Ambiguity Function Research Articles

The affine Wigner distribution

We examine the affine Wigner distribution from a quantization perspective with an emphasis on the underlying group structure. One of our main results expresses the scalogram as (affine) convolution of affine Wigner distributions. We strive to unite the literature on affine Wigner distributions and we provide the connection to the Mellin transform in a rigorous manner. Moreover, we present an affine ambiguity function and show how this can be used to illuminate properties of the affine Wigner distribution. In contrast with the usual Wigner distribution, we demonstrate that the affine Wigner distribution is never an analytic function. Our approach naturally leads to several applications, one of which is an approximation problem for the affine Wigner distribution. We show that the deviation for a symbol to be an affine Wigner distribution can be expressed purely in terms of intrinsic operator-related properties of the symbol. Finally, we present a positivity conjecture regarding the non-negativity of the affine Wigner distribution.

An Optimization Method of Ambiguity Function Based on Multi-Antenna Constrained and Application in Vehicle Attitude Determination.

The global navigation satellite system (GNSS)-based multi-antenna attitude determination method has the advantages of a simple algorithm and no error accumulation with time in long endurance operation. However, it is sometimes difficult to simultaneous obtain the fixed solutions of all antennas in vehicle attitude determination. If float or incorrect fixed solutions are used, precision and reliability of attitude cannot be guaranteed. Given this fact, a baseline-constrained ambiguity function method (BCAFM) based on a self-built four GNSS antennas hardware platform is proposed. The coordinates obtained by BCAFM can replace the unreliable real-time kinematic (RTK) float or incorrect fixed solutions, so as to assist the direct method for attitude determination. In the proposed BCAFM, the baseline constraint is applied to improve search efficiency (searching time), and the ambiguity function value (AFV) formula is optimized to enhance the discrimination of true peak. The correctness of the proposed method is verified by vehicle attitude determination results and baseline length difference. Experimental results demonstrate that the function values of error peaks are reduced, and the only true peak can be identified accurately. The valid epoch proportion increases by 14.95% after true peak coordinates are used to replace the GNSS-RTK float or incorrect fixed solutions. The precision of the three attitude angles is 0.54°, 1.46°, and 1.15°, respectively. Meanwhile, the RMS of baseline length difference is 3.8 mm.

A Recursive Inclusion Checker for Recursively Defined Subtypes

We present a tableaux procedure that checks logical relations between recursively defined subtypes of recursively defined types and apply this procedure to the problem of resolving ambiguous names in a programming language. This work is part of a project to design a new programming language suitable for efficient implementation of logic. Logical formulas are tree-like structures with many constructors having different arities and argument types. Algorithms that use these structures must perform case analysis on the constructors, and access subtrees whose type and existence depend on the constructor used. In many programming languages, case analysis is handled by matching, but we want to take a different approach, based on recursively defined subtypes. Instead of matching a tree against different constructors, we will classify it by using a set of disjoint subtypes. Subtypes are more general than structural forms based on constructors, we expect that they can be implemented more efficiently, and in addition can be used in static type checking. This makes it possible to use recursively defined subtypes as preconditions or postconditions of functions. We define the types and the subtypes (which we will call adjectives), define their semantics, and give a tableaux-based inclusion checker for adjectives. We show how to use this inclusion checker for resolving ambiguous field references in declarations of adjectives. The same procedure can be used for resolving ambiguous function calls.

Ambiguity Function Analysis for Orthogonal-LFM Waveform Based Multistatic Radar

Over the last decade, research on multistatic radar waveform design has attracted significant consideration among the radar system designers.Among the waveform design techniques, linear frequency modulated waveforms are widely used, but the problem associated with it is range Doppler coupling and complexity in achieving adequate pulse-to-pulse diversity. In this letter, an orthogonal linear frequency modulated (OLFM) waveform has been used as the transmitted waveforms, which has good correlation properties and large time bandwidth product. The ambiguity function (AF), which is the output of the matched filter, is used to analyze the target resolution capabilities of the OLFM waveforms. The mathematical expressions of the AF with OLFM waveforms for the multistatic architecture are represented based on the target fluctuations observed at the receiver. These expressions allow the ambiguity in resolving target position and velocity vectors for any transmitted waveform in the multistatic topology. The normalized cross-correlation value of the LFM waveforms is compared with the OLFM waveforms, which signify the isolation factor between two transmitted waveforms at the receiver. A performance comparison has been made based on the probability of target detection for various multistatic scenarios under numerous signal-to-noise ratio (SNR) conditions.

Logarithmic Frequency Waveforms

This article introduces a new class of Doppler-tolerant and estimation-capable radar waveforms called logarithmic frequency waveforms (LFWs). These waveforms are created by mapping the magnitudes, phases, and spacing of good digital radar codes onto a set of carriers defined over a logarithmically warped frequency axis. This mapping preserves the digital code's desirable autocorrelation mainlobe-to-sidelobe ratio when matched filtering is done over the warped frequency variable. The logarithm converts the multiplicative Doppler shift into an additive translation of the code's autocorrelation; thus, the location of the matched filter output's maximum is used to estimate the waveform's Doppler shift. This applies even if the code is neither inherently Doppler tolerant nor Doppler detection capable. We introduce two major classes of LFWs, logarithmic frequency codes, and logarithmic frequency rulers. We numerically illustrate their ambiguity functions and compare their Doppler estimation performance in additive white Gaussian noise against the linear chirp (linear frequency modulated) outfitted with the extended matched filter and the hyperbolic-frequency-modulated waveform.

Multidynamic Radar: A Concept for Strategic Air Surveillance System

Multi-Dynamic Radar (MDR) configuration is emerging as state of the art in the field of airborne radar where several airborne radars work in a coordinated manner. MDR systems provide better spatial diversity and resistance to other vulnerabilities in a combat scenario. This provides high performance for operational requirements compared to multi-static, bi-static or mono-static radar. For last few decades, stealth technology has been one of the most widely used approaches for target hide-out from radar-systems. This technology uses the concept of low observable principle to reduce the radar cross section (RCS) in the direction of radar. Earlier, ground based radars like bi-static-radar or multi-static-radar were used to counter such target hideout. Airborne radar has its own advantages for better visibility and coverage for airborne surveillance. In the proposed Multi-dynamic radar, the transmitters (Tx) and receivers (Rx) are placed on airborne platform such as aircraft or on un-manned air-vehicles. The operational challenges for MDR configuration is how to exploit the Tx-Rx sensors dynamics for benchmarking the performance. In this paper, the MDR system-geometry is considered as a set of bi-static-radar at a given time, and simulated with all possible airborne Tx-Rx pair combinations for evaluating MDR performance. The simulation is also done for a proposed orthogonal frequency-division-multiplexing linear- frequency-modulated (OFDM-LFM) based waveform design using a derived ambiguity function for the proposed MDR scenario. A design methodology is shown for an MDR to achieve phase synchronization across airborne Tx and Rx platform.

Time‐domain and frequency‐domain exponential ambiguity functions and their application in joint estimation of delay and Doppler shift

The classical ambiguity function (AF) in radar signal processing often has a low and fixed resolution in the presence of adjacent multi-targets. The frequency-domain exponential ambiguity function (FEAF) was previously presented with a variable exponent p introduced in the frequency-domain expression of the AF whose range resolution and output signal-to-noise ratio can be mutually replaced by controlling the exponent p. However, the FEAF mainly improves the range resolution of reference signals, but it cannot improve its velocity resolution. In this study, on the basis of symmetry theory in the time and frequency domains, a time-domain exponential ambiguity function (TEAF) was defined by indexing the time-domain expression of AF. The time-frequency properties of the defined TEAF are similar and symmetric to those of the FEAF. On the basis of the combination of the FEAF and TEAF with multi-exponents and a modified CLEAN algorithm for stepwise elimination and estimation of targets, the authors proposed a new joint estimation algorithm for delays and Doppler shifts of multi-targets. Simulation experiments indicated that the proposed estimation algorithm achieves higher accuracies in delay and Doppler shift estimation in comparison with the traditional AF-based estimation algorithm and the FEAF-based estimation algorithm.

A Novel Composite Coding Method for Incoherent Scatter Radar

Many modern ionospheric studies rely on incoherent scatter radars (ISR) since this kind of radar is able to detect various ionospheric parameters over very long ranges. The performance of ISR significantly depends on its coding system. In recent decades, a new type of coding system, which is the so-called composite coding, was presented. It used to be constructed by using a certain code to modulate alternating code to achieve better detection resolution and anti-noise performance for ISRs. In the present study, a new composite coding system was presented, which is constructed based on complementary codes and alternating codes. In this paper, the performance of the new composite code will be compared with that of several traditional codes to show that the new composite code can help to improve the detection performance of the ISR. According to the analysis based on the ambiguity function, the present composite coding system helps to improve the range resolution and detection range for ISR detections. In addition, numerical tests on anti-noise performance show that the complementary composite coding system has a good anti-noise performance and helps to reduce the necessary times of incoherent integration. As a result, the composite coding system can improve the time resolution.

Counter-Interception and Counter-Exploitation Features of Noise Radar Technology

In defense applications, the main features of radars are the Low Probability of Intercept (LPI) and the Low Probability of Exploitation (LPE). The counterpart uses more and more capable intercept receivers and signal processors thanks to the ongoing technological progress. Noise Radar Technology (NRT) is probably a very effective answer to the increasing demand for operational LPI/LPE radars. The design and selection of the radiated waveforms, while respecting the prescribed spectrum occupancy, has to comply with the contrasting requirements of LPI/LPE and of a favorable shape of the ambiguity function. Information theory seems to be a “technologically agnostic” tool to attempt to quantify the LPI/LPE capability of noise waveforms with little, or absent, a priori knowledge of the means and the strategies used by the counterpart. An information theoretical analysis can lead to practical results in the design and selection of NRT waveforms.

Photonic generation of a dual-chirp waveform with an optoelectronic oscillator based on stimulated Brillouin scattering.

A photonic method to generate a dual-chirp microwave waveform (DCMW) is proposed and demonstrated by utilizing a stimulated Brillouin scattering based optoelectronic oscillator and a frequency scanning laser source (FSLS). There are no radio frequency sources or Mach-Zehnder modulators in the proposed structure, which makes the system simple and stable. In the experiment, an alternative scheme is utilized to replace the FSLS, and the DCMW signal with a central frequency of 4.69 GHz is generated; the bandwidth of the generated DCMW signal is up to 7 GHz and the chirp rate up to ±5.3GHz/µs. The autocorrelation and ambiguity function of the generated signal are also investigated and show good performance of pulse compression and reduction of range-Doppler coupling.

Flexible Multi-band Dual-chirped Microwave Waveforms Generation with Double Bandwidth

ABSTRACT A multi-band dual-chirp linearly frequency modulated (LFM) signal-generation scheme with doubled chirp bandwidth and flexible center frequency and band numbers is proposed. In the scheme, the polarization multiplexed multi-tone optical signals with chirped phases are generated via a dual-polarization dual-parallel Mach–Zehnder modulator (DPMZM) cascaded with a polarization modulator. The two polarization multiplexed lightwaves are recombined as two linearly polarized beams by a polarization beam splitter and are detected by a balanced photodiode with a photocurrent that includes the multi-band dual-chirp LFM signal with doubled bandwidth and multiplied central frequencies but does not have a dc component. Since the orders and numbers of the generated optical sidebands are changed by adjusting the bias voltages of a DPMZM, we can obtain the multi-band dual-chirp LFM signal flexibly. Based on simulation, four-band, six-band and two seven-band dual-chirp LFM signals are generated with four groups of bias voltage configurations. For each case, every dual-chirp LFM signal of the generated multi-band dual-chirp LFM signal has a 5-GHz bandwidth and a time-bandwidth product of 512. The auto-correlations and ambiguity function of the generated LFM signal show good performance of pulse compression and range-Doppler resolution.

Design of QAM-OFDM Radar-Communication Integrated Signal Based on Golay Complementary sequences

The integration of radar and communication is conducive to improving the utilization rate of hardware resources and spectrum. To address the issue that the communication information in orthogonal frequency division multiplexing (OFDM) integrated signal for radar and communication affects detection performance, an OFDM integrated signal for radar and communication based on precoding was proposed. The communication information of the integrated signal was pre-coded by using the Gray Complementary Sequences (GCS) with better correlation performance to improve the performance of the ambiguity function of the integrated signal and its detection ability. Simulation results show that GCS have better correlation performance compared to M and Gold sequences, and the OFDM integrated signal pre-coded by GCS has lower Peak Sidelobes Ratio (PSLR) and comparable radar performance to LFM.

Компенсация нестабильности гетеродинов спутников-ретрансляторов для местоопределения наземных источников радиоизлучения

Unintentional and intentional interference of terrestrial radio sources operating via geostationary relay satellites to legal users of satellite communication systems requires accurate determination of their location. Methods of terrestrial radio sources location are based on the calculation of an cross-аambiguity function by additive mixtures of signals and noise received from relay satellites. In the presence of frequency-phase instability of relay satellites heterodynes the retransmitted signals have phase distortions, which lead to a decrease in the signal-to-noise ratio (SNR) when calculating the cross- ambiguity function. The paper is aimed to study the effect of phase distortions caused by the instability of relay satellites heterodynes on SNR at the correlator output and to develop methods for their compensation based on statistical radio engineering and digital signal processing. The study of the proposed compensation methods was carried out by statistical simulation modeling. The SNR dependences at the correlator output on the duration of correlated signals for the model with a domi-nant frequency noise and frequency random walk have been obtained and a method for compensat-ing phase distortions caused by the instability of the relay satellites heterodynes has been developed. The energy gain has been estimated by applying the proposed compensation method. It has been shown that the developed method of compensation of relay satellites heterodynes instability allows achieving a significant gain in the SNR at the correlator output and contributes to increasing the probability of radio source signal detection from auxiliary relay satellites.

Crossterm-free time-frequency representation exploiting deep convolutional neural network

Bilinear time-frequency (TF) analyses provide high-resolution time-varying frequency characterization of nonstationary signals. However, because of their bilinear natures, such TF representations (TFRs) suffer from crossterms. TF kernels, which amount to low-pass weighting or masking in the ambiguity function domain, are commonly used to reduce crossterms. However, existing fixed and adaptive kernels do not guarantee effective crossterm suppression and autoterm preservation, particularly for signals with overlapping autoterms and crossterms in the ambiguity function. In this paper, we develop a new method that offers high-resolution TFRs of nonstationary signals with desired autoterm preservation and crossterm mitigation capabilities, especially for signals with slowly time-varying instantaneous frequencies. The proposed method exploits a convolutional autoencoder network which is trained to construct crossterm-free TFRs. For the signals being considered, the proposed technique with properly trained networks offers the capability to outperform state-of-the-art TF analysis algorithms based on adaptive kernels and compressive sensing techniques.

Experimental Results of Underwater Acoustic Communication with Nonlinear Frequency Modulation Waveform.

In this paper, we propose underwater acoustic (UWA) communications using a generalized sinusoidal frequency modulation (GSFM) waveform, which has a distinct ambiguity function (AF) and correlation function characteristic. For these reasons, it is more robust in multipath channels than the conventional chirp spread spectrum (CSS) with a linear frequency modulation (LFM) waveform. Four types of GSFM waveforms that are orthogonal to each other are applied for each symbol in the proposed method. To evaluate the performance of the proposed method, we compared the performances of the proposed method and conventional method by conducting diverse experiments: simulations, lake trials and sea trials. In the simulation results, the proposed method shows better performance than the conventional method. The lake trial was conducted with a distance of 300~400 m between the transmitter and receiver. As a result of the experiment, the average bit error rate (BER) of the proposed method is and that of the conventional method is , which shows that the proposed method is superior to the conventional method. The sea trial was conducted at a distance of approximately 20 km between the transmitter and receiver at a depth of 1500 m, and the receiver was composed of 16 vertical line arrays (VLAs) with a hydrophone. The proposed method had a BER of in one channel and was error free in the other.

Chaos Based Frequency Modulation for Joint Monostatic and Bistatic Radar-Communication Systems

In this article, we propose the utilization of chaos-based frequency modulated (CBFM) waveforms for joint monostatic and bistatic radar-communication systems. Short-duration pulses generated via chaotic oscillators are used for wideband radar imaging, while information is embedded in the pulses using chaos shift keying (CSK). A self-synchronization technique for chaotic systems decodes the information at the communication receiver and reconstructs the transmitted waveform at the bistatic radar receiver. Using a nonlinear detection scheme, we show that the CBFM waveforms closely follow the theoretical bit-error rate (BER) associated with bipolar phase-shift keying (BPSK). We utilize the same nonlinear detection scheme to optimize the target detection at the bistatic radar receiver. The ambiguity function for both the monostatic and bistatic cases resembles a thumbtack ambiguity function with a pseudo-random sidelobe distribution. Furthermore, we characterize the high-resolution imaging capability of the CBFM waveforms in the presence of noise and considering a complex target.

The relative exponential growth rate of subgroups of acylindrically hyperbolic groups

Abstract We introduce a new invariant of finitely generated groups, the ambiguity function, and we prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate lim n → ∞ ⁡ | B H X ⁢ ( n ) | n \lim_{n\to\infty}\sqrt[n]{\lvert\vphantom{1_{1}}{B^{X}_{H}(n)}\rvert} of a subgroup 𝐻 of a finitely generated acylindrically hyperbolic group 𝐺 exists with respect to every finite generating set 𝑋 of 𝐺 if 𝐻 contains a loxodromic element of 𝐺. Further, we prove that the relative exponential growth rate of every finitely generated subgroup 𝐻 of a right-angled Artin group A Γ A_{\Gamma} exists with respect to every finite generating set of A Γ A_{\Gamma} .

Shedding Light on Primary Donors in Photosynthetic Reaction Centers.

Chlorophylls (Chl)s exist in a variety of flavors and are ubiquitous in both the energy and electron transfer processes of photosynthesis. The functions they perform often occur on the ultrafast (fs–ns) time scale and until recently, these have been difficult to measure in real time. Further, the complexity of the binding pockets and the resulting protein-matrix effects that alter the respective electronic properties have rendered theoretical modeling of these states difficult. Recent advances in experimental methodology, computational modeling, and emergence of new reaction center (RC) structures have renewed interest in these processes and allowed researchers to elucidate previously ambiguous functions of Chls and related pheophytins. This is complemented by a wealth of experimental data obtained from decades of prior research. Studying the electronic properties of Chl molecules has advanced our understanding of both the nature of the primary charge separation and subsequent electron transfer processes of RCs. In this review, we examine the structures of primary electron donors in Type I and Type II RCs in relation to the vast body of spectroscopic research that has been performed on them to date. Further, we present density functional theory calculations on each oxidized primary donor to study both their electronic properties and our ability to model experimental spectroscopic data. This allows us to directly compare the electronic properties of hetero- and homodimeric RCs.

A performance comparison between m-sequences and linear frequency-modulated sweeps for the estimation of travel-time with a moving source.

This manuscript discusses the utility of maximal period linear binary pseudorandom sequences [also referred to as m-sequences or maximum length sequences (MLSs)] and linear frequency-modulated (LFM) sweeps for the purpose of measuring travel-time in ocean-acoustic experiments involving moving sources. Signal design and waveform response to unknown Doppler (waveform dilation or scale factor) are reviewed. For this two-parameter estimation problem, the well-known wide-band ambiguity function indicates, and moving-source observations corroborate, a significant performance benefit from using MLS over LFM waveforms of similar time duration and bandwidth. The comparison is illustrated with a typical experimental setup of a source suspended aft of the R/V Sally Ride to a depth of∼10 m and towed at∼1 m/s speed. Accounting for constant source motion, the root mean square travel-time variability over a 30 min observation interval is 53 μs (MLS) and 141 μs (LFM). For these high signal-to-noise ratio channel impulse response data, LFM arrival-time fluctuations mostly appear random while MLS results exhibit structure believed to be consistent with source (i.e., towed transducer) dynamics. We conclude with a discussion on signal coherence with integration times up to 11 MLS waveform periods corresponding to ∼27 s.

Convolution separation and application of joint diagonalization with optimal parameters on mechanical signals

Blind separation algorithm has the problem of separation stability in the condition of multiple convolution signals. Based on cluster analysis, combined with the optimal distance matrix and the optimal window, the joint diagonalization is improved, which can effectively improve the stability and accuracy of the signal separation. The full frequency divergence is set as the objective function of permutation ambiguity in the convolution separating process, the failure from convoluted blind source separation is solved. Combining with new joint diagonalization and convolution separation is to form a system approach, and is applied to the floor signal from the actual bench, the influence of excitation source in frequency is achieved, the system algorithm can be used as a reference of mechanical vibration analysis.