Feature engineering [
7] is used to process current, thrust, vibration, and audio signals generated during in-flight experiments. In turn, this processing generates feature data. Because the time-domain features used in this article are relatively common, their introduction provided in this paper is short. Conversely, more detailed introductions are provided for the sample and wavelet theories because they are rarely seen in the existing research.
2.1.1. Time-Domain Signal Feature Engineering
The standard deviation (
SD) can display the degree of dispersion between the data and its data group. The degree of dispersion of the vibration signal within the interval can be measured.
where
N is the total number of data points;
xi is a series of data and
μ is its mean value.
The root mean square (
RMS) is used to calculate the mean difference between data and a dataset. It is a statistical method that is often used in the processing of the direct current, sine waves, and normally distributed multi-frequency signals. It can be used as a method to evaluate the energy level of the current and voltage in the dataset interval.
Sample entropy (SE) [
8] is used to measure the complexity of a time-series signal and has been applied in many fields. SE is a modification of approximate entropy (AE) and a powerful method used to analyze time-series signals. SE can measure the degree of chaos of the physical signal when the results are not relative to the size of the sample. In practice, a piece of the data signal is divided into multiple samples; then, the similarity between each sample and the other samples is assessed to determine the confusion degree of the signal in a certain time interval.
A brief introduction of sample entropy is introduced as follows. Let the signal
for
N data points be:
. Then, dividing the data into every m data points yields the
subset. After segmentation, this can be arranged into a template space:
.
First, calculating the largest distance between each template
:
From Equation (4), the largest distance can be determined using the Chebyshev Distance. Then, the Heaviside nonlinear function is used to calculate self-similarity
:
where
r is the threshold, which is usually the proportion of the standard deviation of the signal. Using the standard deviation ensures that the result is not affected by the different scales of fluctuation. Moreover, the sum of all
s divided by
(it is not divided by
because SE does not compare the distance itself).
Finally, by adopting
m + 1 data points to build a new template space, and processing the data from Equations (3)–(6) to obtain
, the final value of sample entropy can be determined:
where
m is the
mth sample,
r is the threshold,
δ is the sample value.
2.1.2. Frequency-Domain Signal Feature Engineering
Wavelet analysis is a time-frequency analysis method [
9] that can not only be used to obtain the power spectrum of a signal, but also to analyze and control the time resolution and frequency resolution [
10,
11]. In this study, data features were extracted from the audio signal using wavelet scattering theory, which can be used to make a further classification assessment.
The Morlet wavelet is based on the use of wavelet transformation for a series of signals:
where
is the wave number. The Morlet wave function can be further dilated and translated:
where
a is the extension (dilation) and
b is the translation. As a result of this property, regardless of whether the frequency is high or low, different time and frequency resolutions can be used for effective analysis. The signal analysis in the time and frequency domains can be obtained by wavelet transform:
Application of the short-time Fourier transform (STFT) to the divided signal can also increase the time resolution of the signal. However, the adjustment of the bandwidth and central frequency of the filter via scaling results in better time and frequency resolutions for the wavelet transform.
Wavelet scattering transformation was proposed by Joakim Anden and Stephane Mallat in 2014 [
12]. The theory is that a wavelet-transformed time-series signal is analyzed using the Convolution Neural Network (CNN) framework, and a wavelet function (high-pass filter) is convolved with the scaling function of a Gabor function (low-pass filter). Equation (11a) is the Gabor function and Equation (11b) is the scaling function of the Gabor function.
where
θ is the center frequency; when its value is zero, it is a Gaussian function. σ is the standard deviation of the data signal.
J is the scale and
Q is a number of wavelet functions per octave; for example, there are 12 notes in an octave for music. The values of
J have several octaves for high-pass decomposition.
Using different scales, convolutional signals, and wavelet functions, from low- to high-frequency information a locally deformed and scale-invariant wavelet conversion coefficient can be obtained:
where
λ is the center frequency of the wavelet function, and the mean value of the wavelet coefficients is zero.
represents the original signal,
is the scaling function, and
is the wavelet function. In order to obtain the characteristic information of the signal, the absolute value is added, which is the modulus. According to [
12], the modulus provides the stability and translation stability features, i.e.,
The modulus is used to compute a lower frequency envelope. In order to obtain stable wavelet coefficients, the scaling function (low-pass filter) is convolved, and we obtain the wavelet scattering characteristic coefficient of the signal:
In the expression of the wavelet scattering framework, a series of wavelet scattering coefficients are analyzed using the deep convolution network architecture.
In order to obtain a low variance and stable features, an averaging function and modulus are applied after the wavelet convolution. There is a sequence of steps to achieve wavelet scattering coefficients as shown in
Figure 1. First, the signal is convolved with the scaling function to obtain the zero-order scattering coefficient of the signal,
. For the second layer, the signal is convolved with the wavelet function,
(first-order filter), and the modulus is determined. This is then convolved with the scaling function to obtain the first-order wavelet scattering coefficient. When determining the next-order wavelet scattering coefficient, the original sequence information must be restored because averaging loses the high frequency information. Therefore, the next wavelet function (second-order filter) needs to be convolved to the original sequence information to ensure that information is not lost. For high-frequency information, in order to obtain a locally deformed and fixed-scale wavelet scattering coefficient, it is necessary to convolve the scaling function to obtain a stable wavelet scattering coefficient.
If the multiple wavelet functions (filters) are set, the network architecture can continue to iterate continuously on the next wavelet function:
A series of wavelet scattering coefficient characteristics can then be obtained:
The wavelet filter banks in multiple wavelet filter libraries are used to analyze the low- and high-frequencies of the complete signal information to obtain frequency domain information. In this study, we used a scaling function (low-pass filter) and two wavelet functions (high-pass filter) to obtain the wavelet scattering coefficients of the signal. Note that, normally, second-order wavelet scattering transform is sufficient because the energy attenuates at each level.