## 1 Introduction  
The direction of arrival (DOA) estimation of a signal source is important for electronic warfare. Detecting and monitoring the direction of high-speed vehicles such as UAVs (unmanned aerial vehicles) is essential to defend allies. Recently, high-speed UAVs have been developed for electronic warfare, because they can be used for attack or surveillance without human intervention [1; 2]. Direction finding systems have wide applications in many military and civilian fields. In recent wars, electronic warfare has become important to defend against enemy attacks, both in aerial and underwater war. In aerial war, an active seeker could transmit and receive pulse signals and detect the DOA, velocity, and range of targets [3; 4; 5]. In an underwater war, the noise of a submarine propeller is used for localization. By using the spatially sparse property, high-resolution direction finding algorithms have been developed for underwater applications [6; 7; 8]. Recently, civilian fields for direction finding have been introduced for autonomous vehicles. Many sensors are required for self-driving. Especially, 77 GHz radar sensors are used for detecting targets that are far away from radar sensors and for crash avoidance [9]. Frequency hopping signals could be used for wireless communication in UAVs. Using frequency hopping signals, information can be protected from various kinds of noise and multi-path distortion, and signal concealment and encryption are possible. Only the receiver having the spreading code can recover the encoded information [10]. DOA estimation of frequency hopping signals was studied in [11; 12; 13; 14]. In frequency hopping (FH) systems, the carrier frequency is continuously changing. In DOA estimation, strictly speaking, the frequency in the array manifold should be identically equal to the frequency of the incident signal. Since getting the instantaneous frequency of the incident FH signal is very hard to implement, nominal frequency should be used in the calculation of the array manifold. Therefore, there is a mismatch between the instantaneous frequency of the incident signal and the frequency in the array manifold, which results in the performance degradation of the DOA estimation algorithm. In this paper, for the interferometer DOA estimation algorithm [15; 16; 17], explicit expressions of the azimuth estimation error and the elevation estimation error due to the frequency mismatch are derived and validated via numerical results. In various previous studies, an analytic error due to phase measurement error for the interferometer DOA estimation algorithm was derived. Its correlation is defined by the difference between the measured phase of each sensor and phase of the array manifold of each sensor for all possible incident angles. The incident angle corresponding to the maximum value of correlation is chosen to be the DOA estimate [18; 19; 20]. In comparison with high-resolution algorithms like MUSIC [21] and ESPRIT [22], the interferometer algorithm is computationally efficient, but the performance in terms of the resolution and accuracy of the interferometer algorithm is worse than that of high-resolution algorithms. Various cost functions could be defined such as least squares, the cosine function, and a correlative method [23; 24; 25]. Differences in the cable length of the sensor array or thermal noise could generate an additive phase error for the phase measurement of the incident signal. The analytic error of the azimuth angle (1D DOA) of the signal incident on the uniform circular array has been derived [26], and the performances of the interferometer algorithms for various cost functions were compared in [27]. In [26] and [27], the authors derived an analytic expression of the error for the azimuth estimation of an interferometer algorithm. In [27], the phase error was assumed to be a Gaussian random variable, and a uniform random variable was introduced to reflect the fact that the phase error could be any value between zero and \(2\pi\).