Beyond directivity index and room response

HRTF analysis of different radiation patterns, focusing on the contribution of side reflections to the diffuse-field response.

How does vertical and horizontal lobing affect the response at the eardrum?
How do reflections from the side walls contribute to the room response at the eardrum? - The relevance of horizontal dispersion.

June 5, 2024

The head related transfer function (HRTF) describes how a signal is filtered by diffraction, scattering and reflection of the head, pinna, and torso before it reaches the eardrum (Head-related transfer function in Wikipedia). Existing measurements focus on free-field and diffuse-field responses, but none of the studies address how radiation in the vertical and horizontal planes affect the eardrum response.

From 5 kHz the ear becomes very sensitive to the quality (directional properties) of the sound field. A small variation in the lateral-to-frontal energy can cause a significant change in the eardrum response even if the response measured with traditional omni-mic remains the same.

This means that above 5 kHz directivity index (DI) and frequency response ("room response") doesn't tell the whole story:

Two loudspeaker systems with different DI and room response may generate similar responses at the eardrum (an example could be a 50mmx25mm tweeter (ribbon or cone) vs a 25mm dome tweeter)
Two loudspeaker systems with same DI and room response may generate different responses at the eardrum. (e.g. 100mmx25mm ribbon tweeter rotated on its axis)

As we will see, our hearing system has an interesting feature: between 5 kHz and 10 kHz the diffuse field response at the eardrum is mainly determined by reflection from the side walls, and the contribution of the other walls is secondary.

Furthermore, if we take the diffuse-field HRTF response as a reference, then a few decibel loss in the room response caused by increasing DI may generate larger loss in the response at the eardrum (for a frontal wave, at the frequency of the pinna notch).

Two tweeters with different vertical but identical horizontal radiation pattern. DI and room responses are also different. How do they sound? Are the eardrum responses the same?

Average HRTF from axis measurements

The average response is calculated from six HRTF responses [1], corresponding to the six main direction. Since top and bottom responses are missing from the original measurements, they are mixed from existing measurements:

Modified diffuse-field response is used for the "top response" (elevation 90). This cheat works because the response from directly above the head is similar to the diffuse-field response. [2,3]
The rear response is used for the bottom response.

Note: since HRTF is omnidirectional at 100 Hz, the level at 100 Hz on the non-normalized HRTF frequency graphs is the normalized room level (measured with an omni-mic), which also corresponds to the room gain.

Average, horizontal plane and vertical plane responses

Sound radiation in horizontal and vertical planes, eardrum responses:

Wide horizontal dispersion, high lateral energy & reflections from the side walls is crucial to achieve diffuse field response between 5 kHz and 10 kHz.

A different calculation. In this case HRTFe00a180 ("rear sound") is left out from the calculation and the level of lateral and vertical axis responses are reduced by 3 dB. Contribution of the lateral axis response to the overall response is still significant:

Modelling vertical beaming is a bit more complex than "switching off" the radiation in a certain axis. Increasing directivity in the vertical plane results in not only the reduction of reflections from the floor and ceiling, but also a certain amount of reduction of reflections from the side walls and even from the wall behind the listener.

Elliptical radiation vs. axisymmetric radiation (with same DI)

An approximation of elliptical radiation is simulated with wide horizontal and narrow vertical radiation. The radiation patterns are modelled with the following attenuation values:

	Elliptical [dB]	Axisymmetric [dB]
HRTFe00a000	0	0
HRTFe00a090	-3	-6
HRTFe00a180	0	0
HRTFe00a270	-3	-6
HRTFe90	-100	-6
HRTFe-90	-100	-6

Notes:

HRTFe00a090: elevation 0 deg, azimuth 90 deg, azimuth 0 deg is front axis.
HRTFe90 is the top response, HRTFe-90 is the bottom response.
Simulated radiation patterns have constant directivity.
Both radiation patterns have the same directivity index, room response (but the response at the eardrum is different).

Simulated elliptical radiation provides a few dB more in the ear between 5kHz and 9kHz.

Practical application?

A practical application could be an optimized dispersion cone tweeter with oval shaped diaphragm. A tweeter with 60 mm x 30 mm diaphragm can provide similar eardrum response as a tweeter with 30 mm x 30 mm diaphragm. Cone tweeters with oval shaped cones could provide high output and good dispersion at low cost. (Beaming sets constraints on the height and width of the diaphragm: 60 mm is a reasonable maximum height, 30mm is a reasonable width.)

Csaba Horváth

References:

[1] "Measuring HRTFs of Brüel & Kjær Type 4128-C, G.R.A.S. KEMAR Type 45BM, and Head Acoustics HMS II.3 Head and Torso Simulators", Snaidero, Thomas; Jacobsen, Finn; Buchholz, Jörg, 2011

[2] "HRTF measurements of a KEMAR dummy-head microphone. Technical Report #280", B. Gardner and M. Keith., MIT Media Lab Perceptual Computing, 1994.

[3] HRTF measurements with blocked ear canal at http://recherche.ircam.fr/equipes/salles/listen/index.html