM, and IAM algorithms.

Average amount of operations for AM, EAM, and IAM algorithms.

In Figures

Table

Furthermore, from Figures

In a huge number of simulations, we have confirmed the efficiency of our proposed EAM and IAM algorithms, while IAM is the best.

In this paper, to reduce the computational complexity of SML estimation, based on the AM algorithm, we propose two more efficient algorithms, that is, EAM and IAM algorithms. The EAM algorithm mainly uses transformation of matrix and properties of spatial projection to divide the SML criterion into two components. One is variable, while the other is fixed. Computational complexity can be greatly reduced since the fixed part can be calculated once in advance and only the varying part should be calculated in each one-dimensional updating process. To avoid numerical instability of EAM (note that because of the numerical instability, wrong estimation of DOA may be got) and to reduce computational complexity further, we derive the irreducible form of EAM, that is, IAM algorithm. The main idea of IAM is to use a uniform linear array and rewrite the EAM criterion into polynomial forms. Then the irreducible form can be got by canceling the common zero factor of the polynomial form. Simulation results show that the IAM algorithm can avoid the numerical instable problem of EAM and reduce computational complexity further.

The authors declare that they have no competing interests.

This paper is supported by the following funds: The National Natural Science Funds, China, with no. 61601519; Fundamental Research Funds for the Central University, China, with nos. 15CX02047A and 15CX05025A; Shandong Provincial Natural Science Foundation, China, with no. ZR2014FM017; Science and Technology Development Plan of Qingdao with no. 15-9-1-80-jch.