We have so far not used any assumptions regarding the microphone/speaker array to be used.
The sampling analysis of §2 on page made use of the far-field assumption in obtaining a spatial aliasing limit that depended only on the source angle and spatial frequency . Generalizing to the near field (arbitrary source distances) means that the sampling analysis is applicable only locally along the array. That is, the wavelength seen by the line array depends on both the source angle and the distance of the source to the array (or equivalently, the relative distance of the source to the array and to the listening point). For example, the line from the source normal to a horizontal array (see Fig.3 on page ) is at angle , which is always oversampled by the array. Points on the array far away from the normal line, however, see an angle approaching degrees to the right and degrees to the left. If a source touches the array at , then all of array points other than the point at see a right angle ( degrees). This behavior means we cannot set a limited stage-angle to avoid spatial aliasing like we did in the far-field case (§2). We can now accept a 180-degree stage, or limit the closeness and layout-width of the sources to obtain a worst-case angle limit (maximally close to the array at the edge of the allowed stage), and treat that as before.