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In this paper, a novel direction of arrival (DOA) estimation algorithm called the Toeplitz fourth order cumulants multiple signal classification method (TFOC-MUSIC) algorithm is proposed through combining a fast MUSIC-like algorithm termed the modified fourth order cumulants MUSIC (MFOC-MUSIC) algorithm and Toeplitz approximation. In the proposed algorithm, the redundant information in the cumulants is removed. Besides, the computational complexity is reduced due to the decreased dimension of the fourth-order cumulants matrix, which is equal to the number of the virtual array elements. That is, the effective array aperture of a physical array remains unchanged. However, due to finite sampling snapshots, there exists an estimation error of the reduced-rank FOC matrix and thus the capacity of DOA estimation degrades. In order to improve the estimation performance, Toeplitz approximation is introduced to recover the Toeplitz structure of the reduced-dimension FOC matrix just like the ideal one which has the Toeplitz structure possessing optimal estimated results. The theoretical formulas of the proposed algorithm are derived, and the simulations results are presented. From the simulations, in comparison with the MFOC-MUSIC algorithm, it is concluded that the TFOC-MUSIC algorithm yields an excellent performance in both spatially-white noise and in spatially-color noise environments.

During the last few decades, DOA estimation, which has been widely applied in the fields of sonar, radar, wireless communication, aeronautics,

Conventional array processing techniques utilize only the second order statistics (SOSs) of the received signal, which may have suboptimal performance due to the transmitted signals, combining with additive Gaussian noise, are often non-Gaussian in real applications, e.g., as in a communications system [

But the conventional MUSIC-like algorithms have high computational requirements as a result of the great number of redundant information contained in the FOC matrix as well as the rigorous requirements of sampling snapshots for the FOC matrix estimation. To mitigate these drawbacks, a fast MUSIC-like algorithm (the MFOC-MUSIC algorithm) is proposed to reduce the computational complexity effectively [

The rest of this paper is organized as follows. Section 2 introduces the system model and the MUSIC-like algorithm. In Section 3, the TFOC-MUSIC algorithm is described in detail. Section 4 presents comparative simulation results that show the effectiveness of the proposed algorithm. Finally, we conclude this paper in Section 5.

Throughout the paper, lower-case boldface italic letters denote vectors, upper-case boldface italic letters represent matrices, and lower and upper-case italic letters stand for scalars. The symbol * is used for conjugation operation, and the notations (^{T}^{H}

Assume that _{l}^{2}

Then, the signal received in time _{i}_{i}

In matrix form, it becomes _{l}_{1}_{N}_{l}^{T}

Then, rewriting _{1}_{N}^{T}_{1}_{M}^{T}_{l}_{M}_{1}_{N}^{T}

The MUSIC algorithm makes use of the covariance matrix of the data received by the sensor array, denoted by
_{S}_{2}

For symmetrically distributed signals, their odd-order cumulants are usually zero. Therefore, even-order cumulants are the main objects of investigation, in particular with the FOC. There exist various definitions about the FOC matrix. For zero mean stationary random process, the _{km}_{4}_{1}_{2}_{3}_{4}_{1}_{2}_{3}_{4}_{4}

The _{2q,x}_{2q,x}_{opt}_{opt}_{4}_{2q,x}_{s}_{4}_{1}_{N}_{2}) are separated into the signal and noise subspaces according to the descending order of the eigenvalues (_{1}_{N}_{2}). The signal subspaces _{S}_{1}_{M}_{1}_{M}_{1}_{1}_{1}_{M}_{+1}, ……, _{N}_{2}) is called the noise subspaces _{N}^{H}_{N}^{2}

As proven in [^{2}

Comparing

The effective number of different sensors
_{max}_{max}_{4}^{2}_{max}

In this section, we describe the MFOC-MUSIC algorithm combined with Toeplitz approximation in detail. To begin with, the MFOC-MUSIC algorithm is described. From _{4}

In light of the above analysis, it can be seen that _{4}_{4}_{4}_{4}_{4}_{4}_{4}

Like in _{4}^{2N}^{−}^{2}^{T}_{4}

In practical applications, we do not have access to true _{4}_{4} in lieu of _{4}_{4} which signifies the estimation value of _{4}_{4}_{2}_{4}_{4}, the Toeplitz approximation, which was primarily presented for DOA estimation of coherent sources [_{4}_{T}_{4}. It is shown that the eigenstructure of _{4}_{T}_{4}_{4}_{T}_{4} to get the signal and noise subspaces representing _{S}_{N}

Since the similar expression between _{2}_{4}_{4}_{T}_{4}_{T}_{4}_{T}_{p}_{(}_{p}_{+}_{h}_{−1)} is the _{4}, _{4}_{T}_{4}_{T}_{4}.

The procedure of the TFOC-MUSIC algorithm is detailed as follows:

Step 1 Estimate _{4} from the received data by array measurements

Step 2 Take out the _{4} in order, and then store these rows in the _{4} matrix.

Step 3 Gain the columns of _{4} via using its conjugate symmetry for reducing computation.

Step 4 Apply Toeplitz approximation to _{4} for _{4}_{T}

Step 5 Remove the redundant items of the expanded steering vector ^{2N}^{−}^{2}^{T}

Step 6 The estimate of DOAs of source directions can be attained by searching the peaks of redefined spatial spectrum

According to the principle of the TFOC-MUSIC algorithm, when compared with the MFOC-MUSIC algorithm, it incurs 2(2^{2} − (2

In this part, we evaluate the performance of the TFOC-MUSIC algorithm with several experiments in spatially-white noise and in spatially-color noise environment, respectively. The FOC-MUSIC, MFOC-MUSIC and TFOC-MUSIC algorithms are compared in terms of spatial spectrum, normalized probability of success, average maximum estimate deviation and average estimate variance of incoming signals with respect to variables such as angle _{i}_{i}_{l}_{l}e^{jωlt}_{l}

Spatial Spectrum

_{4}_{T}_{4}_{4} in the same condition.

Normalized Probability of Success, Average Maximum Estimate Deviation and Average Estimate Variance

The estimated performances of normalized probability of success, average maximum estimate deviation and average estimate variance are plotted in _{4} using Toeplitz approximation which improves the performance of DOA estimation.

Normalized Probability of Success, Average Maximum Estimate Deviation and Average Estimate Variance

_{4} deviates from the ideal _{4}

In addition, the performance curves become stabilized with an increasing data length. Hence, we ascertain that the estimated performance becomes optimal, since the snapshots number goes to infinity. But in the convergence progress, the complexity of the TFOC-MUSIC algorithm is obviously smaller than that of the FOC-MUSIC algorithm, and the convergence speed of the TFOC-MUSIC algorithm is much faster than that of the MFOC-MUSIC algorithm. Complexity reducing benefit from the lower cumulants matrix rank dimension of the TFOC-MUSIC algorithm compared to the FOC-MUSIC algorithm. The cumulants matrix reconstructed using the Toeplitz approximate method is close to the desired _{4} than the MFOC-MUSIC algorithm in the same condition, which helps speed up the convergence.

To sum up, as can be noticed from _{4}, namely, _{4}_{T}

A novel DOA estimation algorithm has been presented in this paper. Its main idea is to utilize the MFOC-MUSIC algorithm in conjunction with Toeplitz approximation. In this way, the effective array aperture of a physical array can be extended that allows the number of estimated signals to be greater than or equal to that of sensors. Moreover, for non-Gaussian sources, in contrast to the MFOC-MUSIC algorithm, the proposed method has lower average maximum estimate deviation and average estimate variance, higher normalized probability of success and angular resolution. And the threshold of snapshots is less than that of the MFOC-MUSIC algorithm to some extent. In addition, the computation of the TFOC-MUSIC algorithm is approximately consistent with that of the MFOC-MUSIC algorithm, while obviously smaller than the FOC-MUSIC algorithm due to estimating DOA of more targets with less sensors. Simulation results show that the proposed method is more effective and efficient than the MFOC-MUSIC algorithm in DOA estimation, both in spatially-white noise and spatially-color noise situations.

This work has been financially supported by the National Natural Science Foundation of China (Grant No. 61101223), the 863 program (Grant No. 2011AA010201), and the Doctoral Foundation from Ministry of Education of China (Grant No. 20110032120087 and 20110032110029).

Spatial spectrum comparisons

Normalized probability of success comparisons

Average maximum estimate deviation comparisons

Average estimate variance comparisons

Normalized probability of success comparisons

Average maximum estimate deviation comparisons

Average estimate variance comparisons