Editing Head-related transfer function (section)

=== HRTF phase synthesis ===
There is less reliable phase estimation in the very low part of the frequency band, and in the upper frequencies the phase response is affected by the features of the pinna. Earlier studies also show that the HRTF phase response is mostly linear and that listeners are insensitive to the details of the interaural phase spectrum as long as the interaural time delay (ITD) of the combined low-frequency part of the waveform is maintained. This is the modeled phase response of the subject HRTF as a time delay, dependent on the direction and elevation.<ref name="Tashev"/>

A scaling factor is a function of the anthropometric features. For example, a training set of N subjects would consider each HRTF phase and describe a single ITD scaling factor as the average delay of the group. This computed scaling factor can estimate the time delay as function of the direction and elevation for any given individual. Converting the time delay to phase response for the left and the right ears is trivial.

The HRTF phase can be described by the [[Interaural time difference#Duplex theory|ITD]] scaling factor. This is in turn quantified by the anthropometric data of a given individual taken as the source of reference. For a generic case we consider ''β'' as a sparse vector

: <math> \beta = [ \beta_1, \beta_2, \ldots, \beta_N ]^T </math>

that represents the subject's anthropometric features as a linear superposition of the anthropometric features from the training data (y{{sup|'}} = β{{sup|T}} X), and then apply the same sparse vector directly on the scaling vector H. We can write this task as a minimization problem, for a non-negative shrinking parameter ''λ'':

: <math> \beta = \operatorname{argmin}\limits_\beta \left( \sum_{a=1}^A \left( y_a - \sum_{n=1}^N \beta_n X_n^2 \right) + \lambda \sum_{n=1}^N \beta_n \right) </math>

From this,
ITD scaling factor value H{{sup|'}} is estimated as:

: <math> H' = \sum_{n=1}^N \beta_n H_n. </math>

where The ITD scaling factors for all persons in the dataset are stacked in a vector ''H'' ∈ '''''R'''''{{sup|''N''}}, so the value ''H''{{sup|''n''}} corresponds to the scaling factor of the n-th person.