## 2 GFB-OFDM Transceiver
### Transmitter  
The GFB-OFDM transmitter is briefly exhibited in Figure 1. The modulated symbols are divided into \(N\) groups, and the symbols in each group are loaded onto its corresponding subband. Then, the first-level (or subcarrier-level) IFFT is performed over each individual subband, and the size of the IFFT is flexible, which aligns with the number of the loaded symbols in the specific subband. The output of each IFFT process is stacked, and the second-level (or subband-level) IFFT is performed over all the subband outputs; thus, the size of the second-level IFFT is the sum of the sizes of all subband IFFT. Then, the polyphase filter is adopted to window the baseband signal. Finally, the digital-to-analog converter (DAC) and the radio frequency (RF) is employed to transmit it out. In summary, the process of dividing modulated symbols into subbands and performing the first-level IFFT is referred to as the subcarrier-level processing module. The process of the second-level IFFT and the polyphase filter are referred to as the subband-level processing module. Figure 1: GFB-OFDM transmitter diagram. The encoded and modulated data are first divided into \(N\) subbands, with each subband undergoing a subcarrier-level IFFT. The resulting data is then processed through a subband-level IFFT along the subband dimension, followed by a polyphase filter. The output is transmitted after passing through the digital-to-analog converter and radio frequency module. By using the two levels of IFFT and the polyphase filters, the large size of the IFFT can be decomposed into two levels of IFFT of smaller sizes. This approach effectively addresses the challenges in the issues of the computational complexity and the hardware implementation. In the following, we formulate the two-level IFFT and the polyphase filtering in the GFB-OFDM transmitter. There are \(K\) modulated symbols to be transmitted over the \(K\) subcarriers. They are divided into \(N\) subbands, and each subband has an identical bandwidth (each subband is able to alter its subcarrier spacing according to different services. For a brief illustration, we assume that each subband has identical subcarrier spacing.). In each subband, \(M\) symbols are mapped to the subcarrier, i.e., \(K=MN\), which can be represented as \(\mathbf{X}[l,k],l=0,1,2,\ldots,M-1\), \(k=0,1,2,\ldots,N-1\). Then, the output of the first-level IFFT \(\mathbf{x}[m,k]\) of \(N\) subbands is calculated as  
\[\mathbf{x}[m,k]=\frac{1}{M}\sum_{l=0}^{M-1}\mathbf{X}[l,k]e^{j2\pi\frac{m}{M}},\quad m =0,1,2,\ldots,M-1 \tag{1}\]  
where \(j=\sqrt{-1}\). Let \(\mathbf{A}\) be the inverse Fourier transform matrix; then, Equation (1) can be rewritten in the matrix form as  
\[\mathbf{x}_{M\times N}=\frac{1}{M}\mathbf{A}_{M\times M}\mathbf{X}_{M\times N}. \tag{2}\]  
Then, the cyclic prefix is added for each subband as  
\[\mathbf{x}_{M\times N}\xrightarrow{\text{Add CP}}\mathbf{x}_{(N_{\text{cp}}+M)\times N} \tag{3}\]  
where \(N_{\text{cp}}\) is the length of the cyclic prefix. Then, the second-level IFFT is performed row-wise over the matrix \(\mathbf{x}\). For each row, \(N\) samples are used as the input of the \(N_{2}\) points double bandwidth oversampled IFFT, i.e., \(N_{2}=2N\), and the input is given as  
\[\mathbf{Y}_{(N_{\text{cp}}+M)\times N_{2}}=\left[0_{(N_{\text{cp}}+M)\times N_{ \text{sp}}},\mathbf{x}_{(N_{\text{cp}}+M)\times N^{\prime}}0_{(N_{\text{cp}}+M) \times N_{\text{sp}}}\right] \tag{4}\]  
where \(N_{s}=\frac{N_{2}-N}{2}\) is the number of zero padding on one side. The operation is equivalent to oversampling in the time domain. After the \(N_{\text{cp}}+M\) second-level IFFT, the \(N_{\text{cp}}+M\) rows are obtained, and each row contains \(N_{2}\) samples. Each row of a sample is a subsymbol data sequence, and the time-domain length \(T\) of a subsymbol data sequence is the reciprocal of the interval between the adjacent subband in the frequency domains, which is also the reciprocal of the width of the subband in the frequency domain.