## 5 Algorithm Simulation Analysis  
To verify the accuracy and performance of the proposed broadband spectral analysis algorithms with a high-frequency resolution, the following simulation analyses were carried out in MATLAB (the version: R2017a). The following parameters were set according to Equations (4)-(6): \(K\) = 1024, \(D\) = 8, \(N_{1}\) = 2048, \(K_{1}\) = 2048, \(D_{2}\) = 8; and the sampling frequency of the real-valued signal \(f_{\rm s}\) = 800 MHz, \(D_{1}\) = 64, \(df\) = 763 Hz, and \(N_{2}\) = 1024. To analyse the algorithm performances, three types of methods were used to calculate the spectrum for the simulated signals. The first involved the direct performance of an ultra-large FFT (direct FFT) to obtain a spectral resolution of 763 Hz. The remaining two methods involved the use of the IWOLA + FFT and IWOLA + WOLA algorithms to achieve the same spectral resolution. First, a real sinusoidal signal with a frequency of 40.00 MHz was inputted to the three algorithms, to evaluate their gains and SNRs, as shown in Figure 17. As can be observed from the figure, the IWOLA + FFT structure exhibited a gain increase of approximately 3 dB when compared with the direct FFT, and a nearly identical SNR. The IWOLA + WOLA structure exhibited a gain decrease of approximately 5 dB when compared with the IWOLA + FFT structure. However, this may not be significant, given that the SNR of the IWOLA + WOLA structure was 6 dB higher than that of the IWOLA + FFT structure. Thus, the gain of the IWOLA + WOLA can be any value, which can then be multiplied by a constant to meet the design requirements only if its SNR is equal or superior to that of the IWOLA + FFT algorithm. Furthermore, to further verify the accuracy of the proposed algorithms, we simultaneously inputted four signals into the three algorithms. The input signal consisted of two real sinusoidal signals with frequencies of 40.00 MHz and 40.000763 MHz, respectively, and two chirp signals, in addition to Gaussian white noise. The starting frequencies of the two chirp signals were 80 MHz and 160 MHz, respectively. Moreover, their bandwidths were 0.78125 MHz, which is equivalent to the subchannel bandwidth of the IWOLA filter bank, and 80 MHz, respectively. The spectral data obtained by the three methods are shown in Figure 18, where (a) presents a spectrum obtained by the direct FFT algorithm, (b) presents a spectrum obtained by the IWOLA + FFT algorithm, and (c) presents a spectrum obtained by the IWOLA + WOLA algorithm. There were two real sinusoidal signals at A, and one chirp signal at B, which are detailed further in this manuscript. As can be observed from the figure, the proposed algorithms demonstrated flat  
Figure 16: Comparison of channelised gain flatness characteristics. gain characteristics. The bandwidth of the second chirp signal occupies multiple sub-bands, but its gain between sub-bands are flat and identical. Figure 18: Comparison of the simulation analysis. Figure 17: Spectrum gain and SNR for the three algorithms (for a sinusoidal signal with a frequency of 40 MHz). There were two real sinusoidal signals at A, with frequencies of 40 MHz and 40.000763 MHz. The magnified views for Point A in Figure 18 are shown in Figure 19. When comparing Figure 19a-c, the resultant spectra obtained by the direct FFT and IWOLA + FFT algorithms for the signal at Point A were the same, whereas the IWOLA + WOLA algorithm demonstrated a superior spectrum. The two proposed algorithms demonstrated the same frequency resolution as the direct FFT algorithm, namely, 763 Hz. The magnified views for Point B in Figure 18 are shown in Figure 20. The chirp signal frequency started at 80 MHz with a bandwidth of 0.78125 MHz, which was equal to each sub-band width of the IWOLA filter bank. In addition, a coherent dispersed pulse was generated in MATLAB as a simulated pulsar signal, with a dispersion measure (DM) of 26.776 pc cm\({}^{-3}\). The frequency ranged from 1200-1600 MHz, and its sampling frequency and observation time were 800 MHz and 35 ms, respectively.