# CPM-OFDM Performance over Underwater Acoustic Channels
## 3 System Architecture
### CPM-OFDM Receiver  
Using a sampling interval of \(\Delta T=T_{s}/N\), the received complex signal is sampled with an analog-to-digital converter (A/D). After A/D, the \(N_{g}\) samples, i.e., the CP of each OFDM symbol, is discarded. This operation removes the effects of ISI provided \(N_{g}\geq N_{h}\). After the removal of CP, each block of \(N\) received samples \(\left\{X_{k},k=0,1,\ldots,N-1\right\}\) is transformed back to the frequency domain using an FFT that is the OFDM demodulation, as shown in Figure 1. The recovered complex data symbols \(x_{k}\) then pass through the CPM de-mapper to recover the data sequence that was transmitted. Figure 3: Constellation diagram for (**a**) \(h=2/5\) (**b**) \(h=1/4\). Figure 2: Trellis diagram for \(h=1/2\). Viterbi Decoder  
Detection of CPM signals is a complex process. However, it becomes less complex and economical if a Viterbi decoder--also called Viterbi algorithm (VA)--is used [21]. This decoding technique was proposed in 1967 by [PERSON] to decode convolutional codes [35]. As was demonstrated by [PERSON] [36], VA is a maximum likelihood decoder. Furthermore, VA has also been used in the detection of signals that have been transmitted over linear models of voiceband channels [37; 38]. Other applications of VA include digital estimation problems. Some examples include voice recognition, recording systems with partial response channels, and optical character recognition [39; 40], etc. As was noted earlier, restricting the modulation index \(h\) to be rational, the number of states in CPM trellis can be kept to a manageable number that also facilitates the use of VA. Figure 4 shows the angle \(\theta\), the respective complex numbers and paths through the trellis for a typical data sequence and an arbitrary subcarrier when \(h=1/2\). Starting from state \(\theta=0\) (\(s_{k}=1+j0\)), a specific path can be traced based on the data sequence. For illustration purposes, a data sequence \(1011\) has been shown using slightly thick lines. The distance between received signal and all the trellis paths entering each state at an instant \(i\) is calculated by VA. This is followed by the removal of those paths that are probably not the candidates for maximum likelihood. For two paths entering the same state, the one having the best metric is selected and it is named the \"surviving path\". For all the states, similar surviving paths are selected. By continuing in a similar manner, VA advances deeper into the trellis and the decisions are made by removing the paths that are least likely [41]. Although such decisions are not maximum likelihood in the true sense, if the decision depth is long enough, these decisions can be almost as good as in the case of true maximum likelihood [42]. The technique for computing the distances between the received and likelihood sequences is as follows. Recall that the received sequence of complex numbers for the \(k\)th subcarrier at the output of FFT is \(x_{k}\) and the transmitted sequence of complex numbers for the \(k\)th subcarrier is \(s_{k}\). It follows that,  
\[ \begin{array}{ll}s_{k}&=u_{k}+jv_{k}\\ x_{k}&=\hat{u}_{k}+j\hat{v}_{k}\end{array}  \tag{6}\]  
Here, \(u_{k}\) and \(\hat{u}_{k}\) are the real parts of the transmitted and received complex numbers, respectively, while \(v_{k}\) and \(\hat{v}_{k}\) are the imaginary parts of the transmitted and received complex numbers, respectively. If \(d_{k}\) denotes the squared distance between two complex number sequences, then,  
\[d_{k}=(u_{k}-\hat{u}_{k})^{2}+(v_{k}-v_{k})^{2} \tag{7}\]  
Figure 4: All the possible paths through the trellis for \(h=1/2\). Solid lines indicate bit 1, broken lines indicate bit 0, and bold lines indicate a typical bit sequence of 1011. The complex numbers beside the figure are the possible CPM mapper output. At each symbol interval, these distances are successively updated. Moreover, all possible state transitions are extended. The path having highest likelihood is retained at next symbol interval, while all others are deleted.