## 2 System Model
### The Conventional Flip-F-OFDM  
In the conventional Flip-F-OFDM transmitter, depicted in Figure 1, the input data flow is digitally modulated and converted into frequency samples. To guarantee the creation of real F-OFDM signals, the input frequency samples to the IFFT bulk are subjected to the HS restriction, which is defined as follows [15; 29]:  
\[X_{Hs}=(X_{0},X_{1},X_{2},\ldots,X_{N-1},X_{N-1}^{*},\ldots,X_{2}^{*},X_{1}^{*}) \tag{1}\]  
The real signal, denoted as \(X_{Hs}\), is obtained from complex frequency samples after applying HS. The HS constraint confirms that the conjugate of \(X\) is appended to the original frequency samples, while setting \(X_{0}=X_{N}=0\) prevents the presence of remaining complex components in the time domain signal [37]. The complex signals after quadrature amplitude modulation (QAM), represented by \(X_{1}\) to \(X_{(N-1)}\), contribute to generating the final real signal.  
As a result, the signal obtained adheres to the requirement of being a real signal. It is to be distinguished that the second half of the input carriers carry the complex conjugate value of the first half. This symmetrical arrangement ensures the generation of a real signal at the IFFT output. However, it is fundamental to note that a sole half of the carriers carry useful data for obtaining the noncomplex signal, resulting in reduced spectral efficiency [38]. To obtain the discrete signal \(x(n)\), the symbol signals \(X(\kappa)\) are applied to the IFFT and are mathematically extracted as follows:  
\[x(n)=\frac{1}{N}\sum_{\kappa=0}^{N-1}X(\kappa)exp^{\frac{2\pi i\kappa n}{N}}, \quad n=0,1,\ldots,N-1 \tag{2}\]  
Figure 1: Conventional Flip\(-\)F\(-\)OFDM transmitter.  
Here, \(N\) signifies the FFT/IFFT length, \(X(x)\) is the subcarrier symbols, and \(j=\sqrt{-1}\). However, due to the imposition of the Hermitian symmetry on the IFFT input signals, the IFFT size turns into \(2N\) points, and hence, (2) can be re-arranged as follows:  
\[x(n)=\frac{1}{2N}\sum_{\kappa=0}^{2N-1}X_{Hs}exp^{\frac{i\text{ terms}}{N}},\quad n=0,1,\dots,2N-1 \tag{3}\]  
where \(X_{Hs}\) represents the real signal. Applying the HS constraint ensures that the time domain signal imaginary part is zero at the IFFT output. This guarantees that the F-OFDM signal is noncomplexly valued, though it is not yet unipolar [38]. To obtain the unipolar signal, a polarity separator is employed to decompose the signal as follows:  
\[x(n)=^{+}x(n)+^{-}x(n) \tag{4}\]  
where the positivity and negativity components are presented as:  
\[^{+}x(n)=\begin{cases}x(n),&\implies\quad x(n)\geq 0\\ 0,&\implies\quad x(n)<0\end{cases} \tag{5}\]  
\[^{-}x(n)=\begin{cases}x(n),&\implies\quad x(n)<0\\ 0,&\implies\quad x(n)\geq 0\end{cases} \tag{6}\]  
In order to change the dipolar signal, \(x(n)\), into a nondipolar signal, the nonnegative portion is conveyed in the first subframe of F-OFDM, while the inverted nonpositive portion is conveyed in the following subframe of F-OFDM, as shown below:  
\[x_{2N}(n)=\begin{cases}^{+}x(n),&n=0,1,..,N-1\\ -^{-}x(n-N),&n=N,..,2N-1\end{cases} \tag{7}\]  
The real nonunipolar F-OFDM signal, on the receiver side, y(n), is reconstructed as shown below:  
\[y(n)=^{+}y(n)-^{-}y(n) \tag{8}\]  
In this context, \({}^{+}y(n)\) and \({}^{-}y(n)\) exemplify the two F-OFDM signal subframes received, respectively. The transmitted signal samples are then retrieved by applying a standard OFDM demodulation process.