Performance of a Three-Dimensional Antenna Array and Its Application in DOA Estimation

Abstract

In this paper, we develop a novel three-dimensional (3D) multiantenna array for enhancing the performance of multiple input multiple output systems. This array is composed of two uniform circular arrays (UCAs). A frequency nonselective Rayleigh fading channel model is introduced to calculate the spatial fading correlation (SFC) and capacity of a multipath channel. Then, the influence of the rotation angle $\phi _0$ and the perpendicular distance h of the upper UCA in a double uniform circular array (DUCA) with reference to the lower UCA and the radius r of the DUCA on the SFC and the capacity are investigated. In addition, the DUCA is applied to a multiple signal classification algorithm in order to estimate the direction of arrival (DOA). Simulation results show that $\phi _0$ which results in the minimum correlation and the maximum capacity depends on the angle of arrival, and the increase in h and r enhances the capacity performance. However, when $h/\lambda >1/2, h$ has little effect on the capacity. Comparing the spatial spectrum and the root mean square error of the DUCA with a general uniform linear array, UCA, and 2-L array, this paper shows that 3D antenna arrays have significant advantages in DOA estimation.

Appendix A

$$\begin{aligned} \rho (m,n)&= {} G_1\int _{\varphi _0 -\Delta _\varphi }^{\varphi _0+\Delta _\varphi }\int _{\theta _0- \Delta _{\theta }}^{\theta _0+\Delta _{\theta }}e^{j\big \{\zeta (cos(\varphi -\alpha _m)-\cos (\varphi -\alpha _n))-k_wh\cos \theta \big \}} \sin \theta {\mathrm {d}}\theta \mathrm {d}\varphi \nonumber \\&= {} G_1\int _{\varphi _0-\Delta _\varphi }^{\varphi _0 +\Delta _\varphi }\int _{\theta _0- \Delta _{\theta }}^{\theta _0+\Delta _{\theta }}e^{j\big \{(Z_1\cos \varphi +Z_2\sin \varphi )\sin \theta +Z_3\cos \theta \big \}}\sin \theta {\mathrm {d}}\theta \mathrm {d}\varphi \nonumber \\&= {} G_1\int _{\varphi _0+\xi -\Delta _\varphi }^{\varphi _0+\xi +\Delta _\varphi }\int _{\theta _0- \Delta _{\theta }}^{\theta _0+\Delta _{\theta }}e^{j(Z\sin \beta \sin \theta +Z_3\cos \theta )}\sin \theta {\mathrm {d}}\theta \mathrm {d}\beta \end{aligned}$$

(26)

where $G_1=1/(4\Delta _\varphi \sin \theta _0\sin \Delta _\theta )$; $\beta =\varphi +\xi$; $Z_1=\frac{2\pi r}{\lambda }(\cos \alpha _m-\cos \alpha _n), Z_2=\frac{2\pi r}{\lambda }(\sin \alpha _m-\sin \alpha _n)$, and $Z_3=-\frac{2\pi h}{\lambda }$ with $Z=\sqrt{Z_1^2+Z_2^2}$, and $\xi =\tan ^{-1}(Z_1/Z_2)$. The real and imaginary part of $\rho (m,n)$ can be derived as

$$\begin{aligned} Re[\rho (m,n)]&= {} G_1\int _{\varphi _0+\xi -\Delta _\varphi }^{\varphi _0 +\xi +\Delta _\varphi }\int _{\theta _0- \Delta _{\theta }}^{\theta _0+\Delta _{\theta }}\cos (Z\sin \beta \sin \theta +Z_3\cos \theta )\sin \theta {\mathrm {d}}\theta \mathrm {d} \beta \nonumber \\&= {} G_1\int _{\varphi _0+\xi -\Delta _\varphi }^{\varphi _0+\xi +\Delta _\varphi }\int _{\theta _0- \Delta _{\theta }}^{\theta _0+\Delta _{\theta }}\big \{\cos (Z \sin \beta \sin \theta )\cos (Z_3\cos \theta ) \nonumber \\&\quad -\sin (Z\sin \beta \sin \theta )\sin (Z_3\cos \theta )\big \} \sin \theta {\mathrm {d}}\theta \mathrm {d}\beta \end{aligned}$$

(27)

$$\begin{aligned} Im[\rho (m,n)]&= {} G_1\int _{\varphi _0 +\xi -\Delta _\varphi }^{\varphi _0+\xi +\Delta _\varphi }\int _{\theta _0- \Delta _{\theta }}^{\theta _0+\Delta _{\theta }} \sin (Z\sin \beta \sin \theta +Z_3\cos \theta ) \sin \theta {\mathrm {d}}\theta \mathrm {d}\beta \nonumber \\&= {} G_1\int _{\varphi _0+\xi -\Delta _\varphi }^{\varphi _0+\xi +\Delta _\varphi }\int _{\theta _0- \Delta _{\theta }}^{\theta _0+\Delta _{\theta }}\big \{\cos (Z\sin \beta \sin \theta )\cos (Z_3\cos \theta ) \nonumber \\&\quad -\sin (Z\sin \beta \sin \theta ) \sin (Z_3\cos \theta )\big \}\sin \theta {\mathrm {d}}\theta \mathrm {d}\beta \end{aligned}$$

(28)

In these formulas, $\cos (Z\sin \beta \sin \theta ), \cos (Z_3\cos \theta ), \sin (Z\sin \beta \sin \theta )$, and $\sin (Z_3\cos \theta )$ can be expressed by the modified Bessel functions as follows [26]

$$\begin{aligned} \sin (x\sin \theta )=2\sum _{k=0}^{\infty }J_{2k+1}(x)\sin ((2k+1)\theta )\end{aligned}$$

(29)

$$\begin{aligned} \sin (x\cos \theta )=2\sum _{k=0}^{\infty }(-1)^{k} J_{2k+1}(x)\cos ((2k+1)\theta )\end{aligned}$$

(30)

$$\begin{aligned} \cos (x\sin \theta )=J_0(x)+2\sum _{k=1}^{\infty }J_{2k}(x) \cos (2k\theta )\end{aligned}$$

(31)

$$\begin{aligned} \cos (x\cos \theta )=J_0(x)+2\sum _{k=1}^{\infty } (-1)^kJ_{2k}(x)\cos (2k\theta ) \end{aligned}$$

(32)

where $J_k$ is the Bessel function of the kth order. Substitute finite series as follows for the Bessel functions

$$\begin{aligned} J_v(Z)=(\frac{Z}{2})^v\sum _{k=0}^{\infty } \frac{(-1)^k}{(k!)\Gamma (v+k+1)}(\frac{Z}{2})^{2k} \end{aligned}$$

(33)

Substituting (29)–(33) for (27) and (28) and using (18) and (19), the closed-form expression of SFC between $a_m$ and $b_n$ can be derived.