Instantaneous Cross-correlation Research Articles

Free Metaplectic Wigner Distribution: Definition and Heisenberg’s Uncertainty Principles

Inspired by a definition of the closed-form instantaneous cross-correlation Wigner distribution (Zhang in IEEE Trans Signal Process 67(21):5527–5542, 2019), we generalize the notion of Wigner distribution to the so-called free metaplectic Wigner distribution (FMWD) through three free metaplectic transforms, in order to tackle a challenge in high-dimensional complex features information processing. We provide some representative special cases for this general form, including the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -dimensional nonseparable affine characteristic Wigner distribution, kernel function Wigner distribution, convolution representation Wigner distribution and instantaneous cross-correlation Wigner distribution. We establish the standard Heisenberg’s uncertainty principles (HUPs) of the real-valued function for the FMWD. We also establish the standard HUPs of the complex-valued function for some specific (i.e., the orthogonal, the orthonormal, the minimum eigenvalue commutative and the maximum eigenvalue commutative) FMWDs. In view of the one-dimensional case of our results, we solve a burning question regarding the limit of time-frequency superresolution triggered by the linear canonical transform free parameters.

Instantaneous cross-correlation function-[formula omitted]-Wigner distribution: Theory and application

To raise the noisy linear frequency-modulated (LFM) signal detection threshold without sacrificing the computing performance, this study employs the well-known τ-Wigner distribution (τ-WD) in extending further the notion of the instantaneous cross-correlation function WD (ICFWD) to the so-called ICF-τ-WD. We extend some essential properties of the ICFWD to those of the ICF-τ-WD. We analyze the computational complexities of the ICFWD and τ-WD, based on which we derive the computational complexity of the ICF-τ-WD. We address an important concern regarding the ICF-τ-WD’s optimal parameters selection for the noisy single component LFM signal processing, in accordance with the output signal-to-noise ratio (SNR) optimization models including the single-objective (expectation-SNR) optimization model and the double-objective (expectation-SNR and variance-SNR) optimization model. We also provide numerical simulations for verifying the advantage of the double-objective optimization model over the single-objective one and the ICF-τ-WD over the ICFWD in the improvement of detection performance.

A Computationally Efficient Optimal Wigner Distribution in LCT Domains for Detecting Noisy LFM Signals

Recently, Wigner distribution (WD) associated with linear canonical transforms (LCTs) is quickly becoming a promising technique for detecting linear frequency-modulated (LFM) signals corrupted with noises by establishing output signal-to-noise ratio (SNR) inequality model or optimization model. Particularly, the closed-form instantaneous cross-correlation function type of WD (CICFWD), a unified linear canonical Wigner distribution, has shown to be competitive in detecting noisy LFM signals under an extremely low SNR. However, the CICFWD has up to nine LCT free parameters so that it requires a heavy computational load. To improve the efficiency of real-time processing, this paper focuses on the instantaneous cross-correlation function type of WD (ICFWD), which has only six LCT free parameters but is not a special case of the CICFWD. The main advantage of ICFWD is that it could be expected to reduce the computational complexity while maintaining detection performance. This paper first proposes an optimization model to the ICFWD’s output SNR with respect to deterministic signals embedded in additive zero-mean noises. It then deduces the model’s solution to a single component LFM signal added with white noise, leading to the optimal selection strategy on LCT free parameters. Simulation results demonstrate that the ICFWD improves almost a doubling of computing speed in comparison with the CICFWD while sharing the same level of detection performance. To be specific, the computing time of ICFWD in sampling frequencies 5 Hz, 10 Hz, 15 Hz, and 20 Hz is about 0.048 s, 0.111 s, 0.226 s, and 0.392 s, respectively, while 0.075 s, 0.233 s, 0.478 s, and 0.821 s for the computing time of CICFWD; the ICFWD and CICFWD have nearly the same output SNR higher than that of the WD.

Instantaneous cross-correlation function type of WD based LFM signals analysis via output SNR inequality modeling

Linear canonical transform (LCT) is a powerful tool for improving the detection accuracy of the conventional Wigner distribution (WD). However, the LCT free parameters embedded increase computational complexity. Recently, the instantaneous cross-correlation function type of WD (ICFWD), a specific WD relevant to the LCT, has shown to be an outcome of the tradeoff between detection accuracy and computational complexity. In this paper, the ICFWD is applied to detect noisy single component and bi-component linear frequency-modulated (LFM) signals through the output signal-to-noise ratio (SNR) inequality modeling and solving with respect to the ICFWD and WD. The expectation-based output SNR inequality model between the ICFWD and WD on a pure deterministic signal added with a zero-mean random noise is proposed. The solutions of the inequality model in regard to single component and bi-component LFM signals corrupted with additive zero-mean stationary noise are obtained respectively. The detection accuracy of ICFWD with that of the closed-form ICFWD (CICFWD), the affine characteristic Wigner distribution (ACWD), the kernel function Wigner distribution (KFWD), the convolution representation Wigner distribution (CRWD) and the classical WD is compared. It also compares the computing speed of ICFWD with that of CICFWD, ACWD, KFWD and CRWD.

Linear canonical Wigner distribution of noisy LFM signals via variance-SNR based inequalities system analysis

In consideration of the latest second-order moment variance type of signal-to-noise ratio (SNR), the so-called variance-SNR, a new inequality model, namely, the variance-SNR inequality between the closed-form instantaneous cross-correlation function type of Wigner distribution and the classical Wigner distribution, is introduced to characterize the detection performance improvement triggered by linear canonical transform (LCT) free parameters embedded. Combining the newly established variance-SNR inequality model with the previous output SNR inequality model gives a stronger inequalities system. Solving it with respect to noisy one-component linear frequency-modulated signals yields more accurate strategies on determination of parameters. Numerical simulations also verify the correctness and rationality of the results of the theoretical deduction, implying that there is a better noise suppression effect by using the newly derived parameters selection strategies.

Linear Canonical Wigner Distribution of Noisy LFM Signals via Multiobjective Optimization Analysis Involving Variance-SNR

The existing output signal-to-noise ratio (SNR) optimization model seems not accurate enough to characterize the closed-form instantaneous cross-correlation function type of Wigner distribution’s optimum detection performance, resulting in an unsatisfactory optimal linear canonical transform free parameters selection strategy. A more appropriate output SNR definition, the so-called variance-SNR, is then proposed to model a new kind of optimization, the variance-SNR optimization model. By integrating these two optimization models, it follows a multiobjective optimization, whose solutions to noisy single component linear frequency-modulated signals are also deduced. The newly derived strategy on determination of the optimal parameters implies the previous one so that there is a better detection performance in terms of noise suppression.

Linear Canonical Wigner Distribution Based Noisy LFM Signals Detection Through the Output SNR Improvement Analysis

In our previous work, we addressed a unified representation problem on Wigner distribution (WD) in linear canonical transform (LCT) domains by introducing a kind of closed-form instantaneous cross-correlation function type of Wigner distribution (CICFWD) that can be regarded as the WD's closed-form representation in the linear canonical domain. We then discussed the application of CICFWD in the detection of linear frequency-modulated (LFM) signals through the numerical simulation analysis approach, and it turns out that there is a causation between LCT free parameters and the detection performance. The main contribution of this paper is to provide a mathematical model for this causality in order to disclose the intrinsic mechanism of non-stationary signal detection performance improvement triggered by LCT free parameters. We first revisit the definition of CICFWD, and establish some related theories including its essential properties and discretization. We then formulate a mathematical theory foundation of the output signal-to-noise ratio (SNR) improvement, an output SNR inequality relation between the traditional WD and the CICFWD for a general noisy signal, and discuss how to solve this inequality to come out LCT free parameters. Within the well-established output SNR improvement analysis framework, we also explore LCT free parameters selection results on LFM signals added with white noise. Finally, numerical experiments are carried out to validate the correctness of strategies on determination LCT free parameters and the feasibility of output SNR improvement analysis approach.

Choi–Williams distribution in linear canonical domains and its application in noisy LFM signals detection

Due to the requirement of signal representation’s flexibility for strong noise interference in non-stationary signal processing, the traditional Choi–Williams distribution (CWD) is extended to the linear canonical domain by using a well-established closed-form instantaneous cross-correlation function (CICF) type of linear canonical transform (LCT) free parameters embedded approach. The derived CICF type of CWD (CICFCWD) unifies all of the linear canonical Choi–Williams distributions, and can be considered as the CWD’s closed-form representation in linear canonical domains. The CICFCWD of a discrete-time signal defined by the sampling equation is discussed for the requirement of digital signal analysis and processing. To break through the limitation that the existing research techniques are limited to quantitative analysis based on numerical simulations, a qualitative analysis to output signal-to-noise ratio (SNR) inequality between the traditional CWD and the CICFCWD for a general noisy signal is introduced, including the inequality modeling and solving. Within this output SNR improvement analysis framework, the strategies on determination LCT free parameters for linear frequency-modulated (LFM) signals added with white noise are developed. Numerical experiments are also carried out to validate the correctness of LCT free parameters selection results and the feasibility of output SNR improvement analysis approach.

Response Coordination Emerges in Cooperative but Not Competitive Joint Task.

Effective social interactions rely on humans’ ability to attune to others within social contexts. Recently, it has been proposed that the emergence of shared representations, as indexed by the Joint Simon effect (JSE), might result from interpersonal coordination (Malone et al., 2014). The present study aimed at examining interpersonal coordination in cooperative and competitive joint tasks. To this end, in two experiments we investigated response coordination, as reflected in instantaneous cross-correlation, when co-agents cooperate (Experiment 1) or compete against each other (Experiment 2). In both experiments, participants performed a go/no-go Simon task alone and together with another agent in two consecutive sessions. In line with previous studies, we found that social presence differently affected the JSE under cooperative and competitive instructions. Similarly, cooperation and competition were reflected in co-agents response coordination. For the cooperative session (Experiment 1), results showed higher percentage of interpersonal coordination for the joint condition, relative to when participants performed the task alone. No difference in the coordination of responses occurred between the individual and the joint conditions when co-agents were in competition (Experiment 2). Finally, results showed that interpersonal coordination between co-agents implies the emergence of the JSE. Taken together, our results suggest that shared representations seem to be a necessary, but not sufficient, condition for interpersonal coordination.

Energy flow between two hydrodynamically coupled particles kept at different effective temperatures

We measure the energy exchanged between two hydrodynamically coupled micron-sized Brownian particles trapped in water by two optical tweezers. The system is driven out of equilibrium by random-forcing the position of one of the two particles. The forced particle behaves as it has an “effective temperature” higher than that of the other bead. This driving modifies the equilibrium variances and cross-correlation functions of the bead positions: we measure an energy flow between the particles and an instantaneous cross-correlation, proportional to the effective temperature difference between the two particles. A model of the interaction which is based on classical hydrodynamic coupling tensors is proposed. The theoretical and experimental results are in excellent agreement.

Leak location in gas pipelines using cross-time–frequency spectrum of leakage-induced acoustic vibrations

The correlation-based location methods are widely used in leak detection of the pipelines assuming that the acoustic speed has been known and constant. In practice, the acoustic speed is frequency-varying due to the dispersions of gas-leak-induced acoustic waves, and thus the assumption is not supported. In this work, a location scheme based on cross-time–frequency spectrum (CTFS) is intended for the gas-leak-induced acoustic waves with frequency-varying acoustic speed. In the scheme, the CTFS is obtained by the one-dimensional Fourier transform of the time domain convolution between the kernel function in correlation domain and the instantaneous cross-correlation of the two spatially separately collected acoustic signals on either sides of a leakage. Then, the time difference of arrival (TDOA) and the corresponding frequency information are extracted simultaneously when the CTFS reaches the maximum value. The resulting peak frequency is used to online determine the frequency-dependant acoustic speed in combination with the known dispersive curve of gas-leak-induced dominated mode. Finally, the gas leakage is located by the TDOA and the frequency-dependant acoustic speed of real-time determination instead of constant acoustic speed. Consequently, for the proposed scheme, the constant acoustic speed is no longer a prerequisite. The proposed scheme has been experimentally validated in leak detection of gas pipelines and results demonstrate that the average relative location errors are reduced by six times compared with the commonly used correlation-based location method.

Hydrochemical changes in a reservoir that receives water contaminated by acid mine drainage

The aim was to characterize hydrochemical changes that take place in the Sancho reservoir, which receives waters contaminated by acid mine drainage (AMD) from the Meca River (Iberian Pyrite Belt, SW Spain). The sampling procedure comprised bimonthly campaigns carried out between October 2011 and May 2012, in the entrance of the reservoir and in the input river waters. In addition to in situ parameters, heavy metals and sulphate were analysed in the laboratory by photometry and atomic absorption spectrometry, respectively. The database was submitted to statistic treatment through the STATGRAPHICS Centurion XVI.I software. The results show a clear relationship between the sampling site in the entrance of the reservoir and in the watercourse, with an almost instantaneous cross-correlation. At the beginning of the sampling period a small rise in pH at both sites was detected, coinciding with a reduction in the concentration of heavy metals. This attenuation results from the dilution process caused by rain as well as the sorption by iron oxyhydroxysulphates that precipitate as a consequence of the above-mentioned rise in pH. The time changes that were detected in this system are due to the influence of input waters from the waste dumps of the Tharsis mines.

Dynamic Structure of Joint-Action Stimulus-Response Activity

The mere presence of a co-actor can influence an individual’s response behavior. For instance, a social Simon effect has been observed when two individuals perform a Go/No-Go response to one of two stimuli in the presence of each other, but not when they perform the same task alone. Such effects are argued to provide evidence that individuals co-represent the task goals and the to-be-performed actions of a co-actor. Motivated by the complex-systems approach, the present study was designed to investigate an alternative hypothesis — that such joint-action effects are due to a dynamical (time-evolving) interpersonal coupling that operates to perturb the behavior of socially situated actors. To investigate this possibility, participants performed a standard Go/No-Go Simon task in joint and individual conditions. The dynamic structure of recorded reaction times was examined using fractal statistics and instantaneous cross-correlation. Consistent with our hypothesis that participants responding in a shared space would become behaviorally coupled, the analyses revealed that reaction times in the joint condition displayed decreased fractal structure (indicative of interpersonal perturbation processes modulating ongoing participant behavior) compared to the individual condition, and were more correlated across a range of time-scales compared to the reaction times of pseudo-pair controls. Collectively, the findings imply that dynamic processes might underlie social stimulus-response compatibility effects and shape joint cognitive processes in general.

Detection of high‐speed and accelerated target based on the linear frequency modulation radar

The high-speed movement of a target may cause range migration and Doppler frequency migration of the radar echo, which has a serious impact on the detection performance of the radar. To resolve the problem of detecting a high-speed target in linear frequency modulation radar, this study analyses the effect on the integration gain caused by range migration and Doppler frequency migration, and proposes a corresponding compensation method according to the different input signal-to-noise ratios (SNRs) of the echo signal. To compensate for range migration in high SNRs, two-dimensional median filtering and constant false alarm rate technology are combined to estimate the speed. For low SNRs, based on coarse valuations, the authors use the discrete Fourier transformation (DFT) to realise the fractional delay cell to improve speed accuracy. Furthermore, to compensate for Doppler frequency migration, an instantaneous cross-correlation method is proposed for high SNRs, which is combined with the fractional Fourier transform method to estimate the acceleration for low SNRs. The input SNR threshold for the different algorithms is then analysed using simulation data, and the theoretical reference value is shown. Finally, the study verifies the effectiveness of the proposed methods through simulation and measured data.

Memory effect and multifractality of cross-correlations in financial markets

We introduce an instantaneous and an average instantaneous cross-correlation function to detect the temporal cross-correlations between individual stocks based on the daily data of the United States and the Chinese stock markets. The memory effect of the instantaneous cross-correlations is investigated by applying the detrended fluctuation analysis (DFA), where the DFA exponents can be partly explained by the correlation function from the common sense. Long-range memory is observed for the average instantaneous cross-correlations, and persists up to a month magnitude of timescale for the United States stock market and half a month magnitude of timescale for the Chinese stock market. In addition, multifractal nature is investigated by a multifractal detrended fluctuation analysis.

Sequential cross‐correlation using higher‐order exponential smoothing method

This paper proposes the sequential cross-correlation function and the sequential cross-correlation coefficient, in which higher-order smoothing is applied to the instantaneous cross-correlation function, providing a basis for the determination of the nonstationary cross-correlation function. Compared to the SSCC (short-span cross-correlation) method, the proposed method gives a smoother transition of the correlation value and expresses the change of the correlation function with a fast and adequate response. This is a property suited to on-line application. It is shown by simulation that the proposed method is better than the SSCC method in terms of detection of change of the correlation function, as well as in noise elimination. It is also shown that there exists an adequate combination of the order of the exponential smoothing and the smoothing constant. Lastly, real data on an economic process are handled, and it is shown that the proposed method can detect the change of the cross-correlation function more effectively than the SSCC method. © 1999 Scripta Technica, ElectronComm Jpn Pt 3, 82(11): 58–64, 1999

Sequential cross-correlation using higher-order exponential smoothing method

This paper proposes the sequential cross-correlation function and the sequential cross-correlation coefficient, in which higher-order smoothing is applied to the instantaneous cross-correlation function, providing a basis for the determination of the nonstationary cross-correlation function. Compared to the SSCC (short-span cross-correlation) method, the proposed method gives a smoother transition of the correlation value and expresses the change of the correlation function with a fast and adequate response. This is a property suited to on-line application. It is shown by simulation that the proposed method is better than the SSCC method in terms of detection of change of the correlation function, as well as in noise elimination. It is also shown that there exists an adequate combination of the order of the exponential smoothing and the smoothing constant. Lastly, real data on an economic process are handled, and it is shown that the proposed method can detect the change of the cross-correlation function more effectively than the SSCC method. © 1999 Scripta Technica, ElectronComm Jpn Pt 3, 82(11): 58–64, 1999

Effect of large eddies on atmospheric surface layer turbulence and the underlying wave field

To improve existing models for air‐sea interaction, a better understanding of the energy transfer across the boundary layer and in particular of the coupling of large atmospheric eddies with the air‐sea interface is needed. Recent investigations have already shown a possible coupling of large structures in atmospheric turbulence and surface ripples. This was done using synthetic aperture radar (SAR) imagery of ocean surface and almost simultaneous advanced very high resolution radiometer (AVHRR) imagery of cloud streets at a cold‐air outbreak. The intent of our study is to further validate this hypothesis in a general case of coastal circulation. For this purpose we analyze a suite of collocated simultaneous records of airflow, radar return, and surface elevations from a coastal platform. We investigate the influence of large eddies (20–60 min) on the turbulent properties of the airflow in the first 2 m above the ocean surface. The analysis shows very prominent peaks in the magnitude of 12‐ to 16‐min fluctuations which are further modulated in 20‐ to 40‐min intervals. These scales and modulations are characteristic for all variables of interest here. The detected scales and their modulation suggest significant interaction of surface layer within the first 1–2 m with large eddies of scales of O(1) and O(10) km. The intermittent structure of turbulence responds by alternating contributions from bursts and sweeps; the frequency of occurrence of bursts and sweeps also reveals the influence of large structures. The instantaneous cross correlation between the shorter scales of momentum flux and radar return, corresponding to the individual burst events, can be 4 times as strong as the overall cross correlation.

Time-Frequency and Ambiguity Function Approaches in Structural Identification

In the identification and control of some structures it is important to be able to resort to special techniques that may exploit environmental excitation in normal serviceability conditions. Hence, preliminary signal processing must be designed to extract significant data, even in the presence of unknown, weak, or transient excitations. In the processing and representation of nonstationary dynamic signals, great importance has been taken on in the literature by the transforms in the joint time-frequency domain. In this context, a method based on the time-frequency representation and some properties of the ambiguity function is proposed to identify the natural frequencies, draw the modal shapes, and detect nonlinearity in such testing conditions. To this end, the performances of the new Choi-Williams exponential kernel are compared with those of the Wigner-Ville transform. Finally techniques based on the instantaneous cross-correlation function are proposed to improve the performance of the identification method, and an application to a real bridge is discussed.

Intermittency of the noise emission in subsonic cold jets

Intermittency of the noise emission has been investigated by means of measurements of instantaneous cross-correlation between the far field acoustic pressure and the velocity component in the jet towards the microphone (pressure signal allowing for the acoustic propagation time). The noise is found to be associated with active periods of the second time derivative of the velocity signal. A conditional sampling procedure with a criterion based on the amplitude of this derivative, has made it possible to show that 50% of the noise is produced during only 10 to 20% of the time. The sections of the jet considered are located between 4 and 10 diameters from the nozzle, a region in which most of the averaged jet noise originates. An attempt to establish a relationship with the large scale structures of the jet has also been made. Examination of various time traces seems to indicate that the noise emission would take place at the upstream side of the large structures where most of the non-turbulent fluid is being caught up and rapidly entrained.