Linear frequency-modulated (LFM) signals are the most significant example of waveform used in low probability of intercept (LPI) radars, synthetic aperture radars and modern communication systems. Thus, interception and parameter estimation of the signals is one of the challenges in Electronic Support (ES) systems. The methods, which are widely used to accomplish this task are mainly based on transformations from time to time-frequency domain, which concentrate the energy of signals along an instantaneous frequency (IF) line. The most popular examples of such transforms are the short time Fourier transform (STFT) and Wigner-Ville distribution (WVD). However, for LFM waveforms, methods that concentrate signal energy along a line in the time-frequency rate domain may allow to obtain better detection and estimation performance. This type of transformation can be obtained using the cubic phase (CP) function (CPF). In the paper, the detection of LFM waveform and its chirp rate (CR) parameter estimation based on the extended forms of the standard CPF is proposed. The CPF was originally introduced for instantaneous frequency rate (IFR) estimation for quadratic frequency modulated (QFM) signals i.e., cubic phase signals. Summation or multiplication operations on time cross-sections of the CPF allow to formulate the extended forms of the CPF. Based on these forms, detection test statistics and the estimation procedure of LFM signal parameters have been proposed. The widely known estimation methods assure satisfying accuracy for high SNR levels, but for low SNRs the reliable estimation is a challenge. The proposed approach based on joint analysis of detection and estimation characteristics allows to increase the reliability of chirp rate estimates for low SNRs. The results of Monte-Carlo simulation investigations on LFM signal detection and chirp rate estimation evaluated by the mean squared error (MSE) obtained by the proposed methods with comparisons to the Cramer-Rao lower bound (CRLB) are presented.

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.

In the paper the parameter estimation of the polynomial frequency law via estimation of parameters A = [a ₁ , a ₂ , a _p ] of a polynomial phase signal (PPS) is addressed. The term “estimation” is used mainly in statistical community, but in this paper the word “estimation” is understood also as a synonym for the word “assessment”. The frequency law is strongly related to the polynomial phase signal, because the instantaneous frequency (IF) is the first derivative of the phase function. There are many parametric and nonparametric estimation methods for computing PPS. The most popular approach is based on the maximum likelihood (ML) estimator, which has the limitation due to a required multi-dimensional search over the parameter space and has numerous local optima making the application of gradient techniques impossible. Instead of the P-dimensional search of the ML estimation, an alternative solution with iterative reduction of the number of PPS coefficients to be estimated is proposed. The reduction of the PPS coefficients can be performed by the Extended Generalized Chirped Transform (EGCT). In the first step of the EGCT algorithm, the first phase coefficient is estimated. The maximum of the EGCT determines the estimated value of this coefficient. Successive de-chirping of the signal with one just estimated phase coefficient allows to formulate the next index function for one-dimensional search of the next coefficient. The proposed one-dimensional maximization process is implemented with initial “coarse” values of parameters obtained from fitting of a polynomial to the IF trajectory acquired from the Spectrogram, the Polynomial Wigner-Ville Distribution (PWVD) and from the Recursive Least Squares filter (RLS). This approach is analyzed mainly on the example of quadratic frequency modulated signals (QFM), but it can be easily extended to higher orders of polynomials.

Radar tomography allows to build a two-dimensional image from one-dimensional functions obtained from radar returns. In this paper the possibility of usage of the reassigned spectrogram as well as different reassigned forms of time-frequency distributions in the Doppler radar tomography imaging is evaluated. The image is obtained from a multi-component signal with time-frequency modulations resulting from rotation. The algorithm of imaging with high resolution is tested on simulated data.