Time-frequency Scattering Convolutional Network Based On Short-time Fractional Fourier Transform

With the rapid development of artificial intelligence,we have already entered into a “intelligence plus" era.Convolutional Neural Network has significant advantages in two-dimensional feature extraction among all the deep learning framework.At present,the performance of CNN is improved by the innovation of network architecture.However,these innovative CNNs not only need huge data to train the model,but lack the mathematical theory interpretability of CNN,which leads to no theoretical supports for interpreting how to set CNN model and hyper-parameter and blocked the development of deep learning in many fields.Scattering convolutional networks is a key to solving the above problems.The scattering convolutional networks use predefined filters as convolution kernels and work without back propagation learning.In addition,it provides theoretical understanding of DCNNs and is insensitive to translation,rotation and deformation.Scattering convolutional networks have already been applied in the fields of digital and texture image recognition,medical image analysis,and financial time series analysis because of its excellent accuracy rate in few-shot learning.However,the current wavelet scattering convolutional network and time-frequency scattering convolutional network are limited by linear time-invariant filters and can’t analyze the fractional frequency spectrum of signals in FRFT domain which lead to ineffective for analysing non-stationary time-varying signals.Therefore,the paper proposes the fractional time-frequency scattering convolutional network shortTime fractional Fourier transform to solve non-stationary few-shot feature extraction.To this end,we must first define the integral kernel,then use the integral kernel to construct the base frame of the space transformation,and define the scattering transform based on the frame,and finally construct the scattering convolutional network according to the scattering transform.The main contents of the paper list as follows:We choose short-Time fractional Fourier transform as the integral kernel of the scattering space.As the limitation of existing STFRFT on local time-fractional-frequency representation and lacking clear physical significance,the paper proposes a new definition of STFRFT and its inverse transform form based on the definition of fractional convolution and short-time Fourier transform.It is proved that the new STFRFT not only retains the basic characteristics of short-time Fourier transform,but also can be realized by fractional low-pass and band-pass filters.In addition,the paper proposes analysis of the fractional time-frequency properties and comparative tests on high resolution spectrograms,which implies that the new STFRFT is effective to processing the non-stationary signal.The paper defines fractional low-pass and band-pass representations as the vector bias which constructs the fractional time-frequency scattering frame.Then the paper gives the definition of fractional time-frequency scattering transform which consists of fractional time-frequency scattering feature operator and the fractional time-frequency scattering propagation operator.The former is defined by the fractional low-pass function,and it represents the general feature of the signal which is robust to rotation,translation,deformation,etc.And the latter is defined by the fractional band-pass functions on which the modulo operation is applied as a non-linear down-conversion.And then the highfrequency component of the signal is gradually modulated to the low-frequency region after multiple iterations.Thereby the high-frequency components can have robustness as the low-frequency ones.So,the nature of the fractional time-frequency scattering transform is converting all the components of the signal into low-frequency components with high anti-interference.By the comparative tests of fractional time-frequency scattering transform and traditional scattering transform on stationary and non-stationary signals,it implies that some energy of non-stationary signal locates in the FRFT domain.And the fractional time-frequency scattering transform solves the deficiency of incapable of analysing in the FRFT domain which improves the performance for feature extraction of non-stationary signal.The paper defines the architecture of the fractional time-frequency scattering convolution network based on the proposed fractional time-frequency scattering transform.Then the fractional time-frequency scattering convolutional network is mathematically proved to have the basic properties of energy conservation,non-expressive,deformation stability and translation invariance according to the properties of traditional scattering convolution network.And the fractional time-frequency scattering convolutional network is a general form of the traditional time-frequency scattering convolution network.It implies that the fractional time-frequency scattering convolutional network not only retains the efficient performance and unsupervised learning characteristics for few-shot feature extraction,but also extends the application fields of scattering convolution network by scattering the signal into the FRFT domain.The paper proposes the discrete form and fast implementation algorithm of the new STFRFT and fractional time-frequency scattering convolutional network.Then the paper proposes a simplified structure of the fractional time-frequency scattering convolutional network according to the energy ratio of each iteration output during feature extracting.At last,we compare the few-shot image classification result among the fractional time-frequency scattering convolutional network,traditional scattering convolution networks,Meta-Baseline network and CNN which further illustrates that the fractional time-frequency scattering convolutional network has the better performance in processing the non-stationary few-shot signals,especially when training two simples.