## 2 GFB-OFDM Transceiver
### Receiver Analysis and Its Extension
Then, we can obtain \[\mathbf{W}[m,b] =(\mathbf{Y}[m,(N_{s}+Q)M]\otimes\text{FIL})e^{-j2\pi\frac{mN_{2}}{N_ {\text{lift}}}b}\] (40) \[=d|\text{FIL}[b]|e^{j2\pi\frac{p_{m}}{M}-j2\pi\frac{mN_{2}}{N_{ \text{lift}}}b}\] (41) \[=d|\text{FIL}[b]|e^{j2\pi m\frac{p_{m}}{M}}\] (42) where \(b=0,1,2,\ldots,N_{\text{lift}}-1\) represents subcarriers in the frequency domain, \"\(\otimes\)\" is the circular convolution symbol, and \"\(|\)\" denotes the module value. 7. Sum the frequency domain data of different subsymbols to obtain \(\mathbf{Z}_{N_{\text{lift}}\times 1}\) \[\mathbf{Z}[b] =\sum_{m=0}^{M-1}\mathbf{W}[m,b]\] (43) \[=d|\text{FIL}[b]|\sum_{m=0}^{M-1}e^{j2\pi m\frac{p-b}{M}}.\] (44)  
Since the purpose of polyphase filtering is to filter data of each subband separately, it can be understood that the frequency domain carrier positions corresponding to the passband bandwidth of the filter are \(b=\left[(N_{s}+Q)M-\frac{M}{2},\ (N_{s}+Q)M+\frac{M}{2}-1\right]\). It should be noted that the positions of the filters differ for different subbands. Additionally, because we have normalized the peak value of our polyphase filtering filters in the time domain, and because their frequency domain bandwidth corresponds to the subband width, we can obtain the frequency domain coefficients of these filters. \[|\text{FIL}[b]|=\frac{1}{M}. \tag{45}\]  
According to (40), it can be similarly derived that, when \(b=(N_{\text{s}}+Q)M-\frac{M}{2}+P\), we have  
\[\mathbf{Z}\bigg{[}(N_{\text{s}}+Q)M-\frac{M}{2}+P\bigg{]}=d. \tag{46}\]  
At this time, it means that the data on this subcarrier are represented as \(d\), while the data on other subcarriers are zero. The time-domain information corresponding to \(\mathbf{Z}_{N_{\text{fit}}\times 1}\), denoted as \(\mathbf{z}_{N_{\text{fit}}\times 1}\), represents the time-domain data after performing IFFT. At the receiver, CP-OFDM can be used to receive data in the frequency domain, and the data on each subcarrier satisfies orthogonality. This is consistent with the conclusion of Theorem 1. More generally, when the length of the filter is \(L(1\leq L\leq\frac{N_{\text{fit}}}{N_{2}})\) and it is a DRRC filter with double roots, (35) becomes  
\[\boldsymbol{y}[m,n_{\text{ifft}}]= \begin{cases}\frac{d}{\sqrt{N_{\text{fitt}}}} e^{\beta 2\pi(\frac{Dm}{M}+n_{\text{fitt}}\frac{N_{\text{s}}+Q}{N_{2}})},&n_{\text{ifft}}< LN_{2}\\ 0,&n_{\text{ifft}}\geq LN_{2}.\end{cases}  \tag{47}\]  
Assuming the filter coefficients of \(LN_{2}\) points, the total time-domain filter coefficients for one OFDM symbol length are given by \(\text{fil}=[\text{fil}0,\text{zeros}(N_{\text{ifft}}-LN_{2},1)]\). Then, the point-wise multiplication result of the symbol \(m\) is subjected to FFT and symbol shifting and we can obtain  
\[\boldsymbol{W}[m,b] =\text{FFT}(\boldsymbol{y}[m,n_{\text{ifft}}]\times\text{fil}(n_{ \text{ifft}}))e^{-2\pi\frac{nN_{2}}{N_{\text{fitt}}}b} \tag{48}\] \[=d\text{e}^{2\pi\frac{(p-b)m}{M}}\text{FFT}(e^{2\pi Mn_{\text{ intt}}\frac{N_{\text{s}}+Q}{N_{2}}}\times\text{fil}(n_{\text{ifft}}))\] (49) \[=d\text{e}^{2\pi\frac{(p-b)m}{M}}\text{FIL}(\left[b-M(N_{\text{s }}+Q)\right]_{N_{\text{fitt}}})\] (50) \[=d\text{e}^{2\pi\frac{(p-b)m}{M}}\left|\text{FIL}(\left[b-M(N_{ \text{s}}+Q)\right]_{N_{\text{fitt}}})\right| \tag{51}\]  
It should be noted that the peak value of the filter in the time domain occurs at \(t=0\) and that its frequency-domain coefficients are real numbers. The data of different subsymbols are then superimposed in the frequency domain.