# Crossband Filtering for Weighted Prediction Error-Based Speech Dereverberation
###### Abstract
The exact relation between each observed subband and the clean signal results from convolutions between all clean subbands and their corresponding crossband filters. These filters capture the interdependencies and interactions between different frequency bands. The influence and information exchange between adjacent frequency components is taken into account by considering the crossband filters, enabling more comprehensive modeling and processing of the observed speech signal. However, the conventional WPE method in the STFT domain neglects the influence of crossband filters. It assumes that each observed subband is solely the result of a convolution between the corresponding clean subband and a convolutive transfer function (CTF), often referred to as the \"band-to-band filter\" [28]. In other words, the CTF approach approximates each observed subband as solely dependent on its corresponding clean subband. This simplification introduces an inherent error in the WPE model when operating in the STFT domain [19]. The approximation error resulting from neglecting the crossband filters can significantly impact the performance of speech processing methods in the STFT domain. Previous studies have explored the effect of this approximation error in the context of system identification methods [28; 29]. However, for WPE-based speech dereverberation, a preliminary study was conducted in [30]. Unfortunately, this study had limited scope and did not provide a comprehensive analysis of the effectiveness of the WPE-based approach incorporating crossband filtering in various real-world scenarios. Given the importance of accurately modeling the crossband filters, further investigation is necessary to understand their impact on the performance of WPE-based dereverberation. A comprehensive analysis of the effectiveness of the WPE approach with crossband filtering in diverse real-world scenarios is essential to shed light on the potential benefits and limitations of incorporating crossband filters into the dereverberation process. Such research will contribute to advancing the understanding and practical implementation of WPE-based methods, facilitating their optimization and broader application in real-world speech processing scenarios. In this paper, we present an extension to the conventional WPE method that enhances the accuracy of the model approximation in the STFT domain by incorporating crossband filters. Our approach aims to capture the interdependencies between adjacent subbands and refine the estimation of the late reflections in the observed speech signal. By considering samples from neighboring subbands, we redefine the WPE observation vector and modify the prediction inverse filter to include both crossband and traditional band-to-band components. We explore two versions of the proposed method to investigate their effectiveness. The first version prioritizes accuracy by improving the model approximation, albeit at the expense of increased computational complexity. In contrast, the second version maintains the same computational complexity as the conventional WPE while enhancing the accuracy. Surprisingly, the second version demonstrates competitive performance compared to the first method. We conduct a series of experiments to validate the performance of our proposed versions. The results confirm that both versions surpass the conventional WPE regarding dereverberation performance. This highlights the significance of incorporating information from neighboring subbands in the STFT domain in improving dereverberation outcomes. Furthermore, our findings suggest that the early samples of the crossband components might offer greater efficacy in mitigating reverberation than the late samples of the band-to-band component. By presenting these experimental results, we provide empirical evidence supporting the effectiveness of our proposed extensions to the WPE method, offering valuable insights into the potential benefits of considering crossband filters for dereverberation tasks in the STFT domain.