## 2 GFB-OFDM Transceiver
### Receiver  
The detection of CP-OFDM can be directly used as the receiver of GFB-OFDM, as illustrated in Figure 3, which is validated by Theorem 1. **Theorem 1**: _For a signal of length \(NM\), the following two operations result in exactly the same output. The first operation: perform a \(NM\)-point IFFT; the second operation: perform \(N\) M-point IFFTs, \(M\) N-point IFFTs and polynomial filtering._  
Proof. : Consider that \(\mathbf{X}[l,k]\), consisting of \(NM\) information-bearing symbols, are transmitted on \(M\) subcarriers and \(N\) subbands. After performing first-level IFFT on \(N\) subbands, the data \(\mathbf{x}[m,k]\) are obtained as  
\[\mathbf{x}[m,k]=\frac{1}{M}\sum_{l=0}^{M-1}\mathbf{X}[l,k]e^{\beta 2\pi\frac{M}{M}} \tag{11}\]  
where \(j=\sqrt{-1}\), \(m=0,1,2,\ldots,M-1\). \(l=0,1,2,\ldots,M-1\) and \(k=0,1,2,\ldots,N-1\) denotes the index of subcarriers and subbands, respectively. Perform \(M\) times of second-level IFFTs on \(N\) points, resulting in \(\mathbf{y}[m,n]\), which can be expressed as  
\[\mathbf{y}[m,n] =\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}\mathbf{x}[m,k]e^{\beta 2\pi\frac{M}{M}} \tag{12}\] \[=\frac{1}{\sqrt{N}}\frac{1}{\sqrt{M}}\sum_{k=0}^{N-1}\sum_{l=0}^{ M-1}\mathbf{X}[l,k]e^{\beta 2\pi\frac{M}{M}}e^{\beta 2\pi\frac{M}{N}}\] (13) \[=\frac{1}{\sqrt{NM}}\sum_{k=0}^{N-1}\sum_{l=0}^{M-1}\mathbf{X}[l,k]e^ {\beta 2\pi(\frac{M}{M}+\frac{M}{N})}. \tag{14}\]  
For each subsymbol, repeat it \(L\) times to yield \(\mathbf{y}[m,n_{\text{ifft}}]\), where \(LN=N_{\text{ifft}}\) and \(n_{\text{ifft}}=0,1,2,\ldots,N_{\text{ifft}}-1\). \[\mathbf{y}[m,n_{\text{ifft}}]=\frac{1}{\sqrt{N_{\text{ifft}}}}\sum_{k=0}^{N-1} \sum_{l=0}^{M-1}\mathbf{X}[l,k]e^{\beta 2\pi(\frac{M}{M}+\frac{M}{N})}. \tag{15}\]  
Here, the analysis should be conducted from a frequency domain perspective. Therefore, an FFT is performed on each symbol to obtain the frequency domain information \(Y[m,n_{\text{ifft}}]\), which is expressed as  
Figure 3: Processing at the receiver. The received time-domain signal is first converted into a digital signal via an ADC, followed by the removal of the cyclic prefix. After undergoing an FFT, the signal is converted to the frequency domain. In the frequency domain, it passes through the symbol detection module, as well as the demodulation and decoding module, resulting in the output information bits. \[Y[m,n_{\text{ifft}}] =\frac{1}{\sqrt{N_{\text{ifft}}}}\sum_{n_{i}=0}^{N_{\text{inft}}-1} \boldsymbol{y}[m,n_{i}]e^{-|2\pi\frac{n_{\text{ifft}}}{N_{\text{ifft}}}} \tag{16}\] \[=\frac{1}{\sqrt{N_{\text{ifft}}}}\frac{1}{\sqrt{N_{\text{ifft}}}} \sum_{n_{i}=0}^{N_{\text{inft}}-1}\sum_{k=0}^{N-1}\sum_{l=0}^{M-1}\boldsymbol{ X}[l,k]e^{2\pi(\frac{n_{i}}{M}+\frac{n_{i}}{N})}e^{-|2\pi\frac{n_{\text{ifft}}}{N_{ \text{ifft}}}}\] (17) \[=\frac{1}{N_{\text{ifft}}}\sum_{k=0}^{N-1}\sum_{l=0}^{M-1} \boldsymbol{X}[l,k]e^{|2\pi\frac{n_{i}}{M}}\sum_{n_{i}=0}^{N_{\text{inft}}-1}e^ {|2\pi(\frac{n_{i}}{M}-\frac{n_{\text{ifft}}}{N_{\text{ifft}}})}\] (18) \[=\frac{1}{N_{\text{ifft}}}\sum_{k=0}^{N-1}\sum_{l=0}^{M-1} \boldsymbol{X}[l,k]e^{|2\pi\frac{n_{i}}{M}}\sum_{n_{i}=0}^{N_{\text{inft}}-1}e ^{|2\pi n_{\text{inft}}\frac{n_{\text{inft}}-n_{\text{ifft}}}{N_{\text{ifft}}}}\] (19) \[\mathbf{y}[m,n] =\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}\mathbf{x}[m,k]e^{\beta 2\pi\frac{M}{M}} \tag{12}\]0  
where \"\([\cdot]_{N_{\text{ifft}}}\)\" denotes the modulo \(N_{\text{ifft}}\) operation and \(\delta[\cdot]\) denotes the Dirac delta function. We can the obtain  
\[\boldsymbol{Y}[m,n_{\text{ifft}}]= \begin{cases}\sum_{l=0}^{M-1} \boldsymbol{X}[l,\frac{n_{\text{ifft}}}{M}]e^{|2\pi\frac{ml}{M}},&n_{\text{ifft }}=kM\\ 0,&n_{\text{ifft}}\
eq kM.\end{cases}  \tag{21}\]  
Then, \(\boldsymbol{Y}[m,p]\) is convolved with the rectangular filter \(\text{FIL}[p]\). At the mean time, the temporal misalignment process is transformed into a phase rotation in the frequency domain.