## 1 Introduction  
Military mobile platforms, such as naval ships and unmanned aerial vehicles, are equipped with multiple antenna systems operating on different frequency bands to perform various missions: radar, electronic warfare (EW), and communications [1, 2, 3]. However, the installation of multiple antennas often raises the radar detection rate, especially when more antennas are protruded from the body of the mobile platforms, due to increased radar cross sections [4, 5, 6, 7]. Thus, active research has been carried out on a shared-aperture antenna (SAA) to integrate multiple antennas into a limited space below the surface for future mobile platforms [8, 9, 10]. Although the SAA allows for a reduced detection rate and various functionalities, there are some inevitable issues that must be resolved to allow its practical use. First, the probability of a system malfunction may increase due to wave scattering and mutual interference between nearby antennas arranged within a restricted aperture area [11]. The interference becomes more significant when operating frequencies, including harmonic resonances, are overlapped with each other, which is unavoidable because the EW antenna occupies a broad bandwidth with a max-min frequency ratio greater than 10:1 [12, 13]. Second, the radiation properties of high-band antennas are often distorted by the electrically large geometry of low-band antennas due to serious platform effects--i.e., wave scattering, reflection, blockage, and diffraction [14, 15, 16]. Third, scan blindness occurs in beamforming operations when radar antennas exhibit strong mutual coupling [17; 18]. Therefore, it is essential to suppress the mutual coupling while preventing undesired pattern distortions caused by platform effects. Note that pattern distortion also occurs when strong mutual coupling exists; however, scattering parameters do not fully represent the degree of pattern distortion, but only the amount of power coupled to other antenna ports [19]. Although there is an alternative approach to a quantitative evaluation of pattern distortion that calculates the average deviation between stand-alone and distorted patterns [20], it is not suitable for estimating phase distortions that are critical for the beamforming operation. Thus, there has been increasing demand for a new figure of merit (FoM) that fully considers the effects of both mutual coupling and pattern distortion. In this paper, we propose a pattern distortion coefficient as a new FoM for quantitatively evaluating both mutual coupling and pattern distortions in multi-antenna systems. The proposed coefficient is defined as a cross correlation between unaffected and affected far-field patterns of an antenna under testing (AUT) for any cause: mutual coupling, platform effects, or both. It is worth noting that the unaffected pattern must be obtained by eliminating only the cause of what we are interested in, herein called \"cause of interest\". For instance, if we are interested in the effect of only a low-band antenna, then the proposed pattern distortion coefficient should be calculated by analyzing the far-field patterns of the AUT with and without the low-band antenna. In our approach, the input patterns were weighted using a Gaussian function whose mean and standard deviation were adjusted to specify the main angular range for the target antenna operation. The proposed FoM was evaluated for a two-antenna system composed of an inverted-F antenna (IFA) operating at 1.5 GHz and a microstrip patch antenna with a resonant frequency of 6 GHz. The standard deviation of the Gaussian weights was adjusted to allow us to better see the tendency, and the results were compared with the conventional mutual coupling by changing the antenna interval and observation frequency. The evaluation was further extended to a single-antenna system mounted beside a conducting wall that generated strong platform effects to provide a quantitative insight into the amplitude and phase distortions of far-field patterns.