Introduction
The relationship between the maximum obtainable directivity and aperture of an antenna is described by:
$$ D = \frac{4\pi A_e}{\lambda^2} $$
Where $D$ is the directivity, $A_e$ the effective aperture and $\lambda$ the used wavelength. The derivation can be found in the following document: pdf.
However, if the antenna does not scan as a sphere but in $x, y$ but not $z$. Following the same path as the above-mentioned document the equation becomes:
$$ D = \frac{2\pi A_e}{\lambda^2} $$
The aperture of a lossless isotropic antenna is then also $A_e = \frac{\lambda²}{2 \pi}$ instead of $\frac{\lambda²}{4 \pi}$
Problem
I cannot manage to use that equation for a microphone array using delay-and-sum beamforming.
For example, if I have an array of 25 microphones spaced as a grid and that my setup has the following properties:
- Smallest distance $d$ between them is $0.01$ meters.
- Amount of orientations $o$ that it looks at is $64$.
- Frequency $f$ to be detected is 8 kHz.
The wavelength $\lambda = \frac{v}{f} = \frac{343}{8000} = 0.0429 $
The aperture $A$ is $A = \frac{\lambda^2}{precision} = \frac{0.0429^2}{\frac{2\pi}{64}} = 0.0187 $
Thus:
$$ D = \frac{2\pi A_e}{\lambda^2} = \frac{2\pi A}{\lambda^2} = \frac{2 \pi \cdot 0.0184}{0.0429^2} = 62 $$
I rounded the numbers for the simplicity of formulating my problem, if you do not round then you get $64$ instead of $62$ which is the number of orientations that it looks at furthermore the results cannot be true because:
- It does not involve the number of microphones used.
- It does not involve the smallest spacing between microphones.
- I probably made a mistake somewhere.
How can I calculate the maximum obtainable directivity for an array of sensors?