## 2 Principle of FOFDM  
The detailed theory of FOFDM used in this paper can be found in [15]. The equations of FOFDM were briefly summarized as follows. It is assumed that N subcarriers are available, and we modulated the k-th subcarrier by \(x(k)\); the time domain signal \(s(n)\), \(n=0,\ldots,N-1\), can be written as  
\[s(n)=\sqrt{\frac{2}{N}}\sum_{k=0}^{N-1}\epsilon(k)x(k)\cos\biggl{[}\frac{\pi}{ N}k\biggl{(}n+\frac{1}{2}\biggr{)}\biggr{]} \tag{1}\]  
where \(\epsilon(k)=\sqrt{0.5}\) for \(k=0\), and 1 for \(k=1,\ldots,N-1\). To enable the use of single-tap equalizers, \(s(n)\) is zero-padded and represented by \(s_{ZP}(n)\), where \(s_{ZP}(n)=s(n)\) for \(0\leq n\leq N-1\), and \(s_{ZP}(n)=0\), for \(N\leq n\leq N+L-1\).  
After passing through an LTI channel, the received signal is written as  
\[r(n)=h(n)*s_{ZP}(n)+z(n),0\leq n\leq N+L-1 \tag{2}\]  
where \(*\) is the convolution operator, and \(z(n)\) is the additive noise.  
At the receiver, \(r(n)\) are zero-padded to obtain \(r_{2N}(n)\). A DFT with a length of 2N is applied to obtain a frequency domain signal \(y(m)\). And there is  
\[\begin{array}{ll}y(m)&=F_{2n}[r_{2N}(n)]=F_{2n}[h_{2N}(n)\cdot s_{2N}(n)]\\ &=F_{2n}[h_{2N}(n)]\cdot F_{2n}[s_{2N}(n)]\\ &=H(m)\cdot S(m)\end{array} \tag{3}\]where \(m=0,1,\ldots,2N-1\),\(F_{2n}[\cdot]\) denotes the DFT with a length of 2N, \(h_{2N}(n)\) is the zero-padded signal of \(h(n)\), and \(s_{2N}(n)\) is the zero-padded signal of \(s(n)\). \(y(m)\) can be written as  
\[y(m)=e^{j\frac{2\pi}{2N}m\cdot\frac{1}{2}}H(m)\cdot e^{-j\frac{2\pi}{2N}m\cdot \frac{1}{2}}S(m)=\hat{H}(m)\cdot\hat{S}(m) \tag{4}\]  
where \(\hat{S}(m)\) can be extended as  
\[\begin{array}{ll}\hat{S}(m)&=\sqrt{\frac{1}{2N}}\sum\limits_{n=0}^{N-1}s(n) e^{-j\frac{2\pi}{2N}m(n+\frac{1}{2})}\\ &=\sum\limits_{k=0}^{N-1}x(k)G(m,k)\end{array} \tag{5}\]  
and there is  
\[G(m,k)=\frac{\varepsilon(k)}{N}\sum\limits_{n=0}^{N-1}\cos\frac{\pi}{N}k\bigg{(} n+\frac{1}{2}\bigg{)}\cdot e^{-j\frac{2\pi}{N}m(n+\frac{1}{2})} \tag{6}\]  
The real part of \(G(m,k)\) was derived as  
\[\begin{array}{ll}G_{R}(m,k)=\frac{\varepsilon(k)}{N}\sum\limits_{n=0}^{N-1} \cos\frac{\pi}{N}k\bigg{(}n+\frac{1}{2}\bigg{)}\cdot\cos\frac{\pi}{N}m\bigg{(} n+\frac{1}{2}\bigg{)}\\ =\left\{\begin{array}{ll}1&m=0\\ 0.5\cdot\delta(m-k)&m=1,\ldots,N-1\\ 0&m=N\\ -0.5\cdot\delta(2N-m-k)&m=N+1,\ldots,2N-1\end{array}\right.\end{array} \tag{7}\]  
To recover the transmitted symbols, the phase can be compensated by multiplying the conjugate of the estimated channel frequency response \(\hat{H}(m)\) to \(y(m)\); then, the real part is  
\[\begin{array}{ll}y^{\prime}(m)=\big{|}\hat{H}(m)\big{|}^{2}\cdot\sum\limits _{k=0}^{N-1}x(k)G_{R}(m,k)\\ =\big{|}\hat{H}(m)\big{|}^{2}\cdot\left\{\begin{array}{ll}x(m)&m=0\\ 0.5\cdot x(m)&m=1,\ldots,N-1\\ 0&m=N\\ -0.5\cdot x(2N-m)&m=N+1,\ldots,2N-1\end{array}\right.\end{array} \tag{8}\]  
To make full use of the above equation, there is a further definition as follows:  
\[\hat{H}_{c}(m)=\frac{1}{2}\Big{(}\big{|}\hat{H}(m)\big{|}^{2}+\big{|}\hat{H}( 2N-m)\big{|}^{2}\Big{)} \tag{9}\]  
So, the transmitted data can be obtained as  
\[x^{\prime}(m)=y^{\prime}(m)-y^{\prime}(2N-m)=\hat{H}_{c}(m)\cdot x(m) \tag{10}\]  
After removing \(\hat{H}_{c}(m)\) by single-tap equalizers, the transmitted data \(x(m)\) can be recovered.