## 2 Proposed CPM-GFDM Transceiver
### CPM-GFDM Transmitter  
The proposed CPM-GFDM transmitter is shown in Figure 1. As shown in this figure, data bits go through the CPM mapper to become converted into CPM-mapped complex numbers, followed by upsampling. Then, \(M\) symbols are collected followed by the pulse shaping filter and IFFT. Cyclic prefix is added next, and the signal is transmitted. Let \(K\) represent the number of subcarriers and let \(M\) represent the number of symbols in each subcarrier. A serial bit stream \(b_{i}\) is converted into a parallel stream of \(K\) symbols where each symbol contains \(\log_{2}J\) bits feeding \(J\)-CPM mappers. These mappers produce \(J\)-ary CPM-modulated complex-valued data symbols at their outputs. The next block upsamples the complex symbol sequence by a factor of \(N\) where \(N\) number of samples are used to represent a time slot. These operations repeat until \(M\) symbols--denoted by a vector \(\overline{d}_{k}\) where \(k=0,1,\ldots,K-1\)--are received at each subcarrier, with \(\overline{d}_{k}=[d_{k,0},d_{k,1},\ldots,d_{k,M-1}]\). Therefore, each of the \(K\) subcarriers transmits \(M\) complex-valued data symbols per GFDM  
Figure 1: Proposed CPM-GFDM Transmitter. frame. Hence, the transmitted signal will consist of \(K\times M\) data symbols that could be represented by a matrix \(\mathbf{d}\) as shown below [36]:  
\[\mathbf{d}= \begin{bmatrix}\overline{d}_{0}\\ \vdots\\ \overline{d}_{K-1}\end{bmatrix} \]  
or  
\[\mathbf{d}= \begin{bmatrix}d_{0,0}&d_{0,1}&\dots&d_{0,M-1}\\ d_{1,0}&\ddots&&d_{1,M-1}\\ \vdots&&\ddots&\vdots\\ d_{K-1,0}&d_{K-1,1}&\dots&d_{K-1,M-1}\end{bmatrix}  \tag{1}\]  
The \(k\)th row in the aforementioned matrix denotes the data symbols sent over the \(k\)th subcarrier, and the \(m\)th column denotes the data symbols sent over the \(m\)th time slot. Each of the elements in this matrix has been modulated by CPM and is given by:  
\[d_{k,m}=\cos\phi_{k}[l]+j\sin\phi_{k}[l] \tag{2}\]  
where \(l\) represents the present symbol index, and the information-bearing phase \(\phi_{k}[l]\) for the \(k\)th subcarrier is given by:  
\[\phi_{k}[l]=\pi\sum_{i=-\infty}^{l-1}h_{k}[i]a_{k}[i]+\pi h_{k}[l]a_{k}[l] \tag{3}\]  
\(\{\alpha_{k}[i]\}\) denotes the sequence of \(J\)-ary information-carrying symbols for the \(k\)th subcarrier and \(i\)th symbol that belong to the set \(\pm 1,\ \pm 3,\ \dots,\ \pm(J-1),\ \alpha_{k}[l]\) is the present symbol for the \(k\)th subcarrier, and \(\{h_{k}[i]\}\) for the \(k\)th subcarrier and \(i\)th symbol are called modulation indices. If \(\forall i,\ h_{k}[i]=h_{k}\), this indicates that \(h\) is the same for all symbols for \(k\)th subcarrier. This scheme is typically known as \"single \(h\)\" CPM. If the value of \(h\) is different from symbol to symbol, then this type of CPM is known as \"multi-\(h\)\" CPM. In multi-\(h\) CPM, the value of \(\{h_{k}[i]\}\) changes in a round robin fashion out of all the \(h\) values. In this work, we use single \(h\) CPM and an identical modulation index for all the subcarriers. The most advantageous aspect of a CPM scheme is that it has memory, which is symbolized by the first term in (3). It is obvious that all symbols up to \((l-1)\) symbols are added together to make up the phase \(\phi_{k}[l]\)[37; 38]. Let \(p\) and \(q\) be two relatively prime integers. Then, the modulation index \(h\) is chosen as the ratio of \(p\) and \(q\) such that \(0<h<1\). By doing so, catastrophic phase situations are prevented and the overall number of points in the CPM constellation are limited to a manageable level [39]. The phase states for even \(p\) are given as,  
\[\phi_{k}=\left\{0,\pi\frac{p}{q},2\pi\frac{p}{q},\dots,(q-1)\pi\frac{p}{q}\right\} \tag{4}\]  
On the other hand, the phase states \(\phi_{k}\) for odd \(p\) are given as,  
\[\phi_{k}=\left\{0,\pi\frac{p}{q},2\pi\frac{p}{q},\dots,(2q-1)\pi\frac{p}{q}\right\} \tag{5}\]  
The signal constellation has \(q\) points when \(p\) is even, and \(2q\) when \(p\) is odd, as shown by (4) and (5), respectively.