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I would like to spend some word on the focalization technique by WFS. The array here represented may be the line one or a section of the disc, non differences since the plotted field is a vertical section od the 3D field.

The array cuses some concave wave fronts, which implode in a focus and then explodes in normal convex wave fronts centered on the focus. So from the point of view of listeners placed below the focus, the focus is a virtual sound source.

The front curvature is achieved with a set of gains and delays, in particular each speaker has an advance in time equal to the fly time which seaparate it from the focus, that is: each speaker signal get to the focus at the same time. This is the formula for the speaker signals, coming from the Rayleigh integral, (for line array)

Sensible parameters of such kind of applicaton are:

The ratio array width on wavelength, which mainly governs the beam and focus width: with wl comparable with the array length the focus will be very wide and the curvature of the fronts will be lost. For wl even greater, the array may loose completely is deirectivity.

The ratio speaker spacing to wl, which rules the phenomenon of spatial alising. Spatial aliasing arises at high frequencies, and in the particular case of focalization can leads to strong side lobes.. It is a good peculiarity of focalization to have the spacial aliasing effect far away from the focus. Hence the phenomenon is bad for the pourpose of spatially limiting the sound, but is not perceived by listeners staying beneath the array. Let’s see the phenomenon alittle it closer in a new slide...

Use a loudspeaker array to reproduced a desired wave field can be seen as a space sampling of the field itself. Just as the time sampling, the space sampling is governed by the Nyquist theorem, which in this case is only a litle more difficult.

Imagine a plane wave inpinging on a discrete speakes array: the speaker spacing is fundamental for a correct sampling; Nyquist says in this case tht the projection of the wavelength on the array must at least be twice the speaker spacing. So this is the formula coming out. Above a certain frequency, the wave must be limited within a certain incidence angle.

Just like in the time domain case, besides a sampling contitions, a reconstruction condition must be respected. It imposes a specific directivity on secondary sources. In pratice if for a given frequency I see a maximum incidence angle, that frequency must not be delivered by secondary sources at angle wider than the limit one.

Take for example the limit case of a plane waves frontally impacting the array: phi is zero so the sampling condition is always verified. But i still need a reconstruction filter, because the secondary fronts are circular, and only in a suffuciently far field their envelope will be flat, whilst in the near field i will experience the spatial aliasing. It’s intuitive to understand that rigid pistons instead of point sources con solve the problem, since the front is flat from the beginning. But we know that a rigid pistons is nothing else than a directional sources, with beamwidth that decreases with frequency. It is not exacly the directivity requested by the formula, because we have lobes for example, but qualitativly it is according with it.

So at the end our choice fell on what was maybe the simples solution one could ever imagine, even without all the discussion made so far. A single diver extended range, which provides for an acceptable coverage of the area and of the frequency spectrum as well. I measured the directivitivity of the speaker, already inserted in the array, and these are the result:

in axis tranfer function: it is not flat, but it radiates sufficiently high at frequencies up to 10-15 KHz. With so many speaker and FIR filtering I think we can reach a transfer function in the focus that is more than flat.

Here we have emission angle vs frequency, and here the polar pattern in octaves, you can see it as section of this surface plot.

This is the curve that describe the reconstruction condition antialiasing. In theory we shoul have yellow inside this “funnel” and blue outside. This doesn’t happens, we have also lobes here.. But there is a general matching which makes us optimistic.

The picture shows a 24 speaker line array which we are using now as a sort of scale model: it’s length is three meters just like the diameter of the designed final structure, and the heigth is 4meters as well. So it can be seen as a diametral section of it.

The choice has of course a logistic reason. The line array is supposed to produce the same kind of focalization of the disc, but limited to a plane containing the array itself. So for example, fo the listener staying here in front of the array, a virtual source will be possible mooved from left to right, but not from front to back, which will be possible instead with the full disc. For the line the focus is represented by a ring going around the array itself, whilst for the disc it is a little sphere. Consequently, the sound inflow in the environment is very different from the case of the disc,sound is much more dispersed and the possibility to create a surrounding silent volume become impossible.

This is the HW driving the array: description...

This is the whole system.

Now the scheme of the processing system which should drive the array. It is still to be implemented, but the idea is to use a linux platform with the Anders torger brutefir running on it. The systems should have 5 -10 inputs, each related to one of the virual source (focus) we want to create. The outputs ars 96 as I told before. This is a matrix of proces which provides for positioning of the vistual source, whilst other filter are put just in the output or in the input.

In particular: the input FIR should be 2048 long, and shoul provide for the WFS filter, the gain increasing with frequency, and for speaker equalization. This two filter have shape which are roughly opposite so it is good to put them toghether for the preservation of the dynamic of the system. More: the total filter will problably be empiric, that is an inverse filter of the focus response in the focus when all the other filter are already set.

The matrix filter, which provides for spatialization, contain the dynamic gain and delay necessary for focalization, and to the anti-alisaing sampling filter.

The output FIRs, one for each out channel, provides for speaker phase matching, i think that 64 taps are enough for it.