# Headphone-To-Ear Transfer Function Estimation Using Measured Acoustic Parameters
## 4 Estimation of HpTFs  
The human ear canal is about \(-27\) mm long and about \(8\) mm in diameter, which can be seen as a slightly bended tube with varying cross-sectional areas. In this paper, for simplicity, the acoustic model of the ear canal is approximated as an \(M\)-sectional tube with each section having the same length \(L\) and variable cross-sectional area \(S_{m},m=1,\dots,M\), and the eardrum impedance \(Z_{ed}(f)\) is at the end of the \(M\)th section. Given the measured headphone acoustic reflectance \(r_{0}(f)\) and Norton equivalent volume velocity source \(U_{0}(f)\), the HpTF can be estimated as [10]  
\[H_{HpTF}(f)=U_{0}(f)Z_{ed}(f)\frac{e^{-jk_{M}L}0.5(1+r_{0}(f))(1+r_{ed}(f))\prod \limits_{m=1}^{M-1}e^{-jk_{m}L}(1+r_{m})}{\left[ \begin{array}{cc}1&r_{0}\ \ (f) \end{array} \right]\left\{ \begin{array}{cc}1\\ r_{m}e^{-jk_{m}2L}&e^{-jk_{m}2L}\end{array} \right\}\right\}\left[ \begin{array}[ ]{cc}1\\ r_{ed}(f)e^{-jk_{M}2L}\end{array} \right]} \tag{16}\]  
where \(k_{m}\) is the wave number in the \(m\)th section, \(r_{m}=(S_{m+1}-S_{m})/(S_{m+1}+S_{m})\), and \(r_{ed}(f)\) is the reflectance of the eardrum \(r_{ed}(f)=(\rho c/S_{M}-Z_{ed}(f))/(\rho c/S_{M}+Z_{ed}(f))\). To simulate the real human ear, the effective eardrum impedance model [11] and ear canal area function [12] measured on human  
Figure 5: (Color online) Measurement results of the acoustic parameters for three different types of headphones (**First row**: Huawei AM12 insert headphone; **Second row**: Beyerdynamic DT880 semi-open circumaural headphone; **Third row**: AKG K550 closed-back circumaural headphone). (**a**) The magnitude responses of the headphone acoustic reflectance; (**b**) The phase responses of the headphone acoustic reflectance; (**c**) The magnitude responses of the headphone Norton equivalent volume velocity source (NEVVS). ears are adopted here for estimating the HpTFs of the Huawei AM12, Beyerdynamic DT880, and AKG K550. The above ear canal area function with fixed ear canal length of 27 mm can be described by the mid locations \(\mathbf{x}_{k}\) that refers to the innermost corner of the ear canal and the corresponding radii \(\mathbf{r}_{k}\). Three different ear canal area functions with fixed \(\mathbf{r}_{k}\) and ear canal lengths of 22 mm, 27 mm, and 32 mm, where the corresponding mid locations are \(22\mathbf{x}_{k}/27\), \(\mathbf{x}_{k}\), and \(32\mathbf{x}_{k}/27\) are chosen to study the effects of various ear canal lengths on HpTFs, and the estimated HpTFs for the AM12, DT880, and K550 are illustrated in Figure 6a. As can be seen, HpTFs vary considerably with headphones and ear canal lengths. Even for the same headphone, HpTFs show large variations between different ear canal lengths. In particular, for AM12, the peak at 6 kHz is caused by the self-resonance of the headphone, not relying on the length of the ear canal. The difference in HpTFs could be even greater if individual eardrum impedances are taken into account. In the same way, to study the effects of various ear canal cross-sectional areas on HpTFs, three different ear canal area functions with fixed ear canal length of 27 mm and ear canal radius of 0.8 \(\mathbf{r}_{k}\), \(\mathbf{r}_{k}\), and 1.2 \(\mathbf{r}_{k}\) are chosen, and the estimated HpTFs for the AM12, DT880, and K550 are depicted in Figure 6b. According to the ear canal acoustic model presented in Equation (16), the effects of the ear canal radii \(\mathbf{r}_{k}\) on HpTFs can be equivalent to the effects of the eardrum reflectance \(r_{cd}\) on HpTFs. As can be seen from Figure 5b, variations of the ear canal radii \(\mathbf{r}_{k}\) have a considerable impact on the overall gains of the HpTFs, while the structures of the HpTFs show slight variations. This is because the resonance frequency of the ear canal is mainly dependent on the length of ear canal rather than the eardrum impedance. On the whole, HpTFs show large variations between different headphones and ear canals.