# A Unified Speech Enhancement System Based on Neural Beamforming With Parabolic Reflector
## 2 Materials and Methods
### Analysis of the Paraboloid with Acoustic Focus  
#### 2.2.1 Principles  
Sten [19] studied the acoustic properties of a paraboloid. Two geometrical characteristics of parabola are illustrated in Figure 1.  
These are essential for its application as an acoustic reflector.  
1. The angle between \(OP\) and the tangent on the point of contact \(P\) equals the angle between \(FP\) and the same tangent. In acoustics, any incident wave route in parallel with the central axis will always be focused into the same position, the focus;
2. For a given line at right angles to the axis, the sum of the lengths of \(OP\) and \(PF\) is a constant. This means that the sound waves which are parallel to the central axis at the same frequency will have the same phase when reflected to the focus.  
and \(a\) are the depth and the focal length of paraboloid. Reflector diameter \(d\) is a function of ratio \(l/a\). \(F\) represents the focus point, \(P\) represents the reflection point, and \(OP\) is a straight line parallel to the central axis.  
As a consequence, the sound pressure is amplified at the focus \(F\). According to [19], the amplification of sounds parallels to the axis is given by Equation (3):  
\[F_{p}=\left\{1+\left[4\pi\frac{a}{\lambda}\ln(1+\frac{l}{a})\right]^{2  
of paraboloid increases with the increase of the ratio \(l/a\). We can intuitively understand the tendency from Figure 2; that is, the amplification performance will increase, along with an increase in either the frequency or the depth of the paraboloid. In addition, as the depth of the parabola increases, the curve becomes smooth and gradually flattens. Therefore, the magnification performance cannot be improved by deepening the paraboloid.  
By means of the indirect boundary element method [20], a series of simulations were implemented in LMS Virtual.Lab software. The results show the convergence of a sound in a parabolic model, which is just the focus with the maximum sound pressure level.  
In order to verify the feasibility of the PR-based method, the results obtained from the experiment performed in a real anechoic chamber were compared with computer simulation results. According to the results presented in Figure 3, both the simulation and the experimental results exhibited a similar trend to theoretical rules in Equation (3). It is also noticeable that there were outliers in the measured data curve. This is because the actual paraboloid was made up of plastic rather than a rigid body (theoretically), which made sound waves partially penetrate the paraboloid. So, the ideal focusing could not be achieved.  
Experimental results verified the effectiveness of the above theories. Based on the corresponding results, the PR system was implemented, and the amplification function of PR was fitted with reference to Equation (3). The ratio was assigned with \(l/a=4\) as a key parameter of the PR system. Other structural parameters of the paraboloid were \(a=40\) mm, \(l=160\) mm, \(d=320\) mm.  
Figure 3: The results of validation.  
Figure 2: Theoretical amplification at the focus when the diameter is constant.