# Acoustic Simulation for Performance Evaluation of Ultrasonic Ranging Systems
## 2 Simulation Setup
A linearized air absorption (slope 39.3 dB/m/MHz, constant term \(-0.26\) dB/m, i.e., about 0.92 dB/m @ 20 kHz and 1.70 dB/m @ 50 kHz) was assumed at around 40 kHz. More in detail, the transducer is a circular and planar piston with a diameter of 8.5 mm. This diameter was shown to be able to cover the entire volume of the room using a 30-50 kHz linear chirp signal [18]. For all the simulations that follow, the transducer is divided into small square elements with sides measuring 0.125 mm by 0.125 mm. The chosen element size in this work is a good compromise between the accuracy of the solution and the computational resources involved in the simulations (see Figure 2). A linear chirp with a bandwidth of 30-50 kHz and a duration of 5.12 ms was used as the emitted signal [11; 25]. One of the major issues of any ultrasonic ranging system is the environmental noise in the signal band employed. The approach here proposed allows us to include in the simulation any kind of noise, even the actual disturbance present in the environment for which the system is being designed. On the other hand, when there are no further specific indications, white noise is mostly used for the study of systems in many fields, as it uniformly covers the entire useful band of the system under examination. The use of white noise here is therefore entirely indicative and simply an example. Uniform white noise was added to the received signal and the reference SNR was calculated at a distance of 1 m from the transducer on its emission axis (see point P in Figure 1). The simulations were carried out in a free space scenario with performances evaluated on the plane section A of \(4\times 3\) m\({}^{2}\), sampling the signal at a frequency \(f_{5}=1\) MHz. Figure 1: Simulation setup: the vertical section of the typical \(4\times 4\times 3\) m\({}^{3}\) room along which the ranging calculations using cross-correlation are computed. The SNR is considered at point P, namely at distance of 1 m from the emitter surface center and on its emission axis. Note that the simulation is carried out in a free space scenario with performances evaluated on the plane section A of \(4\times 3\) m\({}^{2}\). For an efficient use of the computational resources, the acoustic field was calculated on a regular grid of points in the space for the time strictly necessary, i.e., for the duration of the time window compatible with the complete reception of the employed signal [24], here 15.62 ms, including the chirp duration and the traveling time to the furthest corner of section A in Figure 1.  
In a first step, for each considered point in space, the numerical simulation computed the acoustic pressure over the time generated by the complete excitation signal. Subsequently, an ideal receiver, which linearly transduced the pressure signal into an electrical signal, was assumed. This signal was then cross-correlated with the emitted signal at each point and the related ranges were calculated. The simulations were repeated for different SNR levels and the range errors were computed for increasing noise levels.