Head-related transfer function

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.[9]

Let x₁(t) represent an electrical signal driving a loudspeaker and y₁(t) represent the signal received by a microphone inside the listener's eardrum. Similarly, let x₂(t) represent the electrical signal driving a headphone and y₂(t) represent the microphone response to the signal. The goal of the virtual auditory space is to choose x₂(t) such that y₂(t) = y₁(t). Applying the Fourier transform to these signals, we come up with the following two equations:

Y₁ = X₁LFM, and

Y₂ = X₂HM,

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 Y₁ = Y₂, and solving for X₂ yields

X₂ = X₁LF/H.

By observation, the desired transfer function is

T= LF/H.

Therefore, theoretically, if x₁(t) is passed through this filter and the resulting x₂(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 criteria.

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.[10]

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 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

${\displaystyle \beta =[\beta _{1},\beta _{2},\ldots ,\beta _{N}]^{T}}$

that represents the subject's anthropometric features as a linear superposition of the anthropometric features from the training data (y^' = β^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:

${\displaystyle \beta =\operatorname {argmin} \limits _{\beta }\left(\sum _{a=1}^{A}\left(y_{a}-\sum _{n=1}^{N}\beta _{n}X_{n}^{2}\right)+\lambda \sum _{n=1}^{N}\beta _{n}\right)}$

From this, ITD scaling factor value H^' is estimated as:

${\displaystyle H'=\sum _{n=1}^{N}\beta _{n}H_{n}.}$

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

We solve the above minimization problem using Least Absolute Shrinkage and Selection Operator (LASSO). We assume that the HRTFs are represented by the same relation as the anthropometric features.[11] Therefore, once we learn the sparse vector β from the anthropometric features, we directly apply it to the HRTF tensor data and the subject's HRTF values H^' given by:

${\displaystyle H'_{d,k}=\sum _{n=1}^{N}\beta _{n}H_{n,d,k}}$

where The HRTFs for each subject are described by a tensor of size D × K, where D is the number of HRTF directions and K is the number of frequency bins. All H_n,d,k corresponds to all the HRTFs of the training set are stacked in a new tensor H ∈ R^N×D×K, so the value H_n,d,k corresponds to the k-th frequency bin for d-th HRTF direction of the n-th person. Also H^'_d,k corresponds to k-th frequency for every d-th HRTF direction of the synthesized HRTF.