Quantized Angles of Arrival

Since high-quality fractional-delay filtering is expensive, it is worth considering restriction to angles-of-arrival corresponding to integer delays (in samples). If the speaker-to-speaker spacing along a line array is $\Delta_x$ , then the speaker-to-speaker delay for a plane wave at angle-of-incidence $\theta$ is $\Delta_x \cdot\sin(\theta)/c$ , where $c$ denotes sound speed. Thus, an angle-of-arrival $\theta_n$ corresponds to an integer speaker-to-speaker delay $n$ (in samples) when

$$\frac{\Delta_x }{cT} \sin(\theta_n) \eqsp n, \quad n=0,\pm 1,\pm 2,\ldots, \pm N_a \protect$$ (5)

where $T$ denotes the temporal sampling interval in seconds, and

$$N_a \eqsp \left\lfloor \frac{\Delta_x }{cT}\right\rfloor \isdefs$$   floor$$\left(\frac{\Delta_x }{cT}\right).$$

Note that increasing the speaker spacing $\Delta_x$ for a given temporal sampling rate $f_s=1/T$ gives more integer-delay angles $\theta_n$ . However, doing this also decreases the stage-width (or supported bandwidth) by the same factor.

It is clearly inaudible to shift the location of each virtual source $x_i$ so that the time delay to the nearest speaker is an integer number of samples. Then having an integer number of samples for each inter-speaker delay makes all the delays integer. Finally, this can all be implemented as a single delay line with a tap (non-interpolating) for each speaker signal. For moving sources, to avoid clicks, moving taps should be cross-faded from one integer delay to the next in the usual way (Smith, 2010).⁹

Solving Eq.(5), the collection of angles $\theta_n$ corresponding to integer inter-speaker delays $n$ (in samples) is

$$\theta_n = \sin^{-1}\left(\frac{cT}{\Delta_x }n\right), \quad n=0,\pm 1,\pm 2,\ldots\,.$$ (6)

For example, with $\Delta_x =0.1$ m ( $\approx4''$ ), $c=343$ m/s, and $f_s=48$ kHz, the first 11 available angles are

$$\theta_n^\circ \in \pm [0,\, 4.1,\, 8.1,\, 12,\, 16,\, 21,\, 25,\, 30,\, 35,\, 40,\, 45] \protect$$ (7)

degrees, to two digits precision. Azimuth perception is accurate to approximately 1 degree at center-front.¹⁰

Figure 4 depicts the available geometric rays of plane-wave propagation for this example. Thicker rays are drawn for 0 degrees (directly in front) and $\pm90$ degrees (full left and right).

Figure 4: Rays of propagation toward a listener in the center for the available plane-wave angles from a line-array (four-inch spacing) at a sampling rate of 48 kHz. These are the source angles requiring no interpolation--just pure integer delay from one speaker to the next.

If the 21 angles-of-arrival across a $90^\circ$ stage listed in Eq.(7) are deemed sufficient, then PBAP is essentially free: just provide the appropriate integer adjacent-speaker delays for each source in the sum for each speaker. As is well known, an integer delay is an ${\cal O}(1)$ computation, requiring only a single read, write, and circular-buffer pointer-increment each sampling instant (Smith, 2010).¹¹