HRTFs for left and right ear (expressed here as HRIRs) describe the filtering of a sound source (x(t)) before it is perceived at the left and right ears as x_L(t) and x_R(t), respectively.

The head-related transfer function (HRTF), more recently termed the anatomical transfer function (ATF), describes how a given sound wave input (parameterized as frequency and source location) is filtered by the diffraction and reflection properties of the head, pinna, and torso, before the sound reaches the transduction machinery of the eardrum and inner ear (see auditory system). Biologically, the source-location-specific prefiltering effects of these external structures aid in the neural determination of source location, particularly the determination of source elevation. Thus, HRTFs can also be used in acoustics to create the impression of surround sound without multiple speakers. This technology is still on the outskirts of becoming commercially viable. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... Diffraction is the apparent bending and spreading of waves when they meet an obstruction. ... The term reflection (also spelt reflexion) can refer to several different concepts: In mathematics, reflection is the transformation of a space. ... For other uses of the word head, see head (disambiguation). ... The pinna is the visible part of the ear that resides outside of the head. ... The torso is the human body without the head and limbs. ... In physiology, transduction is the conversion of a stimulus from one form to another. ... The tympanum or tympanic membrane, colloquially known as eardrum, is a thin membrane that separates the outer ear from the middle ear. ... For an alternative meaning, see ear (botany). ... The auditory system is the sensory system for the sense of hearing. ... Sound localization is a listeners ability to identify the location of origin of a detected sound or the methods in acoustical engineering to simulate the placement of an auditory cue in a virtual 3D space (see binaural recording). ... Elevation has several related meanings: Geography The elevation of a geographic location is its height above mean sea level (or possibly some other fixed point). ... Surround sound is the concept of expanding the spatial imaging of audio playback from 1 dimension (mono/Left-Right) to 2D or 3D. This is often performed for a more realistic audio environment, actively implemented in cinema sound systems, technical theatre, home entertainment, video arcades, computer gaming, and a growing...

Linear systems analysis defines the transfer function as the complex ratio between the output signal spectrum and the input signal spectrum as a function of frequency. Blauert (1974; cited in Blauert, 1981) initially defined the transfer function as the free-field transfer function (FFTF). Other terms include free-field to eardrum transfer function and the pressure transformation from the free-field to the eardrum. Less specific descriptions include the pinna transfer function, the outer ear transfer function, the pinna response, directional transfer function (DTF) or what is commonly termed the head-related transfer function (HRTF). A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ... The tympanum or tympanic membrane, colloquially known as eardrum, is a thin membrane that separates the outer ear from the middle ear. ... For an alternative meaning, see ear (botany). ...

The transfer function H(f) of any LTI system at frequency f is:

H(f) = Output (f) / Input (f)

One method used to obtain the HRTF from a given source location is therefore to measure the head-related impulse response (HRIR), h(t), at the ear drum for the impulse δ(t) placed at the source. The HRTF H(f) is the Fourier transform of the HRIR h(t). The Fourier transform, named after Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...

Even when measured for a dummy head of idealized geometry, head-related transform functions are complicated functions of frequency and the three spatial variables. For distances greater than 1m from the head, however, the HRTF can be said to attenuate inversely with range. It is this far field HRTF, H(f, θ, φ), that is normally measured. Sine waves of various frequencies; the lower waves have higher frequencies than those above. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... In the study of diffraction and antenna design, the near field is that part of the radiated field that is within a small number of wavelengths of the diffracting edge or antenna. ...

HRTFs are typically measured in an anechoic chamber to minimize the influence of early reflections and reverberation on the measured response. HRTFs are measured at small increments of θ such as 15° or 30° in the horizontal plane, with interpolation used to synthesize HRTFs for arbitrary positions of θ. Even with small increments, however, interpolation can lead to front-back confusion, and optimizing the interpolation procedure is an active area of research. Humans are less sensitive to changes in the azimuth, φ, and HRTFs are often measured only on the horizontal plane or with 45° increments in the median plane. An anechoic chamber is a room that is isolated from external sound or electromagnetic radiation sources, sometimes using sound proofing, and prevents the reflection of wave phenomena (reverberation). ... When sound is produced in an enclosed space multiple reflections build up and blend together creating reverberation or reverb. ... This article is about interpolation in mathematics. ...

In order to maximize the signal-to-noise ratio in a measured HRTF, it is important that the impulse being generated be of high volume. In practice, however, it can be difficult to generate impulses at high volumes and, if generated, they can be damaging to human ears, so it is more common for HRTFs to be directly calculated in the frequency domain using a frequency-swept sine wave or by using maximum length sequences. User fatigue is still a problem, however, highlighting the need for the ability to interpolate based on fewer measurements. The phrase signal-to-noise ratio, often abbreviated SNR or S/N, is an engineering term for the ratio between the magnitude of a signal (meaningful information) and the magnitude of background noise. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... A maximum length sequence (MLS) is a pseudo-random sequence used for measuring impulse responses (e. ...