## 2 Proposed Speech Enhancement Methods for LDVs
### Sft-Based Speech Enhancement  
Figure 2 shows the power spectrum of clean speech (a), observed speech (b), amplitude response of the observed object (c), and phase difference between the observed speech and clean speech (d). As shown in Figure 2c, the amplitude response tends to decrease at high frequencies. From the left figures of Figure 2a,b, when the amplitude of the speech component is sufficiently smaller than the noise, the speech component in the frequency band of 4-8 kHz becomes unobservable. As shown in the right figures of Figure 2a,b, the phase spectrum has a complicated structure, so it is difficult to directly learn the mapping between the phase spectrum of observed speech and clean speech. As shown in Figure 2d, the phase difference between the observed speech and clean speech is strongly related to the amplitude of the observed speech. Therefore, with the STFT-based method, the power spectrum of enhanced speech is first estimated from the power spectrum of the observed speech, then the phase difference between the enhanced speech and observed speech is estimated using the power spectrum of the observed speech. Most speech components above 4 kHz are unvoiced, and the human ear is not sensitive to the phase of such unvoiced components, similar to white noise. Therefore, to increase the phase-restoration accuracy of the voiced components, only the 0-4 kHz frequency band is processed during phase restoration. For frequencies above 4 kHz, the phase of the observed speech is applied. Figure 3 shows the processing procedure of the STFT-based method. Both the power and phase spectra are processed using DNNs. In the learning stage, the logarithmic power spectrum \(|\mathbf{X}^{\text{LPS}}|\in\mathbb{R}^{k\times m}\), \(|\mathbf{X}^{\text{LPS}}|\in\mathbb{R}^{k\times m}\) (\(k,m\) are frequency and frame indices, respectively) and the phase spectrum \(\mathbf{\varphi}_{x}\in\mathbb{R}^{k\times m}\), \(\mathbf{\varphi}_{y}\in\mathbb{R}^{k\times m}\) are first extracted from clean speech \(\mathbf{x}\) and observed speech \(\mathbf{y}\) by Fourier transform. In the DNN for amplitude-spectrum reconstruction, the input and target of the network are \(|\mathbf{Y}^{\text{LPS}}|\) and \(|\mathbf{X}^{\text{LPS}}|\), respectively, and optimized by minimizing the mean squared error (MSE) between them. The structure of this DNN consists of two long short-term memory (LSTM) layers and three fully-connected layers, as shown in Figure 4. The activation function uses a rectified linear unit (ReLU). The input of the DNN for phase-spectrum estimation is \(|\mathbf{Y}^{\text{LPS}}|\) and the target is the phase difference \(\mathbf{\phi}^{\text{PD}}\in\mathbb{R}^{k\times m}\) obtained from \(\mathbf{\varphi}_{x}\) and \(\mathbf{\phi}_{y}\). Since the phase spectrum has a period of 2\(\pi\), the loss function used for learning is defined using Equation (1), i.e., the sum of the cosine distances between the input phase difference and target phase difference for each frame at a low-frequency band of 0-4 kHz. \[\text{Loss}_{\text{PD}}(m)=\sum_{k=0}^{\frac{\xi}{2}+1}(1-\cos(\mathbf{\phi}^{ \text{PD}}(k,m)-\mathbf{\hat{\phi}}^{\text{PD}}(k,m))) \tag{1}\]  
Figure 1: Power spectrum of observed speech acquired from different irradiated objects, i.e., (**a**) clean speech, (**b**) PET bottle, (**c**) cardboard box, (**d**) printing paper, (**e**) aluminum sheet, and (**f**) plastic plate. All objects were placed about 0.1 m from the sound source, and the output sound pressure was 85 dB(A). As shown in Figure 5, the structure of the DNN for phase difference estimation consists of five convolutional layers. The activation function uses gated linear units (GLU) [18]. Both DNNs used Adam [19] to calculate the gradient.