Parametric Study of Various Direction of Arrival Estimation Techniques

Abstract

The adaptive antenna array system consists of a number of an antenna array element with the signal processing unit, which can adapt its radiation pattern to provide maxima towards the desired user and null towards the interferer. Hence to locate the desired user Direction of Arrival (DOA) of the signal needs to be estimated. This paper presents a parametric study of various DOA estimation algorithms on the uniform linear array (ULA) and also a comparative performance regarding resolution, Signal-to-Noise Ratio (SNR), the number of snapshots, separation angle, etc. It starts with traditional methods of Minimum Variance Distortionless Response (MVDR) algorithm. The subspace-based techniques use the eigenstructure of data covariance matrix. The different subspace-based techniques are Multiple Signal Classification (MUSIC), Root-MUSIC, and Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT). However, simulation results show that the antenna array elements, SNR, the snapshots, coherent nature of signals, and separation angle between the two sources can affect the DOA estimation results. The result shows that MUSIC algorithm has comparatively better resolution than MVDR, Root-MUSIC and ESPRIT algorithms.

1 Introduction

The Direction of Arrival (DOA) estimation techniques are the emerging applications of many actual domains, such as radar, speech signal processing, mobile communication and so on [1,2,3]. DOA estimation technique is an essential part of Smart Antenna. The antenna array which collects data from impinging signals on it with a combination of spatial information by receiving signals, this received data used to estimate the angle of arrival with higher resolution algorithms [1, 4].

There are four different types of DOA estimation technique conventional, maximum likelihood, subspace based and integrated techniques [2, 5]. Beamforming techniques were used in conventional methods, and it requires a large number of the antenna element for getting better resolution. Maximum likelihood technique requires large computation. The integrated technique is a combination of property restoral technique and subspace-based technique. The sum and delay or Bartlett and MVDR or Capon etc. are spatial correlation type method. The DOA estimation techniques such as MVDR, MUSIC, Root-MUSIC, and ESPRIT have been analyzed.

The paper is organized as follows. The uniform linear array structure is presented in Sect. 2 with mathematical formulation required for DOA estimation algorithms. Section 3 details explanation of various DOA estimation algorithms. Next Sect. 4 shows the simulation result and discussion. Section 5 presents conclusion on the basis of simulation results.

2 Array Sensor System

Figure 1 shows the system consists of Uniform Linear Array (ULA) [1, 3]. Consider the D-number of signals impinges on ULA antenna elements, the received input data vector at M-elements, separated by a distance d can be represented as a combination of the D-incident signals and noise $n\left( t \right)$. The signal vector $u\left( t \right)$ is defined as

Fig. 1.

Uniform linear array

$$ u\left( t \right) = \sum\limits_{l = 0}^{D - 1} {a\left( {\phi_{1} } \right)} s_{1} \left( t \right) + n\left( t \right) $$

(1)

$$ u\left( t \right) = \left[ {a\left( {\phi_{0} } \right) \ldots a\left( {\phi_{D - 1} } \right)} \right]\left[ \begin{aligned} \begin{array}{*{20}c} {s_{1} \left( t \right)} \\ \vdots \\ \end{array} \hfill \\ s_{l} \left( t \right) \hfill \\ \end{aligned} \right] + n\left( t \right) $$

(2)

$$ u\left( t \right) = As\left( t \right) + n\left( t \right) $$

(3)

Where, $s\left( t \right)$ is the vector of incident signals, $n\left( t \right)$ is the noise vector, and $a\left( {\phi_{j} } \right)$ is the array steering vector. In terms of the above data model, the input covariance matrix $R_{uu}$ expressed as [3].

$$ R_{uu} = AE\left[ {ss^{H} } \right]A^{H} + E\left[ {nn^{H} } \right] $$

(4)

Where $R_{ss}$ is the signal correlation matrix of $E\left[ {ss^{H} } \right].$

3 DOA Estimation Algorithms

3.1 MVDR Algorithm

The concept of MVDR algorithm is to minimize the received power in all direction of every impinging signal to nullify the effect of interference, also maintain a unity gain in desired direction [6, 7]. The condition apply on this algorithm is given as

$$ \mathop {\hbox{min} }\limits_{w} E\left[ {\left| {y\left( k \right)} \right|^{2} } \right] = \mathop{\min}\nolimits_{w} w^{H} R_{uu} w{\text{ subject}}\,{\text{ to}}\, w^{H} a\left( {\phi_{0} } \right) = 1 $$

(5)

By solving (5), the obtained $w$ weight vector is (MVDR) beamformer weights. Applying Lagrange multiplier to Eq. (5) which is a constraint optimization,

$$ w = \frac{{R_{uu}^{ - 1} a\left( \phi \right)}}{{a^{H} \left( \phi \right)R_{uu} a\left( \phi \right)}} $$

(6)

The output power spectrum of MVDR for DOA estimation is given as,

$$ P_{MVDR} \left( \phi \right) = \frac{1}{{a^{H} \left( \phi \right)R_{uu}^{ - 1} a\left( \phi \right)}} $$

(7)

3.2 MUSIC Algorithm

The Multiple Signal Classification algorithm was proposed by Schmidt [8] in 1979.

The array correlation matrix $R_{uu}$ is given as follows,

$$ R_{uu} = AR_{ss} A^{H} + \sigma_{n}^{2} I $$

(8)

Next step is to find the correlational matrices eigenvalues and their associated eigenvectors. Then produce $D$ and $M - D$ eigenvectors associated with the signals and noise respectively. Select the eigenvectors having smallest eigenvalues [9, 12]. Then construct the $M \times \left( {M - D} \right)$ dimensional subspace composed of the noise eigenvectors such that,

$$ V_{n} = \left[ {q_{1} ,q_{2} \ldots q_{M - D} } \right] $$

(9)

The steering vectors of array and eigenvectors of noise subspace are orthogonal to each other, $a^{H} \left( \phi \right)V_{n} V_{n}^{H} a\left( \phi \right) = 0$ at an angle of arrival. Hence the MUSIC spectrum can be calculated as

$$ P_{music} \left( \phi \right) = \frac{1}{{a^{H} \left( \phi \right)V_{n} V_{n}^{H} a\left( \phi \right)}} $$

(10)

The condition of orthogonality will give rises to peaks in the spectrum of MUSIC defined in Eq. (10) by minimizing the denominator.

3.3 Root-MUSIC Algorithm

To increase resolution performance and decrease the computational complexity various modifications have been proposed to the MUSIC algorithm. The Root-MUSIC algorithm developed by Barabell [10] which is based on an idea of polynomial rooting giving higher resolution, but it works only if a uniform spaced linear array is present. For the case of a uniform linear array with the spacing between each element is $d$, the $M^{th}$ element of the antenna steering vector $a\left( \phi \right)$ which can expressed as,

$$ a_{m} \left( \phi \right) = \exp \left( {j2\pi \left( {\frac{d}{\lambda }} \right)\cos \phi } \right){ ; }m = 1,2, \ldots M. $$

(11)

The MUSIC spectrum is given by

$$ P_{music} \left( \phi \right) = \frac{1}{{a^{H} \left( \phi \right)C \, a\left( \phi \right)}},{\text{ Where }}C = V_{n} V_{n}^{H} $$

(12)

$$ P_{MUSIC}^{ - 1} \left( \phi \right) = \sum\limits_{l = - M + 1}^{M + 1} {C_{1} \exp \left( { - j2\pi \left( {\frac{dl}{\lambda }} \right)\cos \phi } \right)} $$

(13)

$$ D\left( z \right) = \sum\limits_{l = - M + 1}^{M + 1} {C_{1} \, Z^{ - 1} } $$

(14)

Evaluating the MUSIC spectrum becomes equivalent to evaluating the roots of the polynomial given by $D\left( z \right)$ on the unity circle, and therefore peaks of the spectrum [13]. In true sense, for the noiseless condition, the roots of the polynomial at the denominator of the function is called as poles and will be expressed on the unity circle which will determine DOA of impinging signal. In other words, a pole of $D\left( z \right)$ at $z = z_{1} = \left| {z_{1} } \right|\exp \left( {j\,{ \arg }\left( {z_{1} } \right)} \right)$ will leads to obtain a peak in the Root-MUSIC spectrum at

$$ \cos \phi = \left( {\frac{\lambda }{2\pi d}} \right)\arg \left( {z_{1} } \right) $$

(15)

3.4 ESPRIT Algorithm

Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT) algorithm involves decomposing an N element array into two identical subarrays each with S element [11]. The objective of ESPRIT algorithm is to estimate the direction of arrival by determining the rotation operator. The relation of $u_{0} \left( t \right) , { }u_{1} \left( t \right)$ are received signals, and $n_{0} \left( t \right) , { }n_{1} \left( t \right)$ are additive noise is given as:

$$ u_{0} \left( t \right) = As\left( t \right) + n_{0} \left( t \right){\text{ and }}u_{1} \left( t \right) = A\left( \phi \right)s\left( t \right) + n_{1} \left( t \right) $$

(16)

$$ {\text{ and, }}\phi = diag\left\{ {e^{{j\psi_{1} }} ,e^{{j\psi_{2} }}, \ldots ,e^{{j\psi_{M} }} } \right\}\!{;}{\text{ where }}\psi {\text{ = kdsin}}\theta_{m} {\text{ and }}m = 1,2, \ldots ,M $$

(17)

$$ \therefore \, u\left( t \right) = \left[ {\begin{array}{*{20}c} {u_{0} \left( t \right)} \\ {u_{1} \left( t \right)} \\ \end{array} } \right] = \bar{A}s\left( t \right) + n\left( t \right) $$

(18)

As per input covariance matrix given in Eq. (8), the Eigen decomposition becomes:

$$ \hat{R}_{uu} = V \wedge V^{H} $$

(19)

Where $\wedge = diag\left\{ {\lambda_{0} \ldots \lambda_{M - 1} } \right\}{\text{ and }}V = \left[ {q_{0} \ldots q_{M - 1} } \right]$ represent the eigenvalues and eigenvectors. Estimation of the number of signals $\hat{D}$, by using the theorem of multiplicity, $K$ of the smallest eigenvalue $\lambda_{\hbox{min} } ,\hat{D} = M - K.$ Obtain, the estimation of the signal subspace $\hat{V}_{s} = \left\lfloor {\hat{V}_{0} \ldots \hat{V}_{L - 1} } \right\rfloor$ and decomposes it into sub-array matrix, and compute the eigendecomposition $\left( {\lambda_{0} \ldots \lambda_{2i} } \right)$ and partition $V\text{into} \, \hat{D} \times \hat{D}$ sub-matrices,

$$ \hat{V}_{01}^{H} \hat{V}_{01} = \left[ {\begin{array}{*{20}c} {\hat{V}_{0}^{H} } \\ {\hat{V}_{1}^{H} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\hat{V}_{0}^{H} } & {\hat{V}_{1}^{H} } \\ \end{array} } \right] $$

(20)

$$ V = \left[ {\begin{array}{*{20}c} {V_{11} } & {V_{12} } \\ {V_{21} } & {V_{22} } \\ \end{array} } \right] $$

(21)

Calculate the eigenvalues of $\psi_{1} = - V_{12} V_{22}^{ - 1}$ as,

$$ \hat{\phi }_{K} = {\text{eigenvalues of}}\left( { - V_{12} V_{22}^{ - 1} } \right),{\text{Where }}K = 0 \ldots \hat{L} - 1 $$

(22)

ESPRIT gives the DOA estimation in terms of the eigenvaluee, therefore estimate the DOA as,

$$ \theta_{K} = \cos^{ - 1} \left[ {\frac{{\arg \left( {\hat{\phi }_{K} } \right)}}{\beta d}} \right] $$

(23)

4 Simulation Results

DOA estimation techniques simulated using Matlab 2014a. In these simulations, 30 antenna elements of ULA are considered which are equally separated by the distance of λ⁄2. The noise is ideal Gaussian white noise. The simulation has been run for two independent narrow band signals with equal amplitude, and 1000 numbers of snapshots. The performance has been analyzed with different values array elements, and Signal to Noise ratio. The closely spaced and coherent type of nature of signals, also considered for the evaluating the performance of theses algorithms.

4.1 MVDR Algorithm

Case 1: Effect of signal to noise ratio and number of array elements on MVDR

The effect of changing the signal to noise ratio with three different values SNR₁ = 10, SNR₂ = 20 and SNR₃ = 30, with the DOA is 20⁰ and 60⁰, it is clear from Fig. 2(a), as the value of SNR increases, the beamwidth of power spectrum becomes narrower. The resolution of DOA estimation can be increased by increasing SNR. Now, the three different values of the number of array elements are M₁ = 10, M₂ = 50, M₃ = 100. It is found that the width of the spectral beam becomes narrower as the number of array elements increases as shown in Fig. 2(b).

Fig. 2.

MVDR spectrum (a) Effect of SNR; (b) Effect of antenna elements

Case 2: Effect of closely spaced in resolution and coherent signals on MVDR

The effect of closely spaced signals on MVDR algorithm with DOA is 20⁰ and 24⁰ i.e. with 4⁰ resolution of the signals. It is clear from Fig. 3(a), as the signals are very close, MVDR gets fails to resolve the signal and observes very poor beamwidth in power spectrum. Now, if the two narrow band signals has large resolution with coherent in nature, and processes by MVDR. It is clear from Fig. 3(b), spectrum becomes oscillating. The auto covariance matrix rank degrades effectively when the signals are coherent in nature as well as in closely spaced in nature, and the MVDR shows poor performance.

Fig. 3.

MVDR spectrum (a) Closely spaced signals; (b) Effect of coherent signals

4.2 MUSIC Algorithm

Case 1: Effect of signal to noise ratio and number of array elements on MUSIC

The effect of changing the signal to noise ratio with three different values SNR₁ = 10, SNR₂ = 20 and SNR₃ = 30, with the DOA is 20⁰ and 60⁰, it is clear from Fig. 4(a), value of SNR increases, the spectral beam width becomes narrower the direction of the signal becomes clearer. The accuracy of DOA estimation can be increased by increasing SNR. Also, the three different values of the number of array elements are M₁ = 10, M₂ = 50, M₃ = 100. Figure 4(b), shows that as the value of array elements increases, the spectral beam width becomes narrower.

Fig. 4.

MUSIC spectrum (a) Effect of SNR; (b) Effect of antenna elements

Case 2: Effect of closely spaced in resolution and coherent signals MUSIC

The effect of closely spaced signals on MUSIC algorithm with DOA is 20⁰ and 24⁰ i.e. with 4⁰ resolution of the signals. It is clear from Fig. 5(a), as the signals are very close, MUSIC gets resolve the signal but with less spectral beamwidth and observes satisfactorily performance in upper power levels in output spectrum. Now, if the two narrow band signals has large resolution with coherent in nature processes by MUSIC with the different values of SNR. It is clear from Fig. 5(b), spectrum becomes oscillating. The auto covariance matrix rank degrades effectively, when the signals are coherent in nature, and the MUSIC shows poor performance. It requires the computation of the matrix inverse which can be expensive for large arrays.

Fig. 5.

MUSIC spectrum (a) Closely spaced signals; (b) Effect of coherent signals

4.3 Root-MUSIC Algorithm

The simulation has been carried out for four independent narrow band signals with a DOA of 14⁰, 23⁰, 35⁰, and 55⁰. The performance of each independent signal has been analyzed for different values of snapshots (10, 50, 100, and 200) expressed in Table 1. Also, the performance of each independent signal has been analyzed for different values of SNR in dB (0, 10, 20, and 30) expressed in Table 2.

Table 1. Root-MUSIC spectrum: DOA vs. number of snapshots

Table 2. Root-MUSIC spectrum: DOA vs. SNR

As the value of number of snapshots and signal to noise ratio increases the Root-MUSIC gives an accurate estimation of the direction of arrival.

4.4 ESPRIT Algorithm

The simulation has been carried out for four independent narrow band signals with a DOA of 14⁰, 23⁰, 35⁰, and 55⁰. The performance of each independent signal has been analyzed for different values of snapshots (10, 50 100, and 200) expressed in Table 3. Also, the performance for different values of SNR in dB (0, 10 20, and 30) expressed in Table 4.

Table 3. ESPRIT spectrum: DOA vs. number of snapshots

Table 4. ESPRIT spectrum: DOA vs. SNR

Hence, the number of snapshots and SNR can be increased for accurate DOA estimation of the signal. As seen from simulation results of various DOA estimation techniques, as a number of snapshots, signal to noise ratio and a number of antenna elements increases, the resolution goes on increasing and results are more sharpen. The reason is that simulation environment is ideal i.e. 30 antenna elements, 1000 snapshots, and positive value of SNR. In practical situations, such as multiple signal receptions, aliasing, zero or negative value of SNR, with less number of snapshots, and fewer number of an antenna elements was degraded the performance of these algorithms. The remedy and scope for future modifications of these existing techniques are to develop the pre-processing as well as post-processing techniques to inherit the performance in real and practical environment.

5 Conclusions

This paper presents comparative performance simulation study of various DOA estimation techniques and their dependence of various parameters. The different DOA estimation techniques are MVDR, MUSIC, Root-MUSIC and ESPRIT algorithms. The algorithms were found to be responsive to the array elements, snapshots, SNR. The simulation result also shows that the resolution of the algorithm increases with increasing the number of array elements, the number of snapshots, SNR. The MUSIC algorithm is accurate as compared to the MVDR, Root-MUSIC, and ESPRIT algorithm, but has wide radiation pattern. The MUSIC algorithm is showing good performance in closely spaced signals compared to the MVDR, Root-MUSIC, and ESPRIT algorithm. Also, for coherent sources, MUSIC algorithm is not as efficient as Root-MUSIC and ESPRIT algorithm.