Cosine Phase Research Articles

Quantitative analysis of onset dominance in lateralization

A statistical method for measuring the weight which an observer gives to each component of a multicomponent signal [COSS Analysis; Berg, J. Acoust. Soc. Am. 86, 1743–1746 (1989)] was applied to precedence. Signals consisted of trains of dichotic clicks whose time envelopes were Gaussian and centered at the cosine phase of a 6-kHz carrier. The interaural delay of each click was perturbed by adding or subtracting a small Gaussian random variable. In a single-interval task, subjects were required to determine whether the lateralized image was to the right or left of the head. Using COSS analysis, weights were determined for each click of the multiclick stimulus. For short interclick intervals (ICI=1.8 ms) as the number of clicks increased (n=2 to 16), the relative weight given to the onset click diminished. The weights estimated for all clicks following the first were nearly the same. Only the relative weight of the first click varied as stimulus parameters were altered. Frequency-channel effects and the idea of restarting in binaural adaptation were also examined. [Work supported by NIH and AFOSR.]

Visual processing delays alter the perceived spatial form of moving gratings

This study shows that there are delays in processing high spatial frequencies relative to low frequencies, and that these may affect the perceived brightness profile of drifting waveforms. The stimuli were complex waveforms consisting of 2-3 sinusoidal components, either drifting or stationary. The phase of the components was varied until the brightness profile of the waveform appeared as a square, triangle, ramp or bar. The results indicate that stationary waveforms are perceived veridically, but drifting waveforms are not. The harmonics of a drifting complex wave must be phase advanced, relative to the fundamental, in order to cancel motion-induced waveform distortions. This suggests that during visual processing the harmonics must be phase delayed, indicating that they are being processed more slowly than the fundamental. The most significant delays appear to be those between the fundamental and its second and third harmonic. Furthermore, the results show that the magnitude of the delays is dependent on the phase relationship between the components at perceptually significant points in the waveform: delays are less when the components are in sine phase than when they are in cosine phase. Separate experiments show that the detectability of phase shifts is least when the components are in sine phase. Together, these results may explain why drifting "sharp-edged" stimuli are not perceptually distorted: the human visual system appears to be relatively insensitive to phase shifts around square-wave phase and may therefore tolerate differences in the processing times of certain harmonics. A discussion of the possible origin of these processing delays is presented, together with the hypothesis that frequency dependent delays may reflect the spatiotemporal inseparability of cortical visual units.

What is the octave of a harmonically rich note?

A sound composed of the harmonics of 131 Hz produces the note C3 with abuzzy timbre if the components have equal amplitude and are in cosine phase. Notes with the same tone chroma and a tone height between C3 and C4 can be produced by 1) attenuating the lower harmonics of the sound, 2) attenuating the odd harmonics, or 3) shifting the phase of the odd harmonics. The effects of the manipulations were measured in an octave experiment: a note was chosen at random and used to play a brief melody on the notes around C; the octave varied from C1 to C6 and the listeners judged the octave of each melody on a scale from C0 to C7. The results show that waves with the same period can lead to average octave judgements that differ consistently by more than half an octave, and that a substantial component of many timbre differences (e.g. that between a piano and a harpsichord) is actually a tone-height difference. The effects of manipulations 2) and 3) are difficult to explain with traditional hearing theories because ...

The global structure of the annual and semiannual sea surface height variability from Geosat altimeter data

The global structure of the annual and semiannual sea surface height variability is constructed from the first 2 years of the Geosat Exact Repeat Mission (ERM) altimeter data. The GEM‐T2 orbits available for the first 2 years of the ERM have an accuracy of 40 cm RMS. Residual orbit error is modeled as one cycle per orbital revolution and is removed from collinear differences of arcs made of data from one full orbital revolution of the satellite. The error in the Schwiderski M2 tidal model must be estimated due to the fact that the tidal variability aliases to 1.15 cycles per year (cpy), and this frequency is not separable from 1 cpy with the 2 years of data. An estimate of the M2 tidal error is made based on the particular temporal and spatial aliasing of the tide. With this error removed, a least squares fit of sine and cosine waves with annual and semiannual frequencies is made to the time series at every point along the ground track of the satellite. This produces sine and cosine coefficients at the ground track points which are interpolated to a regularly spaced ½° grid over the globe. From the sine and cosine maps, amplitude and phase may be obtained. Interpolation of the sine and cosine coefficients using different spatial scales is done to better observe the large‐scale phase changes and to remove small‐scale noise. Results show the phase relationships between major current systems, large‐scale variations near the equator in the Intertropical Convergence Zone (ITCZ), a 180° phase difference between the northern and southern hemispheres for the annual variability, large‐scale westward propagating waves, and other large‐scale gyre features.

Curvature and separation discrimination at texture boundaries.

Visual discrimination of contour curvature was investigated by using contours defined by the locus of points at which the phase of a square-wave grating was shifted by 180 degrees (a texture boundary). Curvature-increment thresholds were measured for contour curvatures from 0.31 to 10.65 deg-1, for grating spatial frequencies of 4.0 and 16.0 cycles per degree (cpd), and for gratings in either sine or cosine phase at the point of maximum curvature. Thresholds for these conditions were compared with curvature discrimination at black-white edges. Grating phase had no effect on performance at any curvature or grating frequency, but 16.0-cpd gratings produced a threshold elevation at all curvatures by an average factor of 2.4. Two-line separation discrimination was also measured for lines defined by texture boundaries. These data can be predicted by a model incorporating end-stopped complex cells of a type reported physiologically in primate area V2.

Dynamical Properties of the Field Phase in the Jaynes-Cummings Model

Abstract Phase properties of a coherent field interacting with a two-level atom in an ideal cavity are studied using the new hermitian phase formalism of Pegg and Barnett. It is shown in particular that phase properties of the field reflect the collapse and revival phenomena. The effects of finite detuning and atomic coherences are treated. The results for the variance of the phase cosine are obtained and compared with those based on earlier approaches.

Phase Properties of Self-squeezed States Generated by the Anharmonic Oscillator

Abstract The phase properties of the self-squeezed states generated during the evolution of the anharmonic oscillator are discussed from the point of view of the new phase formalism of Pegg and Barnett. The phase distribution, the expectation values and the variances of the Hermitian phase operator are obtained and their evolution illustrated graphically. The mean values for the phase cosine and sine functions as well as their variances are also calculated. The results are compared to the Susskind-Glogower formalism results and the results based on the measured phase concept. The relation between squeezing and the phase properties of the field is discussed briefly.

Frequency selectivity and pitch discrimination in young and elderly subjects with cochlear hearing loss

Pure and complex tone pitch discrimination data were obtained for two groups of hearing‐impaired subjects, young and elderly, and for normally hearing subjects. Auditory filter shapes were also estimated for center frequencies of 100, 200, 400, and 800 Hz using a modified notched‐noise method [B. J. Glasberg and B. C. J. Moore, Hear. Res. 47, 103–138 (1990)]. Frequency DLs for pulsed tones were measured for frequencies from 50–4000 Hz. DLs for the fundamental frequency (F0) of complex tones were measured for F0s of 50, 100, 200, and 400 Hz, for complexes containing harmonics 1–12, 6–12, 4–12, and 1–5. The components were added in either cosine phase or in alternating sine‐cosine phase. Auditory filters for the young and elderly impaired subjects were similar; both groups had broader filters than the normal subjects. Complex tone DLs were larger for the impaired subjects. Complex tone DLs were especially large for the tones with harmonics 1–5 and 1–12 at F0s of 50 and 100 Hz. These DLs were reduced (i.e., performances improved) when the lower harmonics were removed. Complex tone DLs were affected by the relative phases of the components for some but not all of the hearing‐impaired subjects. The implications of the results for pitch theories will be discussed. [Research supported by the Andrus Foundation.]

Perception of transient noise at the abrupt onset of a pure tone

The transient noise caused by the instantaneous onset of a pure tone depends upon the phase of the tone at onset. Listening experiments show that if the frequency of the tone is low, the onset noise is perceived to be least if the onset is in “sine phase.” If the frequency of the tone is high, the onset noise is least if the onset is in “cosine phase,” a choice that maximizes the discontinuity in the waveform. These results find a simple physical explanation in terms of the power spectra of the abrupt tones. Experimental psychometric functions show that the transition from low‐frequency perception to high‐frequency perception is sharp but rather strongly subject dependent. There is good correlation between transition frequency and high‐frequency thresholds as measured by individual audiograms. The transition frequency can also be manipulated experimentally by using electroacoustical transducers that have different frequency responses. [Work supported by the National Institutes of Health.]

Is a resonator that contains a phase-conjugating mirror a resonator?

Unlike conventional resonators that exhibit a transfer function that is similar to an Airy’s formula, a resonator that contains a phase-conjugating mirrror (PCMR) shows a cosine phase dependence. But a cosine phase dependence is the characteristic of two-beam interferometers. Does this mean that a PCMR is a two-beam interference device? We show that in a certain sense this is the case. However, for a PCMR this does not necessarily contradict the notion of resonator, which implies multiple-beam interference. Applying the discrete-time technique, we demonstrate that a PCMR can be decomposed into a serial connection of a feedback loop, corresponding to an always-tuned resonator, and a two-beam interferometer. It is this interferometer that causes the cosine phase dependence.

Spatio-temporal interactions and the spatial phase preferences of visual neurons

We recorded single neuron responses in the cat's lateral geniculate nucleus (LGN) and visual cortex to compound stimuli composed of two sinusoidal gratings in a 2:1 frequency ratio. To probe visual receptive field symmetry, we varied the relative spatial phase of the two components and measured the effect on neuronal responses. We expected that on-center LGN neurons would respond best to gratings combined in positive cosine (bright bar) phase, while off-center LGN neurons would respond best to gratings combined in negative cosine (dark bar) phase. When drifting stimuli were used, cells' phase preferences were roughly 90 deg away from the expected values; when stationary, contrast-modulated stimuli were used, phase preferences were as originally predicted. Computer simulations showed that this discrepancy could be explained by taking into account the cells' temporal properties. Thus, tests using drifting stimuli confound the spatial structure of visual neural receptive fields with their temporal response characteristics. A small sample of data from cortical neurons reveals the same confound.

Coding of phase structure in complex sounds by bullfrog auditory‐nerve fibers

This experiment explored how the peripheral coding of the spectral and temporal structure of a complex communication signal is affected by the phase relations between individual frequency components in the sound. Complex sounds with 21 components, all of which were harmonics of a common, low‐frequency fundamental (100 Hz), were digitally synthesized. The frequencies and amplitudes of the individual components matched those of the bullfrog's species‐specific advertisement call. For one stimulus, the 21 components were generated in cosine phase, and for the other stimulus, the 21 components were in random phase. Most amphibian papilla (AP) and basilar papilla (BP) fibers synchronized to the envelope only of the phase‐coherent stimulus, and the complex spectral structure of the stimulus was not accurately encoded by population firing patterns. Envelope synchronization to the random‐phase stimulus was not seen in the responses of some mid‐frequency‐sensitive AP and BP fibers, although low‐frequency‐sensitive AP fibers responded similarly to both stimuli. Behavioral experiments suggest that bullfrogs can discriminate between in‐phase and random‐phase versions of their advertisement calls.

Timbre and tone height

A sound composed of all harmonics of 131 Hz in cosine phase produces the note C3 with a buzzy timbre. One can produce similar notes that are all C's but that sound “higher” than the original C, in at least three different ways. One can (a) attenuate the lower harmonics, (b) attenuate the odd harmonics, and (c) shift the phase of either the odd or the even harmonics. Traditionally, these perceptual changes would be regarded as timbre changes because tone chroma is fixed, but these manipulations also affect the tone height of the sound. Accordingly, experiments were performed to investigate the relationship between timbre and tone height. The stimuli were presented at six octaves with fundamentals ranging from 32–1024 Hz. The listeners judged the octave of the sounds on a scale from 1 to 6. The results show that waves with the same period can produce tone-height judgments that consistently differ by more than an octave, and that a substantial component of many presumed timbre differences (e.g., between a piano and a harpsichord) is actually a tone-height difference.

Difference limens for phase in normal and hearing-impaired subjects.

These experiments measure the ability to detect a change in the relative phase of a single component in a harmonic complex tone. Complex tones containing the first 20 harmonics of 50, 100, or 200 Hz, all at equal amplitude, were used. All of the harmonics except one started in cosine phase. The remaining harmonic started in cosine phase, but was shifted in phase half-way through either the first or the second of the two stimuli comprising a trial. The subject had to identify the stimulus containing the phase-shifted component. For normally hearing subjects tested at a level of 70 dB SPL per component, thresholds for detecting the phase shift [i.e., phase difference limens (DLs)] were smallest (2 degrees-4 degrees) for harmonics above the eighth and for the lowest fundamental frequency (F0). Changes in phase were not detectable for harmonic numbers below three or four at the lowest F0 and below 5-13 at the highest F0. The DLs increased slightly for the highest harmonics in the complexes. The DLs increased markedly with decreasing level, except for the highest harmonic, where only a small effect of level was found. Subjects reported that the phase-shifted harmonic appeared to "pop out" and was heard with a pure-tone quality. A pitch-matching experiment demonstrated that the pitch of this tone corresponded to the frequency of the phase-shifted component. For the highest harmonic, the phase shift was associated with a downward shift of the edge pitch heard in the reference (all cosine phase) stimulus. When the phases of the components in the reference stimulus were randomized, phase DLs were much higher (and often impossible to measure), the pop-out phenomenon was not observed, and no edge pitch was heard. Subjects with unilateral cochlear hearing impairment generally showed poorer phase sensitivity in their impaired than in their normal ears, when the two ears were compared at equal sound-pressure levels. However, at comparable sensation levels, the impaired ears sometimes showed lower phase DLs. The results are explained by considering the waveforms that would occur at the outputs of the auditory filters in response to these stimuli.

Evidence for edge and bar detectors in human vision

The structure of receptive fields of human visual detectors was investigated by studying their phase response. Observers were required to discriminate between pairs of periodic stimuli that differed in phase by 180° (reversed in contrast). The stimuli comprised 256 harmonics, smoothly filtered in amplitude, and congruent in phase at the origin. Reversal discrimination thresholds were measured as a function of the phase of the harmonics. Thresholds were slightly higher for phases around 45°, consistent with the idea that all discriminations were mediated by independent detectors with 0 or 90° phase response (assuming probability summation between them). Discrimination thresholds were also measured with a pedestal stimulus, of phase complementary to that of the test gratings. For discriminations between 0 and 180° (cosine phase), or 90 and 270° (sine phase), the complementary pedestal had little effect, implying independence of detectors in sine and cosine phase. However, for discrimination between 45 and 225° (stimuli containing both sine and cosine components) the complementary pedestal, which also contained both sine and cosine components, facilitated greatly discrimination thresholds. The results suggest that there exist two classes of detectors, one with a Fourier phase spectrum of 0, the other with a Fourier phase spectrum of 90°. This implies that the receptive fields are symmetric, one class having even-symmetry (line-detectors), the other odd-symmetry (edge-detectors).

Contrast modulated maskers: Test of a late nonlinearity hypothesis

The contrast threshold for detecting a low frequency test grating in the presence of a contrast-modulated high frequency masker is heavily dependent upon the phase relations between the gratings: test stimuli in + or −cosine phase are much more detectable than those in + or −sine phase. The present study tests a new hypothesis of these and related phenomena. It is based on the report by Derrington (1987) that recordings from cat geniculate on-center and off-center X-cells exhibit significant point nonlinearities. The major empirical finding of this study is that for highly practiced observers, sine-phase test stimuli and cosine-phase test stimuli are similar in two important respects: (1) there exist phase uncertainty effects; that is, detection thresholds for test stimuli differing in phase by 180° are elevated when they are intermixed within the same block of trials, rather than being presented in separate blocks; (2) phase identification thresholds for test stimuli differing in phase by 180° are the same as their detection thresholds. The new nonlinearity hypothesis cannot account for the results obtained with sine-phase test stimuli, though it gives a better account of the results with cosine-phase stimuli than does the early nonlinearity hypothesis which was tested and rejected by Nachmias and Rogowitz (1983).

Current source density analysis of linear and non-linear components of the primate electroretinogram.

1. We have used the method of current source density analysis to locate the generators of harmonic electroretinogram (ERG) responses to contrast-modulated pattern and uniform-field stimuli in the primate retina. 2. Sinusoidal steady-state analysis was used, with a stimulus temporal frequency of 8 Hz. Fundamental and second-harmonic response components were measured for the uniform-field response. The second harmonic of the average of contrast-reversal pattern responses obtained at a series of spatial phases was also determined in the same experiments. In addition, retinal tissue resistance was measured. All of these measurements were obtained at a series of equally spaced depths in the retina. 3. Retinal resistivity was not observed to vary systematically with depth. In addition, any plausible undetected inhomogeneities of resistivity with depth were found to slightly affect the relative magnitudes of estimated current sources and sinks, but to have little effect on their localization. 4. In a given penetration, the phase lag of each harmonic component was relatively constant with depth in most cases; however the magnitude of this phase lag sometimes varied in different penetrations. To compare data from different penetrations, the constant phase lag for each harmonic was estimated, and the response data phase-shifted so as to bring all data into a standard (cosine) phase. 5. The resulting current source density analyses were found to be quite consistent in overall form for different penetrations and in different animals. These data were averaged to obtain a final estimate of the depth profiles for generators of different ERG components. 6. The uniform-field fundamental response was found to have a predominant source-sink pair in the distal half of the retina (receptor layer to outer plexiform layer). The pattern (second-harmonic) response generators had a quite different depth profile, consisting mainly of a source-sink pair in the proximal 20% of the retina (encompassing the nerve fibre layer to the middle of the inner plexiform layer). The uniform-field second-harmonic response showed a current source density (CSD) depth profile with multiple sources/sinks, as if it contained contributions from the other two.

Difference limens (DLs) for phase in normally hearing subjects and subjects with unilteral cochlear hearing loss

Complex tones containing the first 20 harmonics of 50, 100, or 200 Hz, all at equal amplitude, were used as stimuli. All of the harmonics except one were added in cosine phase. The remaining harmonic started in cosine phase, but was shifted in phase halfway through either the first or the second of the two stimuli used in a 2‐AFC trial. The task of the subject was to identify the stimulus containing the phase‐shifted component. An adaptive procedure was used to meaure DLs for phase for three normally hearing subjects and four subjects with unilateral cochlear impairments. When the normal ears were tested at a level of 70 dB SPL per component, DLs were smallest (2°–4°) for harmonics above the eighth and for the lowest fundamental frequency (F0). Changes in phase could not be detected for harmonic numbers below four at the lowest F0 and below 8–13 at the highest F0. The DLs increased markedly with decreasing level, except for the highest harmonic, where only a small effect of level was observed. The DLs for the impaired ears were generally larger than those for the normal ears when the comparison was made at equal SPL but the discrepancy was reduced when the comparison was made at equal SL. For low harmonic numbers, a few cases were found where DLs were smaller for the impaired than for the normal ear of a given subject, even at equal SPL. This effect may be explicable in terms of the broader auditory filters of the impaired ears. The results will be compared to measures of temporal resolution and pitch discrimination using the same subjects. [Work supported by the Medical Research Council, U.K.]

A joint probability distribution of invariants for all space groups

A probability distribution is derived in the form of an infinite sum of cosine phase invariants between any number of structure factors and is applicable to any space group. No approximations are made. With the use of only the first-order terms, and the assumption that the reflection magnitudes are small and that all atoms are equal, the distributions reduce to the known invariant distribution forms. The effect of neighbourhood reflections is implicit in the form of the distribution.

Mapping auditory phase sensitivity with timbre discriminations

A flat‐spectrum periodic sound composed of 31 equal‐amplitude harmonics was used to investigate the effects of phase on timbre. Listeners discriminated a pulse train, wherein the harmonics start in cosine phase from (1) nonmonotonic alternating‐phase waves wherein successive harmonics have fixed phases p and q, and (2) monotonic, decelerating‐phase waves wherein successive harmonics are phase shifted by an ever decreasing amount. Both phase manipulations produce detectable timbre changes for pulse rates below 400 Hz but monotonic phase increments have to be much larger than alternating phase differences for equivalent performance. For periodic sounds, the output of an auditory filter with center frequency above the fourth harmonic is a modulated sine wave and the modulator's characteristics appear to determine performance. Alternating‐phase waves produce a local maximum in the modulator trough which enables the discrimination to be made within each frequency channel. (These waves sound an octave above the...