Identical Power Spectra Research Articles

Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and the power spectrum

We construct several examples of distinct asymmetric-symmetric pulse pairs with identical or essentially identical intensity autocorrelations and power spectra. From these examples, we infer that pulse retrieval methods based on these two data sets alone produce ambiguous solutions. Furthermore, we used the constructed pulse pairs as test cases to assess the degree of difference in the corresponding interferometric autocorrelations. In several cases, we found that the differences in the interferometric autocorrelations are sufficiently small that they might be quite challenging to distinguish in a practical experimental context.

Construction of cosmic string induced temperature anisotropy maps with CMBFAST and statistical analysis

We use the publicly available code CMBFAST, as modified by Pogosian and Vachaspati, to simulate the effects of wiggly cosmic strings on the cosmic microwave background (CMB). Using the modified CMBFAST code, which takes into account vector modes and models wiggly cosmic strings by the one-scale model, we go beyond the angular power spectrum to construct CMB temperature maps with a resolution of a few degrees. The statistics of these maps are then studied using conventional and recently proposed statistical tests optimized for the detection of hidden temperature discontinuities induced by the Gott-Kaiser-Stebbins effect. We show, however, that these realistic maps cannot be distinguished in a statistically significant way from purely Gaussian maps with an identical power spectrum.

The topology of the IRAS Point Source Catalogue Redshift Survey

We investigate the topology of the new Point Source Catalogue Redshift Survey (PSCz) of IRAS galaxies by means of the genus statistic. The survey maps the local Universe with approximately 15 000 galaxies over 84.1 per cent of the sky, and provides an unprecedented number of resolution elements for the topological analysis. For comparison with the PSCz data we also examine the genus of large N-body simulations of four variants of the cold dark matter (CDM) cosmogony. The simulations are part of the Virgo project to simulate the formation of structure in the Universe. We assume that the statistical properties of the galaxy distribution can be identified with those of the dark matter particles in the simulations. We extend the standard genus analysis by examining the influence of sampling noise on the genus curve and introducing a statistic able to quantify the amount of phase correlation present in the density field, the amplitude drop of the genus compared to a Gaussian field with identical power spectrum. The results for PSCz are consistent with the hypothesis of random-phase initial conditions. In particular, no strong phase correlation is detected on scales ranging from 10 to 32 h−1 Mpc, whereas there is a positive detection of phase correlation at smaller scales. Among the simulations, phase correlations are detected in all models at small scales, albeit with different strengths. When scaled to a common normalization, the amplitude drop depends primarily on the shape of the power spectrum. We find that the constant-bias standard CDM model can be ruled out at high significance, because the shape of its power spectrum is not consistent with PSCz. The other CDM models with more large-scale power all fit the PSCz data almost equally well, with a slight preference for a high-density τCDM model.

Intrinsic two-dimensional features as textons.

We suggest that intrinsic two-dimensional (i2D) features, computationally defined as the outputs of nonlinear operators that model the activity of end-stopped neurons, play a role in preattentive texture discrimination. We first show that for discriminable textures with identical power spectra the predictions of traditional models depend on the type of nonlinearity and fail for energy measures. We then argue that the concept of intrinsic dimensionality, and the existence of end-stopped neurons, can help us to understand the role of the nonlinearities. Furthermore, we show examples in which models without strong i2D selectivity fail to predict the correct ranking order of perceptual segregation. Our arguments regarding the importance of i2D features resemble the arguments of Julesz and co-workers regarding textons such as terminators and crossings. However, we provide a computational framework that identifies textons with the outputs of nonlinear operators that are selective to i2D features.

Numerically Simulating Non-Gaussian Sea Surfaces

A technique to simulate non-Gaussian time series with a desired (“target”) power spectrum and bispectrum is applied to ocean waves. The targets were obtained from observed bottom pressure fluctuations of shoaling, nonbreaking waves in 2–9 m water depth. The variance (i.e., frequency integrated spectrum), skewness, and asymmetry (i.e., frequency integrated bispectrum) of the simulated time series compare favorably with the observations, even for highly skewed and asymmetric near-breaking waves. The mean lengths of groups of high waves from non-Gaussian simulated time series are closer to observed values than those from Gaussian simulations. The simulations suggest that quadratic phase coupling between waves (of different frequencies) in shallow water results in longer wave groups than occur with linear, uncoupled waves having the identical power spectrum.

Spread of excitation produced by maskers with damped and ramped envelopes

Three experiments compared the ability of pairs of modulated narrow-band sounds with identical power spectra to mask signals above, below, and at their center frequency. In the first experiment, forward-masked psychophysical tuning curves (PTCs) were measured using a 1600-Hz 10-ms signal and 400-ms ‘‘damped’’ and ‘‘ramped’’ sinusoidal maskers. The damped maskers were produced by repeating a 25-ms sinusoid which had been shaped by an exponentially decaying function with a half-life of 4 ms. The ramped maskers were simply time reversals of the damped maskers. When the masker frequency was below that of the signal, ramped maskers were more effective than damped maskers, whereas the reverse was true for maskers above the signal. Experiment 2 measured thresholds for a 400-ms 1000-Hz damped or ramped sinusoidal signal, in the presence of a simultaneous masker with the same envelope shape, as a function of the masker frequency. Again, ramped maskers were more effective below the signal frequency, and damped maskers more effective above it. Experiment 3 was similar to experiment 2 except that the masker was a damped or ramped narrow-band noise, and the signal was an unmodulated sinusoid. The pattern of results was generally similar to that obtained in experiments 1 and 2. The results of all three experiments are inconsistent with the ‘‘traditional’’ method of calculating excitation patterns, which is based on the power spectrum of the stimulus, and also differ from the predictions of a recent model of auditory processing [R. D. Patterson, J. Acoust. Soc. Am. 96, 1409–1418 (1994)]. It is suggested that changes to the peripheral filtering stage of Patterson’s model may provide a useful first step toward accounting for the data.

All-night sleep EEG and artificial stochastic control signals have similar correlation dimensions

EEG signals have been considered to be generated either by stochastic process or by non-linear deterministic systems exhibiting chaotic behavior. To address this problem, the correlation dimension of the EEG was computed and compared to the correlation dimension of an artificial signal with identical power spectrum. By using a new type of personal super computer we were able for the first time to calculate the correlation dimension for the sleep episode of an entire night as well as for the corresponding artificial signal. The correlation dimension was high in episodes of rapid eye movement (REM) sleep, declined progressively within each non-REM sleep episode, and reached a low level at times when EEG slow waves (0.75–4.5 Hz) were dominant. The correlation dimension of the artificial signal and the EEG changed largely in parallel, although on average the values of the artificial signal were 7.3% higher. These results do not support the hypothesis that the sleep EEG is generated by a chaotic attractor.

Discrimination of time-asymmetric-modulated noise

Modulated noises with asymmetric temporal envelopes have identical long-term power spectra, yet sound very different. A ‘‘ramped’’ noise—constructed by amplitude modulating a wideband noise with a repeating exponential rise function—sounds like a repeating noise, whereas a ‘‘damped’’ noise—constructed with the same modulation function, but time-reversed—sounds like a snare drum struck by brushes. A damped noise has none of the usual timbre associated with a wideband noise. In experiment 1, listeners reliably discriminated a damped noise from a ramped noise, for modulation half-lives from 2 to 64 ms at modulation frequencies of 10 and 20 Hz. At higher frequencies (40 and 80 Hz) performance deteriorated. The maximum noise spectrum level was 52 dB SPL. The experiment was replicated at 32 and 12 dB SPL, and gave essentially the same results. It is proposed that the sounds’ timbre was used by listeners as the decision statistic. [Work supported by an MRC Research Studentship to the first author and by a DRA grant.]

Level-dependent critical bandwidth for phase discrimination

Monaural phase discrimination was evaluated at 1000 Hz in six normal-hearing listeners as a function of the frequency difference between components in three-tone complexes at 40, 60, 80, and 100 dB SPL. The phase of the center component of 100% sinusoidally amplitude-modulated (SAM) waveforms was shifted by 90 degrees to produce quasi-frequency-modulated (QFM) waveforms that had identical long-term power spectra to the SAM waveforms but with different amplitude envelopes and temporal fine structure. At low modulation frequencies, where spectral components were close together and presumably all well within a single auditory filter, normal-hearing listeners could easily discriminate QFM from SAM waveforms. As modulation frequency increased, a point was reached where listeners could no longer distinguish QFM from SAM waveforms, referred to here as the critical bandwidth for phase discrimination (CBphs). Discrimination performance (d') was measured as a function of modulation frequency to yield a psychometric function for phase discrimination. From the psychometric function, CBphs was defined as the modulation frequency corresponding to d' = 1.0. At a carrier frequency of 1000 Hz, CBphs increased with level between 40 and 80 dB SPL according to the relation: CBphs = 34.I(0.136). Above 80 dB SPL, very little change in CBphs with level was seen. The increase with level of the geometric mean CBphs was predicted from level-dependent auditory filter slopes inferred from forward-masked tuning curves, as was the tendency to reach an asymptote above 80 dB SPL. Comparisons with previous work indicate that CBphs values from the best performing subjects were well within the audibility region for cubic difference tones. It is proposed that internally generated cubic difference tones interact with externally generated acoustic components, both limited by a level-dependent auditory filter, to produce an internal excitation envelope that is the basis for discriminating between SAM and QFM waveforms. It is also suggested that individual differences at low levels may be due to internal phase ambiguities.

Oscillations in Ionospheric Virtual Height, Echo Amplitude and Doppler Velocity: Theory and Observations.

Applying perturbation theory to the Appelton-Hartree formulas, a detailed derivation of ionospheric parameters for ionospherically reflected echoes shows that: 1) oscillations of Doppler velocity, virtual height and echo amplitude have the same power spectrum; 2) the temporal variations in Doppler velocity lead or lag those in phase height by 90° and 3) the phase difference between the variations in echo amplitude and those in virtual height is either 0° or 180°. The analysis of experimental data obtained using a digital HF ionosonde (Dynasonde) confirms that oscillations in Doppler velocity, virtual height and echo amplitude have nearly identical power spectra and the phase difference is in good agreement with the derived theory. It is concluded that several different parameters measured by or derived from ionospheric soundings can be used to study ionospheric variations.

Method to distinguish possible chaos from colored noise and to determine embedding parameters.

We present a computational method to determine if an observed time series possesses structure statistically distinguishable from high-dimensional linearly correlated noise, possibly with a nonwhite spectrum. This method should be useful in identifying deterministic chaos in natural signals with broadband power spectra, and is capable of distinguishing between chaos and a random process that has the same power spectrum. The method compares nonlinear predictability of the given data to an ensemble of random control data sets. A nonparametric statistic is explored that permits a hypothesis testing approach. The algorithm can detect underlying deterministic chaos in a time series contaminated by additive random noise with identical power spectrum at signal to noise ratios as low as 3 dB. With less noise, this method can also be used to get good estimates of the parameters (the embedding dimension and the time delay) needed to perform the standard phase-space reconstruction of a chaotic time series.

Topology of large-scale structure in seeded hot dark matter models

The topology of the isodensity surfaces in seeded hot dark matter models, in which static seed masses provide the density perturbations in a universe dominated by massive neutrinos is examined. When smoothed with a Gaussian window, the linear initial conditions in these models show no trace of non-Gaussian behavior for r0 equal to or greater than 5 Mpc (h = 1/2), except for very low seed densities, which show a shift toward isolated peaks. An approximate analytic expression is given for the genus curve expected in linear density fields from randomly distributed seed masses. The evolved models have a Gaussian topology for r0 = 10 Mpc, but show a shift toward a cellular topology with r0 = 5 Mpc; Gaussian models with an identical power spectrum show the same behavior.

Characterization of acoustic emission from an electrolysis cell

Evolution of hydrogen and oxygen from electrodes in an electrolysis cell may be conveniently monitored in a non-intrusive manner via its ultrasonic acoustic emission. The apparatus used in this work was comprised of a nickel anode, a stainless steel cathode, and a saturated calomel reference electrode, all situated in a three chamber cell containing sodium hydroxide solution. The potential necessary for gas bubble evolution was conclusively detected by onset of bursts of acoustic emission. Individual acoustic emission signals, captured using a broadband transducer mounted on the working electrode, contained frequencies from the audible range to as high as 800 kHz. These were correlated with the release of bursts of bubbles from the electrode's surface, both visually and via a chart recorder trace of peak acoustic power vs. time. Trends in several time-domain signal descriptors were observed with increase in the applied voltage. Acoustic power spectra were obtained by averaging spectra from many acoustic signals. Semi-quantitative estimates of rate of emission were made by integration of the peak acoustic level. The effects of applied potential and electrolyte concentration on the multiple bursts of acoustic emission were characterized and are presented as a system response surface. Increasing the applied potential resulted in greater rates of bubble emission, which increased the intensity of acoustic emission, but produced an identical acoustic power spectrum. The extent of acoustic emission at high concentrations and applied potentials was less than expected, which suggested decreased efficiency under these conditions. Electrolysis start-up kinetics were also studied using a digitizer capable of large record lengths.

Generation of a time series having a specified power spectrum with minimum roundoff noise

The problem of determining a shaping filter which minimized the output roundoff-noise variance is discussed. It is shown that in the family of shaping filters of an identical McMillan degree having an identical power spectrum, the shaping filter of minimum phase attains a minimum output roundoff-noise variance in either the case of optimal word lengths or the case of equal word lengths. It is shown out by examples that this is not valid in the family of shaping filters having an identical power spectrum and different McMillan degrees.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Features versus spatial phase in a tachistoscopic laterality experiment.

The abilities of the two hemispheres to instantly discriminate compound grating pairs with identical power spectra and the same separation in phase (180°) were compared in a tachistoscopic laterality experiment. Consistent with earlier observations by Fiorentini and Berardi (1984), a left-visual-field advantage was found for a stimulus pair differing in local contrast magnitude, whereas discrimination was not possible at all for a mirror-symmetric grating pair. This result contradicts the notion of a right-hemisphere advantage in processing spatial phase. Moreover, it is incompatible with the assumption that compound gratings are discriminated with respect to their Fourier components. This implies that discrimination must depend on the processing of local image components or features. Although the right-hemisphere advantage found is related to the analysis of local contrast variability within an image, the relative position of local image components is generally lost in extrafoveal vision.

The effect of stimulus periodicity on frequency discrimination of noise bands

In order to test the effect of stimulus periodicity on frequency discrimination, we measured the just noticeable frequency change in stimuli with identical power spectra but different temporal characteristics. Our stimuli were made by adding two uncorrelated 50-Hz low-pass filtered Gaussian noise samples which were multiplied by 1-kHz sinusoids differing in phase by 5 degrees. The other condition was the same except that the 1-kHz carriers were 90 degrees out of phase. Stimulus periodicity was quantified by comparing histograms of intervals between zero crossings for the two stimuli. In the first condition, stimulus histograms were much more similar to a histogram obtained from a pure tone at the noise-band center frequency than were histograms of stimuli in the second condition. Preliminary results indicate that for a center frequency of 1 kHz, the difference limen (Δf) obtained from normal hearing adults using a two-alternative forced-choice procedure is half as large with the periodic stimulus as with the other condition. This suggests the importance of stimulus periodicity in frequency discrimination.

The discriminability of spatial phase relationships in amblyopia

Previous work has shown that anomalies in the contrast sensitivity functions of amblyopes are often insufficient to explain the degree of visual deficit in more complex tasks. Our stimuli were compound gratings composed of a fundamental and its third harmonic, added in either square-wave or triangle-wave phase. At medium to high spatial frequencies we find that many amblyopes, unlike normal observers, are unable to distinguish between such gratings which have identical power spectra, but different phase spectra. In this frequency range they can, however, easily distinguish between a compound grating and its fundamental component alone. It seems that in amblyopia visual processing occurs over a more truncated frequency range than is implied by detection experiments. Various explanations of this observation are considered.

Spatial nonlinearities in the instantaneous perception of textures with identical power spectra.

In 1962, Julesz observed that texture pairs with identical second-order statistics but different third- and higher-order statistics were usually not discriminable without scrutiny. Since second-order (dipole) statistics determine the autocorrelation functions and hence the power spectra, this observation also meant that in preattentive perception of texture the phase (position) spectra were ignored. In the last two decades many new classes of texture pairs with identical power spectra have been invented that were not effortlessly discriminable; however, recently (Caelli & Julesz 1978; Caelli et al. 1978; Julesz et al. 1978) several counterexamples were found. In these texture pairs with identical power spectra some local structures of 'quasi-collinearity', 'corner', 'closure' and 'granularity' yielded strong discrimination. These features can be regarded as the fundamental building blocks of form, that is, the essential nonlinearities of the preattentive perceptual system. Here, it will be shown that these counter-examples are not independent of each other, but can be described by two elementary units; bars (line segments) and their terminators. Furthermore, the preattentive texture perception system can count the number of terminators but ignores their positions.

Detection of signals with identical energy spectra but different waveforms under conditions of signal uncertainty.

Previous research has shown that a signal which may be one of two pure tones will be less detectable than a signal that is of constant, known frequency. In the present study, the effect of waveform uncertainty was evaluated by using pulse-train signals with identical power spectra but different phase spectra. Results for a group of 11 listeners suggest that when differences in the energy spectra of signals are eliminated, signal uncertainty has, at best, a small effect on detection performance.

Selective adaptation to frequency-modulated tones: Evidence for an information-processing channel selectively sensitive to frequency changes

Exposure to an FM tone elevates FM threshold but not AM threshold. This holds for a wide range of frequency deviations (delta F = +/- 0.4 Hz- +/- 30 Hz at least) provided that modulation frequency is low (fm = 2 Hz), but if fm is somewhat higher (e.g., 8 Hz) the finding only holds for small frequency deviations. FM threshold can rise with time up to an adapting duration of at least 1200 s, through this buildup depends on frequency deviation. Exposure to an AM tone elevates AM threshold, but not FM threshold, over a wide range of modulation depths (at least m = 5%--50%). Quasi-FM (QFM) adapting tones resemble FM adapting tones in their effects upon FM and AM sensitivities, even though QFM and AM adapting tones have identical power spectra. Exposure to a pure tone produces no difference between FM and AM threshold elevations. These data can be explained if the human auditory pathway contains separate information-processing channels for AM and FM signals whose sensitivities do not overlap even with suprathreshold stimuli. We suppose that the FM channel (but not the AM channel) is sensitive to changing differences (or ratios) between signals from different sites along the basilar membrane.