# Performance Analysis of Interferometer Algorithm under Phase Measurement Error
## 1 Introduction  
An analytic MSE (mean square error) due to phase measurement error for interferometer direction-of-arrival (DOA) estimation algorithm is derived in this paper. An advantage of analytic derivation is the dramatic reduction in the computational complexity in comparison with Monte-Carlo simulation-based approach.  
In recent previous studies [1; 2; 3], we have proposed how the closed-form expressions of the MSE of various DOA estimation algorithms can be analytically obtained.  
In [1; 2; 3], closed-form expressions of the MSEs of the estimation errors for the maximum likelihood DOA estimation, the MUSIC algorithm and amplitude comparison monopulse algorithm, respectively, have been derived.  
Various cost functions could be defined such as least-squares, cosine function and a correlative method [4; 5; 6].  
In [7], algorithm for removing phase ambiguity in the interferometer DOA estimation has been proposed and validated for three antenna system and four antenna system. Experimental measurements have also been conducted to validate the proposed scheme.  
In [8], circular array phase interferometer with three antennas is considered. It has been proposed how to solve the problem of phase ambiguity. The scheme is validated via Matlab-based simulation.  
In [9], computationally efficient implementation scheme for a two-dimensional correlation interferometer algorithm is proposed. Two one-dimensional search processes are performed to replace two-dimensional search.  
In [10], the authors adopted machine learning to compensate for disturbances in the direction-finding system for use with interferometer algorithm. More specifically, multioutput least squares vector regression (MLSSVR) model is applied, which results in much improvement in direction-finding accuracy when deviations in the system is large.  
In [11], the authors developed the fast correlative interferometer direction finding. Direct conversion scheme is applied in the implementation. In addition, the authorscompared the performance of the interferometer algorithm with that of the CRLB (Cramer-Rao Lower Bound). Validation using measurement data has been presented.  
In [12], DeSoMP (direction estimation via spectral orthogonal matching pursuit) has been proposed, and the scheme is validated via numerical results. The paper deals with the case where baseline is far greater than the wavelength of an incident signal.  
To the best of our knowledge, no previous studies on analytic derivation of MSEs for simultaneous estimation of azimuth and elevation have been conducted. Explicit expressions of the azimuth MSE and the elevation MSE have been derived and validated using the numerical results. It is shown that the MSE obtained by Monte-Carlo simulations of the interferometer using a cosine function is approximately equal to the analytically derived MSE. In Section 2, we show how to derive the analytic MSE, and in Section 3, agreements between the analytically derived MSEs and the MSEs from the Monte-Carlo simulation are illustrated.  
Various random variables can be adopted to model phase errors. Gaussian random variables are adequate for nonuniform phase error. On the other hand, uniform random variables can be used for modeling random error uniform distributed. Depending on the physical phenomenon of the source of phase error, specific random variables can be adopted.  
For example, uniform random variables are used to model random thermal noise which is distributed uniformly. In addition, phase error due to the discrepancy between actual cable length and nominal cable length can be more adequately modeled as Gaussian random variables since small discrepancies in cable length occurs more frequently than large discrepancies in cable length.