## 1 Introduction
In the case of smartphones used in far-talk (FT) conditions (i.e., when the user holds the device at a distance from her/his face), the algorithm proposed in Reference [18] exploits the spatial properties of the noisy speech and noise signals. This spatial information is used along with a single-channel SPP detector to estimate the noise at the reference channel and the signal-to-noise ratio (SNR), which are needed for the filter design. This proposal was later extended to general multi-channel devices in Reference [19]. While the techniques mentioned above apply a single-channel filtering at the reference microphone, better performance can be achieved when the filter is designed to operate at the beamformer output, what is known as postfiltering [1]. Moreover, the multi-channel Wiener filter can be expressed as an MVDR beamformer followed by a single-channel Wiener filter [6]. Several postfilters based on the Wiener filter have been proposed in the literature [20; 21; 22], mainly differing in the assumptions made about the noise field. The authors of Reference [23] also evaluate the use of non-linear postfilters, showing improvements with respect to the linear approaches. The work in Reference [24] evaluates the performance of a generalized sidelobe canceler (GSC) beamformer along with an SPP estimator and a non-linear postfilter, showing that the SPP information is useful in the postfiltering design. The postfiltering approach has also been studied on dual-microphone scenarios [25; 26]. For example, in Reference [27] we extended our eKF-RTF framework with the use of postfilters along with MVDR beamforming and the SPP estimator of Reference [4] in order to improve noise reduction. These postfilters exploit the SPP and the RTFs previously obtained to estimate the required single-channel statistics. Both linear and non-linear postfilters were evaluated, showing better performance than a standalone MVDR and other state-of-the-art enhancement algorithms intended to be used in smartphones. In this work we analyze and further extend our dual-microphone speech enhancement eKF-RTF framework presented in References [15; 27]. This extension is developed in a threefold way. First, we present a more detailed derivation of the eKF-RTF estimator where we improve the estimation of the _a priori_ RTF statistics without any simplification in the estimation of the covariance matrices. Second, we analyze the _a posteriori_ SPP estimation and the importance of the _a priori_ speech absence probability (SAP). Then, we propose a novel SAP estimator suitable for dual-microphone smartphones, which exploits the spatial structure of the noise and the power differences between microphones, showing improvements with respect to the estimator used in our previous work. Third, we take advantage of the better estimates of SPP, RTFs and single-channel statistics to redefine the postfilters described in Reference [27] and propose new ones. As a result, we end up with a comprehensive dual-microphone speech enhancement system for smartphones that shows a great performance in terms of both speech quality and intelligibility when compared to other state-of-the-art approaches. The rest of the paper is organized as follows: in Section 2, an overview of our dual-microphone speech enhancement system is given, presenting the constituent elements of the algorithm. Next, in Section 3, the eKF-RTF framework is detailed. The estimation of the noise statistics and the SPP along with the new proposals for SAP estimation on smartphones are developed in Section 4. In Section 5, the different postfilters are described and the estimation of the required single-channel statistics is addressed. In Section 6, the experimental framework and results are presented and discussed, while Section 7 finally summarizes the conclusions.