## 4 Implementation of Joint Plans  
### Ufmc Modulation Principle  
UFMC is a communication transmission scheme developed based on the principle of frequency division multiplexing. In UFMC, each symbol divides the signal bandwidth into multiple sub-bands, and each sub-band carries a corresponding number of subcarriers to facilitate communication. The structure of a single UFMC symbol is illustrated in Figure 2.  
In Figure 2, the yellow box represents the subcarriers of a single UFMC symbol, while the red dashed box illustrates the sub-band structure of a single UFMC symbol. A UFMC symbol is composed of a total of \(K\) subcarriers, which are divided into \(N_{Rb}\) sub-bands. Each sub-band contains \(c\) subcarriers, where: \(K=c\cdot N_{Rb}\). In the process of generating UFMC symbols, the input data are first divided into multiple low-rate sub-streams \(\mathbf{X}_{p}^{i}\), such that \(i\) refers to the number of UFMC symbol bits, and \(p\in\{1,2,\cdots,N_{Rb}\}\) represents the number of sub-bands. The data of each sub-band undergo inverse discrete Fourier transform (IDFT) operations, transforming them into a time-domain digital resource block. Subsequently, a time-domain smoothing pulse shaping filter is applied to each resource block to reduce out-of-band leakage in the frequency domain. This method minimizes the out-of-band interference between adjacent sub-bands in neighboring resource blocks. Finally, these resource blocks are overlaid in the time domain to generate the baseband signal, which is then transmitted into the radio frequency (RF) domain. The signal in the time domain for transmission can be represented as:  
\[\mathbf{x}^{i}=\mathbf{F}^{H}\mathbf{V}^{H}\mathbf{X}^{i}, \tag{1}\]  
such that:  
\[\mathbf{X}^{i}=\bigg{[}\Big{(}\mathbf{X}_{1}^{i}\Big{)}^{T},\Big{(}\mathbf{X} _{2}^{i}\Big{)}^{T},\cdots,\Big{(}\mathbf{X}_{N_{Rb}}^{i}\Big{)}^{T}\bigg{]}^ {T}, \tag{2}\]  
\[\mathbf{X}_{p}^{i}=\Big{[}X_{p}^{i}(0),X_{p}^{i}(1),\cdots,X_{p}^{i}(c-1) \Big{]}^{T}, \tag{3}\]  
\[\mathbf{V}^{H}=diag\big{\{}\mathbf{V}_{1},\mathbf{V}_{2},\cdots,\mathbf{V}_{ N_{Rb}}\big{\}}, \tag{4}\]  
\[\mathbf{F}^{H}=\big{[}\mathbf{F}_{1},\mathbf{F}_{2},\cdots,\mathbf{F}_{N_{Rb} }\big{]}. \tag{5}\]  
where \(\mathbf{X}^{i}\) represents the modulation and weighting value of the subcarrier for the \(ith\) symbol, \(X_{p}^{i}(k)\) corresponds to the communication information carried by the \(kth\) subcarrier in the \(pth\) sub-band, \(k\in\{0,1,\cdots,c-1\}\), \(\mathbf{V}^{H}\) represents the inverse fast Fourier transform  
Figure 2: The structure of the UFMC symbol. (IFFT) matrix, \(\mathbf{V}_{p}\) is the IFFT matrix for the \(pth\) sub-band block, and \(\mathbf{F}^{H}\) is the finite impulse response (FIR) filter matrix. \(\mathbf{F}_{p}\) is the filter matrix for the \(pth\) sub-band block and is a Toeplitz matrix with its first column as \(\big{[}b_{p}(0),b_{p}(1),\cdots,b_{p}(L_{FIR}-1),0,\cdots,0\big{]}^{T}\), such that \(L_{FIR}\) is the length of the FIR filter, and the coefficient \(b_{p}(j)\) represents the FIR filter coefficients for the \(pth\) resource block, \(j\in\{0,1,\cdots,L_{FIR}-1\}\). Each filter coefficient is selected to be \(\sum_{j=0}^{L_{FIR}-1}\big{|}b_{p}(j)\big{|}^{2}=1\) for every filter. To evaluate the advantages of UFMC signals compared to OFDM and FBMC signals, a temporal structure analysis was conducted for these three modulation schemes. Modulation in both OFDM and UFMC signals utilizes QAM, while FBMC signals, to maintain a uniform symbol data rate, employ offset quadrature amplitude modulation (OQAM) modulation. The temporal characteristics of these three modulation schemes are illustrated in Figure 3.  
In Figure 3, given the length of the transmitted data (\(T_{D}\)), to avoid ISI, a CP is added at the beginning of each OFDM symbol, thus forming a complete OFDM symbol. The total length of a single OFDM symbol is \(T_{OFDM}^{1}=T_{CP}+T_{D}\), such that \(T_{CP}\) is the CP length.