# Effective Dereverberation with a Lower Complexity at Presence of the Noise
## 1 Introduction  
In real applications, reverberation and noise often cause the performance of speech communication or human-computer interaction to be degraded. Especially, due to the presence of reverberation, the speech signal [1] is often distorted, which seriously affects the localization of the sound sources. With the rapid development of artificial intelligence, human-computer interaction becomes increasingly more important [2, 3]. If the reverberation and noise are not well cancelled in front of the intelligent devices, then speech communication will be greatly impaired. In recent years, microphone array-based methods using spatial information have received extensive attention. The microphone array-based methods, such as the MVDR-based beamformer [4], the linear-constrained minimum variance (LCMV) beamformer [5, 6], and the generalized sidelobe canceller (GSC) [7], are often used for speech enhancement. The microphone array-based beamforming methods can be used to receive signals from the direction of the target speaker and suppress the interference, such as reverberation and noise [8, 9, 10]. In [8], the GSC and a post-filter implemented with the Wiener filter were introduced to eliminate the reverberation and noise. In [9], the LCMV was used for the dereverberation in the case of multiple speech sources, in which the obtained sparse matrix was used to obtain a preliminary estimation of the desired signal and the LCMV was used to obtain the final estimation of the desired signal. In [10], a beamforming estimator based on minimum mean square error (MMSE) was proposed to obtain the early components of the reverberation signal. In MVDR beamforming, a relatively early transferfunction (RETF) was used. The beamforming and post-filter were used to manage the late reverberation and noise. The methods that only use beamforming to reduce reverberation have limited performance and cannot suppress the reverberation very well. Currently, the most effective dereverberation method is based on multichannel linear prediction (MCLP) deconvolution [11; 12; 13; 14; 15; 16], which is also known as the direct inverse filtering algorithm. The MCLP-based methods model the reverberation components as a delay of the speech signal in the time domain, namely the linear prediction components. In these methods, the estimation of the room impulse response (RIR) is not required, and the reverberation is subtracted from the speech signal itself. Thus, this method can be applied to the situation where the RIR cannot be reliably estimated. In [13], a statistical model-based dereverberation method was proposed. It assumes that the speech source follows the time-varying Gaussian model. The reverberation is modeled as the delayed components after linear prediction, where linear prediction coefficients are approximated by estimating model parameters to obtain the late reverberation components. This modeling method is defined as the weighted prediction error (WPE). In [11], by combining WPE and MVDR, the noise and reverberation were well suppressed simultaneously. The methods of using Kalman filter to estimate linear prediction coefficients for obtaining reverberation components were also proposed [12; 14; 15]. In [14], the MVDR and the MCLP were used for simultaneous noise reduction and dereverberation, and the Kalman filter was used to estimate the beamforming coefficients and linear prediction coefficients simultaneously. This kind of dereverberation framework that combines beamforming and the MCLP has attracted increasingly more attention. In [15], two interacting Kalman filters were used to alternately estimate the MCLP coefficients and reverberation signal. In [16], a two-stage algorithm was proposed. In the first stage, the WPE was used to reduce reverberation, and in the second stage, the MVDR was utilized to remove the residual reverberation. This method treats the reverberation as a diffuse noise.