# Measurement of Head-Related Transfer Functions: A Review
## 6 Post-Processing of Measured HRTFs
### Equalization  
The spectral characteristics of the measurement apparatus are included in the measured HRTFs, and should therefore be removed. This can be achieved either with the free-field or the diffuse-field equalization method [2; 35; 177; 178; 35]. The basic idea is to divide measured HRTFs (\(H(\varphi,\theta,r,f)\)) by a reference transfer function (\(H_{ref}(f)\)), which is expressed as (neglecting the subscripts denoting the left and right ears):  
\[H_{eq}(\varphi,\theta,r,f)=\frac{H(\varphi,\theta,r,f)}{H_{ref}(f)}, \tag{27}\]  
where \(\varphi,\theta\) and \(r\) represent the azimuth, elevation, and distance, respectively. For the free-field equalization, [PERSON] [3] and [PERSON] et al. [177] divided each HRTF by a reference HRTF measured at a specific direction (typically is chosen as the frontal direction: \(0^{\circ}\) azimuth, \(0^{\circ}\) elevation) in the same ear (\(H_{ref}(f)=H(0^{\circ},0^{\circ},r,f)\)). According to the definition of free-field HRTFs in [1], the HRTF can be represented as the ratio of the transfer function from the sound source to the blocked eardrum and the transfer function from the sound source to the head center position without the subject being present. It can be seen that the compensation process is already included in this definition. An extra measurement should be conducted to obtain \(H_{\textit{ref}}(f)\), i.e., measuring reference acoustic transfer functions between the loudspeaker and in-ear microphones by placing the microphones at the head center position without the subject being present (sometimes called measurement equalization) [77]. Alternatively, the loudspeaker and in-ear microphone transfer functions can be measured separately with precision calibration instruments, and the HRTFs are equalized through dividing the raw HRTFs by these measured transfer functions [4]. In the case of the diffuse-field equalization, the reference transfer function is represented by the root-mean-square (RMS) of measured HRTFs across all M directions, i.e., \(H_{\textit{ref}}(f)=\sqrt{\frac{1}{M}\sum_{i=1}^{\text{M}}|H(\varphi_{i}, \theta_{i},r,f)|^{2}}\). Therefore, a lot of data points from different directions are required to calculate the reference transfer function. The diffuse-field equalization attempts not only to remove the influences of measurement systems, but also commonalities among a set of measurements, i.e., direction-independent parts. Hence, diffuse-field equalized HRTFs are also called directional transfer functions (DTFs) [8]. The division of two transfer functions in the frequency domain is actually a deconvolution process, which can be done either in the frequency or in the time domain by approximating the inverse filter of the denominator (see Section 2.1). To avoid instabilities caused by inverting \(H_{\textit{ref}}(f)\), a frequency-dependent regularization can be considered [39]. Alternatively, assuming that the main distortion in measured HRTFs is caused by the magnitude characteristics of the measurement apparatus, the minimum-phase representation of the \(H_{\textit{ref}}(f)\) can be used for the equalization to avoid the causality and instability problems by building its inverse function [179]. After the deconvolution process, an appropriate delay should be added to the equalized HRTF. Of course, minor phase distortions caused by the measurement system are retained [35]. One of the important applications of the HRTFs is the synthesis of binaural signals for headphone reproduction. For binaural rendering purposes, an ideal headphone should have a headphone-to-ear transfer function (HpTF) with a flat magnitude spectrum and a linear phase [180]. However, headphones are usually designed to meet a target response (reference sound field, e.g., free- or diffuse-field) depending on the design concept in each company. [PERSON] et al. [181] studied transfer characteristics of 14 headphones on 40 human subjects and pointed out that non of them had a flat frequency response.