## 1 Introduction  
Time-frequency analysis has been introduced for nonstationary signal analysis in many fields, for instance, machine monitoring, speech, the automotive industry and biomedicine [1; 2; 3]. In the last many years, the spatial dimension has been combined with the time and frequency dimensions, including quadratic and higher-order time-frequency distributions (TFDs), which resulted in spatial time-frequency distributions (STFDs) development for nonstationary array signal processing [4; 5; 6; 7]. The relation between the sensor data TFDs and the source TFDs is confirmed by the steering mixing matrix. The sensor data TFDs are similar to the traditional data covariance matrix in array signal processing [8; 9]. This similarity has a guaranteed subspace algorithm to use the instantaneous frequency of the source for signal DOA estimation. The MUSIC [10] and ESPRIT [11] algorithms based on STFDs are superior to their counterparts based on the data covariance matrix when utilized for the DOA estimation of nonstationary signals [4; 12]. Polarization is generally not only used in satellite communications and wireless but also in various radar array signal processing methods [13; 14; 15; 16]. Antenna and target polarization characteristics are generally applied in synthetic aperture radar (SAR) systems and remote sensing [17; 18]. Polarization information is included in airborne and spaceborne platforms and meteorological radars [19]. Moreover, for target identification in the clutter, polarization plays a significant role [20]. Polarization information has also been applied for polarization sensitivity antenna arrays to promote signal parameter estimation performance, which primarily includes time of arrival and DOA [21; 22; 23]. The two significant fields of time-frequency analysis and polarization signal processing have rarely cooperated in the same platform, despite the abundant research executed in each field separately [24; 25]. In this study, we propose SPTFDs based on cross terms for dual-polarization arrays, where the signal T-F and polarized characteristics are simultaneously sufficiently applied. The virtues of the SPTFDs based on cross terms are confirmed by far-field narrow-band point signals incoming to the array. The polarized information increases an additional degree of freedom of the STFDs, resulting in ameliorative signal spatial resolution and DOA estimation accuracy. When strong and weak nonstationary signals simultaneously arise, the weak target signal auto terms are submerged into the background noise in the time-frequency analysis. They are difficult to extract and use directly. The cross terms also contain weak target signal information [26; 27]. Unlike the suppression of cross terms [28; 29], the SPTFD matrix is constructed by the T-F points of cross terms to estimate the weak target signal DOA in this algorithm [30; 31]. SPTFDs are utilized for defining the polarimetric time-frequency MUSIC (PTF-MUSIC) algorithm, combining the source T-F and polarized characteristics for nonstationary signal polarization DOA estimation. The PTF-MUSIC algorithm is superior to the MUSIC algorithms that merely consider polarization or T-F peculiarities. In addition, an ESPRIT-like algorithm is proposed in [32]. The algorithm in this paper combines the characteristics of the MUSIC and the ESPRIT algorithms to ensure success rate and accuracy. Rough estimation and fine estimation are combined to ensure that the algorithm has a high success rate and relatively accurate target direction estimation when strong and weak nonstationary signals simultaneously arise. This paper mainly solves the problem of the poor DOA estimation performance of weak nonstationary polarized signals in the presence of strong nonstationary polarized interference signals and additive white Gaussian noise. Polarization direction of arrival estimation is proposed using dual algorithms based on time-frequency cross terms. Instead of suppressing cross terms, the method uses cross terms.