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#### HOP Stage Angle

An immediate question that arises is how large is the maximum angle-of-arrival for a PBAP array made using such a HOP? From §2 on page , we have

Assuming it is possible to pack the speakers contiguously, and setting m/s, we obtain for a top row using driver diameters 0.03 m as in Fig.13

or 16.6 degrees, giving a 33.2-degree stage at . The same result is obtained for the highest frequency in each band of this example HOP, due to its strict octave scaling. The alias-free angle-of-arrival range is unfortunately narrow, but it is calculated at the high edge of each band. At the bottom of each band, we obtain a much nicer 70 degree sound-stage. At the center-frequency of each band, defined as the geometric mean of the band edges, we find that a 47.7-degree sound-stage is supported without spatial aliasing, which is not too bad. Again, when pushing the angle-of-arrival limits, only certain frequency bands near the top of each octave band start to spatially alias, and the spectrum as a whole is likely to keep the brain fusing everything together at the intended angle of arrival.

To reduce spatial aliasing or increase the stage width of this HOP, it is tempting to shift the operating range of each speaker down by some fraction of an octave, thereby increasing the stage-width while making each speaker a less efficient radiator, a better approximation to a point source, and a finer spatial sample within the array. Decreasing driver diameter is of course equivalent to decreasing the frequency range over which it operates. However, since most people cannot hear frequencies near 20 kHz, it makes sense to push first on downward frequency scaling. This point is pursued further in §4.9 below.

The need to downscale the frequency band served by each speaker driver is a fundamental problem with circular drivers laid out in a row. Alternatives driver geometries include (1) overlapping driver cones (perhaps using a shared membrane across multiple drivers instead of individual cones), (2) staggered packing of two or more identical rows, as in the hexagonal mesh, or (3) square or rectangular horn drivers with contiguous exit-apertures driven by long-throw pistons.

To summarize the situation from an elementary spatial-sampling perspective, the maximum stage width determines the minimum wavelength seen by the array, and we need two or more drivers per wavelength for proper spatial sampling. The maximum width (diameter) of each driver must be less than half the minimum spatial wavelength seen by the array (which can be made arbitrarily large by restricting the stage angle). If wide stage angles are to be supported, then every driver must be smaller than spatial wavelengths it is emitting. They're all approaching simple sources'' emitting omnidirectional radiation patterns.

The crossover frequency 612 Hz for the low end of the 48-cm woofer in Fig.13 is based on the preferred driving frequencies and dispersion pattern for a conventional loudspeaker of that size, as chosen in the Kenwood JL-840W four-way speaker system. We see that stage-width and spatial-aliasing considerations argue for a smaller crossover frequency, perhaps even an octave lower, if adequate undistorted power can be obtained covering that range.

The nominal lowest frequency in the fifth row (612 Hz) is well above the typical subwoofer crossover range of 80-200 Hz. This means we have almost three missing octaves between this particular five-band HOP and the ideal subwoofer taking over at 80 Hz. Even downscaling the frequency bands by an octave (discussed above), two missing octaves remain. Raising the subwoofer cutoff from 80 to 160 gets one more, leaving almost a one-octave gap, which can be reduced further from either side. A sixth row in the HOP would call for a 96 cm speaker diameter (38 inches or 3.15 feet), which does not appear to be practical using conventional driver technology. We therefore presently allow the bottom row to go undersampled, and cross-fade over to VBAP signals, or even simple stereo using the left speakers as a left channel and the right speakers as a right channel (panning each virtual source accordingly in the stereo field). Fewer than speakers can be used for wider separation but less power. Experiments are needed to determine best practices in this (undersampled) frequency range. At frequencies immune to spatial aliasing, we expect to be limited only by driver quality and noise.

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