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This paper investigates one eigenvalue decomposition-based source number estimation method, and three information-based source number estimation methods, namely the Akaike Information Criterion (AIC), Minimum Description Length (MDL) and Bayesian Information Criterion (BIC), and improves BIC as Improved BIC (IBIC) to make it more efficient and easier for calculation. The performances of the abovementioned source number estimation methods are studied comparatively with numerical case studies, which contain a linear superposition case and a both linear superposition and nonlinear modulation mixing case. A test bed with three sound sources is constructed to test the performances of these methods on mechanical systems, and source separation is carried out to validate the effectiveness of the experimental studies. This work can benefit model order selection, complexity analysis of a system, and applications of source separation to mechanical systems for condition monitoring and fault diagnosis purposes.

In many physical systems, the measured signals can be modeled as a superposition of a finite number of the sources with additive environmental noises, and many signal processing methods such as principal component analysis (PCA) [

In the past decades, many researchers have focused their interests on source number estimation methods and their engineering applications, and proposed many approaches to solve this problem. Ye

Unlike the eigenvalue decomposition-based source number estimation method which requires a threshold, information criteria-based methods do not need any parameters for adaptively estimating the number of sources from the mixed signals, and the algorithms are also easier for calculation and perform efficiently in the applications. The key issue on the information-based methods is to find the extremum values of the constructed objective functions based on information criteria, such as Akaike Information Criterion (AIC) [

Therefore, this paper studies comparatively the performances of both eigenvalue decomposition-based and information-based source number estimation methods on mechanical sound signals, and improves BIC as IBIC to make it easier for calculation and efficient for the data with a large sampling length. Both linear superposition and nonlinear modulation are considered in the numerical case studies, and a test-bed with three sources is constructed to test the performances of the eigenvalue decomposition-based and information-based methods on source number estimation for mechanical systems. This study can benefit for the model order selection, complexity analysis of a system, and applications of source separation to mechanical systems for condition monitoring and fault diagnosis purposes.

The remainder of this paper is organized as follows: in Section 2, we introduce the theoretical background and investigate the mathematical mechanisms of the eigenvalue decomposition-based source number estimation method, and information-based source number estimation methods entitled as AIC, MDL, and IBIC. In Section 3, we test the performances of these methods on typical mechanical signals with both a linear superposition and a nonlinear modulation. In Section 4, a test bed with three sound sources is constructed to further test the performances of these methods on real mechanical systems, and the effectiveness of the experimental studies is validated by source separation and spectral analysis. Finally, Section 5 summarizes the conclusions.

Consider _{1}(_{m}(^{T}_{1}(_{n}(^{T}_{1}(_{2}(_{m}(_{ij}}_{m×n}:

As the source signals and mixing mode are normally unknown for many physical systems, a crucial problem associated with this model is to estimate the number _{1}(_{m}(

Constitute _{1}), ⋯, _{N})] from an ^{T}_{1} ≥ λ_{2} ≥⋯ λ_{m}, L(n) which is a log-likelihood function used to estimate the maximum likelihood of source number

Source number estimation based on eigenvalue decomposition:

The benefit of eigenvalue decomposition is that the source number can be estimated just based on the distributions of eigenvalues, and the crucial step is just a reasonable threshold

Now we comparatively introduce and investigate another three information-based source number estimation methods which can determine the source number adaptively.

Akaike Information Criterion (AIC)

The information theoretic criterion for the model order selection or source number estimation, introduced by Akaike [

The first term, −2

Minimum Description Length (MDL) [

Inspired by Akaike's work, Rissanen [

Note that apart from a factor of 2, the first term is identical to the corresponding one in AIC, while the second term has an extra factor of ½ lg

Bayesian information criterion (BIC) [

Minka [_{J}_{j} except for

In practice, it causes overflow in calculating

In this section, we numerically generate typical signals of mechanical systems to comparatively study the effectiveness of the different source number estimation methods. These generated source signals consider the modulation effects of mechanical systems, and the mixed signals are composed of the sources through a linear superposition and a weak nonlinear mixing. The generating functions of the source signals are listed below:

In the numerical case study, _{1}(_{2}(_{3}(_{4}(

Since the number of the mixed signals should be no less than the number of the source signals for an accurate source separation or system identification, and the source number estimation methods based on the information criteria also require more mixed signals, in the numerical case study we provide six mixed signals composed by the given source signals with a linear superposition matrix

_{1} to _{5}, even _{5} = _{6} = 0, which means that there are 4 principal components contained in the mixed signals (from the definition of principal component analysis [

Therefore, it can be concluded that all the four source number estimation methods are effective for the given numerical case study, and the eigenvalue decomposition-based method and IBIC are more robust and reliable than AIC and MDL as they have very wide boundaries to accurately determine the source number. However, the eigenvalue decomposition-based method requires a reasonable threshold

In this section, a nonlinearity mixing factor

The nonlinearity mixing factor

The accuracy rates of the given three information-based source number estimation methods are displayed in

For the

Furthermore, the maximum values of IBIC for

_{5} is up to 0.9791 for the _{5} = 0.0002 causes AIC failure for the _{5} = 0.0003 causes MDL failure for the

Therefore, it can be concluded that the eigenvalue decomposition-based source number estimation method is difficult to carry out without any prior knowledge of the sources, while the information-based methods can adaptively and accurately estimate the source number for the linear superposition cases. However, for the cases with nonlinear modulation effects, IBIC performs more robustly and reliably than AIC and MDL, which reveals more wide engineering applications of IBIC.

In general, it is difficult to directly measure the source signals in most mechanical systems due to the limited accessibility, and thus signal processing is often required to separate and recover the source information from the mixed signals normally measured by remote sensors. Then, these separated source signals can be used for further purposes such as a condition monitoring and a fault diagnosis of mechanical systems. However, a source number estimation from the measured and mixed signals should be carried out for a prior knowledge to source separation or complexity analysis of the systems. In this section, we apply the source number estimation methods mentioned above to a mechanical system shown in

Aiming at vibration and noise source number estimation for mechanical systems, this study designs a test bed based on a shell structure, which is composed by an end cover, a shell, clapboards, and supports. The whole test bed is supported by four rubber air springs, which can reduce the influences of environmental noises. There are three sound sources: two of them are loudspeakers controlled by the signal generators, and the other one is a motor controlled by the frequency converter. The structure and photo of the test bed are shown in

Six sound pressure sensors are used to measure the sound information, and they are installed in different directions of the test bed with a distance of 0.5 m. A HBM Gen2i data acquisition system is applied to collect the sound data from these six sensors. The framework of the measuring system is shown in

The sound source signals are measured with just one source working at the parameters given in

Both eigenvalue decomposition-based source number estimation method and information-based source number estimation methods are applied to estimate the sound source number of the given test bed. The eigenvalues of the covariance matrix for the mixed signals in different parameters are shown in _{4}, _{5} and _{6} are very close to each other.

The results of source number estimation by information-based methods are shown in

The fast ICA algorithm [

The spectra of the source signals and the separated components are displayed in

In general, like the similar values of the eigenvalues, it is very difficult for the eigenvalue decomposition-based method to accurately and robustly estimate the source number without any prior information about the sources. However, all the information-based methods correctly and adaptively estimate the source number of the test bed, which further reveals that the mixing mode of the sound sources tends to be a linear superposition, and thus guarantees that the information-based methods are effective to the sound signals.

This paper investigates both eigenvalue decomposition-based and information-based source number estimation methods, and improves BIC as IBIC to make it more efficient and easier to calculate. Furthermore, their performances on nonlinear modulation effects of mechanical systems are studied comparatively with numerical case studies and experimental studies.

In the numerical case study with a linear superposition case, the eigenvalue decomposition-based method has a wide band to determine the threshold

Generally, IBIC performs more robustly and reliably toward the nonlinear modulation effects than AIC and MDL, while eigenvalue decomposition-based methods normally require prior information about the sources, and becomes confused when the eigenvalues are very close to each other. Furthermore, the results of information-based methods for the test bed also indicate that the mixing mode of the sound sources tends to be a linear superposition. This study can benefit for model order selection, complexity analysis of a system, and applications of source separation to mechanical systems for condition monitoring and fault diagnosis purposes.

This work is supported by the projects of National Nature Science Foundation of China (No. 51305329, 51035007), the China Postdoctoral Science Foundation (No. 2013M532032), the Doctoral Foundation of Education Ministry of China (No. 20130201120040), and the Shaanxi Postdoctoral Scientific Research Project.

The authors declare no conflicts of interest.

The waveforms of the source signals.

The waveforms and spectra of the mixed signals.

Source number estimation by information-based methods.

Accuracy rates of information-based source number estimation methods.

The waveforms of the mixed signals for

Source number estimation by AIC.

Source number estimation by MDL.

Source number estimation by IBIC.

The structure (I) and photo (II) of the test-bed: (

The measuring system of the test bed.

Waveforms of the source signals.

Waveforms of the mixed signals.

Source number estimation by information-based methods.

The waveforms of the separated components by fast ICA algorithm.

The spectra of the source signals.

The spectra of the separated components.

The eigenvalues of the covariance matrix for the mixed signals.

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} | |
---|---|---|---|---|---|---|

Values | 281.31 | 31.21 | 10.43 | 4.42 | 0.00 | 0.00 |

Eigenvalues of covariance matrix with different factor

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} | |
---|---|---|---|---|---|---|

281.31 | 31.21 | 10.43 | 4.42 | 0.0000 | 0.0000 | |

289.56 | 31.25 | 10.48 | 4.37 | 0.0002 | 0.0000 | |

271.56 | 31.27 | 10.41 | 4.37 | 0.0003 | 0.0001 | |

280.06 | 31.63 | 12.66 | 5.26 | 0.9791 | 0.0206 |

The testing parameters of the measuring system.

Sound pressure sensors | 6 |

HBM Gen2i Data acquisition system | 1 |

Sampling frequency | 10,240 Hz |

Sampling length | 10 s |

Frequency of Loudspeaker I with sine wave | _{1} = 1,600 Hz |

Frequency of Loudspeaker II with triangle wave | _{2} = 3,000 Hz |

Rotational speed of motor | 900 r/min (_{3} = 15Hz) |

The eigenvalues of the covariance matrix for the mixed signals.

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} | |
---|---|---|---|---|---|---|

_{1} = 3000 _{2} = 1600 _{3} = 15 |
0.0306 | 0.0076 | 0.0047 | 0.0022 | 0.0010 | 0.0009 |

_{1} = 3000 _{2} = 1600 _{3} = 20 |
0.0386 | 0.0252 | 0.0097 | 0.0067 | 0.0049 | 0.0038 |

_{1} = 3000 _{2} = 1600 _{3} = 25 |
0.0450 | 0.0121 | 0.0116 | 0.0082 | 0.0061 | 0.0045 |