## 1 Introduction
It is noteworthy that all experiment results in [20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39] demonstrate that the single-step approach outperforms the two-step method for a low signal-to-noise ratio (SNR) and small number of samples. Meanwhile, although this kind of localization method may require more computations and communication bandwidth, novel information technology [40; 41; 42] can be used to overcome these difficulties. For example, the cloud computing and cloud storage technology [40; 41] can be used to reduce the computation loads, and the compressive sensing technology [42] is helpful for reducing the communication bandwidth. In the field of array signal processing, super-resolution DOA estimation methods are known to be sensitive to uncertainties in the array manifold. In recent decades, much attention has been paid to the analysis of the sensitivity of classical DOA estimation algorithms to array model errors. In [43; 44; 45; 46; 47; 48; 49; 50; 51], the statistic performance of the multiple signal classification algorithm and its extensions in the presence of array model errors is studied. An analysis of the estimation of signal parameters via rotational invariance techniques under random sensor uncertainties is performed in [52], and a sensitivity analysis of the weighted subspace fitting algorithm under the combined effects of array model errors and finite samples is presented in [53]. The statistical performance of the maximum likelihood algorithm is also investigated in [54; 55] assuming that array calibration errors exist. Additionally, efficient parameter estimation algorithms are proposed with an uncalibrated array [56; 57] or partly calibrated array [58; 59; 60]. Array model errors are typically caused by gain/phase uncertainties, mutual coupling, and sensor position perturbations. Note that all DPD methods presented in [29; 30; 31; 32; 33; 35; 36; 37; 38; 39] also rely on the accurate knowledge of the array manifold and, therefore, it seems reasonable to expect that their localization accuracy is also severely degraded by array uncertainties. Although the estimation performance of the DPD method in the presence of array model errors is rigorously analyzed in [34; 61; 62], these theoretical studies are simply performed for the case where signal waveforms are known. However, this is rarely realistic for non-cooperative communications. In this paper, the location performance of the DPD method in the presence of array model errors is examined when signal waveforms are not known in advance. Our theoretical analysis focuses on the DPD estimator in [29] because of its fundamental role in the field of direct localization. Because the objective function of this DPD estimator is formulated as the maximum eigenvalue of a Hermitian matrix, the theoretical development begins with a matrix eigen-perturbation result, which expresses the perturbation of eigenvalues as a function of the disturbance added to the Hermitian matrix. Subsequently, the first-order asymptotic expression of the localization errors is given, from which the analytical formula for the mean square error (MSE) of the DPD estimator is available. Furthermore, two exact formulations for the calculation of the probabilities of a successful localization are also deduced, which offers another statistical perspective on the study of the estimation performance. Finally, Cramer-Rao bound (CRB) expressions for the position estimation are derived for two cases: (a) array model errors do not exist and (b) array model errors are present and follow a Gaussian distribution. The obtained CRBs provide further insights into how array model errors affect the localization performance. The remainder of this paper is organized as follows. Section 2 lists the notational conventions that will be used throughout the paper. In Section 3, the signal model for direct localization is formulated. Section 4 briefly describes the DPD method, which is first proposed in [29].