The separation of aircraft background acoustic blindness based on EEMD-ICA

3.1. Blind source separation algorithm based on independent component analysis

At present, the research of blind source separation algorithm based on independent component analysis is mainly focused on the following aspects: the determination of the probability density function of source signals, such as the accurate description of the PDF characteristics of Gauss, super Gauss and sub Gauss source signals. According to ICA’s hypothesis, each source signal is statistically independent, so each source signal s _i can be described by its PDF to p _i  (s _i ). It can be seen that the signal can be divided into Gaussian, Super Gaussian, and Sub-Gaussian signals. Many noises have Gaussian characteristics. Nature-based speech signals and certain music signals have super Gauss characteristics, while images in nature have sub-Gaussian characteristic. If the PDF of the prior source signal is known, ICA can achieve better signal separation. Conversely, if the characteristics of the PDF of each signal source are not known, measures should be taken to include p _i  (s _i ) learning during the learning process. So how to infinitely approach the PDF of the source signal or how to avoid the determination of the source signal PDF in learning is an important issue in ICA research [7]. According to the different definitions of the cost function and the different learning algorithms, various independent component analysis algorithms can be constructed [21–28]. The common ICA algorithm usually proposes a cost function with W is a variable, which is a function of measuring the independence of each component of the y (t) signal, and there are many methods for measuring independence, and thus many cost functions are formed. After the cost function is given, a proper learning algorithm can be used to optimize the cost functions, such as the gradient method and the quasi Newton method. However, whatever algorithm is used, the residual error of the steady state should be small and the convergence speed should be fast. Different cost optimization algorithms can be applied to the same cost function, and the same optimization algorithm can be applied to different cost functions. The statistical properties (such as consistency and robustness) of the ICA algorithm depend on the selection of the cost functions, and the algorithm characteristics (convergence speed and stability) depend on the selection of learning algorithms. Therefore, the characteristics of the ICA algorithm depend on the cost function and the learning algorithm. The study of cost function and learning algorithm has always been the central task of ICA researchers [8]. To illustrate the concept of stability, for simplicity, the case of one-dimensional signals is discussed y = wt, where x and y are random variables. In order to find that the cost function J(w) with the one-dimensional variable w is the dependent variable reaches the minimum value, the following iteration formula can be used to calculate: w ( t + 1 ) = w ( t ) + μ j ( w ( t ) )(1)

In the formula, μ is the step size, j (w) = - ∂J (w)/∂w. If w ⌢ is the solution sought, then there should be E [j (w)] = 0 (E [·] denotes averaging the various possible y-sets). w becomes a balance point for the above iteration calculations [9]. However, the balance point is not necessarily a stable equilibrium. Assuming J′ (w) = - ∂J² (w)/∂w², then only if -E [j (w) is greater than 0 near w ⌢ , w ⌢ is a stable equilibrium point. Figure 4(a, b) show examples of stable and unstable equilibrium points, respectively. For the former, when w deviates slightly from w ⌢ , the iterative calculation will return w to w ⌢ ; for the latter, when w is biased to the left of w ⌢ , the iterative calculation will make w deviate further from w ⌢ . Therefore, how to avoid the solution to the unstable equilibrium is also a place that the researchers pay great attention to [10].

The Infomax principle of information maximization proposed by some scholars can be simply described as: when the mutual information of the input and output ends of the network reaches the maximum, it is equivalent to the removal of redundant information between the components of the output terminal. Some scholars have further explained that: For a neural network with a linear link at one output end [21–28], when the information transmitted through the network reaches its maximum, the mutual information between the components of the output will reach a minimum [11]. Applying the Infomax principle to the blind source separation problem is equivalent to maximizing the combined entropy of the separated signals [12]. The vector form of the combined entropy of the separation signal y(t) is: H ( y ) = ∑ i = 1 n H ( y i ) - I ( y )(2)

In the formula, H (y _i ) is the edge entropy of the output components, and I (y) is their mutual information. Therefore, maximizing the joint entropy H (y) of the split signal means that the mutual information of the output component is minimized. Maximizing H (y) is not a direct target for ICA, but H (y) can minimize I (y) by means of statistical independence of each component of the network separation signal. The idea of algorithm is to get an intermediate vector y (t) = Wx (t) through linear transformation for each observation vector x (t) first. Then the vector u (t) is obtained through nonlinear transformation u _i  = g _i  (y _i ). Second problems and third problems in ISODATA algorithm are analyzed [13]. The initial selection method and K value determination of cluster centers are considered. Both the original ISODATA algorithm and the ISODATA algorithm using the standard function have a common feature that first select the K points as the initial cluster center, and then iterate on this basis [14]. It is difficult to get a stable clustering result from clustering randomly selected initial clustering centers. In order to solve this problem, the selection of clustering center is improved, the dependence of initial value selection in clustering algorithm is improved, and the stability of clustering results is improved, and the experimental results are given.

From the above discussion, it is can be seem that the stochastic gradient ∂J (W)/  - ∂W provides the cost function J (W) the fastest rising direction. The stochastic gradient is based on the assumption that the parameter space Q is based on the Euclidean space with the orthogonal coordinate system W, but the more common case is that the parameter space is the Riemann space. Euclidean space is only a special case of Riemann space. Can the stochastic gradient ∂J (W)/  - ∂W in the Riemann space still provide the fastest rising direction of the cost function J (W) ? This is a place worthy of study. Some scholars have studied this problem seriously and gave a reasonable explanation. In Euclidean space, the length of the increment dW of the coefficient vector can be expressed as: ‖ dW ‖ = ∑ i = 1 n ( dW ) i 2(3)

But if the coordinate system is not orthogonal, then the length of dW is expressed as: ‖ dW ‖ = ∑ i = 1 n ∑ i = 1 n g ij dW i dW j(4)

It can be seen that Euclidean space is a special case of Riemann space. Except for Euclidean space, the stochastic gradient can’t provide the fastest rising direction of cost function in Riemann space. Some scholars have also proved that natural gradients can solve the problem of stochastic gradients. In Riemann space, the fastest rising direction of function J (W) is provided by the following: G - 1 ( W ) ∂ J ( W ) ∂ W(5)

However, the solution of matrix G needs to know the concrete structure of the parameter space, which is difficult to realize in practice, and even if the problem is solved, formula (5) involves the inverse of the matrix to increase the complexity of the algorithm. Especially when the dimension of the parameter space is large, the matrix G inversion must have a large amount of computation. It is very difficult to implement, and it is inconvenient to be applied in practice. In order to solve this problem, some scholars have put forward a more concise and effective way to replace G^-1 (W) with W ^T W, so that the natural gradient algorithm becomes further: W ( t + 1 ) = W ( t ) - η ∂ J ( W ) ∂ J W T W(6)

The number of sample points is 500, where the function rem (u, v) represents the remainder of the variable u divided by v. Take random mixing matrix A = rand(size(sig,1)) and perform Monte Carlo analysis for 100 times. As shown in Table 1, the natural gradient algorithm has advantages in terms of stability, separation performance, and algorithm speed. OPEN IN VIEWER

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Fig.2

Comparison of evaluation results after normalization.

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Fig.3

adjacent judgment value.

4.1. Comparative study of separation performance

Selection algorithm Fast ICA, EFICA, joint algorithm, SANG, JADE. The test sample takes 10000 to 80000 and the step length is 10000. The separation performance of the algorithm is measured by two criteria: statistical standard and computation time [15]. It is found in simulation experiments that the separation effect will change with the increase of sample points for each signal. As shown in the figure. For the real aircraft cabin sound signal, as the sample signal has a continuous zero amplitude signal segment, the statistical SIR value will decrease in the low amplitude segment, and the SIR value increases accordingly when the amplitude increases. When the number of sampling points per observation signal is greater than 25000, the SIR value tends to be stable [16]. When the sample number is less than 25000, the SIR value fluctuates. Autopilot alarm S1 and machine speech reminder signal S3 are more regular. As the number of sampling points increases, the fluctuation becomes stable. S2 is a dense, regular continuous signal of APU fire alarm [17]. After the beginning of a SIR value fluctuation, as the amplitude of the signal becomes smaller, the value of SIR becomes smaller. The S4 pilot’s operation engine switch sound signal is relatively sparse, the amplitude of the longer period is close to zero, and the signal changes with time, and the SIR value increases accordingly when the instantaneous amplitude increases.

It can be seen from the diagram that, for real aircraft cabin sound signals, the SIR values tend to be stable after a certain number of sample points (such as 25000), and a satisfactory separation effect can be obtained, and the average SIR value is above 20d B. The combined algorithm has better separation effect for more dense non Gauss signals (such as Y1, Y2, Y3), and for the switching signal Y4 with more zero value, the separation effect has no other advantages. The natural gradient learning algorithm (SANG) with incomplete constraints of adaptive learning rate can still work very well for the source signal amplitude changing with time or at a certain time zero, so the SANG algorithm is more effective for the separation of Y4 such as signals.

The computer hardware is configured as: the operating system Windows XP, CPU P4 3.0G, memory 1GB. Through simulation experiments, the separation index PI and the average time consuming of each algorithm are gotten, as shown in Table 5.2. It can be seen that the joint algorithm and the improved natural gradient algorithm have better separation effect, and the joint algorithm is more general. Fast ICA algorithm and its optimized and extended EFICA algorithm and joint algorithm, SANG algorithm and JADE algorithm are studied for blind separation and processing of CVR non-speech background acoustic signals. Combining these typical algorithms, the blind separation of CVR background sound signals based on independent component analysis is realized by estimating the evaluation function of the separated signal [18]. The simulation shows that it is feasible to separate and study the background sound signals used by the independent component analysis in the CVR record, and the separation effect of the SANG algorithm and the joint algorithm is better. The SANG algorithm has a better separation effect on the amplitude of the source signal at a time zero or non-stationary signal which changes rapidly with time. The combined algorithm makes full use of the advantages of the two algorithms of EFICA and WASOBI, and has the overall optimal separation performance [19].

4.2. Comparative study of multidimensional observation signal separation simulation

The separation performance index and time consumption after the number of signals are changed are compared. The comparison between the PI value and the SIR value of the graph shows that the joint algorithm is optimal in terms of the average separation performance of multiple signals, followed by the EFICA algorithm [20]. From the SIR curve, it can be seen that each algorithm changes in varying degrees when the number of source signals changes. The separation performance becomes worse and the consumption time increases, as shown in the diagram.

As shown in Fig. 4, the sound insulation curve simulated by Simulink is similar to the sound insulation data curve obtained by the impedance tube sound insulation experiment. And the sound insulation curve shows that when the stiffness of the system dynamic vibration absorber connection spring steel piece is small, the effective frequency band in the low frequency region of 100Hz–500Hz is wider, the frequency of the peak of the sound insulation is lower, and the weight of the weight is different. It will cause the peak of the sound insulation to be different. It can be seen from the above figure that in the impedance tube sound insulation experiment, under the action of the random white noise emitted by the power amplifier, the dynamic vibration absorber sample devices with different spring system cooperation modes have significant noise isolation. The role (Fig. 5). The observation signal obtained by mixing the source signals in the figure is compared with the algorithm proposed by using a better performance of the current algorithm. These algorithms are: SANG algorithm, Fast ICA algorithm, EFICA algorithm, EEFICA algorithm, JADE algorithm, joint algorithm and the dynamic piecewise refinement online algorithm (DSF) proposed in this chapter. The segmentation criterion used in this experiment is: the absolute value of the amplitude of 300 consecutive sampling points is less than 2. That is, if the four source signals are all approximately zero (k value is set up for all I), this k is the breakpoint, and then get the k = 66168. In order to perform performance comparison under suitable sampling points, only the segment is taken as the experimental object. It is found that the separation effect will change with the increase of sample points. In particular, there is a continuous signal segment approaching zero amplitude, and the SIR value will decrease in the low amplitude segment. When the number of sampling points per observation signal is greater than 30000, the SIR value tends to be stable. This also provides a basis: the minimum length of a segment should not be lower than this value. At the same time, it is concluded that the performance of the improved DSF is improved by introducing the adaptive selection nonlinear function and the correction of the weight of the separation matrix on the basis of SANG. The separation performance of DSF is slightly better than that of the joint algorithm, EFICA and EEFICA, mainly because SANG is superior to Fast ICA in a large number of near zero values, but EFICA and EEFICA are improved on the basis of Fast ICA. Joint algorithm is a combination algorithm of EFICA and WASOBI. In general, the joint algorithm, EFICA and EEFICA are more stable and excellent algorithms, while the separation of SANG and DSF involves the setting of the learning step, which has a great impact on performance. But the online algorithm makes the separation process a great flexibility, can dynamically set the segmentation points, and a slight time variant mixing also has a very good separation effect. OPEN IN VIEWER

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Fig.5

Sound insulation curve diagram of three hole two piece wide spring block small mass dynamic vibration absorber device.