# Predicting the Colouration between Binaural Signals
## 1 Introduction  
The most definitive method of evaluating the perceptual difference between audio signals is through subjective listening tests using human participants. A number of listening test methodologies have been ratified by the ITU, including but not limited to ABX [1], for small differences between stimuli (as detailed in ITU-R BS.1116-3), and multiple stimulus test with hidden reference and anchor (MUSHRA), for medium to large differences between stimuli (as detailed in ITU-R BS.1534-3). Listening test paradigms typically involve the comparison of test signals and reference signals to determine either the existence of a perceivable difference or the magnitude of that perceived difference. Through subsequent statistical analysis of the results, it is possible to then determine the statistical significance, and thus the likelihood of reaching the same conclusions if the test were repeated with different participants. However, listening tests are demanding and costly to run as they require time to set up and conduct, as well as the participation and organisation of human subjects. It is therefore desirable to consider objective quality metrics for audio signals as an alternative. These are especially useful in algorithm development and product prototyping phases, which often require fast and repeatable analysis. In virtual and augmented reality applications, which require a realistic binaural audio experience, accurate timbre (or lack of colouration) is considered highly important [2, 3, 4, 5]. In binaural reproduction, colouration can be deduced from spectral difference. One basic way to numerically estimate the colouration between audio signals is to calculate the difference between the magnitude values of frequency bands obtained using fast Fourier transform (FFT) operations on the signals, as used in [6, 7, 8, 9]. This is herein referred to as a Basic Spectral Difference (BSD) calculation. However, BSD is not a highly accurate metric forhuman perception, as the human auditory system differs greatly in sensitivity depending on relative amplitude, frequency and temporal aspects [10]. When comparing the spectra of signals destined for human listening, the perceptual relevance of these differences should be considered. Building on the BSD method leads to perceptual models which can approximate the loudness of a particular sound by taking into account the sensitivity of the auditory system. [PERSON] and [PERSON] [11; 12] presented work that explored the varying loudness perception of different frequencies at equal intensities, often referred to as equal loudness curves. [PERSON] [13] published work on the non-linear link between sound intensity and perceived loudness, and [PERSON] [14; 15; 16] developed models for approximating the summation of loudness across frequency. These models have since been revised [17; 18]. Although such models can incorporate single or multi-band analysis, typical applications, including broadcast and music production, tend to require a single loudness value output that describes the entire wide-band stimulus as a whole. This does not provide a judgement on the predicted colouration between two stimuli other than their overall perceived amplitude, which would likely in any case be normalised during a reproduction stage. A more complex model is Perceptual Evaluation of Audio Quality (PEAQ) [19], which uses a peripheral ear model followed by a feature extraction stage. The features are then mapped to a perceptual quality scale based on subjective data. This model was originally designed for degraded signals, intended for use in signal compression and bitrate reduction applications. The Composite Loudness Level (CLL) is a perceptual loudness model which has been used as a measure of colouration in numerous studies [20; 21; 22; 23], which uses half-wave rectification and 1 kHz low-pass filtering to simulate the hair cell and auditory nerve behaviour, as well as equivalent rectangular bandwidth (ERB) weightings to account for linear FFT frequency sampling, and a Phon calculation [24].