Complex Analytic Signal Research Articles

The unified theory of n-dimensional complex and hypercomplex analytic signals

The unified theory of n-dimensional complex and hypercomplex analytic signals

The New Insight into the Theory of 2-D Complex and Quaternion Analytic Signals

The New Insight into the Theory of 2-D Complex and Quaternion Analytic Signals

Extraction of chemical information from complex analytical signals by a non-negative independent component analysis

Extraction of chemical information of the components from a complex analytical signal has been a great challenge in chemometrical studies for complex sample analysis. Independent component analysis (ICA) has been widely applied in complex signal separation, including the multicomponent overlapping signals in analytical chemistry. Difficulties, however, still exist in the application of ICA in chemical signal processing because chemical signals of different components are generally correlated and non-negative, instead of independence as hypothesized in ICA. In this study, a non-negative ICA method is proposed by means of a post rotation of the independent components (ICs) and applied to the extraction of the chemical information of the components from the signals of complex samples. Raman spectra of pharmaceutical tablets and gas chromatography-mass spectrometry (GC-MS) data of cigarette smoke are qualitatively analyzed. The results show that the Raman spectrum of the active substance in the pharmaceutical tablets and the mass spectra of the components in the overlapping GC-MS signal can be effectively and accurately extracted by using the proposed method.

Estimation of the location and depth parameters of 2D magnetic sources using analytical signals

This study presents a different approach to the use of analytic signals of magnetic profile data in the estimation of the polygonal cross-sectional geometry of 2D bodies. The proposed approach uses a technique which aims to estimate the horizontal and vertical coordinates of the corners of a 2D body. To realize this aim, the first step is to estimate the horizontal coordinates of the corners by using the peak amplitudes of the second or higher horizontal derivatives of the analytical signal. The next step is to determine the depths of corners by using a complex formulation of the analytic signal in the frequency domain. In the second step, the complex analytical signal in the frequency domain is solved by an exponential approximation method. Model studies on triangular, trapezoidal and pentagonal cross-sectional bodies yield satisfactory results. To check the performance and applicability conditions, the technique was also applied to magnetic data obtained from the Cataldere magnetite ore deposit in the Bala district of Turkey. The noisy field data prevented us from obtaining more successful results, as in the case of synthetic data applications.

Complex analytic signals applied on time delay estimation

In this paper, we present the concept of the time delay estimation based on the transformation of real sensor signals into analytic ones. We analyze the differential time delay values obtained using real seismic signals, simulated complex analytic signals and simulated complex analytic signals with real parts coming from real seismic signals. The simulation results indicate that the application of complex analytic signals leads to reliable computation of the differential time delay. The influence of specific signal parameters on spectral coherence threshold in systems for passive localization and proposed methods for lowering the threshold is analyzed. The computation of all differential time delays with respect to the reference sensor (geophone) is based on the application of Generalized Cross-Correlation (GCC) applied on corresponding analytic signals. The difficulties to select a peak of cross-correlation function that corresponds to true differential time delay when dealing with real signals are significantly reduced if GCC is applied on analytic signals. The efficiency of the proposed technique on differential delay estimation is performed on deterministic and real-life signals.

A neural network approach based on gold‐nanoparticle enzyme biosensor

AbstractGold nanoparticles have demonstrated to be a very useful material for the construction of stable and sensitive glucose oxidase (GOx) amperometric biosensors. However, as for other enzyme electrodes, the lack of specificity for glucose limits their practical applications. Coupling biosensor responses with chemometric tools can be used to solve complex analytical signals from mixtures of species with similar properties. In this work, an amperometric biosensor based on a colloidal gold—cysteamine—gold disk electrode with the enzyme GOx and a redox mediator, tetrathiafulvalene (TTF), co‐immobilised atop the modified electrode, was used for the simultaneous determination of glucose and its common interferences, ascorbic acid and uric acid, in mixtures. Analytical data obtained from cyclic voltammograms generated with the biosensor were processed using an artificial neural network (ANN), and the separate quantification of the analytes over a range of 0.1–1 mM each was performed without any pretreatment. In all cases, the correlation coefficients obtained were higher than 0.99 and the mean prediction error was less than 1.7%. Copyright © 2007 John Wiley & Sons, Ltd.

A complex orthogonal decomposition for wave motion analysis

A method is presented for decomposing wave motion into its principle components. The basic idea is a complex generalization of proper orthogonal decomposition. The method involves the representation of real oscillatory signals as complex analytic signals. The relationship between complex modes and wave motion is explored. From an ensemble of complex signals, a complex correlation matrix is formed, and its complex eigensolution is the basis of the decomposition (like a complex singular value decomposition). The complex eigenvectors contain standing and traveling characteristics. A traveling index is proposed to quantify the relative degree of traveling and standing in a waveform. A method of dissecting a wave mode into its traveling and standing parts is also proposed. From the complex modes and modal coordinates, frequencies, wavelengths, and characteristic wave speeds can be obtained. The method is applied to traveling and standing-wave examples.

Representing sound energy, phase, and interference using three-dimensional signals

As graphic representations of vibrations/waves, sound signals capture only selected attributes of the phenomenon they represent. Assuming equivalence between signals and sound waves obscures the fact that 2-D signals are unfit to a) represent wave-energy quantities consistently across frequencies, b) account for the phase flip and alternating positive/negative amplitude values in modulated waves with AM-depth >100%, and c) represent the energy content of interference. The proposed sound-signal representation is based on the complex equation of motion describing a wave. It results in spiral sine signals and twisted-spiral complex signals, similar to complex analytic signals, with the imaginary component of the complex equation of motion representing the signal envelope’s argument (phase). Spiral sine signals offer a consistent measure of sine-wave energy across frequencies, while twisted-spiral complex signals account for the negative amplitudes observed in modulated signals, mapping the modulation parameters onto the twisting parameters. In terms of interference, 3-D signals illustrate that amplitude fluctuations and the signal envelopes that describe them are not just boundary curves but waves that trace changes in the total instantaneous energy of a signal over time, representing the oscillation between potential and kinetic energies within a wave. Examples of 3-D animations illustrating the proposed signals are presented.

Reply to the discussion

First, we thank Xiong Li for drawing to the attention of readers our lapse in not referencing Miller and Singh (1994) as the originators of the tilt angle. He also correctly points out that Miller and Singh (1994) first stated the automatic gain control (AGC) nature of the tilt angle. Thurston and Smith (1997) introduce the local wavenumber, from Nabighian's (1972) complex analytic signal, as an edge detection and characterisation tool, and are also worthy of reference.

Optical vortex metrology for nanometric speckle displacement measurement

As an alternative to correlation-based techniques widely used in conventional speckle metrology, we propose a new technique that makes use of phase singularities in the complex analytic signal of a speckle pattern as indicators of local speckle displacements. The complex analytic signal is generated by vortex filtering the speckle pattern. Experimental results are presented that demonstrate the validity and the performance of the proposed optical vortex metrology with nano-scale resolution.

Instantaneous frequencies of signals obtained by the analytic signal method

In this paper, the instantaneous frequency is defined as the one obtained by converting a real time signal to a complex analytic signal and by differentiating the time-dependent phase with respect to time. Theoretical expressions of instantaneous frequencies for signals given as combinations of sinusoidal components are presented. Those are compared to the results obtained by numerical methods using the discrete Fourier transform in order to confirm the validity of the expressions and to check accuracies of the numerical methods. A reason for the existence of negative instantaneous frequencies is given by a vector representation of signal components. The instantaneous frequencies of frequency- and amplitude-modulated signals are also discussed. For a periodic sinusoidal burst signal, it was found that the instantaneous frequency stays zero during the period when the signal is zero and takes a value equal to twice that of the sinusoid at the onset of the signal.

Propagation properties of ultrashort pulsed beams with constant waist width in free space

Based on the Rayleigh diffraction integral and complex analytical signal representation, the strict propagation expression and its Fourier spectrum for ultrashort pulsed Gaussian beams with constant waist width are derived, which have the general applicable advantage, because the paraxial approximation, pulsed envelope representation and far-field approximation are not invoked. The far-field and paraxial expressions are obtained as the special cases of our general results. Numerical calculation results are given to illustrate the spatiotemporal propagation properties of few- and single-cycle pulsed Gaussian beams with constant waist width. A comparison with the paraxial results and previous work is also made.

Propagation properties of ultrashort chirped pulsed Gaussian beams in the free space

Based on the Rayleigh diffraction integral and complex analytical signal representation, the free-space analytical propagation equation and its Fourier spectrum for ultrashort chirped pulsed Gaussian beams with constant diffraction length are derived. The effect of chirp parameter on the spatiotemporal and spectral properties is illustrated with analytical formulas and numerical calculation results. It is shown that the axial spectra of ultrashort chirped pulsed Gaussian beams become broadened with increasing chirp parameter. For single optical cycle, the transversal intensity distribution is affected by increasing chirp parameter, but almost not affected for several optical cycles. Moreover, the positive or negative sign of the chirp parameter has no effect on the spectral distribution and intensity distribution.

Pseudophase information from the complex analytic signal of speckle fields and its applications Part II: Statistical properties of the analytic signal of a white-light speckle pattern applied to the microdisplacement measurement

To provide a theoretical background for the superiority of the signal-domain phase-only correlation (SDPOC) technique proposed here for microdisplacement measurement, we study the first- and the second-order statistical properties of the complex amplitude of an analytic signal of a white-light speckle pattern, under the assumption of a Gaussian random process, and give a formula for the autocorrelation function of the pseudophase associated with the complex analytic signal. Based on these results, we show mathematically that SDPOC has a performance advantage over conventional intensity-based correlation techniques.

Pseudophase information from the complex analytic signal of speckle fields and its applications Part I: Microdisplacement observation based on phase-only correlation in the signal domain

As an alternative to the intensity correlation used in conventional speckle metrology, we propose a new technique for displacement observation that is based on spatial-signal-domain phase-only correlation and that makes use of the pseudophase of the complex analytic signal generated from a Hilbert-filtered speckle pattern. Experimental results are presented that demonstrate the validity and the advantage of the proposed signal-domain phase-only correlation technique over both the conventional intensity correlation technique and the spatial-frequency-domain phase-only correlation technique.

Comparison of Lorentz pulsed beams obtained by using complex amplitude envelope representation and complex analytic signal representation

By using complex amplitude envelope representation and complex analytic signal r epresentation, propagation expressions of Lorentz pulsed beams are derived. Prop agation properties of them with different bandwidths are illustrated by numerica l calculations. It is found that they are the same in the neighbourhood of z _axis for narrowband pulses, but in the case of broadband pulses, singularities emerge near z_axis for pulsed beams obtained by using complex amplitude enve lope representation and only complex analytic signal representation is applicabl e. The conditions to decide what representation should be chosen to study pulsed beams are given. And the origin that results in the emergence of singularity is given.

The far-field properties of ultrashort pulsed beams with constant waist width in free space

By using the angular spectrum and complex analytic signal representations, the analytic propagation expression and its Fourier spectrum for ultrashort Gaussian pulsed beams with constant waist width in the far field are derived. The far-field properties of few- and single-cycle ultrashort Gaussian pulsed beams with constant waist width are studied, and compared with those of ultrashort Gaussian pulsed beams with constant diffraction length. The properties of pulse broadening, spectrum narrowing, off-axis spectrum redshift and on-axis spectrum blueshift are found for ultrashort Gaussian pulsed beams with constant waist width in the far field.

Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement

Rather than regarding the phase singularities as obstacles or nuisances in phase unwrapping, we explore new possibilities of making use of the phase singularities in optical metrology. Instead of intensity correlation techniques used in conventional speckle metrology, we propose a new technique of displacement measurement that makes use of the density of phase singularities in the complex analytic signal of the speckle pattern, which is generated by Hilbert filtering. Experimental results and theoretical analysis are presented that demonstrate the validity of the proposed technique.

Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam

Analytic propagation expressions of pulsed Gaussian beam are deduced by using complex amplitude envelope representation and complex analytic signal representation. Numerical calculations are given to illustrate the differences between them. The results show that the major difference between them is that there exists singularity in the beam obtained by using complex amplitude envelope representation. It is also found that singularity presents near propagation axis in the case of broadband and locates far from propagation axis in the case of narrowband. The critical condition to determine what representation should be adopted in studying pulsed Gaussian beam is also given.

Digital time-delay beamforming with interpolated signals

In this paper, a general treatment of digital time-delay beamforming with interpolated signals is given. It covers beamforming for low-pass and band-pass signals, and popular digital beamformer structures are included in this approach. Two different topics are combined in the investigation: the first is reconstruction of complex envelope and analytic band-pass signals from a finite number of samples, the second topic applies the first to digital beamforming. Closed form expressions for the digital interpolation beamformer output and its beam-pattern are derived, and the effect of sampling and interpolation is discussed. Numerical simulations underline the power of the general approach of choosing the right sampling rate and interpolation method to realize digital beamformers with very little distortion of their beam-pattern.