## 2 GFB-OFDM Transceiver
### Receiver Analysis and Its Extension  
In this section, we discuss a special case of Theorem 1 where oversampling and different filter lengths are considered. Similar to previous discussions, the total number of IFFT points is \(N_{\text{ifft}}\) and the data bandwidth is divided into \(N\) subbands as shown in Figure 4, each with \(M\) subcarriers. \(N_{2}\) represents the second-level IFFT point number, where \(N_{\text{ifft}}\), \(M\), and \(N_{2}\) are all positive integer power of two. The length of the filter is \(LN_{2}\). Firstly, let us analyze the case where \(LN_{2}=N_{\text{ifft}}\) for filter length selection. Due to the characteristic of orthogonal frequency domain subcarriers, we only need to track the process experienced by data on one subcarrier in the GFB-OFDM waveform at its transmission end in order to easily extend it to the transmission processes of all subcarriers. Assuming that one data \(d\) is transmitted on subcarrier \(P\) in subband \(Q\), then we can analyze the transmission data processing of GFB-OFDM according to the following steps:  
(1) The data \(\mathbf{X}_{M\times N}\) across \(N\) subbands and \(M\) subcarriers can be expressed as  
\[\mathbf{X}[m,n]= \begin{cases}d,&m=P\text{ and }n=Q\\ 0,&m\
eq P\text{ or }n\
eq Q.\end{cases}  \tag{32}\]  
(2) After performing first-level IFFT on \(N\) subbands, the data \(\mathbf{x}_{M\times N}\) is obtained as  
\[\mathbf{x}[m,n]=\text{IFFT}(\mathbf{X}[m,n])= \begin{cases}\frac{d}{\sqrt{M}}\text{e}^ {\beta 2\pi\frac{Pm}{M}},&n=Q\\ 0,&n\
eq Q.\end{cases}  \tag{33}\]  
Figure 4: The locations of the \(N\) subbands in the frequency domain. 3. Perform \(M\) times of second-level IFFTs on \(N_{2}\) points, resulting in \(\mathbf{y}_{M\times N_{2}}\). The position of the subbands determines the mapping location for the data in the second-level IFFT, where \(n_{2}=0,1,2,\ldots,N_{2}-1\). Then, \(\mathbf{y}_{M\times N_{2}}\) can be expressed as \[\mathbf{y}[m,n_{2}]=\frac{d}{\sqrt{N_{\text{lift}}}}e^{j2\pi(\frac{p_{m}}{M}+n_{ 2}\frac{N_{4}+Q}{N_{2}})}.\] (34) 4. For each subsymbol, repeat it \(L\) times to yield \(\mathbf{y}_{M\times LN_{2}}\). According to this assumption, \(LN_{2}=N_{\text{lift}}\) can be written as \(\mathbf{y}_{M\times N_{\text{lift}}}\), where \(n_{\text{lift}}=0,1,2,\ldots,N_{\text{lift}}-1\). \[\mathbf{y}[m,n_{\text{lift}}]=\frac{d}{\sqrt{N_{\text{lift}}}}e^{j2\pi(\frac{p_{m }}{M}+n_{\text{lift}}\frac{N_{4}+Q}{N_{2}})}.\] (35) 5. Here, the analysis should be conducted from a frequency domain perspective because it is believed that the frequency domain information after undergoing second-level IFFT should still contain the information from the original subcarriers. Therefore, an FFT is performed on each symbol to obtain the frequency domain information \(\mathbf{Y}_{M\times N_{\text{lift}}}\), which is expressed as \[\mathbf{Y}[m,n_{\text{lift}}] =\text{FFT}(\mathbf{y}_{M\times N_{\text{lift}}})\] (36) \[=\frac{d}{\sqrt{N_{\text{lift}}}}\sum_{n_{l}=0}^{N_{\text{lift}}-1 }e^{j2\pi(\frac{p_{m}}{M}+n_{l}\frac{N_{4}+Q}{N_{2}})}\frac{1}{\sqrt{N_{\text{ lift}}}}e^{-\frac{|2\pi n_{\text{lift}}n_{\text{lift}}}{N_{\text{lift}}}}\] (37) \[=\frac{d}{N_{\text{lift}}}\sum_{n_{l}=0}^{N_{\text{lift}}-1}e^{j2 \pi\frac{p_{m}}{M}}e^{j2\pi n_{l}(\frac{(N_{4}+Q)M}{N_{2}}-\frac{n_{\text{lift }}}{N_{\text{lift}}})}.\] (38) We can obtain \[\mathbf{Y}[m,n_{\text{lift}}]= \begin{cases}de^{j2\pi\frac{p_{m}}{M}},&n_{\text{ lift}}=(N_{s}+Q)M\\ 0,&n_{\text{lift}}\
eq(N_{s}+Q)M.\end{cases} \] (39) 6. \(\mathbf{Y}_{M\times N_{\text{lift}}}\) is convolved with the rectangular filter FIL. At the mean time, the temporal misalignment process is transformed into a phase rotation in the frequency domain.