# Frequency-Domain Filtered-x LMS Algorithms for Active Noise Control: A Review and New Insights
## 4 Delayless Frequency-Domain ANC Algorithms
### [PERSON]'s Delayless Algorithm  
The delayless frequency-domain ANC is presented in Figure 4. To avoid the delay in the signal path, the calculation of the adaptive filter output can be directly implemented using the time-domain convolution [40; 41; 42; 43]  
\[\mathbf{y}(k+1)=\left[ \begin{array}{c}\mathbf{x}^{T}(kL+1)\mathbf{w}(k+1) \\ \cdots\\ \mathbf{x}^{T}(kL+L)\mathbf{w}(k+1)\end{array} \right]. \tag{35}\]  
If the filter length \(N_{\mathrm{w}}\) is small, the complexity required in (35) may be acceptable. However, when \(N_{\mathrm{w}}\) is large, the computational complexity may be too high for a resource-limited system. To resolve this problem, a hybrid fast convolution scheme is adopted to calculate the control filter output [44]  
\[\mathbf{y}(k+1) = \left[y(kL+1),...,y(kL+L)\right]^{T} \tag{36}\] \[= \left[ \begin{array}{c}\mathbf{\bar{x}}^{T}(kL+1)\mathbf{w}_{0 }(k+1)\\ \cdots\\ \mathbf{\bar{x}}^{T}(kL+L)\mathbf{w}_{0}(k+1)\end{array} \right]+\mathbf{ \Theta}(k+1),\]  
where \(\mathbf{\bar{x}}(n)=[x(n),x(n-1),\ldots,x(n-L+1)]^{T}\), and  
\[\mathbf{\Theta}(k+1)=\mathbf{Q}^{01}\mathbf{F}^{-1}\sum_{p=1}^{P_{\mathrm{w}} -1}\mathbf{X}_{p}(k+1)\mathbf{W}_{p}(k+1). \tag{37}\]  
Please note that the calculation of \(\mathbf{\Theta}(k+1)\) should be completed between \(kL\) and \(kL+1\), and the first term of the right-hand side of (36) is calculated in the time domain on a sample-by-sample basis. The update equation is  
\[\mathbf{W}_{p}(k+1)=\mathbf{W}_{p}(k)-\mu\mathbf{G}^{10}\mathbf{\Lambda}^{-1} (k)\mathbf{V}_{p}^{H}(k)\mathbf{E}(k). \tag{38}\] \begin{table}
\begin{tabular}{l l} \hline
**Operation** & **Complexity** \\ \hline (1) Compute the frequency-domain reference matrix \(\mathbf{U}_{0}(k)\) using (29) & One FFT \\ (2) Estimate the control signal at the error microphone \(\mathbf{\hat{z}}(k)\) using (28) & One FFT and \(4 LP_{\mathrm{s}}\) multiplications \\ (3) Estimate the disturbance signal vector \(\mathbf{\hat{d}}(k)\) using (30) & 0 \\ (4) Compute the frequency-domain reference matrix \(\mathbf{X}_{0}(k)\) using (8) & One FFT \\ (5) Compute the filtered reference signal vector \(\mathbf{v}(k)\) using (11) & One FFT and \(4 LP_{\mathrm{s}}\) multiplications \\ (6) Compute the FFT of \(\mathbf{v}_{0}(k)\) using (15) & One FFT \\ (7) Commutate the pseudo frequency-domain error vector \(\mathbf{\hat{E}}(k)\) using (31) & 2 FFT and \(4 LP_{\mathrm{w}}\) multiplications \\ (8) Update \(\mathbf{\Lambda}(k)\) using (17) & 4\(L\) multiplications \\ (9) Update the weight vector using (34) & \(2P_{\mathrm{w}}\) FFTs and \(6 LP_{\mathrm{w}}\) multiplications \\ (10) Compute the control filter out using (25) & One FFT and \(4 LP_{\mathrm{w}}\) multiplications \\ (11) Compute the cancelling signal using (9) \(u(n)=y(n-2L)\) & 0 \\ \hline \end{tabular} Note: Steps (1)–(10) should be completed between \(kL\) and \(kL+L\); Average and peak multiplications: \((8+2P_{\mathrm{w}})A/L+14P_{\mathrm{w}}+8P_{\mathrm{s}}\).  
\end{table} 
Table 5: FPBMFxLMS algorithm [35]. The calculation of \(\mathbf{w}_{0}(k)\) does not require extra computational cost and is expressed by  
\[\mathbf{w}_{0}(k)=[\mathbf{I}_{L}\quad\mathbf{0}_{L}]\mathbf{F}^{-1}\mathbf{W}_ {0}(k). \tag{39}\]  
Because the calculation of \(\mathbf{\Theta}(k+1)\) uses \(\mathbf{W}_{p}(k+1)\), the weight update in (38) should be completed between \(kL\) and \(kL+1\), which means the update-first approach must be used. By using this approach, the delay in the signal path is removed. However, this algorithm is not practical because (36) and (38) should be completed within one sampling period, which requires \((5+2P_{\text{w}})A+10 LP_{\text{w}}+4 LP_{\text{s}}+L\) multiplications per sample. In addition, the algorithm requires \(L\) multiplications per sample between \(kL+2\) and \(kL+L\). Thus, the complexity of the algorithm is not evenly distributed over time, i.e., its peak complexity is rather high.