## 3 Proposed Frequency-Dependent Amplitude Panning for Stereophonic Image Enhancement
### Frequency-Dependent Amplitude Panning  
This subsection describes how the estimated direction in each frequency bin is used for stereophonic image enhancement. Figure 6 illustrates the concept of the process described in this subsection. As shown in Figure 6a, a sound source is located at an angle of \(\theta\). However, the close spacing between the stereo microphones could mean that it is perceived as being at a lesser angle--_i.e._,  
Figure 4: A-F planes and estimated direction for white noise in an anechoic room: (**a**) \(AF(k,\theta)\); (**b**) \(AF_{s}(k,\theta)\); (**c**) estimated direction using \(AF_{s}(k,\theta)\).  
\(\theta_{p}<<\theta\). Thus, we have to increase the perceived angle by applying frequency-dependent amplitude panning such that \(\theta_{0}\approx\theta>>\theta_{p}\).  
Many panning methods have been reported [19; 25; 26]. Among them, the stereophonic law of sines [19] has been popularly used to reproduce a source using two loudspeakers, and it is realized as:  
\[\frac{\text{sin}\theta}{\text{sin}\theta_{0}}=\frac{g_{L}-g_{R}}{g_{L}+g_{R}}, \tag{11}\]  
where \(\theta_{0}\) is the physical angle between stereo loudspeakers and \(\theta\) is the desired angle at which the sound source should be located in terms of perception. Thus, \(g_{L}\) and \(g_{R}\) become the respective scale factors that are multiplied with the sound source according to the desired angle, as:  
\[y_{L}(n)=g_{L}s(n), \tag{12}\]  
and  
\[y_{R}(n)=g_{R}s(n), \tag{13}\]  
where \(s(n)\) is the sound source, and \(y_{L}(n)\) and \(y_{R}(n)\) are respectively the panned signals of the left and right channel.  
In this paper, we extend the stereophonic law of sines so that it is applied in the frequency domain. For a given direction at the \(k\)-th frequency bin \(\hat{\theta}(k)\), as described in Section 3.2, the frequency-dependent scale factors, \(g_{L}(k)\) and \(g_{R}(k)\), are obtained using the following equation:  
\[\frac{\text{sin}(\hat{\theta}(k))}{\text{sin}\theta_{0}}=\frac{g_{L}(k)-g_{R} (k)}{g_{L}(k)+g_{R}(k)}, \tag{14}\]  
Figure 5: A-F planes and estimated direction for white noise in a reverberant room with \(RT_{60}\) = 230 ms: (a) \(AF(k,\theta)\); (b) \(AF_{s}(k,\theta)\); (c) estimated direction using \(AF_{s}(k,\theta)\).  
Figure 6: Illustrations of stereophonic image enhancement: (a) Original sound source; (b) perceived sound source without any enhancement technique; and (c) perceived sound source after applying the proposed method.  
where \(\theta_{0}\) is also the physical angle between stereo loudspeakers, as described in Equation (10). As in Equations (11) and (12), the scale factors to Equation (13) are multiplied to the \(k\)-th spectral magnitude of the sound source as:  
\[Y_{L}(k)=g_{L}(k)S(k), \tag{15}\]  
and  
\[Y_{R}(k)=g_{R}(k)S(k). \tag{16}\]  
Here, while \(S(k)\) should be separated from \(\mathbf{X}\) according to Equation (4), the spectral magnitude of the sound source is approximated as the mid signal of the recorded sound. That is \(S(k)\approx(X_{L}(k)+X_{R}(k))/2\). Finally, by applying an inverse STFT followed by the overlap-add method, the output signal with an enhanced stereophonic image is obtained.