Acoustical source separation and identification using principal component analysis and correlation analysis

2.1. Fundamental theories

The inner production of random vectors $\mathbf{x}\left(k\right)$ and $\mathbf{v}$ is defined as the projection of random vector $\mathbf{x}$ with zero-mean and $m$ dimensions on the unit vector $\mathbf{v}$ with $m$ dimensions:

The projection vector $\mathbf{y}$ is also a random vector, and it has zero mean and variance ${\sigma }^2$:

where ${\mathbf{R}}_{\mathbf{X}\mathbf{X}}=E\left\{\mathbf{x}\right(k\left){\mathbf{x}}^T\right(k\left)\right\}$ is the covariance matrix of $\mathbf{x}$.

From Eq. (2), the variance of $\mathbf{y}$ is a function of unit vector $\mathbf{v}$:

If the unit vector $\mathbf{v}$ makes the variance function $\varphi \left(\mathbf{v}\right)$ extreme, there will have:

where $\delta \mathbf{v}$ is a small disturbance of $\mathbf{v}$:

Combining Eq. (3) with Eq. (5), and ignoring the third part, there has:

From Eq. (4), $\varphi \left(\mathbf{v}\right)$ can be replaced by $\varphi (\mathbf{v}+\delta \mathbf{v})$:

From the theory, any disturbance of $\delta \mathbf{v}$ is not allowed. But in real calculation, small disturbance of $\delta \mathbf{v}$ is allowed in the condition that:

Considering Eq. (2), the disturbance vector $\delta \mathbf{v}$ must be:

Therefore, the disturbance vector $\delta \mathbf{v}$ must be orthogonal with the vector $\mathbf{v}$, which means the disturbance on the vertical direction is allowed.

To make vectors have the same scale, a scale factor $\lambda$ is used. Thus there has:

The sufficient and necessary condition of Eq. (11) is:

From the basic theories, it can be concluded that the eigenvector ${\mathbf{v}}_i$ of the covariance matrix ${\mathrm{R}}_{\mathbf{X}\mathbf{X}}$ for the random vector $\mathbf{x}$ with zero-mean represents the principal direction, and the variance function $\varphi \left({\mathbf{v}}_i\right)$ has the extreme value in this principal direction. Furthermore, the scale factor or eigenvalue ${\lambda }_i$ in this direction is the extreme value of variance function $\varphi \left({\mathbf{v}}_i\right)$.

2.2. Principal component analysis

As the unit vector $\mathbf{v}$ has $m$ possible solutions, the random vector $\mathbf{x}$ will have $m$ possible projections. Specially, ${\mathbf{y}}_i$ is the projection of $\mathbf{x}$ on the unit vector ${\mathbf{v}}_j$:

The vector ${\mathbf{y}}_i$ that satisfying Eq. (13) is defined as a principal component of the random vector $\mathbf{x}$. All the principal components can be expressed as:

The reconstructed vector $\mathbf{x}$ thus can be denoted as:

2.3. Correlation analysis

For mechanical systems, some information of the sources normally can be known priori by the theory study or instructions. Therefore, correlation analysis can be used to evaluate the separation performances of PCA algorithm and identify the sources. For discrete signals $\mathbf{s}$ and $\mathbf{y}$, the correlation coefficient $\rho$ is defined as:

where:

and $T$ is the data length.

4.1. Introductions of the test bed

A test bed with shell structures is constructed to test the separation performance of PCA, which composes of four components: an end cover, a shell, two clapboards and supports. Rubber air springs are used to reduce the effects of the ground vibrations and environmental noises. Three acoustical sources are: two loudspeakers and one motor. The structures of the test bed is shown in Fig. 5.

Fig. 5The structures of the test bed: a) End cover; b) Loudspeaker I; c) Left clapboard; d) Loudspeaker II; e) Shell; f) Motor; g) Right clapboard; h) Rubber springs; i) Supports

Six sound pressure sensors are used to measure the acoustical signals, and they are located in six directions of the test bed with a distance of 500 millimeters. HBM Gen2i data acquisition system is applied to collect the acoustical data from these six sensors. The framework of the measuring system is shown in Fig. 6, and the testing parameters are shown in Table 1.

Fig. 6The framework of the measuring system

4.2. Acoustical signals of the test bed

All the mixed signals are measured as all the sources are working together, and the acoustical signals from sensor 1, 3 and 5 are used as the mixed signals so as to satisfy the certainty solutions of PCA: the number of the mixed signals should be no less than the number of the source signals. Furthermore, the directions of these sensors represent a diversified mixing mode of all the sources.

Fig. 7Waveforms of the mixed signals

Fig. 8Spectrums of the mixed signals

The waveforms and spectrums of the mixed signals are shown in Fig. 7 and Fig. 8 respectively. From waveforms in Fig. 7, it is difficult to identify the waveform features of the source signals except some periodic and sine waves, which indicate that the waveforms of the mixed signals are complicated, and normally signal processing method is required to extract the desired features. From spectrums in Fig. 8, some major components of 443, 682, 1126, 1331, 1500, 1950, and 2050 Hz are remarkable, which represent the characteristic features of the sources as the experimental settings combining with natural frequencies of the shell structure. Generally, the independent information of the source signals cannot be directly identified from the measured mixed signals as all the waveforms of source signals are coupling together.

4.3. Acoustical source separation

The PCA algorithm is applied to separate the mixed signals into unrelated components, and 3 independent components are extracted from the given mixed signals. To reduce the effects of noises, low pass filter is applied to remove the noisy components of the principal components ${\mathbf{y}}_2$and ${\mathbf{y}}_3$, the cut-off frequency of ${\mathbf{y}}_2$ and ${\mathbf{y}}_3$ are 1300 Hz and 1700 Hz, respectively. The waveforms and spectrums of the filtered principal components are shown in Fig. 9 and Fig. 10, respectively. Fig. 9 clearly shows that the waveform of the principal component ${\mathbf{y}}_1$ has typically oscillating features, and the basic components are sine waves with different amplitudes, which are normally caused by the eccentric vibration of mechanical systems. The spectrums in Fig. 10 also show that the principal component ${\mathbf{y}}_1$ has major components of 443, 682, 1126 and 1331 Hz. The waveform of the principal component ${\mathbf{y}}_2$ has typical features of sine waves with amplitude modulation, and its spectrum also clearly shows the characteristic frequencies of 1450, 1500 and 1600 Hz. The waveform of the principal component ${\mathbf{y}}_3$ has an obvious features of sine waves with amplitude modulation, and its spectrum contains three major components of 1950, 2000, and 2050 Hz.

Fig. 9Waveforms of principal components

Fig. 10Spectrums of principal components

Comparing the spectrums of the principal components with the parameters of the experimental settings, it can be speculated that the principal component ${\mathbf{y}}_1$ represents the typical feature of the source 1 from the motor, while the principal components ${\mathbf{y}}_2$ and ${\mathbf{y}}_3$ represent the typical features of the source 2 and 3 from the loudspeaker I and II, respectively. However, this is just based on the parameters of the experimental settings and it is still not convincing.

4.4. Acoustical source identification and validation

To intelligently identify the sources and validate the effectiveness of PCA in real mechanical systems, acoustical source signals are measured independently by the closest sensors in the condition that only one source is working with the given parameters. The independent source waveforms from sensor 1, 4 and 6 in the condition that only the motor, loudspeaker I, or loudspeaker II is working with the given experimental settings are shown in Fig. 11, and their spectrums are shown in Fig. 12.

Fig. 11Waveforms of the source signals

Fig. 12Spectrums of the source signals

Comparing the waveforms and spectrums of the source signals with that of the filtered principal components, the waveforms of the filtered principal components are similar to that of the related source signals. However, due to the time delay and effects of the mixing mode and structural transmission, the waveforms of the principal components still cannot exactly recover the complete waveform information of the related source signals. Therefore, spectrum analysis is used to further validate the effectiveness of the principal components. Comparing Fig. 10 with Fig. 12, it is clear that the principal component ${\mathbf{y}}_1$ recovers the major characteristic frequencies of 443, 682, 1126 and 1331 Hz contained in source signal 1; the principal component ${\mathbf{y}}_2$ recovers the major characteristic frequency of 1500 Hz contained in source signal 2, and also has two other components of 1450 and 1600 Hz; the principal component ${\mathbf{y}}_3$ recovers the major characteristic frequencies of 1950, 2000, 2050 Hz contained in source signal 3. Generally, all the major components of the source signals have been effectively separated by PCA with low pass filtering, and the major components of each sources can be traced by the spectrum analysis.

To quantitatively and intelligently identify the sources, the waveform correlation analysis is used to evaluate the similarity between the principal components and the sources. All the filtered principal components are made correlation analysis with all the sources, and the correlation coefficients are listed in the correlation coefficient matrix ${\mathrm{\Omega }}_3$:

The correlation matrix ${\mathrm{\Omega }}_3$ shows that the correlation coefficients between the principal components and the related sources are 0.72, 0.64 and 0.74, which indicate relatively high correlation coefficients and high similarity between the principal components and the related sources (Liu [23] obtained waveform correlation coefficients of 0.77±0.03 for ECG signals with noises, and Farila [24] obtained correlation coefficients of 0.70±0.09 for non-stationary surface myoelectric signals); while the correlated coefficients between the principal components and the unrelated sources are less than 0.07, which indicates that all the principal components are unrelated to each other. Therefore, a threshold $\gamma$ can be set as $\gamma \in \text{(0.07,}\text{0.64)}$ (in practice $\gamma \in \text{(0.45, 0.55)}$) to intelligently identify and trace the acoustical sources, and the wide range of the threshold $\gamma$ indicates that the acoustical sources can be well identified in real mechanical systems by PCA with low pass filtering.

After an effective source separation and tracing, the major features of the measured mixed signals can be extracted and traced, and then the related sources also can be traced and identified. With the source identification and tracing information, vibration and noise reduction, monitoring and control can be carried out. Furthermore, the pure source information from source separation can provide important features for machinery condition monitoring and fault diagnosis.