## 3 Simulation Results
### Complexity Analysis  
In general, the complexity of a multiplier is significantly higher than that of an adder. Hence, we focus on the comparison of the multiplication between CP-OFDM and GFB-OFDM. In addition, considering the flexibility in the subcarrier spacing adjustment, the CP-OFDM necessitates the inclusion of the filters for each subband. These filters are positioned differently to correspond to the frequency range of each subband respectively. Note that, since all subbands are the multi-carrier subbands, it illustrates the transmission process of CP-OFDM in Figure 12. Both GFB-OFDM and CP-OFDM require adding CP with a similar complexity. Thus, we can ignore the presence of CP during the comparison. Figure 11: The BLER of CP-OFDM and GFB-OFDM is evaluated in single-carrier and multi-carrier mixture scenarios. The rate-\(\frac{3}{4}\) LDPC channel coding, the 64 QAM modulation, and the AWGN channel are employed. The result demonstrates that the BLER performance of GFB-OFDM is superior to that of CP-OFDM. Figure 10: Subband data-processing procedure for a single-carrier and multi-carrier mixture scenario with the number of modulated symbols \(K=8192\), the number of subbands \(N=8\) and the number of subcarriers in each subband \(M=1024\) or \(M=512\). First, we analyze the number of multiplications involved in the IFFT and the filtering of GFB-OFDM. The GFB-OFDM employs the two-level IFFTs, where it can be deduced from the processing procedure that the first-level IFFT is of M points. Hence, the overall number of multiplications in \(N\) subbands for first-level IFFT processing is \(\frac{N\text{M}}{2}\text{log}M\)[25]. The number of multiplications in \(M\) second-level IFFT points is \(\frac{N_{\text{fft}}}{M}\), so the total number of multiplications in \(M\) second-level IFFT is \(\frac{N_{\text{fft}}}{2}\text{log}\frac{N_{\text{fft}}}{M}\). In addition, the number of multiplications in the polyphase filtering process of GFB-OFDM can be determined using the following calculation. Considering that a second-level IFFT has \(\frac{N_{\text{fft}}}{M}\) points, the length of time-domain data after IFFT is \(\frac{N_{\text{fft}}}{M}\). Therefore, by repeating these data \(L\) times and performing point multiplication and windowing operations, the total number of multiplications becomes \(\frac{LN_{\text{fft}}}{M}\). Consequently, for one OFDM symbol, the overall number of multiplications required for point multiplication and windowing operations is \(\frac{M\times LN_{\text{fft}}}{M}=LN_{\text{fft}}\). Hence, the total number of multiplications involved in IFFT and filtering processing for GFB-OFDM is as follows:  
\[\mathcal{O}(\text{GFB-OFDM})=\frac{NM}{2}\text{log}M+\frac{N_{\text{fft}}}{2} \text{log}\frac{N_{\text{fft}}}{M}+LN_{\text{fft}}. \tag{55}\]  
We then analyze the number of multiplications involved in the IFFT and filtering process of CP-OFDM. The number of multiplications in the IFFT processing of CP-OFDM is \(\frac{N_{\text{fft}}N}{2}\text{log}N_{\text{fft}}\), which is determined based on \(N_{\text{fft}}\)-point IFFT with \(N\) subbands. The calculation process for the number of multiplications in the time-domain convolution filtering of CP-OFDM is as follows. The number of time-domain data points for one OFDM symbol is \(N_{\text{fft}}\), and the number of points included in the time-domain length of the filter function is \(\frac{LN_{\text{fft}}}{M}\). Therefore, the total number of multiplications for \(N\) subbands processed by time-domain convolution filtering is \(\frac{NLN_{\text{fft}}^{2}}{M}\). Hence, the total number of multiplications in the IFFT and filtering processing for CP-OFDM is  
\[\mathcal{O}(\text{OFDM})=\frac{N_{\text{fft}}N}{2}\text{log}N_{\text{fft}}+ \frac{NLN_{\text{fft}}^{2}}{M}. \tag{56}\]  
The calculation process above ignores cyclic prefix because its length has a minimal impact on the overall complexity. It is easy to observe that (36) is much smaller than (37).