## 5 Results and Discussion  
In order to verify the stability analysis performed in this paper, simulations of a feedback ANC system with various delay errors in the estimated secondary path model are performed. The stability bounds obtained from Equations (23) and (24) are compared with the simulation results and the original Nyquist stability condition. In Figure 6, the stability delay error bounds are obtained with the proposed Equation (23) and compared with the stability bounds obtained with the Nyquist method. The results are obtained for different adaptive filter lengths \(N\) = 50, 100, 200 and 400. The proposed stability condition agrees well with the original Nyquist condition showing only few differences in the valid frequency range \(\omega_{0}>\pi/N\), i.e., above 80 Hz for \(N\) = 100, and above 160 Hz for \(N\) = 50. The bounds of the Nyquist stability condition are obtained by the trial and error method. Many frequency response plots of \(W(z)[z^{-\Delta}-z^{-\Delta}]\) are drawn repeatedly on the complex-plane by computer calculation until one of the plots encloses the Nyquist point (\(-1\), \(0\)). The complex plots were drawn for various delay errors in secondary path model and various adaptive filter responses for different lengths of adaptive filter and frequencies of primary noises. As shown in Figure 6, the proposed results do not exceed the Nyquist results, though the proposed equation's results give tighter bounds than the bounds of original Nyquist stability condition. For the cases of \(N\) = 50 and 100, the large differences below the valid frequency range \(\omega_{0}>\pi/N\) are caused by overlapping of the two main lobes of the digital filter responses, which are shown as main two terms in Equation (8). Simulations are performed with the feedback ANC algorithm based on the filtered-x LMS (FxLMS). Sampling frequency is set to 8 kHz. Delay length of secondary path is set to 10 ms, which is 80 samples  
Figure 6: Comparison of stability bounds obtained by Nyquist criterion (dotted line) and the proposed Equation (23) (solid line). in the corresponding sampling frequency. The ANC filter length is set to 200 taps. The tested primary noises are single-tone sinusoidal noises whose frequencies are 50-400 Hz. The normalized least mean square (nLMS) algorithm [18] is used to update the adaptive filter \(W(z)\). To verify the usage of pure delay model in the proposed analysis, simulations are performed for both cases of pure delay secondary path model and practical secondary path model having delay and spectral distortion. For the practical secondary path model and its estimated model, a spectral distortion component is measured from commercial reference quality headphones and applied to a pure delay model as \(S(z)=S^{\prime}(z)z^{-\Delta}\). For the simulation of various delay error of secondary path model, pure delays around 80 samples are additionally inserted to the measured impulse response, which originally includes 4-sample delay caused by the acoustic propagation between the headphone speaker and the error microphone and the AD/DA converting processing. Figure 7 shows the measured impulse response \(s^{\prime}(n)\) and its frequency response \(S^{\prime}(z)\). In the frequency response in Figure 7, the roll-off distortion shown in sub bass frequency range lower than 100 Hz is mostly caused by the physical characteristic of the headphone's speaker and the resonance of the cavity in the headphone ear-cup. Except for the low frequency roll-off, the commercial headphones provide moderate frequency response in most of the ANC system's operation frequency range. Simulation is conducted for different number of the adaptive filter lengths and the results are shown in Table 1. Frequency of primary noise is set to 150 Hz. For the cases of the filter numbers larger than 100 taps, each bound obtained from the proposed equation and simulation exactly agrees with the original Nyquist results.