Clipped Correlation Research Articles

Accuracy analysis of measurements on a stable power-law distributed series of events

We investigate how finite measurement time limits the accuracy with which the parameters of a stably distributed random series of events can be determined. The model process is generated by timing the emigration of individuals from a population that is subject to deaths and a particular choice of multiple immigration events. This leads to a scale-free discrete random process where customary measures, such as mean value and variance, do not exist. However, converting the number of events occurring in fixed time intervals to a 1-bit 'clipped' process allows the construction of well-behaved statistics that still retain vestiges of the original power-law and fluctuation properties. These statistics include the clipped mean and correlation function, from measurements of which both the power-law index of the distribution of events and the time constant of its fluctuations can be deduced. We report here a theoretical analysis of the accuracy of measurements of the mean of the clipped process. This indicates that, for a fixed experiment time, the error on measurements of the sample mean is minimized by an optimum choice of the number of samples. It is shown furthermore that this choice is sensitive to the power-law index and that the approach to Poisson statistics is dominated by rare events or 'outliers'. Our results are supported by numerical simulation.

Measurement of aerosol particles by dynamic light scattering-I. effects of non-Gaussian concentration fluctuation in real time photon correlation spectroscopy

Effects of non-Gaussian concentration fluctuations in real time photon correlation spectroscopy were investigated with several sizes of mono-disperse polystyrene latex particles. The non-Gaussian correlation model for a full or a quasi-full (scaled) correlator showed good performance for particle sizing in the case of scaled correlation. The clipped correlation showed an equivalent performance with the scaled correlation except for the case of non-Gaussian fluctuation, where the measured diameter became inaccurate more rapidly with decreasing particle concentration in the scattering volume. The cause of such an instability of PCS in the clipped mode is attributed to baseline fluctuations of the correlogram induced by information loss of the clipped correlator.

Speckle clipping correlation using various logical operation

The clipping method is used to carry out the fast calculation of a correlation function of speckle intensity in several applications. Using this technique, the peak position of the cross-correlation function can be determined more accurately than that for the nonclipped one. There are several possible logical operations for calculating clipped speckle intensity. Several methods of the logical operation of the intensity clipping are studied to evaluate the accuracy of the peak detection of the clipped correlation function. Both the theoretical consideration and experimental results are given.

Accuracy of peak detection in speckle clipping correlation

Intensity binarization or clipping method is often used to carry out the fast calculation of a correlation function of speckle intensity. The shape of a clipped correlation function becomes sharp compared with a nonclipped one. It is expected that we can determine the peak position of a clipped cross-correlation function more accurately than that of a nonclipping one. The accuracy of the peak detection in speckle correlation is studied by using a computer simulation. The computer simulation for certain optical systems shows that the clipping technique is suitable for the accurate measurement of velocity and displacement based on the peak detection of speckle correlation.

Maximum dynamic range of clipped correlation of integrated laser speckle intensity

The relationship between the dynamic range and the clipping threshold of a double-clipped correlation of integrated laser speckle intensity signals is discussed. In particular, the maximum dynamic range of the clipped correlation is evaluated. For α = 1 (α being the number of the integrated speckles), the maximum dynamic range occurs for the clipping threshold of In 2〈I〉, and the clipping threshold approaches the mean intensity 〈I〉 as the number of integrated speckles increases. On the other hand, the maximum dynamic range does not change and has the same value of Δ = 1/4 (Δ being defined as the dynamic range of the clipping correlation normalized by that of the nonclipping correlation), even if the number of the integrated speckles varies.

Correlation measurements using clipped laser speckle

Clipped laser speckle is used to measure the correlation of time-varying laser speckle. As with standard intensity correlation, clipped speckle is correlated to determine the magnitude of the correlation coefficient. The simple nature of clipped speckle permits high-speed measurements of correlation and velocity. A closed form approximation to the clipped correlation function valid for arbitrary clipping thresholds is presented. The approximation is used to make correlation measurements on speckle from a rotating diffuser using a 2-D random access detector array.

Particle discrimination and background suppression in photon-correlation laser velocimetry

Selective data acquisition controlled by a particle-identification circuit is demonstrated here to improve the performance of photon-correlation laser-Doppler velocimetry (LDV) under unfavorable background-light conditions (e.g., near a window surface). A gated cross-correlation technique is used to accumulate data only when a photon-counting particle discriminator indicates the presence of a light-scattering particle in the LDV measurement volume. Background-light contributions between particle transits are suppressed. The particle discriminator is also shown to facilitate measurements of temporal statistics of turbulence (e.g., time scale, energy spectrum) using particle recurrence-rate correlation. Further applications of the particle discriminator to Rayleigh and Raman scattering are suggested, an adaptation of the circuit for higher-velocity flows is outlined, and the relation of particle-discriminated photon correlation to clipped correlation is discussed.

Self-scaling of clipped photon correlations

The estimates of photoemission rate correlations that are obtained by clipped, e.g., one-bit, correlation techniques are in general distorted by the clipping, unless scaling techniques are used. It is shown that the Pois-son statistics of photoemission processes are such that the processes are inherently self-scaling when the range of signal fluctuations is contained in the range of inherent noise fluctuations. Self-scaling occurs, for example, when the total mean emission rate is large compared with the time varying signal rate. It permits clipped correlation to be used without distortion.

Clipped correlation of integrated intensity fluctuations of thermal light with an arbitrary spectrum

An approximate formula is obtained for the single clipped correlation function of integrated intensity fluctuations of thermal light with an arbitrary spectrum. This formula is compared to the exact function known for light with a Lorentzian spectrum.

The Inversion Problem for Two Fold Photoelectron Counting Statistics

The extraction of spectral information from measurements of the full or clipped photoelectron correlation function is treated. Unlike previous treatments, the discussion here is not limited to counting times short compared to the coherence time of the light. The method of singular value decomposition is applied to the case where there is no prior knowledge of the spectrum. Parameter estimation is applied when the general structure of the spectrum is known.

Two-fold photoelectron counting statistics: the clipped correlation function

Two-fold photoelectron counting statistics are studied for quasimonochromatic light of arbitrary spectral profile. A numerical scheme is developed for the evaluation of the double generating function from which the joint probabilities, the photoelectron correlation function, and the clipped photoelectron correlation function can be evaluated. The time need not be short compared to the coherence time as in previous studies. Numerical results are presented for the photoelectron correlation function and for the clipped correlation function for the spectrum which would be observed in a Brillouin scattering experiment.

Measurements of the Vertical Coherence of the Sound from a Near-Surface Source in the Sea and the Effect on the Gain of an Additive Vertical Array

Using a ship as a source of broad-band sound, measurements have been made of signal coherence between two deep hydrophones separated vertically by 500 ft. The results show that the phase coherence, in terms of the clipped correlation coefficient, falls off rapidly with horizontal range of the source. This decorrelation can be accounted for by multipath interference of bottom-surface reflections. The measured rate of falloff with range indicates a bottom loss between zero and 4 dB, increasing with frequency over the range 44–350 Hz. The effect of multipath interference is to degrade the signal-to-noise gain of an array. It is easily shown that when a conventional, linear, additive array is steered in one of the multipath directions, the gain is reduced by the ratio of the intensity of the sound coming from that direction to the intensity of the total sound field.

Measurements of the vertical coherence of the sound from a near-surface source in the sea and the effect on the gain of an additive vertical array

Using a ship as a source of broad-band sound, measurements have been made of signal coherence between two deep hydrophones separated vertically by 500 ft. The results show that the phase coherence, in terms of the clipped correlation coefficient, falls off rapidly with horizontal range of the source. This decorrelation appears to be accounted for by multipath interference by bottom-surface reflections. The rate of falloff with range indicates a bottom loss between zero and 4 dB, increasing with frequency over the range 44 to 350 Hz. Expressions for the degradation of the gain of a linear vertical array produced by multipath interference are obtained.

Comment on the letter by Jakeman et al. 'Correlation of scaled photon-counting fluctuations'

Jakeman et al. (see Abstr. A73542 of 1972) have presented a method for measuring the full photocount correlation function which might be called single channel scaling (SCS). The advantage of SCS is that it can be used in conjunction with a clipped correlator to measure directly the full photocount correlation rather than the clipped correlation function. The authors point out an equally simple way of performing the same measurement and finite sum by uniformly and sequentially clipping (USC) incoming photocounts. With this method one simply clips sequentially from some maximum clipping level k=l during each of l+1 sampling periods, then starts over again at k=l. In this way the uniformity of the sampling is automatically ensured and can therefore be used as a technique to test experimentally the clipping uniformity of SCS. They performed an extensive set of measurements using both SCS and USC over a wide range of scaling and sequential clipping levels and found both methods lead to exactly equivalent results.

Light scattering from independent particles-nonGaussian correction to the clipped intensity correlation function

The authors consider Rayleigh scattering of laser light from an assembly of independent particles. They allow the particle number M within the scattering volume to fluctuate around a certain mean value (M). They derive a general formula for the intensity moments and find the nonGaussian correction to the single-clipped-at-zero photocount correlation function to order 1/(M)2.

Coherence of Signals, Noise, and Reverberation in the Sea

Separated hydrophones in the sea show varying degrees of coherence, or waveform similarity, depending on separation, frequency, and kind of sound observed. Over the past 10 years, we have measured the phase coherence of different kinds of signals, noise, and reverberation between vertically separated hydrophones. Such information is basic for the optimum design of arrays. This paper summarizes these measurements. When plotted against normalized separation the coherence as measured by the clipped correlation coefficient is found to lie, as it should, between the value of unity for a plane wave and a curve for isotropic noise; it depends on the vertical angle within which the sound arrives at the two hydrophones. The most coherent sounds observed were signals in convergence zones; the least coherent, the reverberation received from great depths in the deep sea.

Scaled photocount correlation of non-Gaussian scattered light

The photocount autocorrelation of non-Gaussian scattered laser light is discussed as a tool for the study of both very dilute or highly correlated scattering systems. A scaled photocount correlation technique is introduced and shown to produce a function directly proportional to the correlation function of the integrated scattered intensity, independent of the field statistics. This technique overcomes the serious problems of interpretation that can arise when the commonly used clipped correlation technique is applied to non-Gaussian fields. The scaled correlation technique is demonstrated experimentally with a simple scattering system, a small number of particles under uniform translational motion.

A method for the clipped intensity correlation measurement

The authors propose an alternative scheme for the clipped light intensity correlation measurement which is complementary to the usual scheme as proposed by Jakeman and Pike (1969). The new scheme gives essentially the same statistical accuracy for the linewidth measurement and furthermore, it has some practical advantages when the mean number of counts per sampling time is sufficiently large.

Correlative Properties of Deep-Water Ambient Noise at Bermuda

Time-delay correlograms of the ambient noise picked up by various hydrophone pairs of the deep vertical array at Bermuda have been obtained. These show, as a function of time delay, the clipped correlation coefficient of the noise for the selected hydrophone pairs. In the two 1-oct bands 200–400 and 1000–2000 cps, and for wind speeds between 4 and 30 kt, different types of correlograms have been found for different combinations of frequency and wind speed. In the lower-frequency band and at low wind speeds, they show a strong peak at zero delay, consistent with the findings of Fox [J. Acoust. Soc. Am. 36, 1537–1540 (1964)] and Axelrod et al. [J. Acoust. Soc. Am. 37, 77–83 (1965)] that the noise arrives horizontally, probably from distant shipping. In the higher band and at high wind speeds, the correlograms are what would be expected for noise generated at the sea surface. The correlograms permit computation of the array gain of any arbitrary vertical array steered at any angle to the horizontal. Considerable degradation of array gain relative to uncorrelated noise is to be expected under certain conditions. A simple example is given.