An adaptive matching pursuit algorithm for linear frequency modulation signal compressive sampling

Signal integration, as an effective method of detecting weak target, is widely used in areas of radar, sonar, etc. In previous studies of long-time coherent integration, researchers usually established a multi-pulse echo model with linear frequency modulation (LFM) signal due to its good compression performance and large Doppler tolerance. Then, perfect analytical formula can be deduced to compensate for range migration and Doppler spread, which is helpful in analyzing the mechanism of long-time coherent integration in depth. However, besides LFM, a wide variety of signal waveforms are also used in modern sonar and underwater guidance system to meet the requirements for diverse applications. For instance, continuous wave (CW) pulse is often used in signal detection, high resolution direction of arrival (DOA) estimation, and velocity estimation, while large time-bandwidth product waveforms such as modulated signal, coded signal, and pseudo-random signal are utilized for special tasks like anti-interference detection, channel matching, and active concealed detection. Therefore, the formulas and corresponding instructive conclusions deduced by LFM have no generality when other sonar waveforms are used in pulse integration. In this paper, we focus on long-time coherent integration for arbitrary signal reflected by underwater target moving with a uniform velocity and propose a motion-compensated coherent integration method for arbitrary complex envelop signal. A kind of general ambiguity function (GAF) for transmitted signal is defined to present a unified expression based on GAF for the output of the matched filter. The operation not only helps us to describe and calculate the pulse compression form of the arbitrary complex envelop by using a general mathematical model, but also provides information about the range migration and Doppler frequency shift of the multi-pulse echo, which is needed in pulse range alignment and FFT integration. For the matched filter output expressed by the GAF, Keystone transform is utilized to correct the complex envelop of the multi-pulse echo and eliminate the range walk. Then, Doppler frequency shift is compensated for by performing FFT transform, and the long-time coherent integration for arbitrary complex envelop is realized. To verify the correctness of the proposed method, we carry out the computer simulation on both signal integration and detection performance by using four sonar waveforms, i.e., CW signal, LFM signal, m-sequence phase-coded signal, and Costas frequency hop coded signal. The simulation results show that the proposed motion-compensated coherent integration method is applicable to arbitrary complex envelop signal. We also design an anechoic water tank experiment scheme which can successfully obtain the multi-pulse echoes of constant moving target. The motion-compensated coherent integration of the experimental data of the above-mentioned four waveforms further validates the effectiveness of the proposed method.

Background: For the traditional Fourier Transform (FT), it cannot effectively detect the frequency of non-stationary signals with time. Analyzing the local characteristics of time-varying signal by using FT is hard to achieve. Therefore, many time-frequency analysis methods which can meet different needs have been proposed on the basis of the traditional Fourier transform, like the Short Time Fourier Transform (STFT), the widely used Continuous Wavelet Transform (CWT), Wigner-Ville Distribution (WVD) and so on. However, the best time and frequency resolution cannot be achieved at the same time due to the uncertainty criterion. Methods: From the point of view of optimizing time-frequency performance, a new Generalized S Transform (GST) window function optimization method is proposed by combining time frequency aggregation with an improved genetic algorithm in this paper. Results: In the noiseless condition, the Linear Frequency Modulation (LFM), Nonlinear Frequency Modulation (NLFM) and binary Frequency Shift Keying (2FSK) signals are simulated. The simulation results prove that the method can improve the time-frequency concentration and decreasing Rényi entropy effectively. Compared with other four traditional time-frequency analysis methods, the improved GST has more advantages. Conclusion: In this paper, a new method of optimizing the window function in GST based on improved GA is proposed in this paper. Experiments on LFM, NLFM and 2FSK signals show that the proposed method not only has the advantages of high resolution, but also determines the optimal parameters via the time frequency focusing criterion and the Rényi entropy. Compared with the other four kinds of time-frequency analysis methods, the optimized GST based on improved GA in this paper has the best time-frequency focusing.

The sparse representation using time-frequency atoms is a useful analysis method for radar signal processing. Based on Time-Frequency theory, time-frequency atoms can match radar signals and represent it sparsely. Traditional atomic dictionary can only match the linear frequency modulation signal, while a novel time-frequency atomic dictionary proposed in this paper can not only match the linear time-frequency modulation signal properly but also match the non-linear time-frequency relationship appropriately. It is because that the proposed atomic dictionary has a sinusoidal factor which improves the atoms' ability to match and represent the linear and non-linear frequency modulation signals. Compared with traditional atomic dictionaries, the simulation result shows that the novel atomic dictionary can represent radar intra-pulse modulation signals more effectively and sparsely.

The fractional Fourier transform (FRFT) can transform the linear frequency modulation (LFM) signal into delta function for a proper orders, since the kernel of FRFT has a property of LFM. Using this property, the paper proposes the filter for LFM signal with noise, which avoids cross-term of quadratic distribution due to the linearity of FRFT. The method can reduce noise by optimizing the parameter p in (p, u) plane. The filtering performance is regulated by the correlation coefficient, and evaluated by mean square error (MSE). Computer simulations show that the proposed algorithm can filter the signals well compared to the pseudo Wigner Ville distribution for avoiding the cross-term problem. The statistical analysis of the MSE is also given for different signal to noise ratio, which illustrates the proposed algorithm has a rather good performance.

A linear frequency modulation (LFM) signal and its corresponding de-chirp operation are one of the basic methods for wideband radar signal processing, which can reduce the burden of the radar system sampling rate and is more suitable for large-bandwidth signal processing. More importantly, most existing methods against interrupted sampling repeater jamming (ISRJ) are based on time–frequency (TF) or frequency domain analysis of the de-chirped signal. However, the above anti-ISRJ methods cannot be directly applied to multiple-input multiple-output (MIMO) radar with multiple beams, because the angular waveform (AW) in mainlobe directions does not possess the TF properties of the LFM signal. Consequently, this work focuses on the co-optimization of transmit beampattern and AW similarity in wideband MIMO radar systems. Different from the existing works, which only concern the space–frequency pattern of the transmit waveform, we recast the transmit beampattern and AW expressions for wideband MIMO radar in a more compact form. Based on the compact expressions, a co-optimization model of the transmit beampattern and AWs is formulated where the similarity constraint is added to force the AW to share the TF properties of the LFM signal. An algorithm based on the alternating direction method of multipliers (ADMM) framework is proposed to address the aforementioned problem. Numerical simulations show that the optimized waveform can form the desired transmit beampattern and its AWs have similar TF properties and de-chirp results to the LFM signal.

A novel in-band OSNR monitoring technique based on fractional Fourier transform of LFM signal

This paper considers the localization of an emitter with an unknown linear frequency modulation (LFM) signal in the passive radar systems based on the direct position determination (DPD) method. To improve the localization accuracy, a novel strategy, wherein the position of the emitter and the unknown transmitted signal are jointly estimated, is proposed. As a result, a six-dimensional estimation problem is formulated, which is, however, computationally intractable for any general application scenario. By combining the DPD algorithm with short time Fourier transform (STFT), a fast and practical suboptimal algorithm, i.e., DPD-STFT, is also designed here to achieve the joint estimation for the emitter position and the unknown parameters of the LFM signal. Simulation results show that the localization performance of the DPD-STFT method is much better than the existing DPD algorithms which do not estimate the transmitted signal.

In this letter, we experimentally demonstrate multi-band linear frequency modulation (LFM) signal generation subjected to multi-input optical injection in an external cavity-based Fabry-Pérot laser diode (ECFP-LD). The ECFP-LD is a specially designed FP-LD with a self-injected single longitudinal mode tuned up to 10 nm. The non-linear dynamics in period-one oscillation in ECFP-LD subjected to multi-input optical injection shows unique characteristics for multi-band linear frequency modulation. In the proposed scheme, four external input beams are injected to ECFP-LD, among which one is power modulated with the saw-tooth like waveform, and others are with the constant power. The varying power beam red-shifts the self-injected mode of ECFP-LD and mode-hopping on attaining a particular power level. Hence, multi-band LFM signals are generated by beating self-injected and multi-input injected beams with a large time-bandwidth product (TBWP). The total bandwidths of 12GHz (LFM1: 33-36 GHz, LFM2: 29-32 GHz, LFM3: 25-28 GHz, LFM4: 22-25 GHz) for four-band linear frequency modulation signals with a TBWP of 12000 is measured. The generated multi-band LFM signal provides flexibility on central frequency tunability and dynamic chirp rate for a multifunctional radar system.

This paper analyses the main factors affecting the estimation accuracy of linear frequency modulation (LFM) signal parameters based on fractional Fourier transform (FRFT). By simulation and comparing we can come to the conclusion that the estimation accuracy of initial frequency is improved along with the search interval decreasing while the modulation frequency slope is improved along with the sample points increasing. Based on which we can improve the estimation accuracy of LFM signal parameters according to different cases at the same computational cost. Simulation results verify the correction of the conclusion.

For pulse compression characteristics and not easily affected by noise, linear frequency modulation signal are widely used in underwater acoustic communication. This paper analyzes the characteristics of hyperbolic frequency modulation signal over underwater acoustic channels. Compared with linear frequency modulation signal, hyperbolic frequency modulation has the same performance of strong anti-noise and anti-multipath, what’s more, hyperbolic frequency modulation signal is better resist the influence of doppler. And discussed the influence of doppler on signal, simulation results show that the hyperbolic frequency modulation signal detection rate is better than linear frequency modulation signal in the doppler environment. DOI: http://dx.doi.org/10.11591/telkomnika.v11i3.2183 Full Text: PDF

Attention has been focused on the moving target detection in heavy sea clutter. On the basis of detection model of moving target with fluctuant amplitudes, a novel adaptive algorithm in fractional Fourier transform (FRFT) domain is proposed, which combines statistic-based and FRFT-based detection method. FRFT has good energy concentration property on linear frequency modulation (LFM) signal with the optimal transform angle, which is determined by calculating spectral kurtosis (SK) in FRFT domain. Grading iterative search method is used for good accuracy of parameter estimation and fast calculation speed. A novel adaptive line enhancer (ALE) in FRFT domain is proposed to suppress sea clutter and improve signal-to-clutter ratio (SCR), which provides less error and faster convergence. Leakage factor is introduced into the update equation of weight vector to reduce ‘memory effect’ and step size is normalised by the power of input signal with better convergence characteristic. In the end, both X-band and S-band real sea clutter is used for verification and the results present that the proposed algorithm has good convergence property and small mean square error (MSE). Weak moving target in low SCR environment (SCR=−6 dB) can be well detected and estimated, which indicates the effectiveness of the algorithm.

Linear frequency modulation (LFM) signal is widely used in pulse compression radars, whereas faced with the time delay bias caused by Doppler shift of radar echoes, which must be accounted for in some applicative occasions where the linear frequency modulation ratio is very low or Doppler shift is very large. The method of time delay estimation based on cross-ambiguity function (CAF) could solve the problem but need two-dimensional search around the entire time delay and Doppler shift plane, which increases computational complexity heavily and brings great challenge to real-time processing. An effective time delay estimation method is proposed in this paper based on the relationship of time delay and Doppler shift, which can reduce two-dimensional search to one-dimensional search along LFM signal energy line, and diminish the computation complexity greatly with the same performance of parameters estimation. Simulations proved the validity of the method proposed.

Robust Connectivity in Time-Frequency Dual Fading Channels for the 6G Satellite IoT: Chirp and FRFT

This paper focuses on the wake-up signal detection design for underwater acoustic communication (UAC) terminals. A wake-up signal detection unit can considerably reduce the power consumption of the terminals. Compared with terrestrial wireless counterparts, the wake-up signal detection design for UAC terminals is challenged by the severe underwater acoustic channels, which is characterized as doubly selective fading and low signal-to-noise ratio (SNR). This paper proposes a wake-up signal detection approach called channel-adaptive detection and location-assisted joint decision (ChAD-LaJD), for UAC terminals. ChAD-LaJD applies a group of linear frequency modulation (LFM) signals as a wake-up signal. In order to increase the detection probability while keeping a low false alarm rate, ChAD-LaJD consists of two procedures: channel-adaptive detection (ChAD) and location-assisted joint decision (LaJD). Besides a pre-determined threshold, ChAD procedure defines two special parameters which reflect instantaneous channel states to detect wake-up signals adaptively. LaJD procedure further exploits the location relationships of LFM signals detected by ChAD to achieve a joint decision. The simulations and field experiments are conducted to evaluate the performance of ChAD-LaJD. The results show that ChAD-LaJD outperforms the conventional methods that consider a fixed threshold (FixTh) and/or constant false alarm rate (CFAR).

Referring to the coherent wideband linear frequency modulation (LFM) signals, direction of arrival (DOA) estimation based on fractional Fourier transform (FRFT) is proposed. In this method, the observed signals are first transformed to fractional Fourier (FRF) domain, using the superior time-frequency localization, time-variant direction matrix is changed into time-invariant one. By dint of the forward/backward (FB) spatial smoothing techniques, the corresponding correlation matrix is constructed. Then the DOA of multicomponent coherent wideband LFM signals can be estimated by the traditional MUSIC algorithm simultaneously. Simulation verifies the effectiveness of the novel method.