Amplitude-weighted phase

Figure 10 shows the amplitude-weighted phase difference (AWPD) and phase structure (AWPS) functions 1.37 measured using PALS for the aqueous CdS particles (144] discussed in Fig. 9. The conditions are identical (24 C and u = 24 ) with the experiment employing a moving real fringe setup [1.37] and a. 30 Hz, 1.37 V/mm sine wave electric field. The autotrack results utilize a feature of the experimental analysis software [140] that corrects for other convective effects such as sedimentation. The mobility of the particles can be determined by analyzing the data in Fig. 10 with appropriate models for AWPD and AWPS functions (137], Analysis of the data in Fig. 10 yielded AWPD and AWPS...

FIG. 10 Amplitude-weighted phase difference (AWPD) and phase structure (AWPS) functions measured using PALS with autotrack on and off for aqueous CdS particles 1144] at 24 C and 0 = 24 subject to 30 Hz, 1.37 V/mm sine wave electric field. 

Take a blank Data Sheet B. Enter the plane number. Plaee a trial weight at any radius and any angle in that plane. Enter these values on the sheet. Now, operate the maehine at the balaneing speed, and measure the vibration amplitude and phase in eaeh plane. Repeat the proeedure for eaeh plane. (Plaee only one trial weight in only one plane at a time.) When finished, you should have as many Data Sheets B as the number of planes. 

The preceding analysis views the problem of solving for the sine-wave amplitudes and phases in the frequency domain. Alternatively, the problem can be viewed in the time domain. It has been shown that [Quatieri and Danisewicz, 1990], for suitable window lengths, the vectors a andJ3 that satisfy Equation (9.75) also approximate the vectors that minimize the weighted mean square distance between the speech frame and the steady state sinusoidal model for summed vocalic speech with the sinusoidal frequency vector . Specifically, the following minimization is performed with respect to a andJ3... 

The term is a constant offset, or the average of the waveform. The b and c coefficients are the weights of the wth harmonic cosine and sine terms. If the function is purely even about t = 0 (this is a boundary condition like that discussed in Chapter 4), only cosines are required to represent it, and only the b terms would be nonzero. Similarly, if the function is odd, only the terms would be required. A general function Fper(0 will require sinusoidal harmonics of arbitrary amplitudes and phases. The magnitude and phase of the mth harmonic in the Fourier series can be found by ... 

One of the attractive features of the Opto-VLSI-based tunable time delay architecture is its ability to generate multiple RF delays without the need for RF splitters. Furthermore, the amplitude weight of each generated RF delay sample can simultaneously be controlled. This architecture offers excellent flexibility in applications such as phased-array null steering because multiple true-time RF delays for each antenna element can simultaneously be synthesized using computer generated holograms. 

Figure 6.29. AWPD and amplitude-weighted square phase difference (A WPS) of a PSL sample in distilled water. Circle, experimental data point solid lines, the best fit according to the theory (by permission of Academic Press) [55].

AWPS Amplitude-weighted square phase difference... 

In Sect. 6.2.4,we saw that the total scattering ampUtude can be calculated by simply summing all other amplitudes, with the weighted phase factor (Eq. 6.6). For a system of N particulate inhomogeneities, each at a position vector, P , Eq. (6.6) can be modified to calculate the total scattering amplitude of these inhomogeneities by the equation ... 

From Equation (17-4), one will find that the phase lag is a function of the relative rotating speed lu/lu and the damping factor (See Figure 17-1.) The force direction is not the same as the maximum amplitude. Thus, for maximum benefit, the correction weight must be applied opposite to the force direction. 

Johnson and Borisy first showed that the lag phase in the plot of turbidity (i.e., polymer weight concentration) versus time accounted for only 5—10% of the entire amplitude obtained upon completion of the polymerization process. By fitting the elongation phase to a single exponential process, these investigators arrived at the correct conclusion that microtubule number concentration becomes relatively stable within the first minutes... 

In practice, recombination of structure factors involves first weighting of the phases of the modified structure factors in a resolution dependent fashion, according to their estimated accuracy or probability. Every phase also has an experimental probability (determined by experimental phasing techniques and/or molecular replacement). The two distributions are combined by multiplication, and the new phase is calculated from this combined probability distribution. The measured associated structure factor amplitude is then scaled by the probability of the phase, and we have our set of recombined structure factors. 

Theory for the self- and tracer-diffusion of a diblock copolymer in a weakly ordered lamellar phase was developed by Fredrickson and Milner (1990). They modelled the interactions between the matrix chains and a labelled tracer molecule as a static, sinusoidal, chemical potential field and considered the Brownian dynamics of the tracer for small-amplitude fields. For a macroscopically-oriented lamellar phase, they were able to account for the anisotropy of the tracer diffusion observed experimentally. The diffusion parallel and perpendicular to the lamellae was found to be sensitive to the mechanism assumed for the Brownian dynamics of the tracer. If the tracer has sufficiently low molecular weight to be unentangled with the matrix, then its motion can be described by a Rouse model, with an added term representing the periodic potential (Fredrickson and Bates 1996) (see Fig. 2.50). In this case, motion parallel to the lamellae does not change the potential on the chains, and Dy is unaffected by... 

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