# Headphone-To-Ear Transfer Function Estimation Using Measured Acoustic Parameters
## 2 The Microphone Array Method for Measuring Acoustic Reflectance  
### Theory  
For the transfer function method [18; 19], two measurement positions are used to measure the reflectance of an acoustic load, and the measurement frequency range is determined by \(0.05c/s\ <\ f\ <\ 0.45c/s\), where \(c\) is the speed of the sound, and \(s\) is the distance between the measurement positions. It is clear that two measurement positions cannot achieve satisfactory measurement accuracy over a wide frequency range. Assume \(N\) microphones located at positions \(x_{1}\) to \(x_{N}\) are simultaneously used in an impedance tube with a pinna simulator, as shown in Figure 1. Let \(p_{i}(f)\) and \(p_{r}(f)\) denote the frequency responses of the incident and reflected sound pressures at \(x=0\), respectively. Then, the output of the \(n\)th microphone \(V_{n}(f)\) at position \(x_{n}\) can be expressed as  
\[V_{n}(f)=[p_{i}(f)e^{jkx_{n}}+p_{r}(f)e^{-jkx_{n}}]\cdot M_{n}(f),\quad n=1,2, \ldots,N, \tag{1}\]  
where \(M_{n}(f)\) is the frequency response of the sensitivity of the \(n\)th microphone, \(f\) is the frequency, \(k\) is the wave number, and \(j=\sqrt{-1}\). If tube attenuation is considered, the wave number can be determined as [20]  
\[k=\frac{2\pi f}{c}-j\frac{0.0194\sqrt{f}}{D_{0}c}, \tag{2}\]  
where \(D_{0}\) is the diameter of the tube. Considering that different microphones have different sensitivities, this paper introduces a complex compensation function \(J_{n}(f)\) to compensate the mismatch between microphone sensitivities. With the frequency response of the sensitivity of the 1 st microphone \(M_{1}(f)\) as reference, the compensation function \(J_{n}(f)\) is defined as  
\[J_{n}(f)=\frac{M_{n}(f)}{M_{1}(f)},\quad n=1,2,\ldots,N. \tag{3}\]  
As shown below, the compensation functions can be obtained by calculating the ratio between the frequency response of the 1 st microphone and that of the \(n\)th microphone measured at the same position in the impedance tube. Based on Equations (1) and (3), the outputs of \(N\) microphones measured at positions \(x_{1}\) to \(x_{N}\) can be written as  
\[\left[ \begin{array}{c}V_{1}(f)\\ V_{2}(f)\\ \vdots\\ V_{N}(f)\end{array} \right]=M_{1}(f)\left[ \begin{array}{cccc}J_{1}(f)&0& \cdots&0\\ 0&J_{2}(f)&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&J_{N}(f)\end{array} \right]\left[ \begin{array}{c}e^{jkx_{1}}&e^{ -jkx_{1}}\\ e^{jkx_{2}}&e^{-jkx_{2}}\\ \vdots&\vdots\\ e^{jkx_{N}}&e^{-jkx_{N}}\end{array} \right]\left[ \begin{array}{c}p_{i}(f)\\ p_{r}(f)\end{array} \right]. \tag{4}\]  
Or, in matrix form  
\[\mathbf{V}_{mic}=M_{1}\mathbf{J}\mathbf{A}\mathbf{P}, \tag{5}\]  
where \(\mathbf{V}_{mic}\) is the output vector of the microphone array, \(\mathbf{J}\) is the matrix of compensation functions, \(\mathbf{A}\) is the propagation matrix determined by the measurement positions, and \(\mathbf{P}\) is the vector containing the unknown incident and reflected sound pressures in the impedance tube. Then, for \(N\geq 3\), the least-squares solution of Equation (5) can be obtained via pseudo-inversion  
\[\widetilde{\mathbf{P}}=\left(\mathbf{A}^{H}\mathbf{A}\right)^{-1}\mathbf{A}^ {H}(M_{1}\mathbf{J})^{-1}\mathbf{V}_{mic}, \tag{6}\]  
where \(\mathbf{A}^{H}\) is the Hermitian transpose matrix of \(\mathbf{A}\), and (\(\sim\)) represents the estimated value. With some algebraic manipulations, Equation (6) can be rewritten as  
\[\left[ \begin{array}{c}\widetilde{p}_{i}(f)\\ \widetilde{p}_{r}(f)\end{array} \right]=\frac{1}{M_{1}(f)}\left(\mathbf{A}^{H} \mathbf{A}\right)^{-1}\mathbf{A}^{H}\mathbf{J}^{-1}\mathbf{V}_{mic}. \tag{7}\]  
Figure 1: The microphone array method for measuring the acoustic reflectance in an impedance tube with a pinna simulator.