## 2 Basic Algorithm Principle
After the decimation by \(D_{1}\), the data rate of each sub-band decreases to a value twice its bandwidth, which is critical for solving the problem of interference from the neighbouring sub-channels. This leads to the maximal conservation of the computing resources, given that two moderate FFTs are used instead of an ultra-large FFT. The overlapping interferences between the adjacent sub-bands can be eliminated by appropriately selecting part of the output data from the \(N_{1}\)-point FFT. After the application of \(N_{1}\)-point FFT to the decimated data of Sub-bands 0-\(K\)/2, the resulting frequency resolution \(df\) for each sub-band can be expressed as follows:  
\[ \begin{array}{l}df=f_{s}/D/D_{1}/N_{1}\\ =2f_{s}/K/N_{1}\end{array}  \tag{5}\]  
To eliminate the interference between the adjacent sub-bands, a part of the FFT bins of each sub-band is selected for the bin combination. The method of bin selection and  
Figure 2: Aliasing effect in MDFB and NMDFB (\(f_{s}\) represents the sampling frequency and \(D\) represents a decimating factor. The aliasing effect is avoided in the NMDFB). the combination is shown in Table 1. The number of selected FFT bins \(N_{2}\) for Sub-bands 0-\(K\)/2-1 can be expressed as follows:  
\[ \begin{split} N_{2}&=f_{\rm s}/K/(f_{\rm s}/D/D_{1}/N_ {1})\\ &=D\times D_{1}\times N_{1}/K\end{split}  \tag{6}\]  
It is necessary for \(N_{2}\) to be an integer, which can be readily obtained. Sub-bands 0-\(K\)/2-1 are all complex signals, which represent the entire frequency range of real-valued signals. As shown in Table 1, the bins from 0-\(N_{2}\)/2-1 and \(N_{1}\)-\(N_{2}\)/2 to \(N_{1}\)-1 are selected separately from the \(N_{1}\) FFT bins for each of the sub-bands from 0-\(K\)/2-1. The selected bins \(N_{1}\)-\(N_{2}\)/2 to \(N_{1}\)-1 from each of the sub-bands from 0-\(K\)/2-1, respectively, are reversed and re-ordered. The selected bins in Sub-bands 0-\(K\)/2-1 are combined into one group along the order of the channel numbering. This represents the frequency-domain data for the real-valued signals within \(f_{\rm s}\)/2, and the signal power can then be calculated. Based on the analysis presented above, the algorithm splits an ultra-large-scale FFT into a two-stage process, in which two smaller-scale FFTs are required. The \(N\)-point FFT requires a magnitude order of \(O(N{\rm log}_{2}N)\) operations. Therefore, if spectral processing with an ultra-high-frequency resolution (\(N\) is extremely large) is implemented in the FPGA, the algorithm reduces the operations to the magnitude order of \(O(K{\rm log}_{2}K)+O(N_{1}{\rm log}_{2}N_{1})\) (\(K\) and \(N_{1}<<N\)). This conserves hardware resources and compensates for the lack of a commercially available enormous point FFT IP core. As mentioned before, the WOLA filter bank is equivalent to the PFB. They are both efficient implementation of the channelization processing. Therefore, their computational and hardware complexity is equivalent. In comparison with the expressions of the IWOLA filter bank and the WOLA filter bank (2018), the expression of the IWOLA filter bank involves two terms more: \((-1)^{l}\) and \(e^{i\pi\eta/K}\). It is obvious that the first term does not increase the computational and hardware complexity; the second term only necessitates one complex multiplication operation and two block RAMs (BRAMs) in the FPGA. The capacity of the two extral BRAMs is \(K\times 16\) bit, given that each data of the second term is expressed with 16 bits in the FPGA. The function of the data decimation is to select or discard data, which does not increase the computational and hardware complexity.