# Big-Delay Estimation for Speech Separation in Assisted Living Environments
## 1 Introduction  
Selective Hearing (SH) is described in [1] as the ability of listeners to focus on a specific sound source within a competing speaker scenario in their auditory environment. The machine emulation of this process via SH technology involves source localization, separation, and enhancement techniques, as well as unifying frameworks that integrate these methods. Machine emulation of SH seeks to enhance the intelligibility and quality of a target speaker's voice at the ear of a listening-impaired person. Speech enhancement focuses on improving the perceptual quality of speech that has been degraded by additive noise [2]. These algorithms work to reduce or suppress background noise and are oftenreferred to as noise suppression algorithms [3]. In conjunction with interference suppression (IS), these types of problems are closely related to the problem of source separation (SS) [4]. For better interference reduction performance, researchers often introduce the assumption that each audio signal in a multi-audio scenario is in proximity to at least one microphone channel, that predominantly captures the target signal [5; 6; 7; 8; 9]. These methods have extensive applications in assisted living (AL) scenarios where hearing aids (HA) are used [9; 10; 11], in voice-controlled smart homes [12; 13; 14] for elderly people and in other speech AL technologies [15; 16; 17]. Audio SS involves extracting individual sources from a combined audio mixture. This technique has applications in speech enhancement [18], speech recognition [19], and HA devices [20; 21]. This paper is motivated by applications in multi-microphone-based auditory localization systems. In the scenario illustrated in Figure 1a, each mixture captures a different version of the speaker, characterized by variations in spatial location, relative attenuation, and delay parameters. These mixtures are processed by powerful speech demixing algorithms capable of isolating a specific voice of interest [22]. The demixed speech is then transmitted in real-time to a wireless HA using Bluetooth, infrared, or other wireless transmission systems [23]. In these systems, the strategic placement of microphones has proven highly effective in addressing challenges such as noise, interference, and reverberation, particularly in the framework of microphone array processing. Advances in Voice-over-Wireless Sensor Network (VoWSN) systems [24; 25; 26; 27] and Acoustic Sensor Networks [28; 29; 30; 31] are making it increasingly feasible to enable audio communication through microphones and other AL devices [22; 32; 33], which are wirelessly connected and strategically placed throughout large rooms. The performance of audio processing tasks--including speech enhancement, separation, diarization, and recognition--benefits greatly from the spatial distribution of microphones across a large area in a room [34]. Assistive hearing devices encompass a wide range of listening technologies designed to enhance access to speech signals or to amplify speech of interest for individuals with hearing impairments. These devices address challenges where traditional ear-mounted HAs [35] may be insufficient. Examples include environments with significant background noise, large speaker-to-listener distances, interference, or reverberation. By wirelessly linking to a user's HA, these devices ensure the delivery of clear, high-quality demixed target speech signals, ensuring a clear and high-quality broadcast of the demixed target speech signal [32]. The prevalence of wireless devices with microphones, including smartphones, laptops, and hands-free kits, is a common feature of modern life. These devices can also be integrated into the network to improve noise reduction in HAs [36]. State-of-the-art solutions often rely on telecoil-based amplification systems. These systems employ wireless remote microphones (RM) paired with HAs [37], using telecoil technology to amplify and transmit the speaker of interest's voice via Bluetooth, infrared, light waves, or other proprietary wireless technologies [23].