Designing microphone arrays is an arduous task that often requires simulations in order to make sure the array can reach the specified performance objectives. While simulations offer valuable insight in array behavior, they consume a lot of time and computational resources.
An approach to reduce this complexity and provide a good starting point for the design process is to use the formula introduced by Sijtsma. In short, this formula is a rule of thumb that approximates the beamwidth of a microphone array for conventional beamforming. The power of this formula lies in its simplicity as it utilizes only the most basic design parameters of the array: array diameter, distance between the array and the source, and the frequency to analyze. Therefore, this approach allows manufacturers to speed up the array design process while retaining an acceptable level of accuracy in predicting the beamwidth.
The beamwidth formula is
\begin{equation} \text{BW} = \frac{425\frac{m}{s}\cdot x}{D \cdot f}\,, \end{equation}
in which \(BW\) is the estimated beam width in meters, \(x\) is the distance from the source to the microphone array in meters, \(D\) is the array diameter also in meters, and \(f\) is the frequency of interest in Hz.
To assess the effectiveness of the formula, we provide the table below listing the formula's outcomes and compare it to simulated data. Three different arrays were used in the simulations: a ring array consisting of 48 microphones with a diameter of 0.70 meters, a Fibonacci array with 72 microphones and a diameter of 0.63 meters, and an Octagon array that consists of 192 microphones spread out over a diameter of 0.75 meters. Finally, the last important parameter - the distance between microphone array and sound source - is set to 1 m.
| Beamwidth (m) | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Array | Ring 48 mic. | Fibonacci 72 mic. | Octagon 192 mic. | ||||||
| Frequency (Hz) | Sijtsma | Simulation | Diff (%) | Sijtsma | Simulation | Diff (%) | Sijtsma | Simulation | Diff (%) |
| 1000 | 0.61 | 0.38 | 38 | 0.67 | 0.58 | 13 | 0.57 | 0.52 | 9 |
| 1250 | 0.49 | 0.30 | 39 | 0.54 | 0.45 | 17 | 0.45 | 0.41 | 9 |
| 1600 | 0.38 | 0.23 | 40 | 0.42 | 0.35 | 17 | 0.35 | 0.32 | 9 |
| 2000 | 0.30 | 0.19 | 37 | 0.34 | 0.28 | 18 | 0.28 | 0.25 | 11 |
| 2500 | 0.24 | 0.15 | 38 | 0.27 | 0.22 | 19 | 0.23 | 0.20 | 13 |
| 3150 | 0.19 | 0.12 | 37 | 0.21 | 0.18 | 14 | 0.18 | 0.16 | 11 |
| 4000 | 0.15 | 0.09 | 40 | 0.17 | 0.14 | 18 | 0.14 | 0.13 | 7 |
| 5000 | 0.12 | 0.07 | 42 | 0.13 | 0.11 | 15 | 0.11 | 0.10 | 9 |
| 6300 | 0.10 | 0.06 | 40 | 0.11 | 0.09 | 18 | 0.09 | 0.08 | 11 |
| 8000 | 0.08 | 0.05 | 38 | 0.08 | 0.07 | 13 | 0.07 | 0.06 | 14 |
| 10000 | 0.06 | 0.04 | 33 | 0.07 | 0.06 | 14 | 0.06 | 0.05 | 17 |
To sum up, the ring array outperformed the estimations of the Sijtsma formula throughout the simulations: In average, the beamwidth was 38 % smaller leading to an even sharper acoustic image. A possible explanation for these results is that the formula is based on the assumption that the microphones of the array are, at least in good approximation, spread out evenly over the area of the circular aperture of the given diameter which clearly is not the case with the ring array. However, this is true for the other two arrays. In their case, the formula overestimates the simulation results only by 16 % (Fibonacci) and 11 % (Octagon). Taking into account that the simulations are based on perfect conditions (exact microphone positions, exact temperature, monopoles) that are almost never met in real-world measurements, Sijtsma’s formula can be considered a good and handy approximation.
P. Sijtsma.: “Experimental techniques for identification and experimental techniques for identification and characterisation of noise sources characterisation of noise sources”, Advances in Aeronautics & Applications, Belgium, 2004.
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