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3.5 Mark Polettiís high-directivity virtual microphones

During the rotation of the microphonic assembly, the two cardioids employed for ORTF recordings also describe a small circumference, with a radius of approximately 110 mm, as shown in Fig. 20.

Figure 20: geometry of ORTF microphones

Looking for simplicity to just one of the two microphones, it samples 36 impulse responses during its complete rotation. From this set of data, it is possible to derive the responses of a set of various-orders coincident microphones, ideally placed in the center of rotation, making use of a modified version of the Polettiís theory [8].

The basis of this method is to define a class of multileaf-shaped horizontal directivity patterns of various orders. The order 0 is an omnidirectional, order 1 are two crossed figure-of-eight microphones (as in horizontal-only Ambisonics); then order 2 and 3 are added, with directivity patterns corresponding respectively to the cosine of twice and three times the angle:

The second explanation suggests a simple way of computing the required responses: the signals coming from the 36 microphones are simply multiplied for a set of 36 weighting factors, obtained by the eqn. 8 above, and summed.

This of course does not provide the wanted frequency-independent, linear-phase result: as clearly demonstrated by Poletti, these ìrawî virtual microphones will exhibit strongly uneven magnitude and phase response, which can however be compensated afterwards.

Poletti also derived the theoretical expressions of the transfer functions, which can be used for creating the proper equalizing filters. However, a more clever and practical solution is simply to measure these ìrawî transfer functions in an anechoic chamber, and then derive, for each virtual microphone, the proper inverse filter by means of the Kirkeby inversion method [18]. This has the added advantage of compensating also for the specific response of the microphone employed, and for its frequency-dependent directivity pattern (which will only roughly correspond to the theoretical cardioid pattern).

Once the response of the high-order microphones are obtained, they can be employed as convolution filters applied to the mono dry signals corresponding to the discrete source positions. After mixing of the results, an high-order Ambisonics decoder is required for deriving the feeds for a multichannel regular array of loudspeakers (typically arranged regularly around a circle surrounding the ìsweet spotî), which provides much better localization and channel separation than ìstandardî (1st order) Ambisonics.

A second possible way of employing these high-order signals is to drive a standard 5.1 ITU array, by synthesizing 5 proper asymmetrical directivity patterns, as suggested in [27].

3.6 Circular WFS approach

The 36 B-format measurements made along the 1mradius circumference are exactly the set of data required for employing the WFS method described in [6].

The basis of this method is the Huygens principle: knowing the sound pressure and particle velocity on a closed surface makes it possible to recreate inside it the same sound field which was present in the original space, employing a suitable array of loudspeakers exactly corresponding to the positions of the microphone. The theory, however, also allows to ìexpandí or ìshrinkî the geometry of the transducer array, provided that the soundfield is decomposed in traveling wavefronts.

The WFS is a 2D reduction of this general theory, where the microphones are placed along a closed curve around the listening area, and consequently the expansion/shrinking can only be done in the horizontal plane. This also limits the amount of ìmovementî which can be applied. However, starting with a 1mradius array, it is quite easy to derive the feeds for a loudspeaker array suitable for a medium-sized listening room, and to ìstretchî the array so that the loudspeakers are arranged in 4 linear arrays instead of in a circular array. The next figure (partially taken from [27]) shows a schematic of this process.

Figure 21: WFS processing scheme

The ìspatial processingî required for deriving the reproduction impulse responses from the measured impulse responses is not trivial, and can be understood only after a deep study of the material published (and unpublished) at the Technical University of Delft. Till now the authors were not yet able to create a simple plugin for performing easily this spatial transformation, although this development is planned for the future.

Of course, this theory requires a little spatial step between consecutive microphone positions, for reducing the spatial aliasing which occurs when sampling the wavefronts. As in this case the number of microphone positions is quite limited (36), this translates in a severe limitation of the frequency range which does not cause spatial aliasing. Above this threshold (which is around 1 kHz for the geometry employed here), it is not possible anymore to reconstruct faithfully the wavefronts. For avoiding artifacts and coloration, it is then advisable to randomize the phases, so that the summation of the output of the various loudspeakers constituting the array does no longer cause interference, and reduces to simple energy summation (as in Ambisonics).

The phase randomization can be obtained by convolution of the signal driving each loudspeaker with a different burst of white noise, or by employing phaseincoherent loudspeakers (distributed-mode loudspeakers).

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