Uniform and nonuniform V-shaped planar arrays for 2-D direction-of-arrival estimation

Authors


Abstract

[1] In this paper, isotropic and directional uniform and nonuniform V-shaped arrays are considered for azimuth and elevation direction-of-arrival (DOA) angle estimation simultaneously. It is shown that the uniform isotropic V-shaped arrays (UI V arrays) have no angle coupling between the azimuth and elevation DOA. The design of the UI V arrays is investigated, and closed form expressions are presented for the parameters of the UI V arrays and nonuniform V arrays. These expressions allow one to find the isotropic V angle for different array types. The DOA performance of the UI V array is compared with the uniform circular array (UCA) for correlated signals and in case of mutual coupling between array elements. The modeling error for the sensor positions is also investigated. It is shown that V array and circular array have similar robustness for the position errors while the performance of UI V array is better than the UCA for correlated source signals and when there is mutual coupling. Nonuniform V-shaped isotropic arrays are investigated which allow good DOA performance with limited number of sensors. Furthermore, a new design method for the directional V-shaped arrays is proposed. This method is based on the Cramer-Rao Bound for joint estimation where the angle coupling effect between the azimuth and elevation DOA angles is taken into account. The design method finds an optimum angle between the linear subarrays of the V array. The proposed method can be used to obtain directional arrays with significantly better DOA performance.

1. Introduction

[2] DOA estimation has many applications in radar [Haykin, 1985], sonar, seismology [Bohme, 1995], and ionospheric research [Black et al., 1993]. The performance of the direction finding (DF) system is significantly dependent to sensor array geometry. Therefore, the design of optimum array geometry for the best two-dimensional (2-D) DOA estimation performance is an important problem. This problem is investigated in previous works for the most general parameter settings. In these works, Cramer-Rao Bound (CRB) on error variance is used as the performance measure and objective functions for the desired performance are minimized. The desired performance can change according to the application. In the work of Baysal and Moses [2003], the goal is to find planar and volumetric arrays for uniform DOA performance in all directions. In the work of Oktel and Randolph [2005], DOA of interest is an angular sector and the goal is to find optimum array geometry for this scenario. It is seen that optimum array geometry for DOA estimation depends on many parameters including the number of sensors, number of sources and their DOA angles [Gershman and Bohme, 1997]. Furthermore it is not easy to find a single optimum geometry since the cost function changes depending on the number of sources and DOAs.

[3] In this paper, the array geometry is fixed as V-shaped in order to simplify the design and use certain advantages of V-shaped arrays which are not well known in the literature. V-shaped planar arrays can be designed for good directional DOA performance. It can also be designed for isotropic response such that the DOA performance is uniform for all directions. When the array intersensor distance is fixed to half of the wavelength, V-shaped array has a larger aperture than circular array. The number of sensors in V-shaped arrays can be decreased when the sensors are placed nonuniformly for each subarray. Furthermore, it is possible to apply forward-backward spatial smoothing [Pillai and Kwon, 1989] for each subarray in order to deal with multipath signals. Fast algorithms can be applied for these subarrays and the results can be combined as in the work of Hua et al. [1991]. It is also shown that joint accuracy of two subarrays is better than each subarray's accuracy [Hua and Sarkar, 1991].

[4] V-shaped arrays are not fully investigated in the literature. In the work of Gazzah and Marcos [2006], V-shaped arrays are considered with limited scope. Statistical angle coupling between the azimuth and elevation angle estimation is ignored and V angle for uniform DOA performance is determined only for infinite number of sensors. Until now there is no known method and expression for finding the isotropic V angle. Furthermore the directional characteristic of the V-shaped arrays is not fully exploited.

[5] In this paper, closed form expressions are presented for the V angle in order to obtain isotropic DOA response. The UI V-shaped array and UCA are compared for the same number of sensors and intersensor distances. The comparison is done in terms of sensor position errors, source signal correlation, and mutual coupling between antennas. It is shown that the DOA accuracy of the UI V-shaped array is better than UCA. V-shaped arrays and UCA have similar robustness for the sensor position errors. The effect of source signal correlation is similar for both arrays while the performance of UI V array gets better as the correlation increases. A similar observation is done for the mutual coupling. The performance of UI V array is better than UCA in case of multiple sources. Different nonuniform V-shaped isotropic arrays are considered where the numbers of sensors at each subarray can be different. It is shown that the DOA performance can be improved significantly when the isotropic nonuniform V-shaped arrays are used. A design procedure for the directional uniform V-shaped arrays is presented. The directional V-shaped arrays are also compared with UCA for different DOA scenarios.

[6] The contribution of this paper for 2-D DOA estimation with V-shaped planar array geometry can be summarized as follows. Closed form expressions for the isotropic V angle are presented. The expressions are given for both uniform and nonuniform V-shaped planar arrays. The performances of V-shaped arrays, including uniform isotropic and nonuniform arrays, are analyzed for different cases. These involve correlated signals, mutual coupling, and sensor position errors. It is shown that V-shaped arrays perform better than the UCA for different types of error sources which is not well known in the literature. A design procedure is presented for uniform directional V arrays which allow one to trade off the isotropic characteristics for the better DOA performance for a given angular sector. The optimization of the V angle is done by defining a cost function over the CRB on DOA error variance which takes into account the coupling effect of azimuth and elevation angles. The proposed design considers two regions, namely, focused and unfocused region. The limits of the regions determine the angular accuracy and the performance of the V array. It is shown that optimum V angle can be found easily with only a limited search due to the monotonic characteristics of the cost function for the worst and best levels specified in the design parameters of the regions. This design procedure can also find the V angle for isotropic DOA performance numerically [Filik and Tuncer, 2008a, 2008b].

[7] The paper is organized as follows. In section 2, we describe the model of the array signals and CRB expressions are presented for 2-D DOA estimation. In section 3, closed form expressions for uniform and nonuniform V-shaped arrays for isotropic azimuth response are given. We present the directional V-shaped planar array design procedure in section 4. In section 5, the effect of mutual coupling between array elements is considered. The performances of the designed V-shaped arrays and UCA are presented in section 6.

2. Problem Formulation

2.1. Data Model

[8] We consider an array of M sensors located at the positions [xl, yl], l = 1, …, M. We assume that there are L (L < M) narrowband signals impinging on the array from the directions Θi = [ϕi, θi] i = 1, …, L, where ϕ and θ are the azimuth and elevation angles, respectively, as shown in Figure 1. If the sensors are identical omnidirectional and far-field assumption is made, the sensor output, y(t), can be written as,

equation image

where N is the number of snapshots. It is assumed that the noise, n(t), is both spatially and temporally white with variance σ2. It is also uncorrelated with the source signals. A(Θ) = [a1, θ1)…aL, θL)] is the M × L steering matrix for the planar array and the vectors a(ϕ, θ) are given as,

equation image

The output covariance matrix, R, is

equation image

where (.)H denotes the conjugate transpose of a matrix, Rs is the source correlation matrix and I is the identity matrix.

Figure 1.

Coordinate system for 2-D angle estimation and V-shaped array.

2.2. CRB for 2-D DOA Estimation

[9] CRB shows the ultimate performance of an unbiased estimate for a given array geometry. When 2-D DOA estimation is considered, there is statistical coupling between the azimuth and elevation DOA performances in general. The existence of coupling depends on array geometry. Some of the array geometries like circular arrays are uncoupled. V-shaped arrays show coupling effects and therefore coupling should be taken into account for the CRB. The proposed V-shaped array design method uses the CRB in the cost function. Therefore, a review of the angle coupling effect for the CRB is considered in this section. The inequality for the variance of the parameters is given as,

equation image

where the mnequation image element of the Fisher information matrix, F, is given by Weiss and Friedlander [1993] as

equation image

For 2-D angle estimation, the unknown parameter vector is defined by p = [ϕ, θ]. Fisher information matrix (FIM) is given by

equation image

where

equation image
equation image

Fequation image, can be written similar to (7). Fequation image = Fequation image and

equation image

If the off-diagonal term, Fequation image is zero, the estimates of the azimuth and elevation angles are uncoupled. But for arbitrary array geometries, this off-diagonal term, Fequation image, is nonzero. For 2-D angle estimation, the CRB defined by Mirkin and Sibul [1991] and Nielsen [1994] takes the coupling effect into account and the CRB for the azimuth and elevation angles are given as,

equation image
equation image

where

equation image

If ρ2 = 1, the estimates are said to be perfectly coupled and the unknown parameters (ϕ, θ) cannot be estimated simultaneously. If ρ2 ≠ 0, uncertainty in one parameter degrades the other parameter's accuracy. ρ2 = 0 is required for uncoupled 2-D DOA estimation. Therefore it is important to consider the coupling effect when 2-D DOA estimation is done. The constraints on array sensor locations for uncoupled DOA angle estimation are reviewed in the following part.

[10] In order to have 2-D uncoupled DOA angle estimation, the off-diagonal terms of FIM must be zero, i.e., Fequation image = 0. The required conditions for uncoupled DOA estimation according to the array sensor locations are derived by Nielsen [1994] as,

equation image

where Pxx, Pyy and Pxy depend on the sensor coordinates,

equation image
equation image

and xc, yc given as,

equation image

(xl, yl) is the lth sensor position and (xc, yc) are array center of gravity. Therefore (13) should be satisfied in order to have uncoupled DOA angle estimation for a planar array.

3. Isotropic Planar Array

[11] CRB for an isotropic planar array in case of a single source is uniform for all azimuth DOA angles. In the works of Baysal and Moses [2003] and Gazzah and Marcos [2006], it is shown that the conditions for an isotropic array are the same as the conditions for an array to have uncoupled azimuth and elevation estimation which is given in the previous part.

[12] In the following part, we present the closed form expressions which return the V angle for uniform and nonuniform V-shaped isotropic planar arrays.

3.1. Isotropic Uniform V-Shaped Array

[13] Let M be an odd number for simplicity and k = equation image is the index of the reference sensor at the origin. The sensor positions for uniform V-shaped array in Figure 2 can be expressed as,

equation image

It is assumed that the sensor positions are symmetric according to the y axis and sensors are separated with a distance which is an integer multiple of a distance d. In order to design isotropic uniform V-shaped array, equation (13) should be satisfied. Since the sensor positions are uniform and symmetric according to the y axis, xc = 0 and Pxy = 0 for all γ angles. The condition Pxx = Pyy should be satisfied for the isotropic angle γiso. The derivation of γiso formulation is presented in Appendix A. The closed form expression for γiso for a uniform V-shaped array is given as,

equation image
Figure 2.

Uniform V-shaped array geometry.

3.2. Isotropic Nonuniform V-Shaped Array

[14] In this case, the distance from the reference sensor is nonuniform for the sensors in the array as shown in Figure 3. It is known that nonuniform arrays can perform better than the same number of element uniform linear array (ULA), in a variety of cases [Tuncer et al., 2007]. There are M1 sensors at the left nonuniform linear sub array and M2 sensors at the right nonuniform linear sub array and a reference sensor at the origin. We can express the sensor positions for the nonuniform V-shaped array as,

equation image

where dl is a real positive number. We have to place the sensors to satisfy (13) for isotropic performance. In order to have Pxy = 0, we need to satisfy the following equations.

equation image

The details of the derivation of the isotropic angle are presented in Appendix B. The closed form expression which returns the V angle for isotropic performance, γiso, for nonuniform V-shaped arrays is given as,

equation image

Both (20) and (21) should be satisfied in order to obtain isotropic performance for a nonuniform V-shaped array.

Figure 3.

Nonuniform V-shaped array geometry.

[15] We can design nonuniform V-shaped arrays for isotropic DOA performance in two steps. In the first step, the sensor locations are selected to satisfy (20). Then the isotropic angle for the nonuniform V-shaped array is obtained as in (21).

4. Directional V-Shaped Planar Array Design

[16] In the previous part, we derived analytic expressions for the design of isotropic V-shaped arrays. While it is useful to have isotropic response in many cases, directional arrays perform better when the array is constrained to look more sensitively to a certain angular sector. In this part we present a design procedure for the directional V-shaped arrays.

[17] The design procedure finds the optimum γ° angle to obtain the best DOA performance. Before going through the design steps we need to understand the characteristics of the V-shaped arrays. CRB for different DOA angles is shown in Figure 4 for different V angles. The characteristic is periodic by 180 degrees. The best performance is seen at 90 and 270 degrees and the worst performance is seen at 0 and 180 degrees. This characteristic is observed when the V-shaped array is configured as shown in Figure 1, where the subarrays are placed symmetrically with respect to y axis. Note that such kind of configuration can always be realized by defining x and y axis appropriately. When we change the V angle, γ, the best and worst performance levels and the width of these regions are changing. We need to find the best V angle for the desired directional response.

Figure 4.

DOA performance of nine-element V-shaped arrays with different γ angles for a single source which is swept between all azimuth angles with 256 snapshots and 20 dB SNR.

[18] In the design procedure, two regions are specified as shown in Figure 5. Focused region is the angular sector where the best possible DOA accuracy is desired. Unfocused region is an angular sector where a DOA accuracy below a certain level, H1, is acceptable. If a focused region different than the one shown in Figure 5, is desired, array and coordinate axis can be rotated appropriately. Note that focused region is centered at 90 degrees where the array shows the best performance. The azimuth angles α1 and α2 determine the focused region limits. While α1 and α2 can be arbitrary in general, the best performance is obtained if α1 + α2 = 180°. Note that in this case α1 and α2 are symmetrically placed with respect to the 90 degrees. The target is to find γ° given the parameters α1, α2 and H1.

Figure 5.

The design regions and parameters for V-shaped planar array geometry.

[19] In Figure 6 the best and the worst performance levels are plotted with respect to γ angle when the coupling effect of the azimuth and elevation angle estimation is taken into account. Note that if the coupling effect is not taken into account, design of V array can converge to a degenerate case, such as, a linear array (γ = 180 degrees). In this case H1 level goes to infinity and therefore both azimuth and elevation angles cannot be resolved simultaneously. As it is seen, H1-γ and H2-γ curves are monotonic. Once H1 is specified, the corresponding angle in Figure 6 is an upper bound for the best performance. Therefore optimum V angle, γ°, should be less than this angle. The proposed design method has the following steps:

Figure 6.

The best and worst performance levels of the azimuth CRB versus V angle, γ, when α1 and α2 are 90 degrees.

[20] Step 1: H1, α1, and α2 values are specified (Figure 5). We assume that α2 = 180 − α1 for simplicity.

[21] Step 2: From Figure 6, γ angle (γ1) corresponding to H1 is found.

[22] Step 3: CRB expression in (10) is evaluated for the α1 azimuth angle corresponding to the V angle, γk, namely CRB(α1, γk). The cost for γk is e(k) = CRB(α1, γk).

[23] Step 4: Decrease γk angle by Δ, γk+1 = γk − Δ, and repeat step 3 for k = 2, …, K. Δ is the step size and K = (γ1γiso)/Δ

[24] Step 5: Find the minimum e(k) and the corresponding γk angle as the optimum V angle, γ°

equation image

5. Analysis of Mutual Coupling Effects

[25] Mutual coupling between antenna elements is an important factor which degrades the DOA estimation performance. CRB for unknown mutual coupling matrix (MCM), C, is the fundamental tool in order to quantify the DOA performance [Friedlander and Weiss, 1991; Ye and Liu, 2008]. In this paper, the CRB formulation of Ye and Liu [2008] is implemented in order to compare the UCA and isotropic V-shaped arrays. The array output in case of mutual coupling can be expressed as,

equation image

The target in this part is to compare the DOA performances for UCA and V-shaped arrays in a fair manner. In order to achieve this target, both arrays are constructed by employing dipole antennas with λ/2 size and 50 ohm load in FEKO [EM Software and Systems S.A. (Pty) Ltd., 2008]. The radius of the dipole is selected as 1.5 × 10−3λ and the operating frequency is 30 MHz. There are 9 antennas and the intersensor distance is set to λ/2 for both arrays. FEKO is an electromagnetic simulation tool which can model the antenna elements with sufficient accuracy and close to the practical situation. Tables 1 and 2 present the distance between array elements for UCA and UI V-shaped arrays, respectively. The mutual coupling between two antennas depends on the distance between antennas. As the distance increases, the magnitude of the coupling coefficient decreases. In the literature, MCM for UCA is usually represented with only one coefficient [Friedlander and Weiss, 1991]. In addition, the coefficients for the antennas with a distance greater than 0.707λ are ignored [Ye and Liu, 2008]. In this paper, we ignored the coefficients when the distance between antennas is greater than λ in order to have a more accurate evaluation. Tables 3 and 4 show the MCM matrices and the mutual coupling coefficients for the two arrays. It can be seen that UI V-shaped array uses seven coefficients whereas the UCA array uses only two coefficients. In addition, coupling coefficients for the same distance may be different for the V-shaped array due to the different interaction between antennas. The coupling coefficients for two arrays are given in Table 5.

Table 1. Distance Between Sensors for Nine-Element UCA in Terms of λ
UCA123456789
100.50.9391.2661.4391.4391.2660.9390.5
20.500.50.9391.2661.4391.4391.2660.939
30.9390.500.50.9391.2661.4391.4391.266
41.2660.9390.500.50.9391.2661.4391.439
51.4391.2660.9390.500.50.9391.2661.439
61.4391.4391.2660.9390.500.50.9391.266
71.2661.4391.4391.2660.9390.500.50.939
80.9391.2661.4391.4391.2660.9390.500.5
90.50.9391.2661.4391.4391.2660.9390.50
Table 2. Distance Between Sensors for Nine-Element UI V-Shaped Array in Terms of λ
V123456789
100.511.521.7531.6271.6491.815
20.500.511.51.2721.2191.3611.649
310.500.510.8140.9081.2191.627
41.510.500.50.4540.8141.2721.753
521.510.500.511.52
61.7531.2720.8140.4540.500.511.5
71.6271.2190.9080.81410.500.51
81.6491.3611.2191.2721.510.500.5
91.8151.6491.6271.75321.510.50
Table 3. Mutual Coupling Matrix for Nine-Element UCA
UCA123456789
11c1c2    c2c1
2c11c1c2    c2
3c2c11c1c2    
4 c2c11c1c2   
5  c2c11c1c2  
6   c2c11c1c2 
7    c2c11c1c2
8c2    c2c11c1
9c1c2    c2c11
Table 4. Mutual Coupling Matrix for Nine-Element UI V-Shaped Array
V123456789
11v4       
2v41v2      
3 v21v3 v7v6  
4  v31v5v1v7  
5   v51v5   
6  v7v1v51v3  
7  v6v7 v31v2 
8      v21v4
9       v41
Table 5. Mutual Coupling Coefficients of UCA and UI V-Shaped Array
UCAUI V Array
c1 = 0.1534 + 0.1019iv1 = 0.1334 + 0.2059i
c2 = −0.0347 − 0.0960iv2 = 0.1386 + 0.1198i
 v3 = 0.1549 + 0.0924i
 v4 = 0.1268 + 0.1210i
 v5 = 0.0876 + 0.1482i
 v6 = 0.0124 − 0.1490i
 v7 = 0.0722 − 0.0915i

[26] The real and imaginary parts of the elements of the MCM contribute to the Fisher Information Matrix (FIM). Therefore as the number of coefficients increases, the size of the FIM and its condition number increases. It may be no longer well conditioned [Svantesson, 1999]. This also disturbs the smoothness of the CRB characteristics. As a result, the increase in the number of coupling coefficients decreases the accuracy of DOA performance.

[27] For a single source, the number of unknowns is large compared to the number of equations for UI V-shaped array in (23). When some of the unknowns are ignored and MCM is estimated, the DOA accuracy decreases. As a result, the DOA performance of UI V-shaped array is worse than the UCA for a single source. It is also observed that its performance gets better than the UCA when the number of coupling coefficients is decreased. As the number of sources increases, the number of equations increases and MCM can be estimated accurately. In our simulations, we have found that UI V-shaped arrays perform better than UCA when there is more than one source. The comparisons of the performances of the two arrays are presented in the following section.

6. Simulation Results

[28] In this section, we consider the isotropic and directional V-shaped arrays in order to show the characteristics of the V array for different cases. Examples of the isotropic uniform and nonuniform V arrays are considered and compared with UCA. Furthermore the effect of sensor position error is investigated for both V-shaped and circular arrays by using the MUSIC algorithm.

[29] In simulations, source angles are considered in degrees where azimuth angles are between 0 and 360 degrees and elevation angles are between 0 and 90 degrees (Figure 1). There are 1000 trials for each experiment and the number of snapshots is 256.

6.1. Simulations for Uniform Isotropic V-Shaped Arrays

[30] UI V-shaped planar arrays can be easily designed from equation (18) for a specified number of sensors, M. For example if M = 9, γiso is 53.9681°. The performance of this V-shaped array is compared with the UCA in Figure 7. There are three sources at the azimuth angles ϕ1 = 60, ϕ2 = 100 and ϕ3 = 120 degrees and elevation angles are fixed at θ = 90 degrees for all sources. Source signals are uncorrelated. As it is seen from Figure 7, UI V-shaped array shows better performance than the circular array when both arrays have the same number of sensors and intersensor distances. In Figure 7, the performances of V-shaped array and UCA are outlined when there is an error in sensor positions denoted by pe. pe is an error with respect to the intersensor distance, d = λ/2 where λ is the wavelength. Therefore % 2 position error corresponds to equation image = 0.02. Error displacement is on a circle with radius ∣pe∣ and the circle center is at the true sensor position. Figure 7 shows that both the UI V array and UCA have similar robustness for the various position errors (% 2, % 1, and % 0.2). Also it is evident that the UI V array has better performance for each of the position errors.

Figure 7.

Azimuth DOA performance for three sources at 60, 100, and 120 degrees, respectively, when UI V-shaped array and UCA are used without and with sensor position errors (% 2, % 1, and % 0.2).

[31] Figure 8 shows the DOA performance of UI V array and UCA for correlated source signals. There are two sources at the azimuth angles ϕ1 = 80 and ϕ2 = 85 degrees, respectively, and the elevation angle is fixed at θ = 90 degrees for each source. SNR is set to 15 dB for the equi-power sources. The source covariance matrix, Rs is taken as,

equation image

where ρ is selected as a positive real value in [0,1] for simplicity. It turns out that the UI V array has better performance for the correlated sources signals. The difference between V array and the UCA increases as the value of ρ increases especially for the values close to ρ = 1. Note that ρ = 1 corresponds to the coherent source case.

Figure 8.

Azimuth CRB DOA performance of nine-element UI V array and UCA for two sources when the sources are correlated with the correlation coefficient ρ. Sources are at 80 and 85 degrees, and elevations are fixed at 90 degrees. SNR is equal to 15 dB.

[32] Figure 9 shows the DOA performance when there are two sources fixed at 161 and 180 degrees and the third source is swept between 0 and 360 degrees. Figure 9 shows the CRB characteristics with and without unknown mutual coupling. The SNR is fixed at 20 dB. It can be easily seen that the coupling decreases the DOA performance. However the DOA performance for UI V-shaped array is better than the UCA for all of the DOA angles.

Figure 9.

CRB DOA performance with and without unknown mutual coupling of UI V-shaped array and UCA for two sources when one source is swept between 0 and 360 degrees while the other source is at 161 degrees. Elevation angles are fixed to 90 degrees.

[33] Figure 10 shows the SNR performance of the UCA and UI V-shaped array for three sources at 60, 100 and 120 degrees, respectively, with and without unknown mutual coupling. It can be seen that the DOA performance degrades due to mutual coupling but the performance of UI V-shaped array is better than the UCA.

Figure 10.

CRB DOA performance with and without unknown mutual coupling for three sources at 60, 100, and 120 degrees, respectively, when UI V array and UCA are used. Elevation angles are fixed to 90 degrees.

6.2. Simulations for Nonuniform Isotropic V-Shaped Arrays

[34] In case of nonuniform V-shaped array, we select the left arm as a nonredundant nonuniform linear array (NLA) for simplicity. The sensor locations for the NLA with respect to d = equation image are dNLA = [0, 1, 4, 6]. The right arm can be adjusted to have M1 = M2 or M1M2. Sensor positions of the right subarray are selected in order to satisfy (20). Then γ°iso is determined from (21). Some of the examples for isotropic nonuniform arrays are presented in Table 6. CRB levels of the designed isotropic nonuniform arrays are given in Figure 11. Figure 11 shows that DOA accuracy can be significantly improved with nonuniform V-shaped arrays for the same number of sensors. Note that this result is obvious due to the fact that array aperture is increased. However, NLA still returns unambiguous solutions since there is at least two sensors with the intersensor distance less than equation image.

Figure 11.

CRB DOA performance of nonuniform isotropic (NUI) V array and UCA for a single source is swept between 0 and 360 degrees when M = 7, M = 10 and elevation angles are fixed to θ = 90° and SNR = 20 dB.

Table 6. Isotropic Nonuniform V-Shaped Design Examples for M1 = M2 and M1M2
Nonuniform Sensor Positionsγisoo (deg)
equation image61.0530o
equation image61.0530o
equation image57.0976o

6.3. Simulations for Directional Uniform V-Shaped Arrays

[35] In the directional case, sources are assumed to be localized in an angular sector. We choose design parameters as α1 = 80o, α2 = 100o and H1 = 0.5°. Angular step size is Δ = 1° for M = 9 sensors and the number of snapshots N = 256. If the design procedure is applied for these parameters, the best DOA performance is obtained for γ° = 119°. In Figure 12, there are two sources at ϕ1 = 81° and ϕ2 = 98° degrees. Figure 12 shows that designed directional uniform (DU) V-shaped array has better DOA performance than UCA and L-shaped array (γ = 90°). The DOA performance for the elevation angle is shown in Figure 13 for ϕ1. As it is seen from Figure 13, elevation performance of the directional V-shaped array changes depending on the azimuth angle. Circular array has uncoupled azimuth and elevation angle response. Figure 14 shows the DOA performance when there are two sources fixed at 83 and 99 degrees and third source is swept between 0 and 360 degrees in one degree resolution. SNR is fixed at 20 dB. Figure 14 shows that DU V-shaped array has significantly better resolution and DOA performance than UCA.

Figure 12.

CRB DOA performance for two sources at ϕ1 = 81°, ϕ2 = 98°, respectively, when DU V array and UCA are used (elevation angles are fixed to θ = 90°).

Figure 13.

The elevation CRB for DU V array and UCA with different azimuth angles (for ϕ1).

Figure 14.

CRB DOA performance of DU V array and UCA for three sources when one source is swept between 0 and 360 degrees while the other sources are at 83 and 99 degrees. Elevation angles are fixed to 90 degrees.

7. Conclusion

[36] We have investigated the uniform and nonuniform isotropic and directional V-shaped planar arrays. Closed form expressions for the isotropic performance are presented for both uniform and nonuniform V arrays. V-shaped isotropic arrays are compared with UCA. The comparison is done for a variety of cases which include correlated sources, sensor position errors and mutual coupling. It turns out that the isotropic V-shaped array has better performance than UCA for the same number of sensors and intersensor distance. The source signal correlation and sensor position error do not change the superiority of the UI V array. In case of mutual coupling, UI V-shaped array has better performance for multiple sources. It is shown that DOA performance can be improved significantly when isotropic nonuniform V-shaped arrays are used.

[37] A design method for directional uniform V-shaped array is proposed. The proposed method finds the optimum V angle, γ°, for the specified design parameters. When the sources are in an angular sector, DU V-shaped array performs significantly better compared to UCA. It turns out that V-shaped arrays have the better performance for the same number of sensors and interelement distance due to its effective aperture.

Appendix A:: Isotropic V Angle for Uniform V-Shaped Arrays

[38] In this Appendix A, we derive the closed form equation (18) which returns isotropic V angle for uniform V-shaped arrays. The array center of gravity (xc,yc) for uniform and symmetric V-shaped arrays, is given as

equation image
equation image

For isotropic V-shaped arrays Pxy, must be zero. Since xc = 0,

equation image

If we open this equation,

equation image

where equation image(lk)∣lk∣ = 0 and equation image(lk) = 0, so Pxy = 0. Pxx must be equal to Pyy for isotropic response. We can find Pxx as,

equation image

which gives

equation image

Then we need to find Pyy

equation image

where

equation image

and

equation image
equation image

Therefore if we combine the expressions in (A7), (A8) and (A9), with (A6), we get Pyy as,

equation image

Using the equations (A5) and (A10) in order to satisfy (13), V angle is found as,

equation image

Appendix B:: Isotropic V Angle for Nonuniform V-Shaped Arrays

[39] In this part, the derivation of (20) and (21) for nonuniform V-shaped isotropic planar array is presented.

equation image
equation image

Pxy must be zero for isotropic response.

equation image

The above equation is satisfied only if

equation image

So xc becomes zero and Pxy = 0. We need to equate Pxx to Pyy in order to get isotropic response.

equation image

If (B4) is satisfied, Pxx and Pyy can be written as,

equation image
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where

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Therefore if we substitute (B8), into (B7), we get Pyy as,

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If we equate (B6) and (B9) in order to satisfy isotropy condition (Pxx = Pyy), we get γiso as in (21).

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