We focus on the decomposition problem for nonstationary multicomponent signals involving Big Data. We propose the kernel sparse learning (KSL), developed for the T-F reassignment algorithm by the path penalty function, to decompose the instantaneous frequencies (IFs) ridges of the overlapped multicomponent from a time-frequency representation (TFR). The main objective of KSL is to minimize the error of the prediction process while minimizing the amount of training samples used and thus to cut the costs interrelated with the training sample collection. The IFs first extraction is decided using the framework of the intrinsic mode polynomial chirp transform (IMPCT), which obtains a brief local orthogonal TFR of signals. Then, the IFs curves of the multicomponent signal can be easily reconstructed by the T-F reassignment. After the IFs are extracted, component decomposition is performed through KSL. Finally, the performance of the method is compared when applied to several simulated micro-Doppler signals, which shows its effectiveness in various applications.
In practical applications such as oceanic investigation [
Signal processing challenges and applications for Big Data.
In the past few decades, many signal decomposition approaches have been developed to capture the accurate monocomponent signals. Commonly, the typical methods can be summarized as time-frequency distributions [
However, most of the decomposition methods are signal-dependent and presume that the signal components meet strict divisible conditions in the T-F domain [
To analyze overlapped signal components, TFA is an important issue which deserves to be mentioned for the extraction of feature information from the components. Currently, the extraction accuracy relies on the energy concentration of TFR generated by TFA methods. There exist a variety of types of TFA methods, and they can be normally divided into two categories: the parametric TFA (PTFA) methods and the nonparametric TFA (NPTFA) methods. PTFA methods, such as polynomial [
Notice that despite all those efforts decomposing overlapped MCSs is still a challenging task. In this paper, we develop the intrinsic mode polynomial chirp transform (IMPCT) which takes into account IFs characteristics and a novel KSL method is proposed in order to obtain a robust decomposition overlapped components in high noise. For a nonstationary signal, instantaneous amplitudes (IAs) and IFs are two of the foremost properties. In [
For the practical implementation of a decomposition method, a very interesting solution is the kernel sparse learning (KSL) approach, which has not yet been developed for the multicomponent decomposition. While we propose the KSL, as an active learning, similar to kernel dictionary learning, which has been successfully applied to classification problems [
The remainder of this paper is organized as follows. In Section
First of all, a numerical example of a noisy environment is applied to demonstrate the motivation of our method proposed. A simulated signal is considered as follows:
TFs representations by (a) SWT, (b) SCT, (c) Iteration=0 IMPCT, and (d) Iteration=50 IMPCT.
SWT
SCT
Iteration=0 IMPCT
Iteration=50 IMPCT
The SWT, the continuous wavelet transform- (CWT-) based SST, sharpens the time-frequency representation of a multicomponent signal but suffers from a lower T-F resolution, as shown in Figure
An intuitive solution stems from a combination of high-energy concentrations that partially result from the different modulation frequencies. The new T-F representations of the IMPCT will have better performance than the SCT, as shown in Figures\ 2(c) and 2(d) with 50 iterations.
Therefore, it is necessary to develop alternative approaches to deal with the challenging cases that there exist components overlapping in the T-F domain. We provide the following definition.
The two components
Equation (
(a) Two components are strictly overlapping. (b) Two components are tangent. Blue line, IF trajectory. Red line, IF trajectory after shift up. Purple line, IF trajectory after shift down.
A signal
Here, to overcome these limitations, a robust technology architecture for overlapped nonstationary signals is proposed, as shown in Figure
Block diagram of the proposed approach.
The Fourier series analysis brings to light the sinusoidal property, so it may be utilized for the reconstruction of some signals with linear phase [
Let us inspect the signal vector
The IMPCT representation explains the motivation for obtaining an initial IFs estimate in the algorithm. The IFs estimate will be based on the value of
There are several methods to implement the IFs estimation. Here, we will adopt a reassignment by the path penalty function to further reduce search space. The set of the IFs estimation is detected in the selected points with the best path penalty function gain. The algorithm is performed recursively until we are no longer able to search out a new point that can reduce values of the path penalty function.
In T-F plane, the frequencies
By selecting paths with shorter chirplets, one can search out chirplets with an increasing function
Illustration of the IFs estimation in the T-F plane.
To form a best chirplet path using the path penalty functions, two key constraints in the connections stage from each point should be satisfied as
Furthermore, we only need to solve the 1D optimization problem in partial optimal path to estimate IFs ridges. Thus, our method can be efficiently used in various real applications and a numerical overlapped signal mentioned in Figure
The relative IFs estimation result of a numerical signal. (a) The spectrogram of IMPCT; (b) the IFs estimation result.
In [
One can rewrite equation (
Considering formula (
To recover the expected components, we need to deal with nonlinear equations in (
In kernel sparse learning, the fixed basis is replaced by a nonlinear combination of the input. In (
For kernel sparse learning, one can notice that the gradients for different terms and updating the coefficients in (
When the amount of samples is much bigger than the dimension of the data, storing and processing the kernel matrix are great difficulty in terms of memory. Consequently, a more efficient scheme needs to be implemented. Using the eigenvalue decomposition, the kernel matrix can be denoted as
Solving (
An agent variable
In the high noise environment, one may choose more complex methods to make sure the number of components is accurate, such as by the split Bregman approach [
In order to achieve parallel computing, through adopting the alternating directions method of multipliers, (
It can be easily seen that the first subproblem (
To test the result of the training samples, we need to generate the features for the sample signal
The dictionary of a test processed the IMPCT by kernel sparse learning.
The question is how to determine the dictionary size hyperparameter
Having estimated the IFs of the signal (
Component separation for the simulated signal by the KSL.
It is known that micro-Doppler effect has been diffusely applied to target detection and recognition in sonar systems [
In this section, the presented approach will be used to decompose both simulated micro-Doppler signals to verify the performance of the proposed method. The maximum number of iterations is set to 80 and we adopt a more suitable
In all simulated examples, a zero-mean Gaussian white noise
Consider an FM signal
The experimental results are observed after every 10 iterations, until the 80th iteration, are shown in Figures
The detected IFs ridges from the TFR of the signal are revealed in Figures
Having estimated the IFs ridges, the separation of the multicomponent signal is executed by the KSL, as shown in Figures
The dictionary of a test processed the IMPCT by kernel sparse learning.
Estimated IFs and component separation for the micro-Doppler signal in Figure
Here, we consider again a poly-harmonic FM signal, expressed as
The TFRs of nonnoisy and noise signal obtained by SWT, shown in Figures
For all the following tests, the MSE versus variable SNR will be calculated (from -10 to 10 dB below, with a 1 dB step). The MSE is expressed as
We consider the IFs estimation from the TFRs of IMPCT, SWT, and SCT, using the T-F reassignment based on the path penalty function. The obtained MSEs are displayed in Figure
TFRs of the signal obtained SWT, SCT, and IMPCT, respectively, and estimated IFs by the proposed T-F reassignment.
IFs estimation MSEs for signals in Experiment 2, using the proposed T-F reassignment.
In this paper, we presented a novel approach which combines the KSL algorithm with the T-F reassignment to decompose seriously overlapped components. The IMPCT is first proposed for the TFRs of a multicomponent nonstationary signal in high noise and can adapt to some practical applications. Comparing with traditional the decomposition approaches, the KSL based on the T-F reassignment can decompose signals with overlapped components. The modified T-F reassignment algorithm by the best path penalty function estimates the main IFs ridges of all the components and then reconstructs the obtained IFs ridges according to the partial optimal path. To estimate performance of the presented method, we analyze the numerical experiments including a real-world signal, which clearly illustrates that it outperforms the IMPCT based on the T-F reassignment and KSL in high noise environments. In addition, it outperforms the state-of-the-art TFRs methods when polynomial phase signal is contaminated by a white Gaussian noise. It should be emphasized that, in essence, the KSL algorithm involves postprocessing of ridges to optimize the IFs estimation of components contaminated by noise or induced by moving targets. Further, the applications in practical scenarios will be the part of our research development.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (61471309 and 61671394) and the Fundamental Research Funds for the Central Universities (20720170044).