# A Review on Head-Related Transfer Function Generation for Spatial Audio
## 2 General Setting and Definitions
### Mathematical Definition of HRTF  
As we will see in Section 2.3 in more detail, the general setting for the definition of the HRTF includes a point source \(S\) and a scatterer, usually the head and torso of a human individual, which are both placed in an anechoic environment. A receiver \(R\) is placed at a point near the entrance of the ear channel of the person as shown in Figure 2. In order to give a precise definition of the HRTF, we need to define a fixed reference system \(Oxyz\) with respect to the head, where the origin \(O\) is a reference point conventionally identified with the centre of the head (for instance, the midpoint between the ears). Usually, the \(xy\) plane is the horizontal plane of the head, with \(x\) pointing in the frontal direction and \(y\) running through the ear channel from the right to the left ear, and \(z\) is such that \(Oxyz\) is right handed. Spherical coordinates \((r,\theta,\phi)\) associated with \(Oxyz\), with \(\theta\) the azimuth angle and \(\phi\) the elevation angle, are also frequently used.  
When a Dirac pulse \(\delta(t)\) is sent by the source \(S\), a signal \(h(t)\), i.e., the HRIR, is received at the receiver \(R\). Clearly, the HRIR is a function of the position of the point source. It is well known from the theory of linear, time-invariant (LTI) systems that, in a time-independent setting (for instance, the person must be in a static position), the HRIR contains all the information needed to reconstruct the signal received at \(R\) for every possible signal emitted by \(S\). As mentioned above, the information encoded in the HRIR is expressed in the  
Figure 2: Reference system \(Oxyz\) associated with the head and related spherical coordinates \((r,\theta,\phi)\). The position of a sound source \(S\) is described by the position vector \(\mathbf{r_{S}}\). The point \(R\) is the position of the receiver.  
frequency domain by the HRTF. The HRTF is usually normalised with sound pressure, which would be measured at the centre of the head \(O\) if the head were not present:  
\[H_{R}(\omega,r,\theta,\phi)=\frac{p_{R}(\omega,r,\theta,\phi)}{p_{O}(\omega,r, \theta,\phi)}, \tag{1}\]  
where \(p_{R}(\omega,r,\theta,\phi)\) and \(p_{O}(\omega,r,\theta,\phi)\) are the values of the pressure field, in the frequency domain, at the receiver \(R\) and at the centre of the head \(O\) (if the head were not present), respectively. The dependency of the HRTF from the position \((r,\theta,\phi)\) of the source is explicitly indicated. The definition (1) also explicitly shows that the HRTF \(H_{R}(\omega,r,\theta,\phi)\) depends on the position of the receiver \(R\). Thus, one usually has two HRTF : one for each ear. When the source is on the opposite side of the head from the ear, it is called the contralateral HRTF. When the source is on the same side of the ear, it is called ipsilateral HRTF. In [22], it is shown that it is possible to transform an ipsilateral HRTF into a contralateral HRTF using a simple model of a spherical head and introducing an error of approximately \(5^{\circ}\) in sound localisation.  
In experimental measurements, the normalisation of Equation (1) allows us to obtain an HRTF independent of the transfer functions of the devices used to reproduce and record the sound. In addition, normalisation allows the contribution of propagation in space from the source to the centre of the head to be separated from the HRTF.