# A Low-Complexity Volterra Filtered-Error LMS Algorithm with a Kronecker Product Decomposition
## 5 Performance Evaluation
### Simulation with Nonlinear Primary Path  
In the first experiment, the situation where the primary path has a nonlinear process is studied. The primary noise recorded by the error microphone is generated by passing the reference signal through a third-order Volterra model with cross terms, which is given by  
\[ \begin{split} d(n)&=u(n-5)+0.8u(n-6)+0.3u(n-7)+0.4u (n-8)\\ &+0.2u(n-5)u(n-6)-0.3u(n-5)u(n-7)+0.4u(n-5)u(n-8)\\ &+0.04u(n-5)u(n-6)u(n-7)-0.02u(n-5)u(n-6)u(n-8),\end{split}  \tag{29}\]  
where the transfer function of the secondary path is \(S(z)=z^{-2}+0.3z^{-3}\). The reference noise signal \(u(n)\) is considered to be two cases; a sinusoidal wave of 500 Hz with a sampling rate of 8000 Hz is utilized in the first case, and a band-limited white Gaussian noise in the second case. The sinusoidal reference signal is given by  
\[u(n)=\sqrt{2}\sin\left(\frac{2\pi\times 500\times n}{8000}\right)+v(n), \tag{30}\]  
where \(v(n)\) is a white Gaussian noise with zero mean and unit variance, and the signal-to-noise ratio (SNR) is set to be 30 dB. The memory length for all algorithms is chosen to be 10. The order of the Volterra kernel is chosen to be 3, and the order of the functional link neural network and even mirror Fourier nonlinear filter is chosen to be 2. The step size for the FELMS algorithm is chosen to be \(\mu_{1}=2\times 10^{-3}\), step sizes for the third-order Volterra filter are set to \(\mu_{1}=2\times 10^{-3},\mu_{2}=1\times 10^{-3},\mu_{3}=6\times 10^{-4}\), step sizes for the 2 OFEEMFNL algorithm and 2 OFEEMFNL algorithm are set to \(\mu_{1}=2\times 10^{-3},\mu_{2}=1\times 10^{-3}\), and step sizes for the 3 ODVMFELMS algorithm are set to \(\mu_{1}=2\times 10^{-3},\mu_{2}=2\times 10^{-2}\), \(\mu_{3}=1\times 10^{-4},\lambda_{2}=0.6,\lambda_{3}=0.04\). Figure 4 shows the NMSE achieved by all five algorithms versus the number of iterations in the sinusoidal noise case. It can be seen from Figure 4 that the reduction of primary noise achieved by the FELMS algorithm, the 3 OVFELMS algorithm, the 2 OFEEMFNL algorithm, the 2 OFEEMFNL algorithm, and the 3 ODVMFELMS algorithm is about 22.5 dB, 25.3 dB, 25.9 dB, 26.2 dB, and 26.3 dB, respectively. All nonlinear algorithms achieve better performance against the FELMS algorithm. The proposed DVMFELMS algorithm performs slightly better in the steady-state performance and the convergence rate in this case. It should be noted that the proposed DVMFELMS algorithm can even outperform the VFELMS algorithm in NMSE, which might be explained by the Volterra model used in the VFELMS algorithm being over-fitting for this simulation case. For results of the broadband noise, the frequency range of the band-limited noise is chosen to be from 50 Hz to 1000 Hz, among which the ANC system usually works well. The signal is normalized to unity, and a white Gaussian noise with zero mean and unit variance with an SNR of 30 dB is also added. All the transfer functions and nonlinearities used are exactly the same as in the first case. The step size for the FELMS algorithm is chosen to be \(\mu_{1}=6\times 10^{-3}\), step sizes for the third-order Volterra filter are set to \(\mu_{1}=6\times 10^{-3}\), \(\mu_{2}=4\times 10^{-3}\), \(\mu_{3}=6\times 10^{-4}\), step sizes for the 2 OVEGFLNM algorithm and 2 OEMFNL algorithm are set to \(\mu_{1}=6\times 10^{-3},\mu_{2}=1\times 10^{-3}\), and step sizes for the 3 ODVMFELMS algorithm are set to \(\mu_{1}=6\times 10^{-3},\mu_{2}=4\times 10^{-3},\mu_{3}=1\times 10^{-3}\), \(\lambda_{2}=0.02,\lambda_{3}=0.02\). Figure 5 shows that the reduction of primary noise achieved by the FELMS algorithm, the 3 OVFELMS algorithm, the FEGFLNN algorithm, the 2 OEMFNL algorithm, and the 3 ODVMFELMS algorithm in the broadband noise case is about 19.8 dB, 23.8 dB, 23.2 dB, 23 dB, and 23.6 dB, respectively. It is worth noting that in the broadband case, the VFELMS algorithm outperforms the 2 OVEGFLNN algorithm and the 2 OEMFNL algorithm in NMSE, which coincides with the results in [19].