## 1 Introduction  
Headphone-based representation of virtual acoustic environments (VAEs) is becoming more and more important due to the growing market of mobile devices, virtual and augmented reality (VR/AR) applications, high-definition television (HDTV), etc. [1]. As shown in Figure 1, the traditional way to synthesize a virtual sound source presented over headphones is by convolving a dry audio signal with a pair of binaural room impulse responses (BRIRs, the time domain representation of binaural room transfer functions (BRTFs)), which can be expressed as [2]:  
\[y_{i}(t)=s(t)*\text{BRIR}_{i}(r,\theta,\varphi,t)=\int_{-\infty}^{\infty}\text {BRIR}_{i}(r,\theta,\varphi,t-\tau)\;s(\tau)\;d\tau,\quad i\in\{\text{left}, \text{right}\}, \tag{1}\]  
where * denotes the convolution operator. \(s(t)\) and \(y_{i}(t)\) are the input audio signal and the synthesized binaural signals, respectively. \(r\) is the source-listener distance. \(\theta\) and \(\varphi\) denote the elevation and azimuth angles of sound sources relative to the listener, respectively. The BRIR represents the direction- and distance-dependent acoustic impulse response from a sound source to the listener's eardrums in a room, consisting of:  
* the direct part, which is the impulse response from the sound source to the listeners' eardrums without room information, as described by the head-related impulse response (HRIR, the time domain representation of the head-related transfer function (HRTF)). * the reverberant part, which contains information about reverberation (early reflections and late reverberation) stemming from the room. In the case of an anechoic chamber, the amount of reverberation is close to zero. Perceived externalization, i.e., out of the head [3], is one of the essential indicators for the construction of immersive acoustic environments. Over the years, some important cues related to the externalization of headphone-based sound images have been discussed and investigated. In anechoic scenarios (free-field conditions), HRTFs (direct parts in BRIRs) are responsible for perceived externalization, which depend strongly on individual anatomies, such as pinnae, head, shoulders, and torso. According to [PERSON] and [PERSON] [3], [PERSON] and [PERSON] [4], the auditory images synthesized by using individual HRTFs could be perceived as externalized, whereas the degree of externalization was reduced if the HRTFs were non-individual. Unfortunately, recording individual HRTFs for each listener is difficult and impracticable in consumer scenarios. Due to this reason, HRTFs from some existing databases, e.g., CIPIC [5], IRCAM [6], and SADIE [7], are commonly used in dynamic binaural rendering systems. [PERSON] and [PERSON] [3] showed that the interaural time differences (ITDs)/interaural phase differences (IPDs) in low-frequencies (<1 kHz), the interaural level differences (ILDs) in the full range of audible frequencies, and the monaural spectral cues from each ear were important for perceived externalization. [PERSON] and [PERSON] [8] have investigated the impact of reduced spectral details in HRTFs on the localization of virtual sound sources at 0\({}^{\circ}\), 45\({}^{\circ}\), 135\({}^{\circ}\), and 180\({}^{\circ}\). The magnitude spectra of HRTFs were expressed as Fourier series (FS), and the spectral details in HRTFs (magnitude spectra of HRTFs) were reduced by truncating the number of FS coefficients. The results of their study demonstrated that the performance of sound localization was unaffected by reducing the number of coefficients in the FS from 512 to 16. Furthermore, all subjects reported that the degree of perceived externalization of virtual sound images remained unchanged (informal listening test); even the number of coefficients was reduced to eight. In non-anechoic scenarios (reverberant conditions), reverberation, especially the early reflection part, facilitates distance perception [9] and is essential for perceived externalization [10].