## 2 Materials and Methods
### Measurements of Heat, Ventilation, and Air-Conditioning Devices  
The company ELCOM, a.s., Ostrava, Czech Republic [26] is focused on designing a wide range of testing platforms, that are implemented in customers' production plants, where a wide range of automotive components are produced. There are, among their many products, HVACs of various parameters and sizes. Of those specific products, each type is tested for acoustic features at the end of the production line inside the specific anechoic chamber, see Figure 1. We used their equipment to acquire a large number of HVAC sound recordings. The measurement was performed with the use of a Briel and Kjaer Type 4966-H-041 microphone, which is characterised by high measurement accuracy in a wide range of volumes (16.5-134 dB), frequencies (6.3-20,000 Hz) and temperatures (\(-\)20-150 \({}^{\circ}\)C). After the measurement, the sound recordings were analysed to depict the pure nature of HVAC sound. The main analysis was undertaken with the use of the discrete Fourier transform:  
\[X_{k}=\sum_{n=0}^{N-1}x_{n}\cdot e^{-\frac{2\pi}{N}kn}=\sum_{n=0}^{N-1}x_{n} \cdot\left[\cos\left(\frac{2\pi}{N}kn\right)-\mathrm{i}\cdot\sin\left(\frac{2 \pi}{N}kn\right)\right], \tag{1}\]  
Figure 1: Testing of the heat, ventilation, and air-conditioning device (HVAC) inside the anechoic chamber, situated in the manufacturing plant of ELCOM, a.s.  
That converts input signal \(x\) from time to frequency domain \(X\)[27]. The length of both sequences is \(N\) and \(k=0,\ 1,\ldots,\ N-1\). [PERSON]'s theorem states that any finite sound signal is composed of a set of single-frequency sound waves; namely, the harmonic functions, sine and cosine [28]. The Fourier transform is used to decompose a signal into its individual frequencies, as well as the amplitudes of those frequencies. The result of the Fourier operation is called the spectrum [29]. As can be seen in Figure 2, the achieved results for HVAC show a recorded sample converted to a spectrum. After analysing thousands of available HVAC sound recordings, we can describe the pure nature of HVAC as a composition of elementary harmonic functions. Each of these elementary sine functions is defined as the n-th harmonic function related to frequency of HVAC system's rotation \(f\) and has its origin in a different component or sub-part of the HVAC [30; 31]. Figure 2b shows four specific peaks that correspond to four sources of noise. These are the only four sources detected from the results of the thousands of diagnosed sound recordings of the HVAC system: the shaft imbalance of the fan, the unequal electromagnetic field created by each coil of the electric motor, which causes a motor failure, defective bearings, and a defective fan blade. A detailed list of established functions with their presumed sources, that correspond to the described failures with their frequencies, is shown in Table 1. The next step is to compare the amplitudes of peaks with limit values which will show us if it can be qualified as a failure or not. These signals make up the base for creating synthetical HVAC sound expressions. For generating such data, we need to define other features of these signals. Since the audibility of these signals is dependent on the factory background noise, we first need to measure \begin{table}
\begin{tabular}{c l} \hline \hline
**Frequency** & **Estimated Source** \\ \hline \(2f\) & Shaft imbalance. \\ \hline \(8f\) & Electric motor failure. Number of coils, that form the \\  & electromagnetic field, is 8. \\ \hline \(13f\) & Defective bearings. Number of balls in the bearing is 13. \\ \hline \(41f\) & Vertilator fault. Number of fan blades is 41. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The list of elementary functions that describe the pure nature of HVAC along with their estimated sources. Figure 2: A sample of recorded HVAC sound (**a**) and its conversion into spectrum with the help of Fourier transform with pointed peaks (**b**).