## 1 Introduction  
Global navigation satellite system (GNSS) applications utilizing antenna arrays are starting to gain significant attention due to their lower hardware and computational costs. Rapid advances in electronic systems, software define radios (SDR) and miniaturization of radio frequency (RF) front-ends and antenna arrays have opened the way for array processing to play an important role in future receivers [1,2]. Anti-jamming and anti-spoofing methods based on antenna arrays are especially effective for both narrow and wideband interference mitigation and, together with signal to noise ratio (SNR) enhancement, make antenna array processing important for civilian and military applications [3, 4, 5]. In general, GNSS applications based on antenna array processing utilize calibrated and non-calibrated array approaches. Methods not requiring calibration do not employ any information about the array configuration, orientation or direction of the incident signals. They do not generally involve the directions of arrival (DOA) of signals in their structure and could be employed as stand-alone methods for anti-jamming and anti-spoofing [5,6]. Although these methods are not complicated in terms of implementation and complexity, they may degrade receiver performance by accidentally nullifying signals. However, in the latter group, the spatial features of GNSS signals are also utilized in the processing. In these methods, the incident directions of the signals are considered in shaping the array beam pattern to not only nullify the undesired signals but also to enhance the SNR of the received signals [7,8]. However, this initially requires an accurately calibrated array to specify the signal DOA, which is referred to as the steering vector or array manifold vector in the literature. Antenna array calibration has been studied for decades and is employed in many applications such as sonar, radar and direction finding [9,10]. The main array uncertainties that must be considered for a full calibration include uneven cable lengths, array configuration and platform orientation perturbations, unknown antenna element phase center variations, coupling between elements, cross talk in the RF front-ends, antenna gain and phase pattern characteristics and the effect of platform scattering on these patterns. Perturbations in antenna elements positions can occur due to turbulence or tilt of the platform during operation or set up preparation. Even a small inclination error of the platform can degrade the performance of calibration and consequently the spatial processing afterwards. Mutual coupling is the behavior of the coupling between the antennas according to the array configurations and becomes more significant in patch antenna arrays that are not properly isolated. The mutual coupling between antennas is generally proportional to the distances between them and depends on the relative antenna geometry. Phase centre variation is the other consideration that should be taken into account in the calibration procedure unless high quality antenna elements are used in which case phase centres remain constant within 1-2 mm. Although individual antenna elements may have identical horizontal and vertical gain patterns, these will possibly change when assembled into an array due to mutual coupling and scattering. Each of these uncertainties causes an estimated steering vector to deviate from the theoretical one as a function of relative distances betweenantenna elements and the signal DOA. It is worth mentioning that these uncertainties do not have to be treated separately since most of them are related to each other and affect the array beam pattern in the same manner. Calibration is traditionally conducted in an anechoic RF chamber such that the antenna array receives a signal from a precisely known direction. The signal direction is changed in incremental steps until all directions of interest are included. Amplitudes and phases are then measured in the receiver as a function of the incident signal direction.