## 1 Introduction  
Speech-related services are pervasively available on mobile devices such as smartphones or tablets. However, reverberant and noisy environments, where these devices are frequently used, often degrade speech signal quality and/or intelligibility [1]. Many current devices include several microphones, so that multi-channel speech processing techniques can be applied to reduce the distortions, which improves the noise reduction performance compared to single-channel approaches. This is our research focus. The most common multi-channel speech processing technique is beamforming [2], which applies spatial filtering to the noisy speech signals captured by several microphones. One of these beamformers is the well-known Minimum Variance Distortionless Response (MVDR) beamformer [3], which has the advantage of being able to reduce the noise power without introducing speech distortion. The performance of MVDR depends on an accurate estimation of the noise spatial characteristics and the acoustic transfer function (ATF) between the target speaker and the microphones. For the estimation of the noise spatial statistics, methods based on multi-channel speech presence probability (SPP) [4, 5] are commonly used when there is no prior knowledge of the signal propagation or the spatial structure of the noise. These techniques update the noise information when speech absence is detected, which allows for the tracking of time-varying noise signals. However, the estimation of ATFs is a more challenging task, as these depend on the speaker's location, room acoustics and microphone responses. One possible solution is based on the estimation of the beamformer weights for a certain reference microphone. As a result, the problem becomes that of estimating a relative transfer function (RTF) between the reference microphone and the rest of microphones [6]. The most common RTF estimators are based on sub-space search, using estimates of the noisy speech and the noise spatial correlations. This category includes techniques such as covariance subtraction [7], covariance whitening [8; 9] and eigenvalue decomposition [10; 11]. A weighted least-squares RTF estimator was proposed in Reference [12] but it is unfeasible for real-time applications. Other approaches based on online expectation-maximization [13; 14] have been analyzed to jointly estimate the RTF and the clean speech and noise statistics under the assumption of a spatially-white noise field. Recently, we proposed an extended Kalman filter (eKF) framework for the estimation of the RTF between two microphones [15] and evaluated its performance on a dual-microphone smartphone under different noisy and reverberant environments. We showed that this technique obtains better estimation accuracy than the sub-space-based estimators while allowing for the tracking of the RTF variability, especially in highly reverberant scenarios. Despite beamforming algorithms are often used in multi-microphone devices, their performance is quite limited on dual-microphone smartphones mainly due to the reduced number of microphones, their particular placement on the device and the short separation between them [16]. Therefore, alternatives to beamforming are necessary to obtain a good performance in these situations. One possible approach is the use of single-channel filters for the reference microphone using statistical information obtained from the dual-channel signal. For example, the power level difference (PLD) algorithm proposed in Reference [17] exploits the clean speech power difference between microphones when the smartphone is used in close-talk (CT) conditions (i.e., when the loudspeaker of the smartphone is placed at the ear of the user). An estimate of the single-channel noise statistics is obtained for both microphones and a Wiener filter is calculated from them.