## 2 System Model of Upgraded Multi-Carrier OGFDM
After the de-multiplexing process, the convoluted subcarriers are transferred to the oversampling level where each utilized frequency subcarrier is down-sampled by \(K\) or \(2K\) according to the oversampling factor of the transmitter. At the allocation level (the first part of upgrading), the doubled set of the complex numbers that belong to the down-sampled subcarriers are gathered in one stream. At the modulation level, where diffident shapes of modulation are applied, the complex numbers of the frequency subcarriers are converted dynamically into a stream of binary digits. From a mathematical perspective, the multi-carrier system can be expressed as follows:  
In the filtration level, the impulse responses of each employed \(l\)th pair of the Hilbert filters for shaping filters (_sf_) are represented as follows [23]:  
\[ \begin{array}{rcl}\mathit{sf_{h}}^{A}(t)&=&g(t)\;cos(2\pi fc_{h}t)\\ \mathit{sf_{h}}^{B}(t)&=&g(t)\;sin(2\pi fc_{h}t)\end{array}  \tag{1}\]  
In addition, the impulse responses of matching (_mf_) are represented as follows [23]:  
\[ \begin{array}{rcl}\mathit{mf_{h}}^{A}(t)&=&\mathit{sf_{h}}^{A}\;(-t)\\ \mathit{mf_{h}}^{B}(t)&=&\mathit{sf_{h}}^{B}\;(-t)\end{array}  \tag{2}\]  
where \(fc_{h}\) indicates the frequency center of the \(l\)th orthogonal pair and, the superscripts \(A\) and \(B\) refer to the in-phase and out-phase of the applied filter. Besides, the \(g(t)\) signifies the baseband pulse as follows [23]:  
\[g(t)=\frac{\sin[\pi(1\;-\;\alpha)\Upsilon]+\;4\alpha\Upsilon\;\cos[\pi(1\;+\; \alpha)\;\Upsilon]}{\pi\Upsilon\left[1\;-\;(4\alpha\Upsilon)^{2}\;\right]} \tag{3}\]  
where \(\Upsilon=t/\Delta t\), roll-off (\(\alpha\)) specifies the filter excess and \(\Delta t\) denotes the sampling interval prior to the oversampling process. Moreover, the output signal of the convolution operation between the shaping and matching filters is expressed as follows [23]:  
\[ \begin{array}{rcl}\mathit{mf_{j}}^{C}(t)\;\otimes\;\mathit{sf_{i}}^{D}(t)& =&\left\{\begin{array}{rcl}\delta(t-t_{0}),\;\mathit{if}\;C=D\;\mathit{and} \;j=i\\ 0,\;\mathit{if}\;C\
eq D\;\mathit{or}\;j\
eq i\end{array}\right.\end{array} \right\} \tag{4}\]  
where \(t_{0}\) states the probable delay, the subscripts \(i\) and \(j\) represent the order of the _fc_, and the superscripts \(C\) and \(D\) indicates either the in-phase or out-phase. Considering that the expanded frequency sampling is equivalent in both sides (transmitter and receiver), the _fc_ of each filter pair is allocated as follows [24]:  
\[\mathit{fc_{h}}=(2h-1)\;(\text{BW/K}) \tag{5}\]  
Figure 2: Receiver design of the OGFDM including the two key parts of the upgraded level. where, \(h\) denotes the position of the Hilbert pair and BW equals to \(F_{DAC/ADC}/2\). Since every applied _fc_ is optimally selected, the utilized Hilbert filters are accommodated orthogonally and distributed sequentially in the available spectrum. The specified BW of each employed filter (\(F_{BW}\)) can be expressed as follows [25]:  
\[F_{BW}=Sub_{S}*(1+\alpha) \tag{6}\]  
where \(1\leq\alpha\leq 0\), and the frequency sampling of subcarrier (\(Sub_{S}\)) represents the size of generated copy for each oversampled subcarrier. In the oversampling level, a flexible oversampling process is applied to decide the required sampling frequency for each frequency subcarrier as follow [25]:  
\[Sub_{S}=F_{DAC}/OV \tag{7}\]  
where \(OV\) refers to the oversampling factor that mostly equals to the number of used subcarriers (_K_). Besides, this important factor (\(OV\)) can be employed to determine the number of generated copies for each utilized subcarrier. Occasionally, the \(OV\) factor is doubled (\(2K\)) to give extra support for the employed filters, nevertheless, this kind of manipulation can impact the overall channel capacity according to Shannon theorem as follows [26]:  
\[Capacity=(BW)*log2\ (1+SNR) \tag{8}\]