# Denoising Directional Room Impulse Responses with Spatially Anisotropic Late Reverberation Tails
## 1 Introduction  
Room impulse response (RIR) convolution has become a widespread method for reproducing room reverberation effects in digital audio processing applications. In the particular case of three-dimensional surround-sound systems, such as those using Higher-Order Ambisonics (HOA), directional room impulse responses (DRIR) measured with spherical microphone arrays (SMA) can reproduce spatialized room reverberation effects through multichannel convolution. However, such measured DRIRs inevitably contain a nondecaying noise floor that may produce an audible \"infinite reverberation effect\" upon convolution with an input sound source. In order for a measured DRIR to be usable in a spatialized convolution reverberation processor, this nondecaying noise floor must therefore be compensated.  
Room reverberation has long been modeled as an exponentially decaying stochastic process [1], and this has been shown to be valid assuming sufficiently high echo density and modal overlap is achieved [2]. These conditions lead to a lower time limit for echo density, known as the \"mixing time\", and a lower frequency limit for modal overlap, known as the \"Schroeder frequency\". Beyond these limits, the late reverberation field is considered to be fully \"diffuse\", i.e., a spatially isotropic distribution of a statistically significant number of stochastically independent [3] and therefore incoherent [4] plane waves. In such a case, the nondecaying background noise present in a measured RIR can be replaced with a zero-mean Gaussian noise filtered by a prolongation of the reverberation tail's exponentially-decaying time-frequency envelope, parameterized by decay rates and initial spectra extracted from the RIR's energy decay relief (EDR) [5]. This idea can be extended to spatializedDRIRs measured with an SMA by performing such tail resynthesis directly in the spherical harmonic domain (SHD) [6], where the orthogonality and spatial symmetry of the SHD components guarantee that the diffuseness properties of the late reverberation will be preserved [7]. It should be noted that other noise compensation techniques have been presented in the literature [8], but these have rarely dealt with reproduction of reverberation effects through convolution (a notable exception being [PERSON] et al. [9]) and even less with the spatialized DRIR case.  
In certain cases, however, the late reverberation field of a DRIR may become spatially incoherent without being energetically isotropic [10]; in other words, the stochastic process describing the plane waves incident on the SMA will have a certain spatial power distribution (or an amplitude density distribution, strictly speaking). This may be due to the particular geometric and/or absorptive characteristics of the measured space (e.g., coupled volumes, semi-open spaces, highly heterogeneous absorptive surfaces, etc.), in combination with the directivity and orientation of the sound source (which in general may not be perfectly omnidirectional), and can be investigated using directional energy decay curves (DEDC) [11] or energy decay deviations (EDD) [12] calculated on a spatial representation of the DRIR. As we show below, such anisotropic spatial distributions of the late reverberation tail cannot be arbitrarily recreated through analysis and synthesis in the SHD, and a different spatial representation must therefore be used for denoising.