Editing Head-related transfer function (section)

=== Localization of sound in virtual auditory space===
A basic assumption in the creation of a virtual auditory space is that if the acoustical waveforms present at a listener's eardrums are the same under headphones as in free field, then the listener's experience should also be the same.

Typically, sounds generated from headphones are perceived as originating from within the head. In the virtual auditory space, the headphones should be able to "externalize" the sound. Using the HRTF, sounds can be spatially positioned using the technique described below.<ref name="Carlile, S 1996"/>

Let ''x''{{sub|1}}(''t'') represent an electrical signal driving a loudspeaker and ''y''{{sub|1}}(''t'') represent the signal received by a microphone inside the listener's eardrum. Similarly, let ''x''{{sub|2}}(''t'') represent the electrical signal driving a headphone and ''y''{{sub|2}}(''t'') represent the microphone response to the signal. The goal of the virtual auditory space is to choose ''x''{{sub|2}}(''t'') such that ''y''{{sub|2}}(''t'') = ''y''{{sub|1}}(''t''). Applying the Fourier transform to these signals, we come up with the following two equations:

: ''Y''{{sub|1}} = ''X''{{sub|1}}LFM, and
: ''Y''{{sub|2}} = ''X''{{sub|2}}HM,

where ''L'' is the transfer function of the loudspeaker in the free field, ''F'' is the HRTF, ''M'' is the microphone transfer function, and ''H'' is the headphone-to-eardrum transfer function. Setting ''Y''{{sub|1}} = ''Y''{{sub|2}}, and solving for ''X''{{sub|2}} yields

: ''X''{{sub|2}} = ''X''{{sub|1}}LF/H.

By observation, the desired transfer function is

: ''T''= ''LF''/''H''.
 
Therefore, theoretically, if ''x''{{sub|1}}(''t'') is passed through this filter and the resulting ''x''{{sub|2}}(''t'') is played on the headphones, it should produce the same signal at the eardrum. Since the filter applies only to a single ear, another one must be derived for the other ear. This process is repeated for many places in the virtual environment to create an array of head-related transfer functions for each position to be recreated while ensuring that the sampling conditions are set by the [[Nyquist rate|Nyquist criteria]].