# Frequency-Domain Filtered-x LMS Algorithms for Active Noise Control: A Review and New Insights
## 6 Evaluations  
In this section, we carry out computer simulations to evaluate the convergence performance of several frequency-domain algorithms in the context of ANC. The sampling rate is \(f_{\rm s}=32768\,\rm Hz\). The length of the primary and secondary path are \(N=1536\) and \(N_{\rm s}=256\), respectively. The optimal solution of the control filter is computed using the MINT method [52] and has a length of \(N_{\rm w}=1536\). Their corresponding time-domain impulse responses are plotted in Figure 7. Figure 7c shows that maximum delay allowed in the signal path is \(D=256\) to guarantee the causality condition of the system. The signal-to-noise ratio (SNR) measured at the error microphone is 30 dB. Two signals, i.e., the narrow band signal between 100-800 Hz and a multi-tone signal at 40, 340 and 700 Hz, are used as the reference. The convergence performance is evaluated by the learning curve given by  
\[A(k)=10\text{log}_{10}\left(P_{e}(k)/P_{d}(k)\right), \tag{48}\]  
where \(P_{d}(k)\) and \(P_{e}(k)\) denote the powers of the disturbance signal \(d(n)\) and the error signal \(e(n)\), which are computed recursively as  
\[P_{d}(k) =\beta P_{d}(k-1)+(1-\beta)\sum_{n=kL-L+1}^{kL}d^{2}(n), \tag{49}\] \[P_{e}(k) =\beta P_{e}(k-1)+(1-\beta)\sum_{n=kL-L+1}^{kL}e^{2}(n), \tag{50}\]  
Figure 6: Peak and average complexities. (**a**) Average multiplications per sample. (**b**) Peak multiplications per sample. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Average Multiplications per Sample** & **Peak Multiplications per Sample** \\ \hline FPBFxLMS I [28; 37] & \((5+2P_{\rm w})A/L+10P_{\rm w}+4P_{\rm s}\) & \((5+2P_{\rm w})A+10 LP_{\rm w}+4 LP_{\rm s}+4L\) \\ FPBFxLMS II [39] & \((5+2P_{\rm w})A/L+10P_{\rm w}+4P_{\rm s}\) & \((5+2P_{\rm w})A/L+10P_{\rm w}+4P_{\rm s}\) \\ FPBMFxLMS [35] & \((8+2P_{\rm w})A/L+14P_{\rm w}+8P_{\rm s}\) & \((8+2P_{\rm w})A/L+14P_{\rm w}+8P_  
where \(\beta\) is the smoothing factor. In the following, we use \(\lambda=0.8\) and \(\beta=0.8\). Figure 8 compares the convergence performance of the filtering-first and update-first approaches in Tables 1-4. The narrow band noise is used as input, and the block length is \(L=64\). The update-first approach is more stable than the filtering-first approach given the same step size, which agrees with the above analysis. In addition, the two methods have the same complexity, and hence the update-first approach is recommended for practical implementation. The effect of the block length on the convergence performance of several frequency-domain FxLMS algorithms is investigated in Figures 9 and 10. In Figure 9, we use the multi-tone signal as the reference, and the step size \(\mu=0.03\). The block length is set to \(L=64,128,256\). All the algorithms are convergent for \(L=64,128\), while they diverge for \(L=256\). We found that all the algorithms can converge using a smaller step size \(\mu=0.001\) for the block length \(L=256\). For the multi-tone input, all the frequency-domain algorithms can converge if a sufficiently small step size is employed. Figure 8: Convergence comparison between the update-first and filtering-first approaches. (a) FPBFxLMS I and FPBFxLMS III. (b) FPBFxLMS II and FPBFxLMS IV. Figure 7: Primary and secondary paths. (a) Primary path. (b) Secondary path. (c) Least square solution of the control filer. The experiment in Figure 9 is repeated using the narrow band noise as input, and the learning curves are presented in Figure 10. The three delayless frequency-domain algorithms are stable and have a similar convergence for narrow band noise input in all the experiments. However, their convergence rate decreases as the block length. The convergence performance of the traditional frequency-domain FxLMS algorithms is dramatically affected by the block length. For \(L=64\), the causality condition is fulfilled for all the algorithm involved, and they exhibit a similar convergence.