## 2 Design of Distance-Rendering Methods
### Envelope-Based Model  
This model is based on a simplified representation of the energy envelope of the impulse response, here divided into two temporal segments: the early energy \(E_{s}\), comprising both the direct sound and early reflections, and the late reverberation energy \(E_{rev}\). The energy of these two temporal segments are modified according to the desired distance. Different perspectives can be considered to demarcate the time limit between these two segments. This transition time can be derived from perceptual considerations, regardless of the room geometry. A time limit of 50 ms has been considered in room acoustics as criteria describing the quality of speech perception, such as _definition_ (\(D50\)) or _clarity_ (\(C50\)). In order to delineate the \"useful sound\" in opposition to the \"detrimental sound\", [PERSON] and [PERSON] [20] consider a weighting function equal to 1 until 35 ms and then linearly decreasing up to 95 ms. This transition time can also refer to the physical properties of the room and of its impulse responses. After reaching a sufficiently high echo density and modal overlap, the room reverberation exhibits an exponentially decaying stochastic behavior. This lower time limit is referred to as the mixing time. Several estimators of the mixing time have been suggested in the literature. The estimation \(t_{m}=\sqrt{V}\) (with \(t_{m}\) the mixing time in ms and \(V\) the volume of the room in cubic meter) was proposed in [21]. Other methods rely on the evaluation of the diffuseness of the sound field from the statistics of the echoes observed in the impulse response. This estimation may either be conducted in the time domain [22; 23], or in the spatial domain, when an SRIR is available [24]. In our case, a temporal estimation of the mixing time was chosen and a value of 15 ms (after the onset delay) was provided. This time will define the temporal limit between what is considered as the early part of the BRIR and the reverberation tail. The design of this model was based on the hypothesis that the early-to-late energy ratio is a relevant reverberation-related distance predictor [7]. [PERSON] tested a predictive model in which he used an early-to-late energy power ratio to compute the perceived distance with success [6]. He also highlighted the role of the spatial distribution of early reflections. In [7], he showed that the early energy definition could be based on a lateralization window (i.e., aggregation of the reflections sharing an ITD close to that of the direct sound) rather than on a time window. Here, the mixing time was chosen as a first estimation of the length of the temporal integration window of the early energy. The model applied to alter the initial BRIR to control its apparent source distance is inspired by previous work from [PERSON] et al. described in [25]. In the proposed approach, the sound source distance is driven through the control of two temporal segments, the direct sound energy \(E_{dir}\) and the reverberation energy \(E_{rev}\). The level of the direct sound according to the source distance is expressed as follows:  
\[E_{dir}(f,d)=\frac{S_{\phi}(f)^{2}\mu(f)^{d}}{4\pi cd^{2}} \tag{1}\]with \(c\) the sound celerity, \(f\) the frequency, \(\mu(f)\) the frequency-dependent sound absorption for a 1-m propagation in the air, and \(S_{\phi}(f)\) the free field transfer function of the source in the direction of the receiver. The level of the diffused part of the reverberation after the mixing time \(\tau\) in the impulse response is expressed as follows:  
\[E_{rev}(\tau,f)=\frac{Tr(f)S_{d}(f)}{13.81*V}exp(-13.81*\frac{\tau+\frac{d}{c} }{Tr(f)}) \tag{2}\]  
with \(Tr(f)\) the reverberation time, \(V\) the volume of the room, \(S_{d}(f)\) the diffuse-field transfer function. The dependence of the reverberation energy with the distance \(d\) agrees with [PERSON]'s revised theory on energy relations in the room response [26]. In the present study, some further simplifications are made. Air absorption is not considered.