# Measurement of Head-Related Transfer Functions: A Review
## 4 Multi-Loudspeaker-Based Fast HRTF Measurement Methods
### Continuous Measurements
#### 4.2.2 Time-Varying Adaptive Filtering  
[PERSON] [80] introduced an adaptive filtering method for continuously recording HRTFs on the horizontal plane with a fixed loudspeaker. The measurement setup can be assumed as \(S_{E}\) in Figure 9. A subject equipped with a pair of in-ear microphones rotates continuously at a constant speed, while a loudspeaker plays back excitation signals at a fixed position. Assuming that the rotating HRIR is a time-varying linear system, the recorded ear signal \(y(n)\) at the discrete time \(n\) can be described as (neglecting the subscripts denoting the left and right ears):  
\[y(n)=\mathbf{h}^{T}(\varphi_{n})\;\mathbf{x}(n)+v(n) \tag{23}\]  
where \(\varphi_{n}\) is an azimuth angle at the discrete time \(n\), and \(v(n)\) describes the measurement noise. \(\mathbf{h}(\varphi_{n})\) represents the HRIR for the azimuth angle of \(\varphi_{n}\) in a vector form. \(\mathbf{x}(n)\) is an input vector consisting of the most recent N samples of the excitation signal, where N is usually the same as the HRIR length. In this model, each time index corresponds to an azimuth angle of \(\varphi_{n}\). It should be noted that this model is valid under the assumption that the time of system changes is larger than the HRIR length [80]. By applying the NLMS recursive equation, the HRIR at the discrete time \(n+1\) can be predicted as:  
\[ \begin{split} e(n)&=y(n)-\mathbf{h}_{\text{est}}^{T}( \varphi_{n})\;\mathbf{x}(n),\\ \mathbf{h}_{\text{est}}(\varphi_{n+1})&=\mathbf{h}_ {\text{est}}(\varphi_{n})+\mu\;\frac{\mathbf{x}(n)}{||\mathbf{x}(n)||_{2}^{2} +\epsilon}\;e(n),\end{split}  \tag{24}\]  
where \(e(n)\) represents the residual error between the estimated and the recorded ear signals at the discrete time \(n\). This adaptation process can alternatively be implemented in the frequency domain [163]. [PERSON] [162] further extended the 1D HRTF estimation approach to measure HRTFs on both horizontal and elevation planes with multi-loudspeaker setups. Different from the method used in [161], the loudspeakers on the elevation plane can simultaneously play back excitation signals, while the subject continuously rotates. The measurement signals reproduced via loudspeakers must be independent of each other, which is an important condition for the simultaneous and unique identification of HRIRs from various directions. At the discrete time \(n+1\), the HRIRs for each elevation plane can be expressed as:  
\[ \begin{split} e(n)&=y(n)-\sum_{\theta_{v}}\; \mathbf{h}_{\text{est},\theta_{v}}^{T}(\varphi_{n})\;\mathbf{x}_{\theta_{v}}(n ),\\ \mathbf{h}_{\text{est},\theta_{v}}(\varphi_{n+1})&= \mathbf{h}_{\text{est},\theta_{v}}(\varphi_{n})+\mu\;\frac{\mathbf{x}_{\theta_ {v}}(n)}{\sum_{\theta_{v}}||\mathbf{x}_{\theta_{v}}(n)||_{2}^{2}+\epsilon}\;e (n),\end{split}  \tag{25}\]  
where the subscript \(\theta_{v}\) denotes different loudspeaker positions corresponding to discrete elevation angles of HRIRs. The excitation signal from each elevation angle (loudspeaker direction) is normalized with a common summing term, \(\sum_{\theta_{v}}||\mathbf{x}_{\theta_{v}}(n)||_{2}^{2}\), and the same residual error is applied to update HRIRs for different elevation angles [162]. [PERSON] et al. [36] verified the system with different loudspeaker numbers and rotational speeds. The measurement accuracy was represented by the error signal attenuation (ESA) with ESA = 10 \(\log_{10}(\sigma_{e}^{2}/\sigma_{y}^{2})\), where \(\sigma_{e}^{2}\) and \(\sigma_{y}^{2}\) denote the variance of error and recorded signals, respectively [36]. In that study, the white noise was used as the excitation signal for the measurement. Overall, the ESA reduces (measurement accuracy increases) with the decreasing rotational speed and number of loudspeakers. For the revolution time of larger than 60 s, the measurement accuracy is almost constant with the increasing rotational speed, and the dependence between the measurement accuracy and the loudspeaker numbers becomes small [36].