# Design of an Underwater Acoustic Waveform and Integrated System for Communication and Detection
## 2 Waveform Design  
### Signal Model  
A constant envelope waveform is considered with CPM data combined with LFM. This is given by  
\[S_{CPM-LFM}(t)=S_{CPM}(t)S_{LFM}(t) \tag{1}\]  
which can be expressed as  
\[S_{CPM-LFM}(t)=A\text{rect}(t/T_{p})exp[j\Big{(}\pi\mu t^{2}+\phi(t,I)+\phi_{0 }\Big{)}] \tag{2}\]  
with  
\[\phi(t,I)=\sum\
olimits_{n=0}^{N-1}\text{rect}(\left(t-nT_{s}\right)/T_{s}) \Big{(}\pi h{\sum\
olimits_{i=0}^{n-I}I_{i}}+2\pi h{\sum\
olimits_{i=n-L+1}^{ n}I_{l}}q(t-iT_{s})\Big{)} \tag{3}\]  
Here, \(S_{CPM}(t)\) represents the expression of a CPM modulated signal, \(S_{LFM}(t)\) represents a linear frequency modulation (LFM) signal, and \(S_{CPM-LFM}(t)\) denotes a composite waveform signal that carries the constant envelope phase information of CPM using the linear frequency modulation signal. \(S_{CPM-LFM}(t)\) is obtained by multiplying \(S_{CPM}(t)\) and \(S_{LFM}(t)\). We employ the equivalent lowpass representation, \(A\) is the amplitude of the LFM signal,is the pulse width, \(\mu\) is the modulation rate, \(\mu=B/T_{p}\), \(B\) is the bandwidth of the LFM signal. \(\phi(t,l)\) is the information-carrying phase, \(\{l_{i}\}\) is the sequence of \(M\)-ary information symbols selected, and \(l_{i}\in\{\pm 1,\pm 3,\cdots,\pm(M-1)\}\). \(N\) is the number of data symbols within an LFM pulse, and \(T_{s}\) is the symbol duration, \(T_{s}=T_{p}/N\), \(h\) is the modulation indices. \(\phi_{0}\) is the initial phase of CPM. The waveform \(q(t)\) is represented in general as the integral of pulse \(g(t)\). Three popular pulse shapes are given in Table 1, and LREC denotes a rectangular pulse with duration \(LT\), where \(L\) is a positive integer, \(T\) is sampling duration. LRC denotes a raised cosine pulse with duration \(LT\). The third pulse is a GMSK pulse with bandwidth parameter \(B\)', which represents the \(-3\) dB bandwidth of the Gaussian pulse. Equation (2) shows that the CPM-LFM waveform differs from CPM only by the term \(\pi\mu^{2}\) in the phase. Thus, it has continuous phase and constant envelope. Figure 1 illustrates the signal generation. Simulation was conducted to obtain the time domain and frequency domain characteristics of the proposed waveform. The parameters considered are \(A=1\), \(B=4\) KHz, \(T_{p}=0.5\) s, \(h=0.5\), \(\phi_{0}=0.5\), \(L=2\), and LREC pulse. Figure 2 presents the time domain characteristics of the LFM and CPM-LFM waveforms. This shows that the superposition of CPM does not alter the constant envelope characteristic of the LFM waveform. Thus, it will be robust to amplifier nonlinearity, so the distortion will be minimal. This makes the waveform compatible with existing sonar systems. \begin{table}
\begin{tabular}{c c c} \hline  & \(g(t)\) & \(q(t)\) \\ \hline LREC & \(g(t)=\left\{\begin{array}{cc}\frac{1}{2 LT}&(0\leq t\leq LT)\\ 0&\text{else}\end{array}\right.\) & \(q(t)=\left\{\begin{array}{cc}1/2,&LT<t\\ \frac{t}{2 LT},&0\leq t\leq LT\\ 0,&0<t\end{array}\right.\) \\ LRC & \(g(t)=\left\{\begin{array}{cc}\frac{1}{2 LT}\left(1-cos\frac{2\pi t}{L} \right)&(0\leq t\leq LT)\\ 0&\text{else}\end{array}\right.\) & \(q(t)=\left\{\begin{array}{cc}1/2,&LT<t\\ \frac{t}{2 LT}-\frac{1}{4\pi}sin\left(\frac{2\pi t}{LT}\right),&0\leq t\leq LT\\ 0&0<t\end{array}\right.\) \\ GMSK & \(g(t)=\left\{\begin{array}{cc}Q\Big{[}2\pi B^{\prime}\Big{(}t-\frac{T}{2} \Big{)}/(ln2)^{1/2}\Big{]}\\ -Q\Big{[}2\pi B^{\prime}\Big{(}t+\frac{T}{2}\Big{)}/(ln2)^{1/2}\Big{]}\end{array} \right\}\) & \(q(t)=\frac{1}{\sqrt{2\pi}}\int_{t}^{\infty}\int\frac{(-\frac{t^{2}}{2})}{2}\,exp\) \\  & where \(Q(t)=\int_{0}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}dt\) & \\ \hline \end{tabular}
\end{table} 
Table 1: Three commonly used CPM pulse shapes. Figure 1: Constant envelope CPM-LFM signal generation. Figure 3 illustrates the frequency domain characteristics of LFM, the proposed waveform, and GMSK-LFM. Note that GMSK is a special case of CPM. This shows that the CPM is within the LFM frequency range. Using LFM as the carrier allows data to be transmitted via phase modulation.