## 1 Introduction
The disadvantage of spatial filtering is that the degrees of freedom are limited by the number of antenna elements; a spatial filter with a limited number of degrees of freedom is not able to sharply separate the received signal directions. In this way, the mitigation of the interfering signal in the spatial domain also results in the suppression of the useful GNSS signals, particularly when they have a direction close to that of the interfering signals. Nowadays, the research topic on time-frequency (TF) transforms adopted in interference detection for GNSS receivers has obtained increasing attention [9, 10]. In many cases, interferences may appear for a limited time and present a very variable behavior in frequency. In comparison to the GNSS signal, interfering signals are extremely different in terms of time and frequency characteristics. Interfering signals usually present a sharp TF energy concentration in a limited region of the two-dimensional TF plane, while the GNSS signal and noise term spread over the entire TF plane. Since the TF characteristics of the interference can be easily distinguished from those of the useful GNSS signal and the noise term in the TF plane, the TF analysis techniques can be used as very effective countermeasures for detecting a large variety of interfering signals in GNSS receivers. Different TF representations (TFRs), such as the spectrogram and the Wigner-Ville distribution (WVD), have been considered in interference detection for GNSS applications [9, 10]. The spectrogram can be generated via a moving window in time or a moving window in frequency, and its characteristics strictly depend on the used analysis window function. The spectrogram approach presents a trade-off between the time and frequency resolutions as a consequence of [PERSON]'s uncertainty principle, providing poor TF localization properties. In order to deal with the TF resolution limitation present in the spectrogram, WVD has been used in interference detection for GNSS receivers [9, 10]. WVD is well known, since it provides nearly the best TF resolution among all of the TF distributions and also satisfies a large number of desirable theoretical properties, but it presents very severe cross-interfering terms without any physical meaning between true signal components (auto-terms), due to the interaction of different frequency components [11, 12]. The presence of cross-terms leads to a distorted interpretation in the TF plane, which makes the TF transform difficult to visually interpret and results in serious instantaneous frequency estimation error for the interfering signals. The WVD approach is seriously impaired by the cross-term problem, which limits the effectiveness of its sharp localization in multi-component signal situations. Therefore, a total or partial removal of such parasite terms of the bilinear TF distributions is required. In order to overcome or attenuate the cross-interfering terms present in the the quadratic TF distributions, the Choi-Williams transform (CWT) has been proposed to detect the sweep interference for GNSS receivers, but the cross-term attenuation with the CWT approach is paid at the price of TF resolution degradation [10]. Additionally, several kernel design methods have been proposed to mitigate the cross-term effect [13, 14, 15]; unfortunately, these techniques need heavy computational complexity when they are applied in a real-time context. The unsatisfactory results obtained with the existing TF distributions justify the search for better tools; one way of achieving this is to start from the general form of quadratic representations. All of the TF distributions of current interest could be subsumed in the general framework of [PERSON]'s class, whose element can be written as the generalized filtered version of the WVD with a two-dimensional smoothing kernel [11]. This smoothing operation is able to mitigate the cross-terms present in the TF distributions; however, it also produces a less accurate TF localization of the signal components.