Complex Analytic Signal Research Articles

Three-dimensional sound signals and their relevance to wave energy quantities and sound interference products

Signals are graphic representations of vibrations/waves and, like every representation, capture only selected attributes of the phenomenon they are meant to represent. The often assumed equivalence between signals and sound waves obscures the fact that two-dimensional signals are not fit to (a) represent wave-energy quantities consistently across frequencies, (b) account for the alternating positive/negative amplitude values of modulated waves with AM-depth>100%, and (c) represent the energy content of interference. An alternative sound-signal representation is proposed, 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. 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 and map 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. a)Work completed while at the Department of Ethnomusicology, University of California, Los Angeles.

Manifestation of classical wave delays in a fully quantized model of the scattering of a single photon

We consider a fully quantized model of spontaneous emission, scattering, and absorption, and study propagation of a single photon from an emitting atom to a detector atom both with and without an intervening scatterer. We find an exact quantum analog to the classical complex analytic signal of an electromagnetic wave scattered by a medium of charged oscillators. This quantum signal exhibits classical phase delays. We define a time of detection which, in the appropriate limits, exactly matches the predictions of a classically defined delay for light propagating through a medium of charged oscillators. The fully quantized model provides a simple, unambiguous, and causal interpretation of delays that seemingly imply speeds greater than c in the region of anomalous dispersion.

Competition between two frequencies for phase synchronization of a chaotic laser.

Competition between two distinct driving frequencies to phase synchronize the intensity dynamics of a chaotic laser has been observed. The phase of the chaotic intensity signal is constructed using the complex analytic signal. Competing frequencies alternately show phase locking and phase slipping. Competition has been quantified by calculating the portion of time the laser phase locks to each of the driving frequencies and their average.

The production and elimination of the spatial singularity for ultrashort chirp ed pulse-beam propagation in free space

We study the propagation of ultrashort chirped pulsed beams in free space. We find that the slowly-varying envelope approximation and complex amplitude envelope are not suitable for ultrashort chirped pulsed beams anymore, because they lead to the spatial singularity that is inconsistent with the physical significanc e of the beam propagation. And the chirp influences the spatial singularity mor e seriously than the width of the pulsed beam does. The spatial singularity can be eliminated by using the complex analytical signal theory. In this paper, we g ive the simulations of spatial singularity about ultrashort chirped pulsed Gauss ian beams with a series of parameters, and the conditions that lead to the spati al singularity etc. We think that the broad spectrum of the ultrashort pulsed be am is the original reason for the spatial singularity.

Spatiotemporal behavior of ultrashort pulses in the far field

By using the complex analytic signal representation and far-field approximation, a solution of the wave equation in free space is found, which enables us to study the spatiotemporal behavior of ultrashort pulses in the far field. The consistency of our result with the previous work is interpreted. Some interesting phenomena of ultrashort pulses propagating in free space, such as the pulse broadening, spectrum narrowing and redshifting, etc., are illustrated with nonsinusoidal Poisson-type pulses.

A multimodel method for depth estimation from magnetic data

The local wavenumber and a multimodel wavenumber are complex attributes derived from a complex analytic signal. These quantities have been used to interpret anomalies arising from contacts, thin sheets, and horizontal cylinders. A new multimodel wavenumber can be used for computing depths of 2-D thick dikes and 2-D sloping steps. These two multimodel wavenumbers have been incorporated into a depth-estimation algorithm based on automatic curve matching. This algorithm works on profile data and has three appealing features: (1) the most appropriate of these five models is selected automatically; (2) the automatic curve matching uses a least-squares technique to reject responses that do not conform to the model assumptions; and (3) interference from distant sources can be accounted for as a base-level shift of the multimodel wavenumber curves. Applying the automatic technique to survey data from the Western Canada sedimentary basin yields four thick dikes between 3400 and 4300 m below sensor. These depths are equivalent to 2.2 and 3.1 km below sea level, which is consistent with the basement depths derived from drillhole information. Using these solutions as a starting point in an iterative forward modeling exercise, the measured data were explained with a geologically reasonable model.

Vector analysis of the propagation of ultrashort pulsed beams in free space

By using a Fourier transform technique and applying the paraxial approximation in the frequency domain, we derive an integral solution for the transverse and longitudinal components of the light field of an ultrashort pulsed beam propagating in free space. In particular, on the basis of the theory of complex analytical signals, the solutions for the components of a pulsed Gaussian-like beam, in rectangular coordinates, are derived and compared with those derived from slowly varying envelope approximation (SVEA). The corresponding numerical investigations are carried out and show that for sub-cycle pulses, the SVEA will cause spatial singularity for both transverse and longitudinal components of the light field.

INVALIDATION OF THE SLOWLY-VARYING ENVELOPE APPROXIMATION AND PRODUCING AND ELIMINATING OF THE SPATIAL SINGULARITY FOR ULTRASHORT PULSED-BEAM PROPAGATION IN FREE SPACE

In this paper we study the propagation of the single cycle or much shorter pulsed-beam in free space. We show the invalidation of the slowly-varying envelope approximation (SVEA) theory because it leads to the spatial singularity of the ultrashort pulsed-beam solution and is inconsistent with the physical significance of the beam propagation. We eliminate the spatial singularity by making use of the complex analytical signal theory (CAS), and then give the solution and simulation of the Gauss, hyperbolic secant and Lorentz pulse beam for SVEA and CAS. The SVEA and CAS solutions are consistent if the pulsed beam becomes long enough.

Space-time description of photon emission from an atom

Starting from the postulate that the electromagnetic field appearing in the transverse set of microscopic Maxwell-Lorentz equations governing field-matter interactions, properly normalized, can be looked upon as describing one-photon-emission and -absorption processes in space and time, a first-quantized initiation of photon emission by a single atom is presented. The wave function for the emerging photon is introduced as a six-vector object constructed from the complex analytical signals of the Riemann-Silberstein vectors belonging to opposite photon helicities. When the atom is no longer electrodynamically active, the emitted photon is described in first quantization by the so-called energy wave function well known for photons in free space. From the momentum representation of the emerging photon wave function a condition on the analytical part of the transverse atomic current density is established which ensures that precisely one photon is emitted. A propagator description of the emerged photon dynamics in the coordinate representation is established. The photon propagator is introduced as a two-component spinor, where upper and lower tensor components are constructed, respectively, from positive and negative helicity combinations of the propagators describing the time-space evolution of the transverse electric and magnetic fields. It is shown that the emission region for the photon coincides with the region in space where the transverse atomic current density is nonvanishing. For a photon emitted in an electric dipole transition the emission region essentially is the near-field zone of the atom, and this zone therefore determines the initial (and best) spatial confinement of the photon. The photon emerging from an atom active for a finite time necessarily is of the polychromatic sort and the associated wave packet essentially is confined between spherical shells moving outwards with the vacuum speed of light. To illustrate the main principles of the fundamental theory in a heuristic fashion we apply it to a study of the emission of a one-photon sinusoidal wavetrain from a pointlike atom. It is found that the atomic current density needed to create just one photon is independent of the oscillation period in the train and thus depends only on the number of periods in the wave train. An explicit expression for the one-photon energy is derived, and it is shown that only for extremely short pulse trains pronounced deviations from the textbook result, $E=\ensuremath{\Elzxh}{\ensuremath{\omega}}_{0},$ occur. The radial energy flow in the coupled atom-photon system in the near-field zone of the atom is investigated, and the cycle-averaged outwards energy transport carried by the emerging photon in a given distance from the atom is determined.

Short-scan interferometric interrogation and multiplexing of fibre Bragg grating sensors

We describe an interferometric technique for the demodulation of serial fibre Bragg grating sensor arrays, yielding absolute measurement of the individual grating mean wavelengths. The composite beam reflected from the array illuminates a scanning Michelson interferometer but, in contrast to spectral measurement by Fourier Transform Spectroscopy, our technique requires an OPD scan far shorter than the coherence length of the grating reflections. The technique is based on the high-resolution measurement of the phases of the complex analytic signals and of the relationship of these phases to interferometric delay. The analytic signals are derived via the Hilbert transform, allowing frequency domain filtering of individual signals within the signal processing. The technique has yielded a resolution of 0.007 nm for an OPD scan of 1.2 mm.

Vocal microtremor assessed by complex demodulation

Vocal microtremor refers to small involuntary fluctuations of fundamental frequency (f0) and sound intensity during attempted constant phonation. Physically such an utterance may be considered as a FM/AM process x(t) with f0 as carrier frequency (after filtering the acoustic signal around f0 to eliminate its harmonics): x(t)=a(t)*cos(f0t+φ(t)) and f(t)=d[φ(t)]/dt. To study the instantaneous changes of f(t) and a(t) as well as their mutual relationships, complex demodulation was applied. First, the complex analytic signal y(t) was formed using the Hilbert operator H: y(t)=x(t)+jH[x(t)]=a(t)*ej[f0t+φ(t)]. Then demodulation was performed by multiplying y(t) with the negative complex carrier, thus shifting the narrow-band spectral content around f0 to zero: z(t)=y(t)*e−jf0t=a(t)*ejφ(t). At least the instantaneous quantities a(t) and f(t) could be recovered from z(t). The analysis was implemented with matlab functions including procedures to estimate modulation indices and the cross-correlation coefficient of the demodulated signals. Both waveforms showed periodic as well as irregular patterns and were positively correlated (about 0.5), indicating physiological or mechanical coupling in the sound generating mechanisms. This analysis might be further useful to study musical vibrato as well as laryngeal and neurological disorders.

Automatic conversion of magnetic data to depth, dip, and susceptibility contrast using the SPI (TM) method

The Source Parameter Imaging (SPI™) method computes source parameters from gridded magnetic data. The method assumes either a 2-D sloping contact or a 2-D dipping thin‐sheet model and is based on the complex analytic signal. Solution grids show the edge locations, depths, dips, and susceptibility contrasts. The estimate of the depth is independent of the magnetic inclination, declination, dip, strike and any remanent magnetization; however, the dip and the susceptibility estimates do assume that there is no remanent magnetization. Image processing of the source‐parameter grids enhances detail and provides maps that facilitate interpretation by nonspecialists. The SPI method tests successfully on synthetic profile and gridded data. SPI maps derived from aeromagnetic data acquired over the Peace River Arch area of northwestern Canada correlate well with known basement structure and furthermore show that the Ksituan Magmatic Arc can be divided into several susceptibility subdomains.

Linear phase cosine modulated maximally decimated filter banks with perfect reconstruction

We propose a novel way to design maximally decimated FIR cosine modulated filter banks, in which each analysis and synthesis filter has a linear phase. The system can be designed to have either the approximate reconstruction property (pseudo-QMF system) or perfect reconstruction property (PR system). In the PR case, the system is a paraunitary filter bank. As in earlier work on cosine modulated systems, all the analysis filters come from an FIR prototype filter. However, unlike in any of the previous designs, all but two of the analysis filters have a total bandwidth of 2/spl pi//M rather than /spl pi//M (where 2M is the number of channels in our notation). A simple interpretation is possible in terms of the complex (hypothetical) analytic signal corresponding to each bandpass subband. The coding gain of the new system is comparable with that of a traditional M-channel system (rather than a 2M-channel system). This is primarily because there are typically two bandpass filters with the same passband support. Correspondingly, the cost of the system (in terms of complexity of implementation) is also comparable with that of an M-channel system. We also demonstrate that very good attenuation characteristics can be obtained with the new system.

In pursuit of accuracy: Nomenclature, assumptions, and standards

Abstract

Speech intelligibility measurements sing TEF analysis

Recent investigations into the measurement of speech intelligibility using large listener groups (three groups of 30 each) in a cathedral, a concert hall, and a classic motion picture theater of the early 1930s suggest that a listener's subjective intelligibility score will be best matched in objective testing by dividing the sound considered to be direct sound from sound considered to be reverberant sound at the highest level return within the first 50 ms. This, in a majority of cases, results in using only the first arrival and no integration of early sound. Evidence will be presented that indicates that the use of impulse squared measurements that fail to process the signal as a complex analytic signal often fail to properly record major reflective signals unless extensive spatial averaging is resorted to. The paper will be supported by actual postprocessing of data taken with a Techron TEF analyzer using the Heyser technique for energy time curve measurement. In such measurements, minor movements of the microphone dramatically affect the impulse and doublet responses while the same change barely affects the energy time curve measurement, thus demonstrating the need for the complex analytic signal in such analysis.

Statistical Analysis of Slow-Drift Responses

The statistical properties of second-order wave-induced response processes are investigated theoretically. Emphasis is placed on the slow-drift components. The assumed forcing waves are irregular with continuous frequency spectra. A spectral analysis of the response of a general system is made. It is shown that the slow-drift components are closely connected to the complex analytical signal and the Hilbert envelope of the wave elevation. A simple mathematical expression exists for the slow-drift components, based on the complex wave signal and the second-order impulse response of the system. By use of this explicit formula, the theoretical probability functions of slow-drift responses are investigated. The analysis is based on the Kac-Siegert method. A similar approach has earlier been applied to study the sum of both the low-frequency and the high-frequency second-order responses. Final calculations of the probability density functions are in general very complicated, but it can be simplified by the use of a simple idealized model for the second-order transfer function. Probability density curves for a few simple cases are presented.

Detection of Coherent Optical Signals in Generalized Gaussian Noise

Results of receiver performance for the detection of a coherent light signal in the presence of background radiation are obtained and compared for two different types of background noise: a Gaussian field and a Gaussian field whose complex analytic signal is not circularly symmetric. It is pointed out that the randomness in the noise radiation does not significantly diminish the probability of correct detection.

Model for successive determination of aluminium and fluoride by combination of fluorimetric and potentiometric methods

The paper represents a non-conventional approach to the successive determination of aluminium and fluoride in the same sample. It is based on the mathematical correction of a complex analytical signal for aluminium in the presence of fluoride (obtained by the fluorescence method). Additionally, a potentiometric determination of the fluoride content is necessary to complete the analysis. A mathematical model to describe the dependence between the complex analytical signal and the concentrations of aluminium and fluoride for a wide range of concentrations of both species is obtained. The approach seems to be useful for continuous monitoring of samples containing both aluminium and fluoride such as tooth paste, cryolite, industrial waste waters etc.

Analogue implementation of analytic signal processing for pulse-echo systems

An alternative to rectification is proposed for detection of an ultrasonic signal. This method is especially useful in medical and non-destructive evaluation (nde) applications. With this method, the magnitude of the complex analytic signal is used to define the envelope of the ultrasonic waveform. The square of this quantity has been shown elsewhere to be equal to the true rate-of-arrival of energy. An earlier study, using digital data processing, has already demonstrated the superior resolvability of closely spaced interfaces obtained with the analytic signal magnitude, as compared to conventional rectification. Here, an analogue implementation is presented which utilizes single-sideband techniques to obtain both quadrature components of the analytic signal and its magnitude. A conventional transducer, pulser, and receiver are used.

Improved ultrasonic detection using the analytic signal magnitude

An alternative to rectification is proposed for detection of an ultrasonic signal for medical or non-destructive evaluation applications. With this technique the envelope of the ultrasonic waveform is obtained by calculation of the magnitude of the complex analytic signal. An implementation, particularly suited for digital data processing, is presented which utilizes a fast Fourier transform for computation of the quadrature component of the received signal using the Hilbert transform. An ordinary transducer, pulser, and receiver are used together with a digital data acquisition system. The results of this processing are compared with conventional rectification, both with and without smoothing. The superior resolvability of two closely spaced interfaces on the A-mode presentation is clearly evident. Similar improvement in B-mode quality will result when an image is required.