## 3 Analysis of the IWOLA + FFT Algorithm Characteristics
### Performance Comparison of the IWOLA + FFT Algorithm with a Direct FFT  
An assessment of the scalloping losses of the IWOLA filter bank, proposed algorithm, and direct FFT is presented below. First, we compared the scalloping loss of the IWOLA filter bank with that of the direct FFT by sweeping the frequency domain with 11 sinusoidal signals from one bin centre (e.g., the first bin) to an adjacent bin centre (the second bin). Thereafter, the maximum magnitude was plotted as shown in Figure 6. The analysis results revealed that the IWOLA filter bank exhibited a gain depression of approximately 0.36 near the border of the adjacent sub-channels. The adverse influence on the IWOLA + FFT algorithm is detailed further in this manuscript, and can be minimized by changing the characteristics of the window function for the IWOLA filter bank. We scanned the frequency characteristics from one bin centre to an adjacent bin centre using 11 sinusoidal signals. The improved result is plotted in Figure 7, which indicates a flatter gain curve of the IWOLA filter bank. This implies that the gain loss problem caused by channelisation was solved, and the scalloping loss of the IWOLA + FFT structure only depends on the succeeding FFT. Furthermore, under the assumption that the frequency resolution of the IWOLA + FFT algorithm is the same as that of the direct FFT, we analysed the scalloping losses using a similar approach to Figure 6. We then plotted the normalized magnitude, as shown in Figure 8. As can be seen from the figure, the proposed algorithm demonstrated the same scalloping loss characteristics as the direct FFT. Figure 5: Diagram of the actual bandwidth and sampling rate for each subchannel of a \(N_{1}\)-point FFT. Figure 6: Gain flatness feature between two adjacent sub-channels of the IWOLA filter bank and the direct FFT. As can be observed from Figure 3, for the channelisation characteristics of the IWOLA filter bank, there was a gain depression near the border between the two adjacent channels due to the channelisation process. Therefore, it had an adverse influence on the characteristics of the proposed algorithm. We obtained the channelised gain flatness characteristics, as shown in Figure 9, by sweeping the frequency domain with 31 sinusoidal signals from the centre of one sub-channel to that of the neighbouring sub-channel and plotting the normalized magnitude. A lower amplitude gain was caused by the IWOLA filter bank near the border of the adjacent sub-channels. With the proposed algorithm, this problem was solved by changing the characteristics of the window function for the IWOLA filter bank, which did not influence the adjacent sub-bands due to the NMDFB. The characteristics of the window function can be improved by increasing its cut-off frequency by 1.4 times. We analysed the frequency channel gain flatness using a similar approach to Figure 9. We then plotted the normalized magnitude, as shown in Figure 10, which indicates that Gain depression near the border of the adjacent sub-channels was improved by changing the characteristics of the window function for the IWOLA filter bank. The improved frequency channel gain flatness characteristics were the same as those of the direct FFT. There was a gain fluctuation of approximately 0.32. Figure 8: Scalloping loss of the IWOLA + FFT algorithm and the direct FFT. Figure 7: Scalloping loss of the IWOLA filter bank and the direct FFT. The results revealed that the IWOLA filter bank exhibited a flatter gain feature than the FFT. The flat channel characteristics of the IWOLA filter bank ensure that the IWOLA + FFT algorithm features the same spectral leakage as the direct FFT algorithm. A sinusoidal signal with a frequency of 45 MHz was inputted to the two algorithms, where \(K=1024\), \(N_{1}=2048\), \(f_{\mathrm{s}}=800\) MHz, \(D=8\), and the frequency resolution was approximately 763 Hz.