## 4 Experiment Results  
In this section, experiments are carried out with a commercial ANC headphone to verify the proposed methods. The experimental system is shown in Figure 2. A self-designed acrylic box is used as the dummy head and the microphone fixed on its right-hand side is used as the dummy head ear. The primary noise is generated by the combination of a loudspeaker as well as a subwoofer. The subwoofer is used as a complement of the loudspeaker to generate noise below 100 Hz. The distance between the dummy head ear and the primary noise sources is about 1 m. A self-designed controller with ADAU1772 as its core is used to implement the feedback control. The internal sampling frequency of ADAU1772 is 192 kHz. The controller can also upload signals sampled by ADAU1772 to the computer, with which acoustic FRs of the experimental system can be tested. First, both S(z) and R(z) are tested with a white noise stimulus to the headphone's speaker. Figure 3a,b show the tested magnitude and phase responses, respectively. It could be seen that the FRs of the secondary paths are in high accordance with each other below 1 kHz, but show large differences in the range from 2 to 6 kHz. This is confirmed by the FR mismatch term R(z)/S(z) for the secondary path, whose responses are shown in Figure 3c,d. Then, a white noise is played by the primary noise sources and the FR of P(z)/Q(z) is tested with sampled signals of both the feedback microphone and the dummy head ear. The results are also shown in Figure 3c,d, from which it could be found that the primary paths also exhibit larger differences above 1 kHz. Finally, the responses of \(\Delta\)(z) are calculated, as shown by the yellow lines in Figure 3c,d. It can be observed that FR differences of both the primary and secondary paths contribute to the FR mismatch term \(\Delta\)(z). In the frequency range below 1 kHz, \(\Delta\)(z) approximately equals to 1, which indicates that the NR performances at the feedback microphone and the dummy head ear would be in high accordance. However, \(\Delta\)(z) could exhibit large values when the frequency gets higher. Consequently, large noise enhancement might arise at the dummy head ear even if it is limited to a low value at the feedback microphone, just as analyzed in Section 2. With the tested acoustic FRs, the cascade biquad filters of the feedback controller could be designed with the proposed methods. In the objective function described by (10), the frequencies \(\mathrm{f_{k}}\) are discretized logarithmically from 20 to 20 kHz and K is set to 300. According to the frequency range where the FR mismatch is small, the target frequency band [\(\mathrm{f_{start}}\), \(\mathrm{f_{stop}}\)] is set to [20 Hz, 1 kHz]. The weight coefficients \(\mathrm{w_{k}}\) are set to 1 for [50 Hz, 500 Hz], within which the NR performance is expected to be strengthened, and set to 0.2 for 20 Hz as well as 1 kHz. All the other \(\mathrm{w_{k}}\) are obtained by linear interpolation. In the design process, the minimum magnitude margin \(\mathrm{L_{min}}\) and the minimum phase margin \(\mathrm{\gamma_{min}}\) are set to 10 dB and 30\({}^{\circ}\), respectively. A constant noise enhancement limitation is chosen, i.e., \(\mathrm{\delta_{k}}=\delta\) for all \(\mathrm{f_{k}}>\mathrm{f_{stop}}\). All the punishment intensity parameters \(\mathrm{\xi_{1}}\), \(\mathrm{\xi_{2}}\), \(\mathrm{\xi_{3}}\) are set to 10,000. In the feedback controller, \(N=5\) biquad filters are used. The filter prototypes  
Figure 3: Tested FRs of the experimental system: the (**a**) magnitude and (**b**) phase responses of the secondary paths, the (**c**) magnitude and (**d**) phase responses of the frequency mismatch. Figure 2: The experimental system. and the corresponding bounds of the parametric vector \(\mathbf{X}\) is shown in Table 1. It can be seen that by setting different values of g within \(\mathbf{Ib}\) and \(\mathbf{ub}\), the biquad filters could be forced into different types even if the same prototype is used.