# Simulating Signal Aberration and Ranging Error for Ultrasonic Indoor Positioning
## 1 Introduction
In the far field, the emission cone semi-angle \(\vartheta\) is approximately described by the well-known relationship [19]  
\[\sin\vartheta=1.22\frac{\lambda}{D} \tag{1}\]  
where \(\lambda\) is the emitted wavelength and D the diameter, or aperture, of the circular planar transducer. Furthermore, it is well-known that, to obtain good results, the receiver must always operate within the emission cone of the emitter [20]. At present, no equivalent formulas are known in the case of any signals, such as chirp. To design a positioning system based on ultrasonic signals, tools to evaluate the spatial coverage of each transducer in terms of quality (amplitude and level of deformation) of the received signal are needed. For any signals, there are no simplified formulas that give a reliable indication; therefore, the usage of numerical tools is mandatory. The use tout court of finite element analysis (FEA), which is a very powerful general use tool, seems excessive for the design of positioning systems and definitively not practical from the computational point of view. In fact, for the systems under investigation, large spatial regions of several cubic meters and time windows of tens of milliseconds should be considered (see, e.g., [9; 10; 13]). From these premises, it was, therefore, decided to use a powerful numerical tool, the academic Field II software [21], for the analysis of ultrasound positioning systems. Field II was developed and is currently very well-known for the simulation of complete ultrasonic imaging systems. A transducer for ultrasonic signals both for transmission and reception, and the formation process of images in the field of medical ultrasounds, are simulated. However, Field II has the numerical characteristics that make it a valid tool also in the field of ultrasonic positioning. Among many other capabilities, which however fall outside the scope of this work, this tool is able to calculate the acoustic pressure field at any point in the space for transducers and signals of any shape, taking into account the attenuation properties of the propagation medium. In other words, Field II allows to modify in any way the transducer shape and size, and the signal applied to the transducer, to evaluate their effects in a trial and error design cycle, if necessary. In this work, using Field II, the effectiveness of cross-correlation-based ranging techniques using a chirp signal when the diameter of the circular plane transducer used as ultrasonic emitter is changed is shown. This work will show that, considering a chirp signal outside a certain emission cone generated by the transducer, the usual ranging technique introduces a significant error in calculating the emitter-receiver distance. In perspective, the main advantages of the proposed approach are the possibility of examining the acoustic field over time and space at each point of the region of interest as a function of the aperture and of the type of signal emitted (e.g., of its bandwidth or shape) and the ability to easily test each algorithm dedicated to estimating the TOA in the various positions and operating situations. This paper is structured as follows. Section 2 describes the proposed simulation setup, while Section 3 shows the simulation results, and Section 4, the discussion. Section 5 draws the paper's conclusions.