# Data Augmentation Method Using Room Transfer Function for Monitoring of Domestic Activities
## 3 Proposed Data Augmentation Method
### Room Transfer Function  
Under room conditions, when the sound source reaches the receiver, it arrives as a combination of direct sound and reflected sound from the perspective of geometrical acoustics [18], as shown in Figure 4a. The direct sound first arrives at the receiver, and its combination with the reflected sound that arrives later produces an impulse response known as the RIR, as shown in Figure 4b. This RIR varies according to the acoustic properties of the room, which are determined by factors such as the size of the room and the materials comprising the walls, floor, and ceiling.  
The variations between a sound source and its specific spatial location can be described by equations in the frequency domain. As demonstrated in (1), the received sound signal \(y(t)\) at a target position can be expressed with the convolution of sound source \(x(t)\) and transfer function \(h(t)\), where \(t\) is the time:  
\[y(t)=x(t)*h(t) \tag{1}\]  
Figure 3: Overall training process.  
Figure 2: Feature extraction process.  
Here, \(h(t)\) denotes the RTFs. When transforming (1) into the frequency domain through the Fourier transform, the convolution operation between the sound source and the transfer function becomes a multiplication, as expressed in (2).  
\[Y(f)=X(f)\times H(f) \tag{2}\]  
The transfer function \(H(f)\) can be expressed as the sum of the real part \(H_{re}(f)\) and imaginary part \(jH_{im}(f)\), as in (3) after the Fourier Transform. The absolute value of this \(H(f)\) is as in (4). We can assume that \(H_{re}(f)\) and \(H_{im}(f)\) follow a Gaussian distribution according to the central limit theorem [23], meaning \(|H(f)|\) is the sum of the square root of the two Gaussian distributions and has a Rayleigh distribution.  
\[H(f)=H_{re}(f)+jH_{im}(f) \tag{3}\]  
\[|H(f)|=\sqrt{\left(H_{re}(f)\right)^{2}+\left(jH_{im}(f)\right)^{2}} \tag{4}\]  
Suppose we have \(y_{1}(t)\) and \(y_{2}(t)\) with the same sound source, although with another transfer function. In other words, although \(y_{1}(t)\) and \(y_{2}(t)\) have the same sound source, they have different transfer functions. In the frequency domain, it can be expressed by (5) and (6), which can be observed in Figure 5.  
\[Y_{1}(f)=X(f)\times H_{1}(f) \tag{5}\]  
\[Y_{2}(f)=X(f)\times H_{2}(f) \tag{6}\]  
Here, if \(Y_{1}(f)\) is divided by \(Y_{2}(f)\), we can observe that multiplying \(Y_{2}(f)\) by the ratio of the two transfer functions as in (7) becomes \(Y_{1}(f)\).  
\[Y_{2}(f)=Y_{1}(f)\times\left(H_{2}(f)/H_{1}(f)\right) \tag{7}\]  
Typically, the magnitudes of frequency characteristics (e.g., spectrum, Mel-frequency spectrum, constant-Q transform) are used as input features for MDA models. Therefore we only use the magnitude and take the absolute value as shown in (8).  
\[|Y_{2}(f)|=|Y_{1}(f)|\times|H_{2}(f)/H_{1}(f)| \tag{8}\]  
To obtain \(|Y_{2}(f)|\) in (8), \(|H_{2}(f)|\) must be known. In general, \(|H_{2}(f)|\), which is the transfer function of the unknown signal, cannot be known exactly. However, the statistical properties of the transfer function \(|H_{2}(f)|\) can be used, which is described in the next section.  
Figure 4: RIRs in the geometric acoustics. The direct sound and reflected sound in (**a**) arrive at the receiver with a time delay, as shown in (**b**).