# Design of an Underwater Acoustic Waveform and Integrated System for Communication and Detection
## 2 Waveform Design
### Performance Analysis  
In an AWGN channel, the received signal can be expressed as  
\[ \begin{split}& S_{r}(t)=S_{CPM}(t)S_{LFM}(t)+n(t)\\ &=\,Arect\big{(}t/T_{p}\big{)}exp\big{[}j\big{(}pmt^{2}+j(t,I)+j_ {o}\big{)}\big{]}+n(t)\end{split}  \tag{4}\]  
where \(n(t)\) of zero means white Gaussian noise. As the phase is used for data transmission, it is necessary to first process the received signal to extract the phase information. For simplicity, in this section, the frequency and phase offsets due to the relative motion between the transmitter and receiver are ignored. Multiplying by the conjugate of the LFM signal gives  
Figure 3: Frequency domain characteristics of three waveforms. Figure 2: Amplitude of LFM and CPM-LFM waveforms. \[ \begin{array}{l}S_{ra}(t)=S_{r}(t)S_{LFM}^{*}(t)\\ =A\text{rect}(t/T_{p})exp[j(\phi(t;I)+\varphi_{0})]+S_{LFM}^{*}(t)n(t)\\ =S_{CPM}(t)+S_{LFM}^{*}(t)n(t)\end{array}  \tag{5}\]  
According to reference [30], the Euclidean distance between \(S_{n}(t)\) and \(S_{CPM}(t)\) for an observation duration of \(NT_{s}\) can be expressed as  
\[d_{ij}=\left[\int_{0}^{NT_{s}}\Bigl{[}s_{i}(t)-s_{j}(t)\Bigr{]}^{2}dt\right]^{ 1/2} \tag{6}\]  
where \(s_{i}(t)=S_{i-m}(t)\) and \(s_{j}(t)=S_{jCPM}(t)\). For simplicity, consider the squared Euclidean distance  
\[ \begin{array}{l}d_{i,j}^{2}=\int_{0}^{NT_{s}}\bigl{[}s_{i}(t)-s_{j}(t) \bigr{]}^{2}dt\\ =\int_{0}^{NT_{s}}\bigl{[}S_{i-CPM}(t)+S_{LFM}^{*}(t)n(t)-S_{j-CPM}(t)\bigr{]} ^{2}dt\\ =\int_{0}^{NT_{s}}\bigl{[}\bigl{(}S_{i-CPM}(t)-S_{j-CPM}(t)\bigr{)}+S_{LFM}^{* }(t)n(t)\bigr{]}^{2}dt\\ =\int_{0}^{NT_{s}}\bigl{[}S_{i-CPM}(t)-S_{j-CPM}(t)\bigr{]}^{2}dt+\int_{0}^{ NT_{s}}\bigl{[}S_{LFM}^{*}(t)n(t)\bigr{]}^{2}dt\\ +2\int_{0}^{NT_{s}}\bigl{[}\bigl{(}S_{i-CPM}(t)-S_{j-CPM}(t)\bigr{)}S_{LFM}^{* }(t)n(t)\bigr{]}dt\end{array}  \tag{7}\]  
The second term is the integral of the product of the energy of the LFM signal and the energy of the noise, and it is not a function of the modulation parameters. The third term is the integral of the product of a deterministic signal and the noise and so is considered to be zero. Thus, the second and third terms have a negligible effect on the performance due to changes in the parameters, so the focus here is on the first term. The shared waveform is affected by the modulation parameters in the same way as continuous phase modulation. Next, we will analyze the first term,  
\[d^{2}=\int_{0}^{NT}[s_{i-CPM}(t)-s_{j-CPM}(t)]^{2}dt \tag{8}\]  
where \(S_{i-CPM}(t)\) and \(S_{j-CPM}(t)\) are two different signals with distinct phase trajectories, which are \(\phi(t,I_{i})\) and \(\phi(t,I_{j})\), respectively. The first symbols of the sequences \(I_{i}\) and \(I_{j}\) are different. Let the symbol for energy be \(\varepsilon\), and the bit energy be \(\varepsilon_{b}=\varepsilon/log_{2}M\), then,  
\[ \begin{array}{l}d^{2}=\int_{0}^{NT_{s}}\bigl{[}S_{i-CPM}(t)-S_{j-CPM}(t) \bigr{]}^{2}dt\\ =2E_{b}\frac{\log_{2}M}{I_{s}}\int_{0}^{MT_{s}}\bigl{\{}1-\cos\bigl{[}\phi(t, I_{i})-\phi\bigl{(}t,I_{j}\bigr{)}\bigr{]}\bigr{\}}dt\end{array}  \tag{9}\]  
where \(\delta_{\tilde{\eta}}{}^{2}\) is defined as the bit energy normalized Euclidean distance, its expression is:  
\[\delta_{\tilde{\eta}}^{2}=\frac{\log_{2}M}{T}\int_{0}^{NT_{\tilde{\eta}}}\bigl{\{} 1-\cos\bigl{[}\phi(t,I_{i})-\phi\bigl{(}t,I_{j}\bigr{)}\bigr{]}\bigr{\}}dt \tag{10}\]  
When utilizing the maximum likelihood estimation (MLE), the bit error rate (BER) of CPM signals is  
\[P_{M}=K_{\delta_{min}}Q(\frac{\sqrt{\mathcal{E}_{b}}}{N_{0}}d^{2}{}_{min}) \tag{11}\]  
where \(K_{\delta_{min}}\) represents the number of paths with the minimum Euclidean distance, and \(d^{2}{}_{min}\) is the minimum Euclidean distance for bits. \[ \begin{array}{l}d_{min}^{2}=\underset{N\to\infty}{lim}\min_{i,j}\delta_{ \tilde{\eta}}^{2}\\ =\underset{N\to\infty}{lim}\min_{j}\{\frac{\log_{2}M}{I_{s}}\int_{0}^{NT_{ \tilde{\eta}}}\bigl{[}1-\cos\phi\bigl{(}t,I_{i}-I_{j}\bigr{)}\bigr{]}dt\}\end{array}