Estimating the Instantaneous Frequency of Linear and Nonlinear Frequency Modulated Radar Signals-A Comparative Study

Instantaneous Frequency Estimation of FM Signals under Gaussian and Symmetric α-Stable Noise: Deep Learning versus Time–Frequency Analysis

Deep learning (DL) and machine learning (ML) are widely used in many fields but rarely used in the frequency estimation (FE) and slope estimation (SE) of signals. Frequency and slope estimation for frequency-modulated (FM) and single-tone sinusoidal signals are essential in various applications, such as wireless communications, sound navigation and ranging (SONAR), and radio detection and ranging (RADAR) measurements. This work proposed a novel frequency estimation technique for instantaneous linear FM (LFM) sinusoidal wave using deep learning. Deep neural networks (DNN) and convolutional neural networks (CNN) are classes of artificial neural networks (ANNs) used for the frequency and slope estimation for LFM signals under additive white Gaussian noise (AWGN) and additive symmetric alpha stable noise (SαSN). DNN is composed of input, output, and two hidden layers, where several nodes in the first and second hidden layers are 25 and 8, respectively. CNN is the content input layer; many hidden layers include convolution, batch normalization, ReLU, max pooling, fully connected, and dropout. The output layer consists of a fully connected softmax and classification layers. SαS distributions are impulsive noise disturbances found in many communication environments such as marine systems, their distribution lacks a closed-form probability density function (PDF), except for specific cases, and infinite second-order statistics, hence geometric SNR (GSNR) is used in this work to determine the effect of noise in a mixture of Gaussian and SαS noise processes. DNN is a machine learning classifier with few layers for reducing FE and SE complexity. CNN is a deep learning classifier, designed with many layers, and proved to be more accurate than DNN when dealing with big data and finding optimal features. Simulation results show that SαS noise can be much more harmful to the FE and SE of FM signals than Gaussian noise. DL and ML can significantly reduce FE complexity, memory cost, and power consumption as compared to the classical FE based on time–frequency analysis, which are important requirements for many systems, such as some Internet of Things (IoT) sensor applications. After training CNN for frequency and slope estimation of LFM signals, the performance of CNN (in terms of accuracy) can give good results at very low signal-to-noise ratios where time–frequency distribution (TFD) fails, giving more than 20 dB difference in the GSNR working range as compared to the classical spectrogram-based estimation, and over 15 dB difference with Viterbi-based estimate.

Estimation and Classification of NLFM Signals Based on the Time-Chirp Representation

A new approach to the estimation and classification of nonlinear frequency modulated (NLFM) signals is presented in the paper. These problems are crucial in electronic reconnaissance systems whose role is to indicate what signals are being received and recognized by the intercepting receiver. NLFM signals offer a variety of useful properties not available for signals with linear frequency modulation (LFM). In particular, NLFM signals can ensure the desired reduction of sidelobes of an autocorrelation (AC) function and desired power spectral density (PSD); therefore, such signals are more frequently used in modern radar and echolocation systems. Due to their nonlinear properties, the discussed signals are difficult to recognize and therefore require sophisticated methods of analysis, estimation and classification. NLFM signals with frequency content varying with time are mainly analyzed by time-frequency algorithms. However, the methods presented in the paper belong to time-chirp domain, which is relatively rarely cited in the literature. It is proposed to use polynomial approximations of nonlinear frequency and phase functions describing signals. This allows for applying the cubic phase function (CPF) as an estimator of phase polynomial coefficients. Originally, the CPF involved only third-order nonlinearities of the phase function. The extension of the CPF using nonuniform sampling is used to analyse the higher order polynomial phase. In this paper, a sixth order polynomial is considered. It is proposed to estimate the instantaneous frequency using a polynomial with coefficients calculated from the coefficients of the phase polynomial obtained by CPF. The determined coefficients also constitute the set of distinctive features for a classification task. The proposed CPF-based classification method was examined for three common NLFM signals and one LFM signal. Two types of neural network classifiers: learning vector quantization (LVQ) and multilayer perceptron (MLP) are considered for such defined classification problem. The performance of both the estimation and classification processes was analyzed using Monte Carlo simulation studies for different SNRs. The results of the simulation research revealed good estimation performance and error-free classification for the SNR range encountered in practical applications.