Simple Sinusoid Research Articles

Analytical Calculation of the DC-Link Current Harmonics in Active Power Filter Considering Coupling Effect

Current harmonic is an important factor in power converters' DC-link design. Its accurate derivation is conducive to estimating the reliability of DC-link capacitors. In active power filter (APF), due to its wide output bandwidth, the DC link suffers current harmonics on a broader spectrum, and the coupling effects among compensated harmonics make the derivation in this application more complex. However, the traditional methods in voltage source inverters (VSIs) usually assume the output current to be a simple sinusoid, or only calculate the root mean square (RMS) values. They cannot readily illustrate the coupling effect, nor discriminate between different modulation strategies. This paper presents a general method for APF to solve the DC-link current harmonics under any operation condition. Using Double-Fourier theory, a new variable called ‘harmonic factor’ is defined to describe the coupling effect among compensated harmonics. Then the DC-link current spectrum is quantitatively derived with the low-frequency compensated harmonics and the switching-frequency ripple currents. Finally, the simulation and experiment of a 2-level 100-A APF verify the theoretical analysis.

Identifying Higher-Order Interactions in Wave Time-Series

Abstract Reliable design and reanalysis of coastal and offshore structures require, among other things, characterization of extreme crest elevation corresponding to long return periods. Extreme crests typically correspond to focused wave events enhanced by wave–wave interactions of different orders—third-order, four-wave interactions dominating in deep water (Janssen, P. A. E. M., 2003, “Nonlinear Four-Wave Interactions and Freak Waves,” J. Phys. Oceanogr., 33(4), pp. 863–884). Higher-order spectral (HOS) analysis can be used to identify wave–wave interactions in time-series of water surface elevation; trispectral analysis is needed to detect third-order, four-wave interactions. Four-wave interactions between Fourier components can involve interactions of the type where f1 + f2 + f3 = f4 and where f1 + f2 = f3 + f4, resulting in two definitions of the trispectrum—the T- and V-trispectrum (with corresponding tricoherences), respectively. It is shown that the T-tricoherence is capable of detecting phase-locked four-wave interactions of the type f4 = f1 + f2 + f3 when these are simulated with simple sinusoids, but such interactions were not detected in HOS model simulations and field data. It is also found that high V-tricoherence levels are detected at frequencies at which four-wave interactions of the type f1 + f2 = f3 + f4 are expected, but these may simply indicate combinations of independent pairs of Fourier components that happen to satisfy the frequency relationship. Preliminary analysis shows that using a cumulant-based trispectrum (Kravtchenko-Berejnoi, V., Lefeuvre, F., Krasnosel'skikh, V. V., and Lagoutte, D., 1995, “On the Use of Tricoherent Analysis to Detect Nonlinear Wave–Wave Interactions,” Signal Process., 42(3), pp. 291–309) may improve identification of wave–wave interactions. These results highlight that caution needs to be exercised in interpreting trispectra in terms of specific four-wave interactions occurring in sea states and further research is needed to establish whether this is in fact possible in practice.

Practical Computation of FFT-based Generalized Two-dimensional Correlation Spectroscopy

A practical computation of fast Fourier transformation (FFT) based generalized two-dimensional (2D) correlation spectroscopy is described. Using simple sinusoids, we tested and confirmed that the method served effectively and properly, invariant to the changes of the number of data points of the time profiles. This computation is applicable to any type of waveforms in a versatile manner.

A Lower Limb Phantom for Simulation and Assessment of Electromyography Technology.

Electromyography signal processing approaches have traditionally been validated through computer simulations. Electromyography electrodes and systems are often not validated or have been validated on human subjects where there is no clear ground truth signal for comparison. We sought to develop a physical limb phantom for validation of electromyography hardware and signal processing approaches. We embedded pairs of wires within a conductive gelatin surrounding an artificial bone such that the antennae could broadcast identified ground truth signals. The ground truth signals can be simple sinusoids or more complex representations of muscle activity. With the phantom and surface electromyography electrodes, we were able to show varying levels of crosstalk between nearby recording electrodes as we altered the amplitude of the antennae signals. We were also able to induce motion artifacts in our recordings by lightly dropping the phantom on a surface while antennae broadcast signals. High-density electromyography recordings of the trials showed that traditional filtering techniques fail to fully eliminate relatively small motion artifacts. The results suggest that the electrical limb phantom could be a valuable tool for testing potential effects of muscle crosstalk and motion artifacts on different electromyography systems and signal processing approaches.

Multisensory perceptual interactions between higher-order temporal frequency signals.

Naturally occurring signals in audition and touch can be complex and marked by temporal variations in frequency and amplitude. Auditory frequency sweep processing has been studied extensively; however, much less is known about sweep processing in touch because studies have primarily focused on the perception of simple sinusoidal vibrations. Given the extensive interactions between audition and touch in the frequency processing of pure tone signals, we reasoned that these senses might also interact in the processing of higher-order frequency representations like sweeps. In a series of psychophysical experiments, we characterized the influence of auditory distractors on the ability of participants to discriminate tactile frequency sweeps. Auditory frequency sweeps systematically biased the tactile perception of sweep direction. Importantly, auditory cues exerted little influence on tactile sweep direction perception when the sounds and vibrations occupied different absolute frequency ranges or when the sounds consisted of intensity sweeps. Thus, audition and touch interact in frequency sweep perception in a frequency- and feature-specific manner. Our results demonstrate that audio-tactile interactions are not constrained to the processing of simple sinusoids. Because higher-order frequency representations may be synthesized from simpler representations, our findings imply that multisensory interactions in the temporal frequency domain span multiple hierarchical levels in sensory processing. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Autocorrelation-based, passive, non-contact, photoplethysmography: Computationally-efficient, noise-tolerant, extraction of heart rates from video

Photoplethysmography (PPG) is the detection of blood flow or pressure by optical means. The most common method involves direct skin-contact measurement of light from an LED. However, the small color changes in skin under normal lighting conditions, as recorded by conventional video, potentially allow passive, noncontact, PPG. Eulerian Video Magnification (EVM) was used to demonstrate that small color changes in a subject’s face can be amplified to make them visible to a human observer. A variety of methods have been applied to extract heart rate from video. The signal obtained by PPG is not a simple sinusoid, but has a relatively complex structure, which in video is degraded by ambient lighting variations, motion, noise, and a low sampling rate. Although EVM and many other analysis methods in the literature essentially operate in the frequency domain, fitting the video data to their model requires extensive preprocessing. In this paper a time-based autocorrelation method is applied directly to the video signal that exhibits superior noise rejection and resolution for detecting quasi-periodic waveforms. The method described in the current work avoids both the preprocessing computational cost and the potential signal distortions.

Echo Shaping Using Sums of Damped Complex Sinusoids

Feedback delay lines are the basis of myriad audio effects and reverberation schemes. The feedback delay line, by itself, is limited to producing an infinite sequence of exponentially decaying echoes. We introduce a new type of linear time-invariant echo effect whose impulse response is a generalized sum of damped complex sinusoids. This permits echo responses to be shaped as simple sinusoids, Fourier-based waveforms, and many other complex, possibly nonperiodic, patterns that would not be feasible with other existing methods. Additionally, because the response is complex-valued, it can be used to produce auto-panning echo effects with many kinds of spatial trajectories. The effect is computationally efficient and straightforward to implement, as it only requires a parallel combination of feedback delay lines.

Numerical Study of the Effect of Lip Surface Roughness on Lubrication of Radial Shaft Seals with a Simple Sinusoidal Model

Numerical Study of the Effect of Lip Surface Roughness on Lubrication of Radial Shaft Seals with a Simple Sinusoidal Model

Multistability of twisted states in non-locally coupled Kuramoto-type models

A ring of N identical phase oscillators with interactions between L-nearest neighbors is considered, where L ranges from 1 (local coupling) to N/2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sign which has a profound influence on its behavior. Without the limitation of the generality, the frequency of the free-running oscillators can be set to zero. The resulting system is of gradient type, and therefore, all its solutions converge to an equilibrium point. All so-called q-twisted states, where the phase difference between neighboring oscillators on the ring is 2πq/N, are equilibrium points, where q is an integer. Their stability in the limit N → ∞ is discussed along the line of Wiley et al. [Chaos 16, 015103 (2006)] In addition, we prove that when a twisted state is asymptotically stable for the infinite system, it is also asymptotically stable for sufficiently large N. Note that for smaller N, the same q-twisted states may become unstable and other q-twisted states may become stable. Finally, the existence of additional equilibrium states, called here multi-twisted states, is shown by numerical simulation. The phase difference between neighboring oscillators is approximately 2πq/N in one sector of the ring, -2πq/N in another sector, and it has intermediate values between the two sectors. Our numerical investigation suggests that the number of different stable multi-twisted states grows exponentially as N → ∞. It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discrete-time translational dynamical system where the space-variable (position on the ring) plays the role of time. The q-twisted states are then fixed points, and the multi-twisted states are periodic solutions of period N that are close to a heteroclinic cycle. Due to the apparently exponentially fast growing number of such stable periodic solutions, the system shows spatial chaos as N → ∞.

VARIATIONS IN THE AXISYMMETRIC TRANSPORT OF MAGNETIC ELEMENTS ON THE SUN: 1996-2010

We measure the axisymmetric transport of magnetic flux on the Sun by cross-correlating narrow strips of data from line-of-sight magnetograms obtained at a 96 minute cadence by the MDI instrument on the ESA/NASA SOHO spacecraft and then averaging the flow measurements over each synodic rotation of the Sun. Our measurements indicate that the axisymmetric flows vary systematically over the solar cycle. The differential rotation is weaker at maximum than at minimum. The meridional flow is faster at minimum and slower at maximum. The meridional flow speed on the approach to the Cycle 23/24 minimum was substantially faster than it was at the Cycle 22/23 minimum. The average latitudinal profile is largely a simple sinusoid that extends to the poles and peaks at about 35° latitude. As the cycle progresses, a pattern of inflows toward the sunspot zones develops and moves equatorward in step with the sunspot zones. These inflows are accompanied by the torsional oscillations. This association is consistent with the effects of the Coriolis force acting on the inflows. The equatorward motions associated with these inflows are identified as the source of the decrease in net poleward flow at cycle maxima. We also find polar countercells (equatorward flow at high latitudes) in the south from 1996 to 2000 and in the north from 2002 to 2010. We show that these measurements of the flows are not affected by the nonaxisymmetric diffusive motions produced by supergranulation.

Array of Josephson junctions with a nonsinusoidal current-phase relation as a model of the resistive transition of unconventional superconductors

An array of resistively and capacitively shunted Josephson junctions with nonsinusoidal current-phase relation is considered for modeling the transition in high-Tc superconductors. The emergence of higher harmonics, besides the simple sinusoid Ic sin ϕ, is expected for dominant d-wave symmetry of the Cooper pairs, random distribution of potential drops, dirty grains, or nonstationary conditions. We show that additional cosine and sine terms act, respectively, by modulating the global resistance and by changing the Josephson coupling of the mixed superconductive-normal states. First, the approach is applied to simulate the transition in disordered granular superconductors with the weak-links characterized by nonsinusoidal current-phase relation. In granular superconductors, the emergence of higher-order harmonics affects the slope of the transition. Then, arrays of intrinsic Josephson junctions, naturally formed by the CuO2 planes in cuprates, are considered. The critical temperature suppression, observed at values of hole doping close to p=1/8, is investigated. Such suppression, related to the sign change and modulation of the Josephson coupling across the array, is quantified in terms of the intensities of the first and second sinusoids of the current-phase relation. Applications are envisaged for the design and control of quantum devices based on stacks of intrinsic Josephson junctions.

The Analysis of Simple and Complex Auditory Signals in Human Auditory Cortex: Magnetoencephalographic Evidence From M100 Modulation

Ecologically valid signals (e.g., vowels) have multiple components of substantially different frequencies and amplitudes that may not be equally cortically represented. In this study, we investigate a relatively simple signal at an intermediate level of complexity, two-frequency composite tones, a stimulus lying between simple sinusoids and ecologically valid signals such as speech. We aim to characterize the cortical response properties to better understand how complex signals may be represented in auditory cortex. Using magnetoencephalography, we assessed the sensitivity of the M100/N100m auditory-evoked component to manipulations of the power ratio of the individual frequency components of the two-frequency complexes. Fourteen right-handed subjects with normal hearing were scanned while passively listening to 10 complex and 12 simple signals. The complex signals were composed of one higher frequency and one lower frequency sinusoid; the lower frequency sinusoidal component was at one of the five loudness levels relative to the higher frequency one: -20, -10, 0, +10, +20 dB. The simple signals comprised all the complex signal components presented in isolation. The data replicate and extend several previous findings: (1) the systematic dependence of the M100 latency on signal intensity and (2) the dependence of the M100 latency on signal frequency, with lower frequency signals ( approximately 100 Hz) exhibiting longer latencies than higher frequency signals ( approximately 1000 Hz) even at matched loudness levels. (3) Importantly, we observe that, relative to simple signals, complex signals show increased response amplitude-as one might predict-but decreased M100 latencies. : The data suggest that by the time the M100 is generated in auditory cortex ( approximately 70 to 80 msecs after stimulus onset), integrative processing across frequency channels has taken place which is observable in the M100 modulation. In light of these data models that attribute more time and processing resources to a complex stimulus merit reevaluation, in that our data show that acoustically more complex signals are associated with robust temporal facilitation, across frequencies and signal amplitude level.

Monte Carlo Simulations for Tomlinson Sliding Models for Non-Sinusoidal Periodic Potentials

It is shown that the velocity dependence of a tungsten tip sliding against a mica surface cannot be fit to a semi-empirical analytical solution of the Tomlinson/Prandtl model using a simple sinusoidal sliding potential. This could be due to invalid assumptions in the model itself. However, if it is assumed that the periodic sliding potential is much sharper than a simple sinusoid, quantitative agreement between the experimental velocity dependence of the sliding force and theory is obtained using a single variable parameter, the height of the surface potential. Sliding is modeled in this case using Monte Carlo theory, and it is found that the height of the potential varies linearly with the normal load.

Incoherent Method of Optical Interference Cancellation for Radio-Frequency Communications

In this paper, we describe a system for cancelling radio-frequency (RF) interference using optical techniques. Specifically, we attempt to receive a weak RF signal-which we assume to be of the order of microwatts-in the presence of high-power local RF interference. This local interference is a signal whose power is of the order of 100 W and is generated in close proximity to the receiver. We wish to emphasize that the nature of the interfering signal is completely known to us in practice, since we are generating it for communications purposes. This knowledge of the interfering signal will prove to be useful in our attempts to cancel it, as will be shown. We refer to this technique as optical interference cancellation, or opto-cancellation. We have demonstrated that this opto-cancellation system can cancel a simple sinusoid at 3 GHz, as well as broadband interference of approximately 100-MHz bandwidth centered at 3 GHz. We have also demonstrated cancellation of sinusoids and broadband signals at other center frequencies as well. In the case of sinusoidal signals, we have demonstrated optical cancellation over 70 dB; and in the case of the ~ 100-MHz signal, we have demonstrated optical cancellation over 30 dB.

“Sequential Order” Rules in Generalized Two-Dimensional Correlation Spectroscopy

The widely used "sequential order" rules in the generalized two-dimensional (2D) correlation spectroscopy were adopted from the mechanical perturbation-based 2D infrared, where dynamic spectral intensity variation must be a simple sinusoid. 2D correlation analysis is fundamentally a form of parametrization of the integrated or overall relationship between two variable quantities. In generalized 2D correlation spectroscopy, however, the dynamic spectral intensity variations are generally nonperiodic and monotonic, and spectral intensity changes are largely instantaneous. The sequential orders in generalized situations are therefore localized. It is naturally necessary and important to testify whether the analysis result obtained by using the sequential order rules is consistent with the local sequential order of events, which reflects the real sequential order in generalized situations. Unfortunately, this test was not done yet. In this report, the sequential order rules have been tested in the generalized situations using simulated spectra with different local sequential orders and assuming the intensity changes of bands take the exponential forms. It has been found that the sequential order rules correctly identify the local sequential order of two events when spectral intensities of two bands increase or decrease at the same direction but fail when spectral intensities change at different directions. In addition, 2D correlation analysis cannot distinguish the local sequential order from the rate difference of events. A theoretical analysis demonstrates that the synchronous and asynchronous spectra in the generalized 2D correlation spectroscopy may also indicate the linear/nonlinear relationship, in addition to the integrated or overall sequential order of events. The synchronous and asynchronous spectra in the generalized 2D correlation spectroscopy do not necessarily provide the information on the local sequential order or rate difference of events.

Calculation of the performance of resonant wave energy converters in real seas

It is well known that the performance of point-absorber wave energy converters (WECs) depends upon resonance with the wave frequency. Indeed, the ideal performance of a resonating point-absorber WEC in a regular sea that can be represented by a simple sinusoid is well known, provided all motions are small and remain in the linear region. However, the performance of such a device in a more realistic, irregular sea that is not represented by a simple sinusoid cannot be so readily calculated. The first difficulty lies in modelling the hydrodynamic behaviour of the device. Recent developments in representing the hydrodynamic diffraction and radiation forces have enabled relatively simple simulation models to be developed, such as those presented and used in this paper. The second difficulty lies in the design of the device itself. In a regular sea with a known wave frequency, the settings of the power take-off system can be defined at well-known optimum values. It is shown in the present paper that, even when the wave frequency is not constant, the local wave frequency can be estimated, and this estimate can be used to adjust the power take-off system settings to maintain quasiresonance and, hence, approach the level of performance in a comparable regular sea. In this manner, for irregular seas it is possible to identify a dominant wave frequency over a relatively short time period and to use this frequency continuously to adjust the power take-off system settings, so as to adapt to the current sea conditions. This is likely, in some sea conditions, to involve the power take-off supplying power over part of the cycle, rather than absorbing it. This will increase the demands placed on the power take-off - particularly on its efficiency when the direction of power flow has to be reversible. The relative performance of such a tuneable point-absorber WEC is assessed in the paper. It is shown that the power converted in irregular seas could be as much as 50 per cent of the rated power, where the latter estimate is equivalent to the power converted in a corresponding regular sea.

Complex magnetic order in the kagomé staircase compoundCo3V2O8

${\mathrm{Co}}_{3}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$ (CVO) has a different type of geometrically frustrated magnetic lattice, a kagom\'e staircase, where the full frustration of a conventional kagom\'e lattice is partially relieved. The crystal structure consists of two inequivalent (magnetic) Co sites, one-dimensional chains of $\mathrm{Co}(2)$ spine sites, linked by $\mathrm{Co}(1)$ cross-tie sites. Neutron powder diffraction has been used to solve the basic magnetic and crystal structures of this system, while polarized and unpolarized single crystal diffraction measurements have been used to reveal a rich variety of incommensurate phases, interspersed with lock-in transitions to commensurate phases. CVO initially orders magnetically at $11.3\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ into an incommensurate, transversely polarized, spin density wave state, with wave vector $k=(0,\ensuremath{\delta},0)$ with $\ensuremath{\delta}=0.55$ and the spin direction along the $a$ axis. $\ensuremath{\delta}$ is found to decrease monotonically with decreasing temperature and then locks into a commensurate antiferromagnetic structure with $\ensuremath{\delta}=\frac{1}{2}$ for $6.9<T<8.6\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. In this phase, there is a ferromagnetic layer where the spine site and cross-tie sites have ordered moments of $1.39\phantom{\rule{0.3em}{0ex}}{\ensuremath{\mu}}_{B}$ and $1.17\phantom{\rule{0.3em}{0ex}}{\ensuremath{\mu}}_{B}$, respectively, and an antiferromagnetic layer where the spine-site has an ordered moment of $2.55\phantom{\rule{0.3em}{0ex}}{\ensuremath{\mu}}_{B}$, while the cross-tie sites are fully frustrated and have no observable ordered moment. Below $6.9\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, the magnetic structure becomes incommensurate again, and the presence of higher-order satellite peaks indicates that the magnetic structure deviates from a simple sinusoid. $\ensuremath{\delta}$ continues to decrease with decreasing temperature and locks in again at $\ensuremath{\delta}=\frac{1}{3}$ over a narrow temperature range $(6.2<T<6.5\phantom{\rule{0.3em}{0ex}}\mathrm{K})$. The system then undergoes a strongly first-order transition to the ferromagnetic ground state $(\ensuremath{\delta}=0)$ at ${T}_{c}=6.2\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. The ferromagnetism partially relieves the cross-tie site frustration, with ordered moments on the spine-site and cross-tie sites of $2.73\phantom{\rule{0.3em}{0ex}}{\ensuremath{\mu}}_{B}$ and $1.54\phantom{\rule{0.3em}{0ex}}{\ensuremath{\mu}}_{B}$, respectively. The spin direction for all spins is along the $a$ axis (Ising-like behavior). A dielectric anomaly is observed around the ferromagnetic transition temperature of $6.2\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, demonstrating that there is significant spin-charge coupling present in CVO. A theory based on group theory analysis and a minimal Ising model with competing exchange interactions can explain the basic features of the magnetic ordering.

SYSTEM IDENTIFICATION USING MEASUREMENTS SUBJECT TO STOCHASTIC TIME JITTER

When the sensor readings are perturbed by an unknown stochastic time jitter, classical system identification algorithms based on additive amplitude perturbations will give biased estimates. We here outline the maximum likelihood procedure, for the case of both time and amplitude noise, in the frequency domain, based on the measurement Discrete Time Fourier Transform (DTFT). The method directly applies to output error continuous time models, while a simple sinusoid in noise example is used to illustrate the bias removal of the proposed method.

The Evolution of LMC X‐4 Flares: Evidence for Super‐Eddington Radiation Oozing through Inhomogeneous Polar Cap Accretion Flows?

We present the results of two extensive Rossi X-Ray Timing Explorer observations of large X-ray flaring episodes from the high-mass X-ray binary pulsar LMC X-4. Light curves during the flaring episodes comprise bright peaks embedded in relatively fainter regions, with complex patterns of recurrence and clustering of flares. We identify precursors preceding the flaring activity. Pulse profiles during the flares appear to be simple sinusoids, and pulsed fractions are proportional to the flare intensities. We fit Gaussian functions to flare peaks to estimate the mean FWHM to be ~68 s. Significant rapid aperiodic variability exists up to a few hertz during the flares, which is related to the appearance of narrow, spiky peaks in the light curves. While spectral fits and softness ratios show overall spectral softening as the flare intensity increases, the narrow, spiky peaks do not follow this trend. The mean fluence of the flare peaks is (3.1 ± 2.9) × 1040 ergs in the 2.5-25 keV energy range, with its maximum at ~1.9 × 1041 ergs. The flare peak luminosity reaches up to (2.1 ± 0.2) × 1039 ergs s-1, far above the Eddington luminosity of a neutron star. We discuss possible origins of the flares, and we also propose that inhomogeneous accretion columns onto the neutron star polar caps are responsible for the observed properties.

Prediction of southern oscillation index using spectral components

The time series of SOI (Southern Oscillation Index, Tahiti minus Darwin sea-level atmospheric pressure difference) was spectrally analysed by a simple method MEM-MRA, where periodicities are detected by MEM (Maximum Entropy Method) and used in MRA (Multiple Regression Analysis) to get the estimates of their amplitudes and phases. From these, the three or four most prominent ones were used for reconstruction and prediction. Using data for 1935-80 as dependent data, the reconstructed values of SOI matched well with observed values and most of the El Niños (SOI minima) and La Niñas (SOI maxima) were located correctly. But for the independent data (1980 onwards), the matching was poor. Omitting earlier data, 1945- 80, 1955-80, 1965-80 as dependent data again gave poor matching for 1980 onwards. When data for 1980 onwards only were used as dependent data, the matching was better, indicating that the spectral characteristics have changed considerably with time and recent data were more appropriate for further predictions. The 1997 El Niño was reproduced only in data for 1985 onwards. For 1990 onwards, only a single wave of 3.5 years was appropriate and explained the 1997 and 1994 events but only one (1991) of the 3 complex and quick events that occurred during 1989-95. The UCLA group of Dr. Ghil has been using the SSA (Singular Spectrum Analysis)-MEM combination for SOI analysis. For the 1980s, they got very good matching, but the 1989-95 structures were not reproduced. For recent years, their SSA-filtered SOI (used for prediction) is a simple sinusoid of ~3.5 years. It predicted the El Niño of 1997 only at its peak and even after using data up to February 1997, the abrupt commencement of the event in March 1997 and its abrupt end in June 1998 could not be predicted.
 
 Using only a 3.5 years wave, an El Niño was expected for 2000-2001. However, a very long-lasting La Niña seems to be operative and there are no indications as yet (September of 2001) of any El Niño like conditions.