## 1 Introduction  
Wireless positioning technologies enable a wide range of application scenarios where positional information provides key points for the development of unattended and non-invasive context-aware services [1, 2]. In recent years, the emerging of paradigms that can benefit from localization in many applications, such as Internet of Things and Smart Environment [3], led to an increasing interest in both the scientific community and industry. As a consequence, such new scenarios have fostered different localization techniques and the investigation of related technologies [4]. Several methods are available in the literature, due to the very different cases of studies. They can be classified according to the specific signal features involved in the position estimation, such as Direction of Arrival (DoA), Time of Arrival (ToA), Time Difference of Arrival (TDoA), and range evaluation by acquiring the received signal strength [5]. These techniques rely on the established radio links between the target to be positioned and two or more spacial references, also called anchors. In this respect, different antenna structures and methodologies were proposed during the last years. In particular, the Switched Beam Antenna (SBA) technology got notoriety due to the optimal trade-off between simplicity and effectiveness [6, 7, 8, 9]. Such architecture allows to achieve the necessary spatial diversity to properly evaluate received signal features and perform estimation methods. The main drawback for SBAs is due to their structural dimensions, because the high directivity requirement for each antenna element leads to significantly large overall dimensions. In turn, it is common that high directivity SBAs have begun to face phase impairment problems caused by increased distance between antenna center of phases [10]. Such issue is well known in Multiple Input Multiple Output (MIMO) antenna arrays design [11], and an example of its effect is a reduction of field of view in radar imaging systems [12; 13]. As well as the mentioned wireless systems, wireless positioning estimation accuracy is also affected by the phase impairing issue, in particular when enabled by maximum-likelihood (ML)-based algorithms [14] such as the ones in [15; 16]. The trivial workaround to overcome the phase impairing problem is to implement DoA estimation relying only on scalar measurements and ignoring phase information [5; 17; 18]. Nevertheless, the reduction of measurement information certainly leads to a worsening of overall accuracy [19; 20], so a methodology to recover corrupted phase information becomes of fundamental importance for a phase-conscious SBA-based positioning system. In this paper, we propose a novel method based on TDoA measurements able to reconstruct a coherent relative phase information using a natively phase-impaired SBA architecture. The phase reconstruction is done exploiting wideband signals, which provides high time-resolution [17], making them well suited for accurate TDoA estimations. Exploiting the TDoA information together with the signal pulse shaping, a relative phase information can be reconstructed given that a time synchronization between different antenna readings is given. Note that dealing with time differences, an absolute time reference between transmitter and receiver is not needed, thus the proposed method does not require any signal coding scheme needed to embed propagation delay information within the transmitted data packet. To validate the proposed technique, in this work we provide a massive campaign of localization experiments carried out in an anechoic chamber, and laboratory instrumentation instead of dedicated electronic systems. This permits the provision of a pure experimental proof of the method based on wideband signals, without any kind of communication protocol and embedded information; furthermore, the use of instrumentation equipments permits reducing significantly measurement uncertainty, thus leading to more reliable conclusions.