# A Low-Complexity Volterra Filtered-Error LMS Algorithm with a Kronecker Product Decomposition
## 5 Performance Evaluation
### Simulation with Nonlinear Primary Path and Nonlinear Secondary Path  
To further demonstrate the effectiveness of the proposed DVMFELMS algorithms in the NANC/NSP case, both primary and secondary paths are considered to have nonlinearity in this experiment. The primary noise recorded by the error microphone is generated by passing the reference signal through a second-order Volterra model given by  
\[\begin{split} d(n)&=u(n-5)+0.8u(n-6)+0.3u(n-7)+0.4u(n- 8)\\ &+0.2u(n-5)u(n-6)-0.3u(n-5)u(n-7)+0.4u(n-5)u(n-8),\end{split} \tag{31}\]  
while the secondary path is considered to be an NLN model used in [27], which is expressed as  
\[\begin{split}& l_{1}=1-0.6z^{-1}+0.05z^{-2},\\ & N(z_{1}(n))=3.3 tanh(0.3z_{1}(n)),\\ & l_{2}=1+0.2z^{-1}+0.05z^{-2}.\end{split} \tag{32}\]  
Figure 6 depicts the NMSE obtained by all five active noise control algorithms versus the number of iterations by using the same sinusoidal signal as the reference signal in (30). The memory length for all algorithms is chosen to be 10, and the order of the Volterra kernel is chosen to be 2. The step size for the FELMS algorithm is chosen to be \(\mu_{1}=2\times 10^{-1}\), step sizes for the second-order Volterra filter are set to \(\mu_{1}=2\times 10^{-1},\mu_{2}=2\times 10^{-1}\), step sizes for the FEGFLNN algorithm are chosen to be \(\mu_{1}=2\times 10^{-1},\mu_{2}=4\times 10^{-3}\), step sizes for the 2 OEMFNL algorithm are set to \(\mu_{1}=2\times 10^{-1},\mu_{2}=1\times 10^{-2}\), and step sizes for the 2 ODVMFELMS algorithm are set to \(\mu_{1}=2\times 10^{-1},\mu_{1}=2\times 10^{-2},\lambda_{2}=0.02\). Figure 6 shows that the reductions of primary noise achieved by the FELMS algorithm, the 2 OVFELMS algorithm, the 2 OFEEMFNL algorithm, and the proposed 2 ODVMFELMS algorithm are about 20.8 dB, 29.7 dB, 29.3 dB, 29.1 dB, and 29.7 dB, respectively. It should be noted that the FEEMFNL algorithm shows bad convergence speed in the NSP case. The proposed DVMFELMS algorithm can still achieve comparable performance with the lowest computational complexity in the NSP case. However, the convergence rate of the proposed 2 ODVMFELMS algorithm is worse in these cases because of the sinusoidal signal and NSP used.  
Figure 5: Performance comparison using the nonlinear primary path with band-limited white Gaussian reference signal.  
For broadband results, the band-limited white Gaussian noise used here is exactly the same as in experiment 2. The SNR is also set to be 30 dB. All the transfer functions are exactly the same as in the first case. Figure 7 shows the NMSE achieved by all five algorithms versus the number of iterations. The step size for the FELMS algorithm is chosen to be \(\mu_{1}=2\times 10^{-1}\), step sizes for the second-order Volterra filter are set to \(\mu_{1}=2\times 10^{-1},\mu_{2}=2\times 10^{-1}\), step sizes for the 2 OFEGFLNN algorithm are chosen to be \(\mu_{1}=2\times 10^{-1},\mu_{2}=4\times 10^{-3}\), step sizes for the 2 OEMFNL algorithm are set to \(\mu_{1}=2\times 10^{-1}\), \(\mu_{2}=1\times 10^{-2}\), and step sizes for the 2 ODVMFELMS algorithm are set to \(\mu_{1}=2\times 10^{-1}\), \(\mu_{1}=2\times 10^{-2},\lambda_{2}=0.02\). One can see from Figure 7 that the reductions of primary noise achieved by the FELMS algorithm, the 2 OVFELMS algorithm, the 2 OFEGFLNN algorithm, and the 2 ODVMFELMS algorithm are about 17.3 dB, 24.1 dB, 23.7 dB, 23.5 dB, and 23.8 dB, respectively. It can be seen that the FEFLNN algorithm and the FEEMFNL algorithm show worse performance in steady-state NMSE and convergence speed than the Volterra filter-based algorithm in the broadband reference signal and NSP case. The proposed DVMFELMS algorithm has performance comparable with the VFELMS algorithm, with a dramatic reduction of computational complexity.  
Figure 6: Performance comparison using the nonlinear primary path and secondary path with sinusoidal reference signal.