## 3 Methods Used for Identification of Tonal Noise Sources  
### Tonal Metrics  
The existence of tonality refers to the presence of a single dominant frequency or pitch in a sound. This dominant tone is usually caused by a single, constant source of vibration, such as a rotating machine that contains a single frequency component due to unbalance. Thus, when the sound produced by this vibration is analysed, a relatively strong and clearly defined tone is apparent. Tonality can be a useful indicator of certain types of problems in mechanical systems. In a centrifugal compressor, for example, tonality in the noise or vibration spectrum may indicate the presence of an unbalanced blade or entire rotor. Similarly, the tonality of the sound generated by a bearing may indicate the presence of a major malfunction. Understanding the tonality of sound can be helpful in diagnosing problems and developing strategies to mitigate or eliminate unwanted noise and vibration. Tonal sound is perceived by humans as unpleasant. Tonality is a psychoacoustic indicator that identifies emerging tones above ambient background noise and determines their contribution to the perception of sound. The more energy there is around a frequency, the less perceptible that frequency is, and, thus, the less tonal that sound is. The presence of tonal sound is therefore significant. In practice, when measuring compressor sound, there are constant small fluctuations in the measured quantities, even under steady-state conditions. For example, compressor speed can fluctuate by tens of revolutions per minute. Thus, periodic averaging must be used for identification. This averaging uses the knowledge of the actual rotor speed according to the speed mark. The idea is to select a short period of time when the speed variation is insignificant and to analyse this period separately. After \(N\) time periods, the entire measured signal is processed. The sound pressure amplitude of the \(N\) time steps is then determined as follows:  
\[p(f)=\mathcal{F}(p(t)), \tag{6}\]  
where \(p(t)\) is the measured sound pressure, \(t\) is the time defined at discrete points \(t=t_{j}\dots t_{j}+\Delta T\) corresponding to the sampling frequency, \(\Delta T\) is a short time interval typically lasting several hundred rotor revolutions and \(\mathcal{F}\) is the discrete Fourier transform operator converting the time signal into the frequency domain. According to Equation (6), the amplitude of the sound pressure in the frequency domain is determined as a function of frequency for each time sub-interval. In practical cases, when a compressor is measured for a long period under steady-state operating conditions, there are constant small fluctuations in the rotor speed and, therefore, in the acoustic parameters. Thus, the analysed frequency component associated with a particular aerodynamic phenomenon occurring in the compressor might continuously vary slightly within the time frame. For this reason, the amplitude-frequency characteristic \(p(f)\) is converted into a dependence of the pressure amplitude on the frequency ratio \(p(\varepsilon_{R})\). However, due to the variability in speed, the frequency ratio must be slightly corrected according to the actual known speed \(f_{R}\). It is worth noting that in rotordynamics there is a group of well-established algorithms called order tracking which allow the measured signal to be transformed from the time domain to the angular domain with respect to the actual shaft speed. These include computed order tracking, Vold-Kalman filters and order tracking transforms. The use of order tracking algorithms is certainly possible, but given the objective of this study, which is mainly to detect dominant tones whose level may be affected to some extent by the impedance of the environment, the procedure used here is fully justified. The tonal metrics for identifying the tonal components of noise include an algorithm that can search for tones in the frequency spectrum of the measured sound pressure.