## 4 HFM-Based AUV Localization
### HFM Overview  
The HFM signal is a kind of chirp signal that use time-varying frequency to present a symbol. The frequency of a chirp signal either increases or decreases from the start frequency \(f_{1}\) to the end frequency \(f_{2}\) during symbol duration \(T\). If \(f_{1}<f_{2}\), it is regarded as up-chirp used to present bit '1', otherwise down-chirp used to present bit '0'. The expression of HFM signal is defined as  
\[s_{HFM}(t)=cos\left[\frac{2\pi}{k}log(1+kf_{1}t)+\theta\right],0<t<T, \tag{13}\]  
where,  
\[k=\frac{f_{1}-f_{2}}{f_{1}f_{2}T}. \tag{14}\]  
The rate of frequency variation is determined by the chirp rate \(k\). Figure 5 gives an example of chirp signal, assuming that the symbol duration is 5 ms, and the start and end frequencies are 1 kHz and 5 kHz respectively.  
Figure 5: An example of HFM signal, assuming that \(T\) is 5 ms, and the used frequency band is [1; 5] kHz.  
In chirp signal-based multiple access system, the multiple signals generated with different chirp rates can be separated even though the signals are mixed. In other words, as long as the ANs generate a HFM signal with a unique chirp rate, the simultaneous transmission can be guaranteed. As shown in (14), the chirp rate \(k\) is determined by \([f_{1},f_{2}]\). This means that the signal generated with a given chirp rate is composed of the frequencies ranging from the start and end frequencies of the chirp. In this example, a sub-channel of which start and end frequencies are 1 and 5 kHz is allocated to a specific user. Therefore, the generation of a set of unique chirp rates may be thought as how to divide the total bandwidth into sub-channels without frequency overlap. This is comparable to the channel assignment in FDMA, different sub channels being assigned to different users. For instance, the available bandwidth is given as [10; 20] kHz and the number of total ANs is five, and then the divided sub-channels are [10; 12], [12; 14], [14; 16], [16; 18] and [18; 20] kHz, and the low and upper frequency in each sub-channel becomes the start (\(f_{1}\)) and end (\(f_{2}\)) frequency respectively. Let \(B_{i}\) and \(k_{i}\) denote the \(i\)th sub-bandwidth and chirp rate generated with \(B_{i}\) respectively, a set of unique chirp rates is \(\mathbf{K}=[k_{1},k_{2},...,k_{N}]\). The \(i\)th chirp rate is assigned to the \(i\)th AN, then the \(i\)th AN uses the assigned chirp rate to generate the HFM signal for the packet transmission. This ensures the simultaneous transmission, leading to remove the MAC delay in localization. The proof of of simultaneous transmission using HFM signal is beyond the scope of this paper, so the detailed knowledge can be acquired the aforementioned references.  
One notable thing in the use of HFM signal is that Doppler effect has to be considered at the receiver. Although the HFM signal is known as a Doppler-invariant signal, the demodulation of a Doppler-distorted signal has a practical problem, the shifting of matched filter outputs. To deal with this problem, the sampling time should be adjusted according to the estimated Doppler factor at the receiver. However, tracking of the Doppler effect is another research area. Instead of adjusting the sampling time, ref. [24] proposes a simple solution that finds the peak of matched filter output not at an exact sampling time but within a certain time interval. We adopt this method in our system to correctly demodulate the a Doppler-distorted HFM signal.