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# Head-related transfer function

In acoustics, a Head-Related Transfer Function (HRTF) is the output from a linear time invariant system, composed of the listener's head, pinna and torso. The output from this system is different from the input due to the diffraction and reflection properties of the head, pinna, and torso. HRTFs can be used to synthesize binaural environments from monoaural sources for 3D audio applications.

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).

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), , at the ear drum for the impulse placed at the source. The Fourier transform of the HRIR, , yields the HRTF, .

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, , that is normally measured.

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, , and HRTFs are often measured only on the horizontal plane or with 45° increments in the median plane.

In practice 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. User fatigue is still a problem, however, highlighting the need for the ability to interpolate based on fewer measurements.