Multiple frequencies Multiplicative noise

The frequency distribution of noise is characterized by a power spectrum. There a two types. First, white noise, whose noise power is independent of the frequency. This noise arises from the statistics of electrons or photons and of the thermal energy of conductors. White noise can be reduced by extending the measuring time. Second, excess low frequency noise, flicker, or 1// noise is due to fluctuations, drift and schlieren. It can be reduced by modulation. All types of noise are reduced by multiplex procedures, multichannel techniques, and multiple recording (Schrader, 1980). 

When a multiple-frequency excitation signal is used, the impedance at all frequencies is obtained at the same time, which is a considerable advantage when studying systems with fairly low stability. If the impedance is measured separately at every frequency the noise-level is usually lower, but the system may change during the time passed from the recording of... 

However, for the same data collection time, interferometry is more sensitive to multiplicative noise (i.e., noise proportional to the signal) than continuous-scan interferometry [591]. To eliminate the multiplicative and 1// noise, phase modulation (at 400 Hz) of IR radiation in conjunction with LIA demodulation is used [591]. Since the LIA and some IR detectors need the IR signal to be modulated at a single carrier frequency, a mechanical chopper, phase modulation (when at each position the fixed mirror is dithered at a fixed frequency), or modulation of absorption of the sample is used to produce a carrier frequency. In this case, the TR measurement is referred to as a synchronous multiple-modulation experiment. Multiple modulation is unnecessary if the so-called dc coupled detector which does not require a varying signal is used. 

In the fast-continuous region, species populations can be assumed to be continuous variables. Because the reactions are sufficiently fast in comparison to the rest of the system, it can be assumed that they have relaxed to a steady-state distribution. Furthermore, because of the frequency of reaction rates, and the population size, the population distributions can be assumed to have a Gaussian shape. The subset of fast reactions can then be approximated as a continuous time Markov process with chemical Langevin Equations (CLE). The CLE is an ltd stochastic differential equation with multiplicative noise, as discussed in Chapter 13. 

The sound absorption of materials is frequency dependent most materials absorb more or less sound at some frequencies than at others. Sound absorption is usually measured in laboratories in 18 one-third octave frequency bands with center frequencies ranging from 100 to 5000 H2, but it is common practice to pubflsh only the data for the six octave band center frequencies from 125 to 4000 H2. SuppHers of acoustical products frequently report the noise reduction coefficient (NRC) for their materials. The NRC is the arithmetic mean of the absorption coefficients in the 250, 500, 1000, and 2000 H2 bands, rounded to the nearest multiple of 0.05. 

By means of this procedure our problem is not only reduced from three to two dimensions, but also is the statistical noise in the scattering data considerably reduced. Multiplication by —4ns2 is equivalent to the 2D Laplacian89 in physical space. It is applied for the purpose of edge enhancement. Thereafter the 2D background is eliminated by spatial frequency filtering, and an interference function G(s 2,s ) is finally received. The process is demonstrated in Fig. 8.27. 2D Fourier transform of the interference function... 

There is significant debate about the relative merits of frequency and time domain. In principle, they are related via the Fourier transformation and have been experimentally verified to be equivalent [9], For some applications, frequency domain instrumentation is easier to implement since ultrashort light pulses are not required, nor is deconvolution of the instrument response function, however, signal to noise ratio has recently been shown to be theoretically higher for time domain. The key advantage of time domain is that multiple decay components can, at least in principle, be extracted with ease from the decay profile by fitting with a multiexponential function, using relatively simple mathematical methods. 

Additionally, with the inclusion of computers as part of an instrument, mathematical manipulation of data was possible. Not only could retention times be recorded automatically in chromatograms but areas under curves could also be calculated and data deconvoluted. In addition, computers made the development of Fourier transform instrumentation, of all kinds, practical. This type of instrument acquires data in one pass of the sample beam. The data are in what is termed the time domain, and application of the Fourier transform mathematical operation converts this data into the frequency domain, producing a frequency spectrum. The value of this methodology is that because it is rapid, multiple scans can be added together to reduce noise and interference, and the data are in a form that can easily be added to reports. 

Burke and his students (11) have published a proposal for solving the non-linearity problem associated with CC and the consequent correlation noise. They used a constant frequency multiple injection signal while this occurred, this frequency was modulated. Before each injection, a random number was generated to determine the magnitude and sign of the deviation from the carrier frequency for the next injection time. Thus, the next... 

This approach needs modification as soon as multiple attracting periodic trajectories exist for a particular set of operating parameters. A conceptually different modification will be necessary to account for attractors which are not simply periodic. Quasi-periodic solutions, characterized by multiple frequencies, are the first type one should expect these are by no means exotic but occur generally in several periodically forced systems. Deterministic chaotic situations, arising from the system nonlinearities (and not the stochastic responses due to random noise) need not be discarded as intractable (Wolf et al., 1986 Shaw, 1981). 

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