## 3 System Architecture
### Continuous Phase Modulation (CPM)  
Two advantages of CPM have led to its widespread application. The transmitted signal's constant envelope and lack of amplitude variations are the first of these. Constant envelope modulations are required in applications where completely saturated, non-linear RF power amplifiers are required due to power supply constraints. Constant envelope modulations are also helpful in situations requiring low-cost, basic transmitters. Bluetooth technologies and digital FM land mobile radios are two instances of this. Overall, CPM is particularly transmitter-friendly because of the modulation's constant envelope feature [51]. The constant envelope of CPM happens in single carrier communication. For the proposed OFDM system, since CPM is employed in the mapper, the transmitted signal does not have a constant envelope. Power efficiency and bandwidth are CPM's other advantages. The size of the data alphabet (binary, M-ary, etc. ), the modulation index or indexes, and the frequency pulse's duration and shape define a specific CPM format. One can regulate the signal's spectrum and power efficiency by carefully choosing these parameters. Because of this, CPM is incredibly spectrum-friendly and adaptable [51]. The transmitter and the transmission medium, which make up the first two components of the communications link, benefit from these advantages (constant envelope, power efficiency, and spectrum efficiency). CPM, however, does not handle the receiver--the last component of the communications link--well. The signal is nonlinear because of its constant envelope, which indicates that it is not a linear function of the transmitted data. This complicates the process of demodulating and synchronizing the signal. Moreover, employing longer, smoother frequency pulses and expanding the size of the data alphabet are common ways to increase spectral efficiency. These also make the receiver more complex. Despite having a complex receiver, CPM still possesses a lot of potential in improving wireless and underwater acoustic communication. In this study, we will utilize complex-baseband notation to represent various signals, operating on the understanding that the outcomes are applicable in a carrier-modulated environment. The complete set of CPM signals can be denoted as follows [20]:  
\[s(t;\alpha)=\sqrt{\frac{E}{T}}exp(j\psi(t;\alpha)) \tag{1}\]Here, \(E\) stands for symbol energy, \(T\) represents symbol duration, and \(\psi(. )\) denotes the phase of the signal. As evident from this equation, the CPM signals exhibit a constant envelope, a characteristic that adds to the appeal of this modulation type. All the information is encoded in the signal phase, which is given by [52]:  
\[\psi(t;\alpha)\equiv 2\pi\sum_{i=-\infty}^{n}\alpha_{i}h_{i}q(t-iT),\ nT\leq t<(n+ 1)T \tag{2}\]  
In this equation, \(\left\{h_{i}\right\}\) is a set of modulation indexes with \(N_{h}\) elements, \(\alpha=\left\{\alpha_{i}\right\}\) represents data symbols drawn from an \(M\)-ary alphabet, and \(q(t)\) signifies the phase pulse. The subscript notation with an underline in Equations (1) and (2) is defined as modulo-\(N_{h}\), i.e.,  
\[\underline{i}\triangleq i\ \text{mod}\ N_{h} \tag{3}\]  
We assume that the modulation indexes are rational numbers of the form [27]:  
\[h_{i}\triangleq\frac{K_{i}}{p} \tag{4}\]  
Here, \(p\) is determined by expressing all modulation indexes as fractions and selecting \(p\) as the smallest common denominator. The case where \(N_{h}=1\) is recognized as single-\(h\) CPM, and the less common scenario where \(N_{h}>1\) is termed multi-\(h\) CPM.