## 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.