## 1 Introduction
Therefore, the uniform distributed phase measurement error is more adequate for the case where the phase measurement error is due to random thermal noise. On the other hand, the Gaussian random variable tends to lie more densely around the mean values. Therefore, the Gaussian distributed phase error can be used for the phase error due to the difference in the RF cable lengths of the array system. Although the nominal lengths of the RF cable of the array system are equal, the actual lengths can be a little different from channel to channel. Note that, in previous studies on the DOA estimation error due to phase measurement error, the possible frequency offset between the actual frequency of the incident signal and the nominal frequency used in the calculation of the array manifold was no explicitly taken into account. In this paper, we are not concerned with performance degradation due to phase measurement error. That is, phase measurement error is not explicitly taken into account in this paper. On the other hand, we address degradation in the accuracy of the interferometer DOA estimation algorithm due to frequency mismatch or frequency offset between the actual frequency of an incident signal and the nominal frequency used for the array manifold. Frequency offset or frequency mismatch occurs when the local oscillator signal for down conversion in the receiver cannot synchronize with the carrier signal contained in the received signal. This phenomenon can be attributed to two important factors: frequency mismatch in the transmitter and the receiver oscillators; and the Doppler effect as the transmitter or the receiver is moving. In direction of arrival estimation, an emitter transmitting the signal is not cooperative with the direction-finding (DF) station. Therefore, it is almost impossible to remove the frequency offset completely in DOA estimation. A large frequency mismatch occurs in the DOA estimation of the frequency-hopping (FH) signal, since the instantaneous frequency of an incident signal changes very rapidly, especially in a fast hopping environment. Frequency mismatch in this case is usually much larger than frequency mismatch due to failure in synchronization between the oscillator in the receiver and the oscillator in the transmitter. In this paper, explicit expressions of the azimuth error and the elevation error are derived and validated using the numerical results. It is shown that the error obtained by simulations of the interferometer using a cosine function is equal to the analytically-derived error. Our contribution in this manuscript does not lie in how much improvement can be achieved by proposing a new interferometer DOA algorithm or by modifying a previously existing interferometer algorithm. Our contribution in this paper lies in the analytic derivation of the azimuth estimation error and the elevation estimation error. It turns out that the azimuth estimation error is equal to zero and that the elevation estimation error can be obtained from the root of a second order equation or a fourth order equation depending on the approximations employed. Our concern in this paper is how much the azimuth estimation error and elevation estimation error occur due to the mismatch between the frequency of an incident signal and the nominal frequency used for the array manifold. The phase at the \(m^{\text{th}}\) antenna relative to that at the origin for an incident signal with a wavelength of \(\lambda_{RF}\) from \((\theta^{(0)},\psi^{(0)})\) is defined as \(\phi_{m}\), which is specified in (1) in Section 2. The corresponding quantity for a wave with a wavelength of \(\lambda\), not \(\lambda_{RF}\), from general direction \((\theta,\psi)\) is given by the first term in the brackets in (2) in Section 2. We are concerned with, given \((\theta^{(0)},\psi^{(0)})\), what value of \((\theta,\psi)\) in (2) in Section 2 maximizes the cost function in (2) when \(\lambda_{RF}\) is different from \(\lambda\). The arguments maximizing (2) are chosen to be the azimuth estimate and the elevation estimate.