Speakers as Spatial Samples

The polar pattern for a microphone or loudspeaker is its gain along a circle of constant radius away from the diaphragm/driver. For a spherical-wave ``point-source'', the polar pattern is simply a constant at each radius, e.g., $p(\theta) =p_1/r$ , where $p_1$ denotes the pressure-scaling at $r=1$ and $r$ denotes the distance from the center of the source.

Since our speaker arrays are typically flat, we need to calculate a slice through the polar pattern along a listening line (or plane) which we will take to be parallel to the array and to the $x$ axis, as shown in Fig.15. The polar-pattern slice is then be considered as one sample (interpolation kernel) used to reconstruct the soundfield at a distance $z_l$ from the array.

Figure 15: Geometry of $x$ -slice through polar pattern to evaluate the effective sampling kernel along a line parallel to the speaker array.