## 3 Losnus
### Signal Processing  
Signal processing for ToF or ToA systems is a compromise along three factors--range, resolution and repetition rate. For resolution, two aspects are important: The first is separation of two nearby targets, which are limited by \(\Delta R\leq c/(2B)\) where c is the speed of wave propagation and \(B\) is the bandwidth [22]. The second aspect is uncertainty of the ToF or ToA estimates where various techniques are available including different type of estimators like classical cross correlation (CC), generalized cross correlation (GCC), maximum likelihood (ML), least mean square adaptive filters (LMS) and others [23]. Furthermore, the sample rate of the system, if low, can require the use of interpolation methods. Simple algorithms include parabolic interpolation of the correlation functions where more complex ones use different types of transforms or up-sampling methods [24]. _LOSNUS_ uses a linear frequency modulated chirp with binary cross correlation at a high sample rate. A linear frequency modulated chirp can be written as the real signal of the analytical waveform according to Equation (1) where T is the duration of the signal. The function rect is defined identical as in [22] and is zero if \(|t|\geq T/2\) and 1 otherwise. The total frequency sweep is \(kT=\Delta\). \[s(t)=rect\left(\frac{t}{T}\right)e^{2\pi i(f_{0}t+kt^{2}/2)} \tag{1}\]  
A very important property for a chirp is the dispersion factor or processing gain \(D=T\Delta\). The processing gain is the gain in the peak of the autocorrelation function for a pulse with equal range resolution. Furthermore it can be shown that for larger values of D the spectral amplitude of the chirp signal is approximately a rectangular window of width \(\Delta\). However, even for values of D as low as 10 about 95% of the energy are contained in the band \(\Delta\)[22]. Binary correlation for a discrete input signal \(r[n]\) with \(0\leq n\leq N\) and a reference signal s[m] with \(0\leq m\leq M-1\) is calculated according to Equation (2). Both input signals can have values of \(+1\) and \(-1\) respectively. The number of computations required is N\(\cdot\)M. As both signals are binary, efficient and fast implementation is possible. \[(s\star r)[n]=\sum_{i=0}^{M-1}s^{*}[i]r[i+n] \tag{2}\]  
The most important properties for a linear chirp with the settings used in _LOSNUS_ are shown in Figure 3a,b. Figure 3a shows the theoretical 1-bit quantized chirp and Figure 3b its binary waveform and autocorrelation function. Figure 2: Locating operation: Transmitting sequence (**a**), Transmitted frames (**b**). _LOSNUS_ in this setup uses broadband electrostatic transducers from Senscomp allowing the transmission of such signals. One problem faced during system design was the coverage area as the broadband transducers have a frequency dependent radiation pattern. Signals resulting from directions outside the mainlobe are heavily changed in phase/amplitude and correlation can drop by more than 50% [25]. For this reason a small cone has been attached to the sensors similar to [26]. The resulting directivity diagrams before and after the modification are shown in Figure 4a,b. With the cone attached the waveform show in Figure 5a was recorded using the linear frequency modulated chirp from Figure 3a with driving amplitude of 160 Vpp and a DC Bias of 100 V. Using the signal of Figure 5a in its 1-bit binary quantized representation its autocorrelation is shown in Figure 5b. Results are still satisfying and the performance gain compared to a burst is easily seen. For comparison, in Figure 5 the burst had equal length to the chirp as a small pulse is more difficult to measure. As the signal length is only 256 \(\upmu\)s and the sample rate is 5 MS FFT, frequency resolution is low and the spectrum of the waveforms in Figure 5a is not shown. Figure 4: **(a)** Directivity diagram without attached cone of Senscomp 600 electrostatic transducer; **(b)** Directivity diagram after application of cone.