## 2 Problems with FR Mismatches  
In a feedback ANC headphone, the feedback loop is generally constructed with the feedback microphone, the controller as well as the speaker. The feedback microphone is used as the error sensor to pick up the residual noise signal inside the earcup, whose output is fed into the controller and then drive the speaker to generate a secondary noise so that the primary noise could be reduced. The schematic diagram of a typical feedback ANC headphone is shown in Figure 1, where C(z), S(z) represent the transfer functions of the controller and the secondary path, M(z) and Y(z) denote for the z-transforms of the outputs of the feedback microphone and the controller, respectively. The primary noise P(z)N(z) is modeled by an outside noise signal N(z) transmitting through the primary path P(z). With the abovementioned definitions, the closed-loop sensitivity function of the feedback control system can be written as:  
\[\mathrm{G_{mic}(z)}=\frac{\mathrm{M(z)}}{\mathrm{P(z)N(z)}}=\frac{1}{12212 \mathrm{C(z)S(z)}} \tag{1}\]  
whose magnitude response represents the NR performance at the feedback microphone. In order for a better NR performance, the FR of the controller C(z) should be enlarged as much as possible within the frequency band where active control of noise is expected to be effective, and should be attenuated as the frequency gets higher so that the noise enhancement could be limited according to the waterbed effect. Various optimization and control methods have been proposed with different kinds of controllers [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14] and generally a good NR performance could be expected at the feedback microphone position. However, the actual NR performance would be perceived by the human ear, whose position is different from the one of the feedback microphone. In Figure 1, the residual noise signal perceived by the human ear is denoted by E(z), which can be written as:  
\[\mathrm{E(z)}=\mathrm{Q(z)N(z)}+\mathrm{R(z)Y(z)}=\left(\mathrm{Q(z)}+\frac{ \mathrm{C(z)R(z)}}{1-\mathrm{C(z)S(z)}}\mathrm{P(z)}\right)\mathrm{N(z)} \tag{2}\]  
where Q(z), R(z) represent the primary and secondary paths for the human ear, respectively. Another sensitivity function can be defined at the human ear position as the ratio between the residual noise E(z) and the primary noise Q(z)N(z)  
\[\mathrm{G_{ear}(z)}=\frac{\mathrm{E(z)}}{\mathrm{Q(z)N(z)}}=1+\frac{\mathrm{ C(z)R(z)}}{1-\mathrm{C(z)S(z)}}\frac{\mathrm{P(z)}}{\mathrm{Q(z)}}=1+ \mathrm{T(z)\Delta(z)} \tag{3}\]  
where  
\[\mathrm{T(z)}=\frac{\mathrm{C(z)S(z)}}{1-\mathrm{C(z)S(z)}},\ \ \ \Delta(z)=\frac{\mathrm{P(z)}}{\mathrm{Q(z)}}\frac{\mathrm{R(z)}}{\mathrm{S(z)}} \tag{4}\]  
Figure 1: The schematic diagram of the feedback ANC headphone. Similarly, the magnitude response of (3) represents the actual NR performance at the human ear position. In (4), T(z) is the complementary sensitivity function corresponding to the feedback microphone, which could be seen as a normalized secondary noise (normalized with the primary noise). When T(z) = \(-1\), the secondary noise has the same amplitude but the opposite phase so that the primary noise could be totally eliminated at the feedback microphone position. However, as shown by (3), at the human ear position this normalized secondary noise is disturbed by \(\Delta\)(z), which represents the FR mismatches between the feedback microphone and the human ear. It can be observed from (4) that FR differences of both the primary and the secondary paths contribute to the FR mismatch term \(\Delta\)(z). If no FR mismatches exist (i.e., \(\Delta\)(z) = 1), we have G\({}_{\text{mic}}\)(z) = G\({}_{\text{ear}}\)(z), which indicates the same NR performances would be obtained at both the feedback microphone and the human ear.