## 4 Primary and Secondary Paths
### Primary Paths Analysis  
The primary path is considered as a path between the input signal to the primary noise source, i.e., the loudspeaker inside the active casing, and the signal acquired by one of the error microphones. The amplitude functions of the frequency functions of the impulse response estimates for the primary paths for both setups are presented side by side. The paths are presented in frequency range up to 500 Hz, in which active noise control methods are effective. In each figure, primary path frequency response functions for all distances (0.1-0.8 m) are presented.  
The loudspeaker used as a noise source starts to transmit sound for frequencies above 40 Hz, which can be observed in Figures 3-5. In Figure 3, the primary paths for the error microphone corresponding to right casing panel are presented (locations R and R\({}^{\prime}\)). The amplitude responses in Setup 1 (Figure 2(a)) are distinctly different comparing to the amplitude responses obtained for Setup 2 (Figure 2(b)). Maximum magnitudes in Figure 2(a) are substantially greater than in Figure 2(b). A general trend observed in Figure 2(a) is an increase of magnitude for decreasing distance between the casing and the corner. Such effect is not evident in Figure 2(b). This means that when decreasing distance \(d_{i}\), the primary noise starts to resonate in the gap between the casing and the wall. Hence, if an error microphone is placed in such a gap, the signal it measures becomes less representative of the general noise emitted by the given casing panel and it more represents a specific resonant effect, which may be misleading for the adaptation of the control algorithm. For the evaluated laboratory setup, this is the case for \(d\leq 0.4\) m.  
In Figure 4, the primary paths for the error microphone corresponding to back casing panel in both setups are presented (locations B and B\({}^{\prime}\)). Analogous behavior can be observed as in primary paths presented for the right error microphone in Figure 3, hence, for the sake of brevity, it is no further discussed.  
Figure 4: The amplitude functions of the frequency functions of the impulse response estimates for the primary paths with the error microphone adjacent to the back casing panel.  
Figure 3: The amplitude functions of the frequency functions of the impulse response estimates for the primary paths with the error microphone adjacent to the right casing panel.  
In Figure 5, the primary paths for the error microphone corresponding to top casing panel in both setups are presented (the location T). The top error microphone placement is the same for both setups, and hence, amplitude responses are very similar for them. This confirms consistency of obtained results. On the other hand, when the distance \(d_{i}\) changes, the sound emission pattern in the room also changes. It is noteworthy that in the frequency range up to about 125 Hz, an effect of magnitude increase for decreasing distance \(d_{i}\) is observed. Above 125 Hz, such effect is not clearly visible. The effect observed below 125 Hz indicates that moving the casing closer to the corner causes slightly stronger emission of low frequencies in the laboratory enclosure, as it could be expected, based on the theory recalled in Section 3.2.  
The paths between the primary noise source and one of the room microphones are given in Figure 6. The magnitudes are in general lower than the ones obtained for error microphones, as the distances from the noise source to the room microphones are larger. Similarly as for the top error microphone, the paths are consistent for both setups. However, it is noteworthy that in the frequency range up to about 125 Hz, a slight effect of magnitude increase for decreasing distance \(d_{i}\) is observed again. It confirms that moving the casing closer to the corner causes stronger excitation of low frequencies in the whole laboratory enclosure, as used for the sound reproducing systems in audio engineering.