# Big-Delay Estimation for Speech Separation in Assisted Living Environments
## 1 Introduction
RMs [38] allow clearer reception of the distant speaker of interest at the receiving end which is otherwise achievable when relying on the sound-field near the listener's ear [39], in other words when the talker speaks close to the listener's ear. Recent developments aim to help HA users better understand speakers located at a distance [10; 11; 37; 40; 41]. Listening can be challenging when there is excessive background noise, interference, reverberation, or a significant distance between the sound source and the person with hearing loss, as these factors can reduce the perceived signal quality [37]. The use of RM has been shown to improve intelligibility in adverse conditions reliably [37; 41]. The wireless RM provides a relatively clean reference for speech recognition systems [42]. Additional spatial cues are often available in the form of the relative delay/interaural phase difference (IPD) and the received signal strength/interauralintensity difference (IID). These cues correspond to the panning coefficient in the pan-pot mixing model and are captured on the far-end microphone relative to the RM [38], as shown in Figure 1. Ideally, an RM in conjunction with other distant microphones in a wireless acoustic sensor network (WASN) setting mimics the sound-field energy present at the speaker's mouth and can transmit it to the listener's ear in a manner resembling free-field anechoic transmission [39]. Speech enhancement in these settings refers to a class of algorithms that improve the quality and intelligibility of speech transmitted from a far-end microphone--where speech is captured as part of a mixture--to the listeners' HA, which ultimately broadcasts the demixed target speech of interest. Spatial cues help us to distinguish sounds arriving at our ears from different talkers and directions separated in space [10]. Understanding how humans interpret these spatial cues can inform the design of microphone array setups that effectively separate sound sources. For instance, a close-talk or near-field microphone array setup, designed to capture sound sources in close proximity, helps the detection of target speech with a relatively high signal-to-noise ratio (SNR). It decreases in effectiveness with increased speaker-to-microphone distance [43; 44]. A significant gap, larger than half the smallest wavelength content, \(\lambda_{\text{min}}\), of the propagating wave, between the RM and the far-end microphone will render the estimation of relative delay cues/IPDs ambiguous [45; 46]. This is known as the phase wraparound effect. Phase wraparound occurs because the higher-frequency components of the signal cannot diffract around the microphones positioned along their propagation path [47]. These high frequencies are corrupted and pose a challenge when using the time-frequency (TF) representation [48]. Typical indoor temperatures are often maintained between 20 \({}^{\circ}\)C and 25 \({}^{\circ}\)C (68 \({}^{\circ}\)F and 77 \({}^{\circ}\)C) in AL environments. At standard atmospheric pressure, and at a typical room temperature of 20 \({}^{\circ}\)C, the speed of sound is 343 m/s. To accurately capture and represent these sound waves digitally, a sufficient sampling rate is required. Given a sampling rate of \(F_{s}=16\) kHz, the maximum frequency content of that signal is \(f_{\text{max}}=8\) kHz. The bound on the inter-microphone gap so that phase wraparound does not occur, is commonly known as the spatial Nyquist criterion  
\[d_{s}<\frac{\lambda_{\text{min}}}{2}=\frac{c}{2f_{\text{max}}}. \tag{1}\]  
Substituting these values into Equation (1), the maximum permissible separation between sensors is 2.14 cm. Under similar conditions, setting the inter-microphone gap to be greater than 1 m, \(d_{s}>1\) m, allows the maximum frequency that will be uncorrupted to equal up to \(f_{\text{aliasing}}=171.5\) Hz. Frequencies above this maximum frequency, \(f_{\text{aliasing}}\), may experience phase wraparound [49]. This corresponds to 97% of the total frequencies being corrupted, resulting in ambiguous delay estimates.