## 2 Background Theory  
### General Signal Models for Pulse Position Modulation  
For wideband guided elastic signals, we adopt a signaling method that is similar to the ultra-wide-band (UWB) signaling in radio communications [19, 22, 29]. A general signaling model can be described as follows. Every transmitted symbol is composed of \(N_{f}\) repeated pulses, called frame. Let \(T_{s}\) and \(T_{f}\) denote the symbol duration and frame duration, respectively. Then, \(T_{s}=N_{f}T_{f}\), which means there is one pulse per frame. Each guided elastic pulse, \(g(t)\), is of a short duration, \(T_{0}<T_{f}\), at the microsecond scale, occupying a wide bandwidth of a few hundred kilo-hertz. Each frame contains \(N_{c}\) chips, each of duration, \(T_{c}\). A general wideband symbol signature waveform transmitted during the acquisition process for a single user can be expressed as a series of pulses:  
\[g_{s}(t)=\sum_{j=0}^{N_{f}-1}g(t-jT_{f}-c_{j}T_{c}) \tag{1}\]  
where \(c_{j}\in\{0,1,2,\cdots,N_{c}-1\},\forall j\in[0,N_{f}-1]\) represents user-specific pseudorandom time hopping (TH) code for user separation [29], which may take on different patterns, such as fast hopping or slow hopping. The pulse amplitude is scaled to satisfy \(\int g^{2}(t)dt=1/N_{f}\), such that the symbol waveform has unit energy, \(\int g_{s}^{2}(t)dt=1\). Furthermore, the data symbols are modulated during transmission. In pulse-position modulation (PPM), different symbols are realized by shifting a pulse to distinct positions in time within the specified symbol duration. For a binary PPM, the modulated symbols transmitted by the desired user can be described as:  
\[s(t)=\sum_{i=-\infty}^{\infty}g_{s}(t-iT_{s}-a_{i}\Delta) \tag{2}\]where \(a_{i}\in\{0,1\}\) represent the data symbols that are modeled as binary-independent and identically distributed (i.i.d.) random variables and \(\Delta\) represents the time shift imposed on all the elastic pulses of a given transmission stream by a unit data symbol. Next, we discuss the elastic wave propagation channel for wideband signaling. Under the linear elastic wave theory, the overall channel response can be modeled by a stochastic tapped delay line:  
\[h(t)=\sum_{l=0}^{L-1}\alpha_{l}f_{l}(t-\tau_{l}) \tag{3}\]  
where \(L\) is the number of taps in the channel response, \(\alpha_{l}\) is the path gain at excess delay, \(\tau_{l}\), corresponding to the \(l\)-th propagation path. We assume that the delay is \(\tau_{0}<\tau_{1}<\cdots<\tau_{L-1}\). The first arrival, \(\tau_{0}\), is normally referred to as the timing information and is measured with reference to the receiver's local clock for demodulation purposes. The function, \(f_{l}(t)\), represents the combined responses of the transmit PZT sensor, the receive PZT sensor, as well as the propagation channel corresponding to the \(l\)-th path of a transmitted pulse. We note that in this paper, the underlying assumption regarding a surface-mounted PZT wafer active sensor is that its response is characterized as a linear time invariant system and, thus, is absorbed in \(f_{l}(t)\). More detailed modeling of PZT wafer active sensors can be found in various literature (see, e.g., [30,31]). For simplicity, we let \(f_{l}(t)=\delta(t)\), an ideal channel impulse response. Hence, we can re-write Equation (3) as:  
\[h(t)=\sum_{l=0}^{L-1}\alpha_{l}\delta(t-\tau_{l,0}-\tau_{0}) \tag{4}\]  
where \(\tau_{l,0}\triangleq\tau_{l}-\tau_{0}\) is the relative time delay of the \(l\)-th path.