Time Series Space Phase Qualitative Analysis and Application
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Published: Fri, 09 Mar 2018
- Dan CHICEA1 and Silviu REI2
ABSTRACT: In a coherent light scattering experiment using a laser beam and a cuvette containing a suspension the interference field has a boiling speckle aspect. Using a detector and a data acquisition system a time series can be recorded. A possible definition for the space phase of a time series is defined. The distribution of the velocities and the trajectory in the phase space are analyzed both for computer simulated samples and for recordings on two suspensions that have the average particle size measured using Dynamic Light Scattering procedure. The results reveal that the distribution of the velocities and the trajectory in the phase space can be used in a qualitative way to characterize the average particle size in suspension.
KEY WORDS: time series, phase space, velocity distributions, suspensions
If coherent light is incident on a medium with scattering centers (SC hereafter) a non-uniformly illuminated image is obtained, currently named speckled image, having a statistical distribution of the intensity over the interference field. The speckled image appears as a result of the interference of the wavelets scattered by the SC, each wavelet having a different phase and amplitude in each location of the interference field. The image changes in time as a consequence of the SC hereafter complex motion of both sedimentation and Brownian motion [1-3]. This complex motion produces fluctuations of the image intensity in each location of the interference field, giving the aspect of “boiling speckles”.
The speckled image can be observed either in free space and is named objective speckle or on the image plane of a diffuse object illuminated by a coherent source and it is named subjective speckle in . The review paper  names the two types of speckled images as far field speckle and image speckle. In this work the objective speckle, respectively the far field speckle is considered, recorded and analysed.
Although light propagation through disperse systems such as biological cells in suspensions or grouped in tissues has been studied extensively, especially using image processing techniques ,  the scattering process modelling and characterization is not straightforward. Recording the far field fluctuations on a certain location using a detector and a data acquisition system will produce a time series. Time series have been extensively analyzed using different procedures. A widely used class of procedures start with the Fourier transform to produce the power spectrum density (PSD). The shape of the PSD depends of two parameters. Finding them using a least squares fit will provide information on the size and size distribution of the particles in suspension. Just some of the papers that describe in detail and report results found using the above mentioned procedure, currently named Dynamic Light Scattering (DLS), are [8-12].
An alternative procedure to analyse time series, whether produced by a detector and a data acquisition system in a light scattering experiment or extracted from a recording using a charged coupled device (CCD) , in a qualitative manner, using the patterns of the time signal in the space phase, is presented in the next section.
2 SPACE PHASE
The space phase (SP hereafter) is the collection of all states for a system. A time series is actually a vector x(i), i=1÷n, containing the collection of values, intensity if we record a far interference field in a light scattering experiment, and n is the total number of recorded values. Typically, the values are recorded at equal time intervals, Δt=1/fs, where fs is the sampling rate of the data acquisition system. We can define a velocity at time t, hence corresponding to the value x(i) of the time series, as the variation rate of the time signal at that moment:
We therefore find a pair of [x(i), v(i)] for each value in the time series, a total of n-1 pairs. Each pair represents a point in the 2 dimensional space, which we can define as the space phase (SP) for the time series.
First a simple algorithm to generate in a realistic manner time series that were recorded during a DLS experiment was established. A time series, as results from the Fourier analysis of the recorded data [8-11], can be modelled as an overlapping of harmonic functions of different frequencies. Moreover, the smaller the particles in suspension are, the bigger the turnover point is  in the PSD. With these facts in mind, a program to generate the time series with a sampling rate fs was written and used. The program generates a certain number nf (50 for the work reported here) of amplitudes ai in the range [1, 10] using a uniform random distribution, with a random initial seed, computed using the system clock. It also generates nf frequencies fi in the range [f1, f2] and initial phases φi in the range [0, 2π] and the vector containing the moments ti with the desired number of values, equally spaced at Δt. Each value xi is computed using equation (2):
In [8-12] it is stated that the smaller the particles are, the bigger the frequencies are, therefore an attempt to simulate experimentally recorded time series for different particles in suspension must have different frequency ranges. Several time series were used in the simulation presented in this work and two of them were selected. We call the first s1, having the 50 frequencies generated in the range 100-200 Hz and the second s2 with the frequencies in the range 2000-2500 Hz. Each series lasted for 2 seconds and fs was 10000 Hz, thus having 20000 values each.
Figure 1:A 0.5 s sequence of each series. The blue line is for s1 and the red line for s2.
A plot of a 0.5 s sequence of each series is presented in Fig. 1. It reveals a faster variation of the s2 time series than in s1, as expected.
The SP for the two time series, s1 and s2 is presented in Figs. 2 and 3. We notice the big difference in the SP shape. S1 that contains slower fluctuations is elliptic closer to a circle for this axes range selection while s2 that contains faster fluctuations is a prolonged ellipsis.
The distribution of the velocity values v, in SP, for the two series, each value calculated with equation (1), is presented in Fig.4. We notice the difference between the two distributions, laying in being wider for the s2 time series containing faster fluctuations. Moreover, we notice that the distribution is symmetrical in both situation and the resemblance with the Gaussian is very good.
Figure 2:The SP for s1.
Figure 3:The SP for s2.
Figure 4:The distribution of the velocity values v, in SP, for the two series. The blue line is for s1 and the red line is for s2.
3 RESULTS and discussion
The results of the qualitative analysis presented in the previous section were compared with experimental results. A water sample from a river was extracted after a heavy raining and was analyzed using the DLS procedure described in details in [8 – 12]. The DLS results reveal that the average diameter was 0.134 μm. We name this time series rw1. After 24 h the sample was analyzed again and the average diameter was found to be 0.635 μm. We name this time series rw2. The average diameter increased because the inorganic suspensions, like sand and silt, which have the biggest concentration and smaller diameter, became sediment, while the organic particles having the density closer to the water density remained suspended.
Fig. 5 reveals the SP for the experimental rw1 time series and Fig. 6 for the experimental rw2 series having bigger particles in suspension. Examining the two imagines we notice that the SP of rw1 appears to have more domains, as there are at least two groups of particles in suspension, the inorganic particles, having bigger concentration and the organic particles, which remained suspended.
Figure 5:The SP for rw1.
Figure 6:The SP for rw2.
The other feature of the SP that appears to present differences was the velocity distribution. The distribution for the velocity distributions of rw1 and rw2 time series recorded during a DLS experiment are presented in Fig. 7. Examining the velocity distributions we notice that the distribution of rw2 is wider than the distribution of rw1. Moreover, we notice that the distributions cannot be viewed as simple Gaussian distributions but as a complex overlap of several distributions and this can be the result of the fact that the particles in suspension, both in rw1 and rw2 have a complex size distribution rather than a mono dispersed distribution, therefore the simple observation from Fig. 4 does not stand in all situations.
Figure 7:The distribution of the velocity values v, in SP, for the two series. The blue line is for rw2 and the red line is for rw1.
The SP was defined for a time series as the collection of all possible states. Several time series were generated using a computer code written for this purpose. The time series were the sum of 50 harmonic functions with random amplitudes, phases and frequencies, the last being generated in a certain interval. Two series with frequencies in the intervals 100 – 200 Hz and 2000 – 2500 Hz were generated starting from the assumption that the frequency of the fluctuations in a DLS time series strongly depends of the suspended particle size. A scatter plot in the SP and a velocity distribution are presented for each of the two samples.
The simulation reveals that the plot of the SP can give a qualitative image about the particles assumed to be in suspension. The smaller the particles are and the narrower the particle size distribution is (reflected in smaller frequency range), the more prolonged the image in the SP is. Moreover, smaller particles (hence bigger frequencies) produce a wider velocity distribution. A narrow frequency interval produces a symmetrical distribution resembling very well a Gaussian.
These qualitative conclusions are confirmed in part by the results of the same procedures applied on experimentally recorded time series during a DLS experiment. Using the two river water samples, we found the shape of the image of the SP appears to exhibit different domains if there are more types of particles in suspension. And more, the velocity distribution appears to be the sum of several distributions, under the same circumstances.
These conclusions suggest a simple procedure that can be used in monitoring the evolution of the particle size in a suspension, as in sedimentation of particles in rivers, colloid aggregation or biological fluids. The shape of the space phase and the width and shape of the velocity distribution can provide qualitative information on the distribution of the average suspended particle diameter. Work is scheduled to simulate the time series in a more realistic manner, considering a complex size distribution of the suspended particles.
Here are mentioned the institutes from where are the authors (Style 1, Times New Roman, 10 points, left, normal)[R1]
1 Professor, Ph.D., Faculty of Sciences, Department of Environmental Sciences, Physics, Physical Education and Sports, University “Lucian Blaga”, Sibiu, Romania
2 Physicist, M.S., Group Leader, Continental Automotive Systems, Sibiu, Romania
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ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 7,ISSUE 1/2009
[R1]Trebuie facut un review la povestea asta !!!
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