# Frequency-Domain Filtered-x LMS Algorithms for Active Noise Control: A Review and New Insights
## 3 Conventional Frequency-Domain ANC Algorithms
### Analysis
In the frequency-domain ANC algorithm, the distance between the reference sensor and the secondary loudspeaker should be large enough to satisfy the casualty condition [30; 31]. However, this is not always achievable because of practical installation constraints. Therefore, the delay in FPBfxLMS algorithms is a major limitation for broadband ANC. At this point, we investigate the effect of the secondary path on the weight update. Using (3), we rewrite the error vector \(\mathbf{e}(k)\) as  
\[\left[ \begin{array}{c}e(kL-L+1)\\ \cdots\\ e(kL)\end{array} \right] =\left[ \begin{array}{c}d(kL-L+1)\\ \cdots\\ d(kL)\end{array} \right]+\left[ \begin{array}{c}z(kL-L+1)\\ \cdots\\ z(kL)\end{array} \right] \tag{22}\] \[=\left[ \begin{array}{c}d(kL-L+1)\\ \cdots\\ d(kL-L+1)\\ \cdots\\ d(kL)\end{array} \right]+\left[ \begin{array}{c}u(kL-L+1)*s\\ \cdots\\ u(kL)*s\end{array} \right]\] \[=\left[ \begin{array}{c}y(kL-L+1-D)*s\\ \cdots\\ y(kL-D)*s\end{array} \right]\]  
where \(*\) denotes the linear convolution. The last element of \(\mathbf{e}(k)\) can be expressed as  
\[e(kL) =d(kL)+y(kL-D)*s \tag{23}\] \[=d(kL)+\sum\limits_{p=0}^{P_{\text{s}}-1}\mathbf{y}^{T}(k-m-p) \mathbf{\tilde{s}}_{p}\] \[=d(kL)+\sum\limits_{p=0}^{P_{\text{s}}-1}\left[\mathbf{Q}^{01} \mathbf{F}^{-1}\sum\limits_{q=0}^{P_{\text{s}}-1}\mathbf{X}_{q}(k-m-p) \mathbf{W}_{q}(k-m-p)\right]\mathbf{\tilde{s}}_{p},\]  
where \(\mathbf{\tilde{s}}_{p}=[s_{(p+1)L-1},...,s_{pL}]^{T}\). From (23), it can be seen that the generation of \(e(kL)\) requires \(P_{\text{s}}\) past weight vector \(\mathbf{W}_{q}(k-m-p),p=0,...,P_{\text{s}}-1\). Similarly, the generation of \(\mathbf{e}(k)\) requires \(\mathbf{W}_{q}(k-m-p),p=0,...,P_{\text{s}}\). Therefore, the following assumption is used implicitly for deriving (12)  
\[\mathbf{W}_{q}(k)\approx\mathbf{W}_{q}(k-1)\approx\cdots\approx\mathbf{W}_{q }(k-m-P_{\text{s}}). \tag{24}\]  
The relationship in (24) holds only when the step size \(\mu\) is very small, and the frequency-domain algorithm used in ANC is very similar to the delayed LMS algorithm [32]. Hence, the maximum step size that can guarantee the stability of the algorithm is reduced, and the fastest convergence speed of the algorithm is lower. Therefore, the frequency-domain ANC algorithm becomes quite different from the standard FDAF algorithm in terms of convergence behavior. The above analysis provides another interesting insight to the implementation of the frequency-domain algorithm. For the two algorithms in Tables 1 and 2, we first calculate the filtering-out using \(\mathbf{W}_{p}(k)\) and then update \(\mathbf{W}_{p}(k)\), which is called the filtering-first approach. We could propose an alternative method, namely, the update-first approach. That is, we first update the weight vector to obtain \(\mathbf{W}_{p}(k+1)\) using (18), and then we use the new weight vector \(\mathbf{W}_{p}(k+1)\) to calculate the filtering-out  
\[\mathbf{y}(k)=\mathbf{Q}^{01}\mathbf{F}^{-1}\sum\limits_{p=0}^{P_{\text{s}}- 1}\mathbf{X}_{p}(k)\mathbf{W}_{p}(k+1). \tag{25}\]By so doing, the delay between the weight update and the observation of the error signal can be reduced, and hence the convergence performance can be improved. We then obtain two algorithms in Tables 3 and 4 by applying the update-first approach to the algorithms in Tables 1 and 2, respectively. The update-first approach exhibits a better convergence behavior than the filtering-first approach, while these approaches have a similar computational complexity.