# Simulating Signal Aberration and Ranging Error for Ultrasonic Indoor Positioning
## 1 Introduction  
Augmented reality (AR) and many other applications based on positioning are emerging technologies that need indoor positioning technology. Mall navigation, path finding in large hospitals or airports, the automatic guidance for unmanned cleaning and maintenance vehicles, surveillance systems, and others require positioning systems capable of operating inside buildings with high positioning accuracy [1; 2; 3; 4; 5]. Many accurate positioning systems use trilateration, a technique that has proven to work well indoors. The trilateration (multilateration) positioning technique requires three (many) range measurements between reference emitters and a sensor to be located. Distances and spatial positions with a high degree of precision at a relatively low cost can be provided by systems based on ultrasonic waves [6; 7; 8; 9; 10]. The most used ranging technique involves the estimation of the time of arrival (TOA) of a suitable ultrasonic signal. Typically, the TOA is estimated by finding in the received signal or in the postprocessed signal a specific that is easy to identify upon arrival. In some cases, the arrival time of the maximum peak of the envelope of an ultrasonic pulse is used. However,despite the simplicity of this technique, it is highly subject to acoustic disturbances, with errors in the order of wavelengths (centimeters) even in the presence of a high signal-to-noise ratio (SNR) [11; 12; 13; 14; 15; 16]. Methods based on cross-correlation estimate the TOA with high accuracy and, in general, good acoustical noise immunity [13; 14; 15; 16]. Digital cross-correlation techniques properly sample and analog-to-digital convert the received acoustical signal to obtain the array C = S \(\bigstar\) T, where S is the numerical array of received signal samples, T is the digital reference signal, previously stored in the sensor processor memory as a numerical array of samples, and \(\bigstar\) denotes the cross-correlation operator. The best aligning of S and T in time is revealed by the peak of the cross-correlation. The inter-signals displacement, or lag \(\tau\), corresponding to the cross-correlation peak (i.e., \(\tau_{\text{MAX}}\)), is proportional to the TOA [17; 18]. The distance measurement accuracy of the order of the current space sampling, which is the distance covered by the ultrasound during the time sample interval and that can be made much smaller than the ultrasonic wavelength, is achieved by estimating the TOA through the cross-correlation peak. In addition, the cross-correlation peak is easily detectable when the signal T is a chirp. In practical systems, it is possible to achieve a range resolution up to the order of one-tenth of the wavelength used. For example, in [16], a range resolution of \(\pm\) 1.2 mm was experimentally achieved using a 15-40-kHz chirp, with a wavelength range 22.86-8.57 mm, considering a sampling frequency of 1 MHz and, thus, space sampling of 0.34 mm with a sound speed of 343 m/s. When the signal and noise are uncorrelated, cross-correlation significantly increases the SNR. It is a known drawback that, in some cases, the cross correlation peak associated with the chirp that travels along the direct path (or line of sight, LOS) is lower than the signals coming from the reflection paths. In the presence of reflections, a number of signals from indirect paths combine to produce a peak higher than that of the direct path signal. The acoustic field generated by acoustic transducers according to the shape and aperture of the transducer has been widely studied in the past considering impulsive or continuous sinusoidal wave signals. The closed-form solution of the generated acoustic field made it possible to derive simple approximate formulas to calculate the emission angle as a function of the wavelength, aperture, and distance from the emitter, the best known of which apply to circular apertures.