Mathematically Built HRTF

We explain what is Mathematically Builte HRTF(Head Related Transfer Function) which is essential for quality of our spatial audio.

Motivation

HRTF obtained by measurement inevitably has measurement errors. In fact, there is something wrong with rendering sound using currently available measured HRTFs. So We tried to express HRTFs mathematically using conceptual deduction without using any measurements. The human brain analyzes the directions of arrival of hundreds of sound elements, including numerous reflections, in real time. We thought there must be a mathematical simplicity in HRTF that makes this possible.

How did we find it?

When trying to find HRTF, there are literally an infinite number of possible expressions. So we tried out all the function forms and values that came to mind intuitively, taking into account physical constraints and boundary conditions. We listened to the sound with the determined values, decided whether to accept the trial or not, and repeated the process. The number of trials probably have exceeded a million. The important thing is "the intuition of what to try." Pure random trials would not bring us any closer to the answer.

What did we get?

After much effort, we were able to express the HRTF in a simple mathematical formula, which delivers the accurate and beautiful sound we expected. In some ways the formula is simpler than expected, and in other ways it is more complicated than expected. Looking at this, we can see how people hear and distinguish sounds from in front of or behind, from above or below, and so on, all of which are expressed in the formula.