## 1 Introduction  
As an important research area about signal processing, passive location has attracted extensive attention recently. It has applications in many fields including sonar, communication and astronomy [1, 2, 3, 4]. With the development of radio technology, the electromagnetic environment becomes more and more complex which makes it hard for the conventional single antenna to extract accurate and useful information [5]. Besides, single antenna can only get the direction information of the targets without distance and other information. Instead, multiple arrays have the advantages of flexible beam control [6], higher spatial resolution [7] and higher signal gain [8]. Original two-step location technology needs to estimate the intermediate parameters from the original data, such as time of arrival (TOA) [9], time difference of arrival (TDOA) [10], angle of arrival (AOA) [11]. Then, the locations of targets can be estimated through the spatial geometric relationship [12]. Instead of estimating the intermediate parameters, DPD can directly obtain the estimation of target locations from the original data [13]. Therefore, DPD avoids the error caused by modeling the intermediate parameters and can acquire better positioning performance than original two-step location [14]. According to the elliptic covariance characteristics of the source, sources could be divided into two categories: circular sources and non-circular sources [15]. At present, a lot of research has been done on DPD algorithm of circular sources. [PERSON] proposed a DPD algorithm of multiple radio-frequency signals in 2004 [16]. Right after this, [PERSON] proposed a DPD algorithm via maximum likelihood (ML) method [17] in 2007. After some time, [PERSON] provided a direct position determination algorithm: using ML and Capon method [18] to improve the estimation accuracy in 2010. Then, [PERSON] provided a DPD algorithm with moving arrays in 2018 [19]. [PERSON] proposed an iterative adaptive direct localization technology to deal with coherent sources [20] in 2019. [PERSON] proposed a DPD technology with a moving coprime array [21] in 2021. However, many NC signals exist in modern wireless communication systems, like amplitude modulation (AM), quadrature phase shift keying (QPSK) and binary phase shift keying (BPSK) signals. Thus, it is necessary to promote the research of direct localization algorithm with NC sources. In recent years, some scholars have extended the idea of direction of arrival (DOA) estimation of NC signals to DPD of NC signals. [PERSON] used the characteristics of NC sources to improve the location performance in [22; 23; 24]. Then, [PERSON] and [PERSON] used the features of Doppler Shifts and sparse array to further improve the location performance in [25; 26; 27]. However, current DPD algorithms of NC signals need to fuse the received data of each observation station for high-dimensional spectral function search, which undoubtedly has a great computational complexity. In addition, the difference of SNRs for each observation station is often large, and the impact of this difference on the estimation performance of the algorithm can not be ignored [28]. As the existing algorithms do not consider the effect of this difference, they are more sensitive to the locations of the observation stations, resulting in insufficient estimation accuracy. Thus, combined with the idea of Euler transformation [29] and elliptic covariance information of NC sources, this paper derives the Euler ESPRIT DPD algorithm for NC sources (NC-Euler-ESPRIT). As SNRs of received signals at each observation station are different, this paper introduces the principle of power allocation [30] and derives weighted Euler ESPRIT DPD algorithm for NC sources (NC-Euler-WESPRIT). The consequence of simulation shows that the proposed NC-Euler-WESPRIT algorithm not only improves the estimation performance, but also greatly reduces the computational complexity compared with SDF technology and NC-ESPRIT algorithm. The main contributions are summarized as follows:  
1.