Chirp Rate Research Articles

Efficient photonics beam forming system incorporating super structure fiber Bragg grating for application in Ku band

We propose and demonstrate a photonic true-time delay beamforming system for a 1 × 4 linear phased array antenna, operating in the Ku-band, based on superstructured fiber Bragg gratings (SFBGs) with linearly rising equivalent chirps. The proposed beam-forming network operates at 18 GHz frequency, which is theoretically analyzed and simplified by simulated results. The SFBG has a distinct number of gratings in each beamforming path, which provides a large and progressive phase delay. The advantage of the proposed technique can be found in terms of its wavelength tunable scheme making the system squint-free. The proposed architecture is a true time delay optical beam forming a network, which enables squint-free beam steering of radio frequency (RF) signals. In this study, a technique known as double sideband modulation of multiple wavelengths is utilized to investigate the reflection quality of SFBG at various chirp rates. Taking into account that each segment of FBG consists of 50, 60, 70, and 80 gratings and phase shift section samples of each segment, the overall length of the SFBGs is calculated to be 2 cm, 2.4 cm, 2.8 cm, 3.2 cm, respectively, are used for calculating group delay. An equation in a closed form representing the corresponding dispersion is derived. SFBG group delay response is analyzed theoretically with required simulation to investigate the reflectance characteristics. All simulation work for designing the beam forming system is done on Optisystem version 19 and designing of Super structure fiber Bragg grating Optigrating version 4.2. The optically process wavelengths are detected by high speed dedicated photo-detector (PD) and delay response is investigated through array factor of each antenna. Errors in the fabricated SFBGs' group delay responses are examined also.

A statistical frequency-chirprate extractor for mode retrieval with crossover instantaneous frequencies

It is challenging to analyze oscillatory signals whose instantaneous frequencies (IFs) are crossover in the time-frequency (TF) domain. Driven by this need, we propose a statistical frequency-chirprate extractor (SFCE) to analyze multi-component non-stationary signals whose IF lines intersect each other at the TF plane. In this paper, an effective sliding real-valued frequency filter is designed to realize the local representation of the instantaneous attribute value in the linear chirplet transform (LCT) inner product space. On this basis, we obtain SFCE by extracting the fixed-point condition of the statistical frequency-chirprate estimator from the time-frequency-chirprate (TFC) domain. Finally, the TFC information is extracted to the estimated IF and chirprate (CR) trajectories by SFCE, thus providing a novel three-dimension synchroextracting chirplet transform (TDSECT), which obtains high precision model retrieval for crossover signals. In addition, we provide a relatively rigorous theoretical analysis of SFCE for multi-component non-stationary signals. Simultaneously error analysis is given. Numerical experiments on two synthetic signals are employed to verify the successfulness of the proposed method.

High-performance transient SBS-based microwave measurement using high-chirp-rate modulation and advanced algorithms.

The transient stimulated Brillouin scattering (SBS) effect, enabled by optical chirp chain (OCC) technology, has already been proposed and demonstrated for microwave frequency identification with high temporal resolution. Through increasing the OCC chirp rate, the instantaneous bandwidth can be effectively extended without loss of the temporal resolution. However, the higher chirp rate results in more asymmetric transient Brillouin spectra, which worsens the demodulation accuracy when using the traditional fitting method. In this Letter, advanced algorithms, including image processing and artificial neural network, are employed to improve the measurement accuracy and demodulation efficiency. A microwave frequency measurement scheme is implemented with 4 GHz instantaneous bandwidth and 100 ns temporal resolution. Through the proposed algorithms, the demodulation accuracy of transient Brillouin spectra under 50 MHz/ns high chirp rate is improved from 9.85 MHz to 1.17 MHz. Moreover, owing to the matrix computations of the proposed algorithm, the time consumption is reduced by two orders of magnitude compared with the fitting method. The proposed method allows a high-performance OCC transient SBS-based microwave measurement, which provides new possibilities to realize real-time microwave tracking for diverse application fields.

Photonic generation of tunable wideband dual-chirp microwave waveforms using a monolithic integrated three-section laser

A method for flexibly generating dual-chirp microwave waveform (DCMW) is proposed and experimentally demonstrated. The scheme uses a monolithic integrated three-section laser, which consists of a front section laser, a phase section and a rear section laser. Under the appropriate control current, the integrated laser works in the period-one (P1) state, and the two beams of different frequencies generated by the integrated laser can produce stable beat frequency. After photoelectric conversion, the generated DCMW has the flexible tunability of the central frequency. Through theoretical analysis and experimental demonstration, the DCMW with time-bandwidth-product (TBWP) of 6 × 105 and chirp rate of ± 0.03 GHz/μs is obtained at different centre frequencies. In addition, the auto-ambiguity function of the generated DCMW is also studied and it is proved that the proposed scheme can improve the range-velocity resolution, so it has application prospects in microwave photonic radar systems.

A Novel Parameter Estimation Method Based on Piecewise Nonlinear Amplitude Transform for the LFM Signal in Impulsive Noise

In a complex electromagnetic environment, any noise present generally exhibits strong impulsive characteristics. The performance of existing parameter estimation methods carried out in Gaussian white noise for the linear frequency modulation (LFM) signal degrades or even fails under impulsive noise. This paper proposes a novel parameter estimation method to address this problem. Firstly, the properties of the piecewise nonlinear amplitude transform (PNAT) are derived. This manuscript verifies that the PNAT can retain phase information of the LFM signal while suppressing the impulsive noise. Subsequently, a new concept known as piecewise nonlinear amplitude transform parametric symmetric instantaneous autocorrelation function (PNAT-PSIAF) is proposed. Based on this concept, a novel method called piecewise nonlinear amplitude transform Lv’s distribution (PNAT-LVD) is proposed to estimate the centroid frequency and chirp rate of the LFM signal. The simulations show that the proposed algorithm can effectively suppress the impulsive noise without prior knowledge of the noise for both the single-component and double-component LFM signal. In addition, two parameters of the LFM signal can be precisely estimated by the proposed method under low generalized signal-to-noise ratios (GSNR). The stronger the impulsive characteristics of the noise, the better the performance of the algorithm.

Deep Learning-Based Link Configuration for Radar-Aided Multiuser mmWave Vehicle-to-Infrastructure Communication

Configuring millimeter wave links following a conventional beam training protocol, as the one proposed in the current cellular standard, introduces a large communication overhead, especially relevant in vehicular systems, where the channels are highly dynamic. In this paper, we propose the use of a passive radar array to sense automotive radar transmissions coming from multiple vehicles on the road, and a radar processing chain that provides information about a reduced set of candidate beams for the links between the road-infrastructure and each one of the vehicles. This prior information can be later leveraged by the beam training protocol to significantly reduce overhead. The radar processing chain estimates both the timing and chirp rates of the radar signals, isolates the individual signals by filtering out interfering radar chirps, and estimates the spatial covariance of each individual radar transmission. Then, a deep network is used to translate features of these radar spatial covariances into features of the communication spatial covariances, by learning the intricate mapping between radar and communication channels, in both line-of-sight and non-line-of-sight settings. The communication rates and outage probabilities of this approach are compared against exhaustive search and pure radar-aided beam training methods (without deep learning-based mapping), and evaluated on multi-user channels simulated by ray tracing. Results show that: (i) the proposed processing chain can reliably isolate the spatial covariances for individual radars, and (ii) the radar-to-communications translation strategy based on deep learning provides a significant improvement over pure radar-aided methods in both LOS and NLOS channels.

Parameters estimation of hybrid modulated signal combining BPSK and LFM Based on Born-Jordan Distribution

Aiming at the problem of the estimation of hybrid modulated signal combining Binary Phase Shift Keying and Linear Frequency Modulation (LFM-BPSK), the existing methods have low estimation accuracy in low SNR. To solve this problem, the algorithm based on Born-Jordan time-frequency distribution (BJD) is proposed, which uses the Hough transform (HT) to estimate the initial frequency and chirp rate of LFM-BPSK signal, and then estimates the code element width by dimensionality reduction time-domain signal. Finally, simulation experiments show that the parameter estimation accuracy of the proposed algorithm under low SNR is significantly better than that of existing algorithms, and the performance is close to MCRLB.

A computationally efficient algorithm to estimate the parameters of a two-dimensional chirp model with the product term

Chirp signal models and their generalizations have been used to model many natural and man-made phenomena in signal processing and time series literature. In recent times, several methods have been proposed for parameter estimation of these models. However, these methods are either statistically sub-optimal or computationally burdensome, especially for two dimensional chirp models. In this paper, we consider the problem of parameter estimation of two dimensional chirp models and propose a computationally efficient estimator and establish asymptotic theoretical properties of the proposed estimators. Moreover, the proposed estimators are observed to have the same rates of convergence as the least squares estimators. Further, the proposed estimators of chirp rate parameters are shown to be asymptotically optimal. Extensive and detailed numerical simulations are conducted, which support the theoretical results of the proposed estimators.

A method for LFM signal parameter estimation based on MWC system and STFT-DPT

A method of estimating linear frequency modulation (LFM) signal parameters using compressed sampling (CS) data of a Modulated Wideband Converter (MWC) system is proposed. The short-time Fourier transform algorithm is used to analyze the CS signal of the MWC system, and the obtained CS signal frequency will present a form of multi-segment distribution. By intercepting the CS signal corresponding to the first segment frequency and analyzing the intercepted segment signal with discrete polynomial transform and Rife algorithm, the chirp rate and initial frequency estimation of the segment CS signal can be preliminarily estimated. According to the random sequence with cyclic displacement, the spectrum shift relationship between the input signal and the output signal can be calculated. Therefore, the parameter of the original signal can be estimated by using single channel CS data. Then, all channels are analyzed with the same method, and a more stable parameter estimation result can be obtained than the single channel output. Finally, combined with the simulation experiment, the proposed method can be used to estimate the initial frequency and chirp rate of the LFM signal using CS data effectively.

ISAR Imaging of Non-Stationary Moving Target Based on Parameter Estimation and Sparse Decomposition

This paper studies the inverse synthetic aperture radar imaging problem for a non-stationary moving target and proposes a non-search imaging method based on parameter estimation and sparse decomposition. The echoes received by radar can be thought of as consisting of chirp signals with varying chirp rates and center frequencies. Lv’s distribution (LVD) is introduced to accurately estimate these parameters. Considering their inherent sparsity, the signals are reconstructed via sparse representation using a redundant chirp dictionary. An efficient algorithm is developed to tackle the optimization problem for sparse decompositions. Then, by using the reconstructed data, adaptive joint time–frequency imaging techniques are employed to create high-quality images of the non-stationary moving target. Finally, the simulated experiments and measured data processing results confirm the proposed method’s validity.

Photonic Generation of Chirp-Rate-Tunable Microwave Waveforms Using Temporal Cavity Solitons with Agile Repetition Rate

Chirp-rate-tunable microwave waveforms (CTMWs) with dynamically tunable parameters are of basic interest to many practical applications. Recently, photonic generation of microwave signals has made their bandwidths wider and more convenient for optical fiber transmission. An all-optical method for generation of multiband CTMWs is proposed and demonstrated on all-fiber architecture, relying on dual temporal cavity solitons with agile repetition rate. In the experiment, the triangular optical chirp microwave waveforms with bandwidth above 0.45 GHz (ranging from 1.45 GHz to 1.9 GHz) are obtained, and the chirp rate reaches 0.9 GHz/ms. The reconfigurability is also demonstrated by adjusting the control signal. This all-optical approach provides a technical basis for compact, multi-band reconfigurable microwave photonics transmission and reception systems.

First steps in the development of next-generation chirped volume Bragg gratings by means of fs laser inscription in fused silica.

The first steps towards the development and characterization of next-generation chirped volume Bragg gratings (CVBGs) by means of fs laser inscription were made. Based on the phase mask inscription technique we realized CVBGs in fused silica with a 3 × 3 mm2 aperture and a length of almost 12 mm with a chirp rate of ∼190 ps/nm around a central wavelength of 1030.5 nm. Strong mechanical stresses induced serious polarization and phase distortions of the radiation. We show a possible approach to solution of this problem. The change in the linear absorption coefficient associated with local modification of fused silica is quite small, enabling utilization of this type of gratings in high average power lasers.

Femtosecond laser fabrication of chirped and tilted fiber Bragg gratings for stimulated Raman scattering suppression in kilowatt-level fiber lasers.

Chirped and tilted fiber Bragg gratings (CTFBGs) are important all-fiber filtering components in high-power fiber lasers for stimulated Raman scattering (SRS) suppression. The fabrication of CTFBGs in large-mode-area double-cladding fibers (LMA-DCFs) by femtosecond (fs) laser is reported for the first time to the best of our knowledge. The chirped and tilted grating structure is obtained by scanning the fiber obliquely and moving the fs-laser beam relative to the chirped phase mask at the same time. By this method, the CTFBGs with different chirp rates, grating lengths, and tilted angles are fabricated, and the maximum rejection depth and bandwidth are ∼25 dB and ∼12 nm, respectively. To test the performance of the fabricated CTFBGs, one is inserted between the seed laser and the amplifier stage of a 2.7 kW fiber amplifier, and an SRS suppression ratio of ∼4 dB is achieved with no reduction in laser efficiency and degradation in beam quality. This work provides a highly fast and flexible method to fabricate large-core CTFBGs, which is of great significance to the development of high-power fiber laser systems.

Multi-band triangular frequency modulation signal generation based on gain-switched laser

A multi-band triangular frequency modulation signal generation method is proposed and verified by simulation and experiment. This is a simple photonics method based on distributed feedback gain switching states of semiconductor lasers without the need for complex external modulators. The carrier frequency and bandwidth of the generated triangular frequency modulation signal are N times that of the fundamental signal (N is the number of comb lines). By tuning the bandwidth and duration of the fundamental signal, the product of the number of frequency bands and the time bandwidth (TBWP) of the generated signal can be easily adjusted. The autocorrelation results of the generated waveform are also given. The triangular frequency modulation signals of six bands from S-band to K-band are realized, and have different chirp rates. The maximum TBWP reaches 24000, and the corresponding range resolution reaches 6.3 cm.

Reduced-Complexity Estimation of FM Instantaneous Parameters via Deep-Learning

Signal frequency estimation is a fundamental problem in signal processing. Deep learning is a fundamental method to solve this problem. This paper used five deep learning methods and three datasets including different singles Single Tone (ST), Linear- Frequency-Modulated (LFM), and Quadratic-Frequency-Modulated (QFM). This signal is affected by Additive White Gaussian (AWG) noise and Additive Symmetric alpha Stable (SαS) noise. Geometric SNR (GSNR) is used to determine the impulsiveness of noise in a Gaussian and SαS noise mixture. Deep learning methods are Gated Recurrent Unit (GRU), Long Short-Term Memory (LSTM), Bi-Direction Long Short-Term Memory (BiLSTM), and Convolution Neural Network (1D-CNN & 2D-CNN). When compared to a deep learning classifier with few layers to get on high accuracy and complexity reduces for Instantaneous Frequency (IF) estimation, Linear Chirp Rate (LCR) estimation, and Quadratic Chirp Rate (QCR) estimation. IF estimation of ST signals, IF and LCR estimation of LFM signals, and IF, LCR, and QCR estimation of QFM signals. The accuracy of the ST dataset in GRU is 58.09, LSTM is 46.61, BiLSTM is 45.95, 1D-CNN is 51.48, and 2D-CNN is 54.13. The accuracy of the LFM dataset in GRU is 82.89, LSTM is 66.28, BiLSTM is 20%, 1D-CNN is 74.79, and 2D-CNN is 98.26. The accuracy of the QFM dataset in GRU is 78.76, LSTM is 67.8, BiLSTM is 69.91, 1D-CNN is 75.8, and 2D-CNN is 98.2. The results show that 2D-CNN is better than other methods for parameter estimation in LFM signals and QFM signals, and the GRU is better for parameter estimation in ST signals.

Radar Maneuvering Target Detection Based on Product Scale Zoom Discrete Chirp Fourier Transform

Long-time coherent integration works to significantly increase the detection probability for maneuvering targets. However, during the observation time, the problems that often tend to occur are range cell migration (RCM) and Doppler frequency cell migration (DFCM), due to the high velocity and acceleration of the maneuvering target, which can reduce the detection of the maneuvering targets. In this regard, we propose a new coherent integration approach, based on the product scale zoom discrete chirp Fourier transform (PSZDCFT). Specifically, by introducing the zoom operation into the modified discrete chirp Fourier transform (MDCFT), the zoom discrete chirp Fourier transform (ZDCFT) can correctly estimate the centroid frequency and chirp rate of the linear frequency-modulated signal (LFM), regardless of whether the parameters of the LFM signal are outside the estimation scopes. Then, the scale operation, combined with ZDCFT, is performed on the radar echo signal in the range frequency slow time domain, to remove the coupling. Thereafter, a product operation is executed along the range frequency to inhibit spurious peaks and reinforce the true peak. Finally, the velocity and acceleration of the target estimated from the true peak position, are used to construct a phase compensation function to eliminate the RCM and DFCM, thus achieving coherent integration. The method is a linear transform without energy loss, and is suitable for low signal-to-noise (SNR) environments. Moreover, the method can be effectively fulfilled based on the chirp-z transform (CZT), which prevents the brute-force search. Thus, the method reaches a favorable tradeoff between anti-noise performance and computational load. Intensive simulations demonstrate the effectiveness of the proposed method.

The W transform with a chirp-modulated window

A high-resolution time–frequency spectrum is desirable for processing and interpreting seismic data. The W transform is a method that effectively preserves the resolution in the low-frequency region of the time–frequency spectrum of nonstationary seismic signals. To further increase the energy concentration of the time–frequency spectrum estimated with the W transform, we propose to combine the W transform with a chirp-modulated window. The chirp rate in the chirp-modulated window can control the rotation of the window in the time–frequency plane to achieve a better match with the time–frequency spectrum. The chirp rate in the modified W transform can be directly determined with the estimated instantaneous frequency. It has been shown that the W transform with a chirp-modulated window maintains the resolution of the time–frequency spectrum and improves the energy concentration around the dominant frequency against noise. To speed up the computational process of the W transform with a chirp-modulated window, we formulate the transform as a matrix-vector multiplication, which can be accelerated by using graphics processing unit computations. The application of the modified algorithm to synthetic and field data indicates that the frequency anomalies can be easily identified with the modified W transform.

Statistical Synchrosqueezing Transform and Its Application to Seismic Thin Interbed Analysis

Synchrosqueezing transform (SST) benefits from an instantaneous frequency (IF) estimator in the time-frequency domain, providing an energy-concentrated time-frequency representation to describe the time-varying frequency of seismic signals. To enhance the concentration performance of SST, this paper theoretically proposes a statistical SST (SSST) by constructing a spectrum-weighted IF estimator in the short-time Fourier transform (STFT) domain. In this SSST, a linear chirp signal is introduced to better capture its chirp rate, and then the quadratic STFT spectra are weighted by a window-related weighting function. On this basis, two equations related to the IF and chirp rate are constructed to derive the spectrum-weighted IF estimator. Finally, the STFT coefficients are squeezed to the estimated IF trajectories by a frequency fixed-point iterative algorithm, thereby providing a more concentrated time-frequency representation for multi-component signals than existing advanced methods, while enabling to retrieve each component. Two synthetic examples and one field seismic data on thin interbeds are utilized to demonstrate the effectiveness of the proposed SSST and show its ability to highlight the time-varying frequency features of seismic signals, which is a promising seismic data analysis tool, such as characterizing thin interbed thickness variations.

Disentangling modes with crossover instantaneous frequencies by synchrosqueezed chirplet transforms, from theory to application

Analysis of signals with oscillatory modes with crossover instantaneous frequencies is a challenging problem in time series analysis. One way to handle this problem is lifting the 2-dimensional time-frequency representation to a 3-dimensional representation, called time-frequency-chirp rate (TFC) representation, by adding one extra chirp rate parameter so that crossover frequencies are disentangled in higher dimension. The chirplet transform is an algorithm for this lifting idea, which leads to a TFC representation. However, in practice, we found that it has a strong “blurring” effect in the chirp rate axis, which limits its application in real-world data. Moreover, to our knowledge, we have limited mathematical understanding of the chirplet transform in the literature. Motivated by the need for the real-world data analysis, in this paper, we propose the synchrosqueezed chirplet transform (SCT) that enhances the TFC representation given by the chirplet transform. The resulting concentrated TFC representation has high contrast so that one can better distinguish different modes with crossover instantaneous frequencies. The basic idea is to use the phase information in the chirplet transform to determine a reassignment rule that sharpens the TFC representation determined by the chirplet transform. We also analyze the chirplet transform and provide theoretical guarantees of SCT.

On estimating parameters of a multi-component Chirp Model with equal chirp rates

Multi-component chirp signal models with equal chirp rates appear in various radar applications, e.g., synthetic aperture radar, echo signal of a rapid mobile target, etc. Many sub-optimal estimators have been developed for such models, however, these suffer from the problem of either identifiability or error propagation effect. In this paper, we have developed theoretical properties of the least squares estimators (LSEs) of the parameters of multi-component chirp model with equal chirp rates, where the model is contaminated with linear stationary errors. We also propose two computationally efficient estimators as alternative to LSEs, namely sequential combined estimators and sequential plugin estimators. Strong consistency and asymptotic normality of these estimators have been derived. Interestingly, it is observed that sequential combined estimator of the chirp rate parameter is asymptotically efficient. Extensive numerical simulations have been performed, which validate satisfactory computational and theoretical performance of all three estimators. We have also analysed a simulated radar data with the help of our proposed estimators of multi-component chirp model with equal chirp rates, which performs efficiently in recovery of inverse synthetic aperture radar (ISAR) image of a target from a noisy data.