Noise theoretical limitations

The noise of an actual resistance is always higher than the theoretical limit. While for metal resistors the noise level is close to the theoretical limit, the noise level in carbon resistors is much higher. The resistance of the tunneling junction, which is parallel to the feedback resistor, should be taken into account when its value is comparable to that of the feedback resistor. 

Vary, 1985] Vary, P, (1985). Noise suppression by spectral magnitude estimation -Mechanism and theoretical limits. Signal Processing, 8(4) 387-400. 

Signal co-addition is the traditional method to increase the signal-to-noise ratio of any NMR experiment. However, in the context of photo-CIDNP this collides with the problem of sample decomposition because chemical turnover is a prerequisite for polarization generation the achievable improvement is thus always smaller than the theoretical limit for n acquisitions, i.e. smaller than vw. Since the CIDNP signal is proportional to the number of reacting molecules and each acquisition incurs the same noise penalty, the only room for improvement left with a given chemical system (i.e. fixed amount of polarization per photon) and a given... 

As mentioned above, it is important not to exclude weak data. However, there is no reason to use data from high-resolution shells when they are all very weak, since these reflections are in fact noise and contain no usable information. Generally, intensities are weaker at higher 2 angles and almost no crystal diffracts to the theoretical limit of dmax = /2. Some care must be taken in the determination of the effective maximum resolution of a dataset. 

Scharfetter, H., Hollaus, K., Rosell-Ferrer, J., Merwa, R., 2006b. Single-step 3-D image reconstruction in magnetic induction tomography theoretical limits of spatial resolution and contrast to noise ratio. Ann. Biomed. Eng. 34, 1786-1798. 

The established method for experimentally estimating the diffusion coefficient of a particle makes use of the linear dependence of its mean squared displacement (MSD) with respect to time (another method that involves a maximum likelihood estimate (MLE) that is optimal with respect to an information-theoretic limit has been proposed recently [3]). However, noise in the measurement of the displacements, due to optical and instmment constraints, complicates this estimation. Frequently though, the mean squared measurement noise (Xmeas(O) IS Well approximated to be additive [4] and satisfies (X t)) = 2Dt + ( meas(O)- Thus, a linear regression model is... 

What is more interesting, at least from the theoretical point of view is that oscillatory behaviour might emerge as a result of the interaction between the system and the external noise applied. This phenomenon was earlier described in a radio-engineering context (Kuznetsov et ai, 1965). Studying the role of multiplicative coloured noise for the catalytic oxidation of CO on a platinum surface, de la Rubia et ai (1982) demonstrated that a limit cycle is induced by external noise. Similarly, Treutlein Schulten (1985) found noise induced limit cycles in the Bonhoffer-van der Pol model of neural pulses (see further Lefever Turner (1984)). 

Finally, the storage capacity in Eq. (124) is plotted as a function of AAivf in Fig. 18. The improvement starts at AAlvf = 100 nm, then the storage capacity increases hnearly with increasing AXlvf, and finally, it asymptotically approaches the theoretical limit of the monochromatic case. Note that the kink observed at AAlvf = 0.06 nm is derived from the minimum function that is to say, from that point on, the storage capacity further increases due to the reduction of the intra-page crosstalk noise SPsojfMax in Eq. (117). 

Thus, for even the very best possible high-gain amplifier with complete inversion (Usp = 1), the noise figure F is never less than 2 (3.01 dB). This is a fundamental result for an IMDD system that can be arrived at more generally using Heisenberg s uncertainty principle as the starting point. While this so-called quantum limit has been approached in other types of optical amplifier (most notably erbium-doped fiber amplifiers), it is never achieved in SOAs because, as Fig. 9 shows, the spontaneous emission factor never reaches the theoretical limit of Usp =. ... 

Silver alloy electrodes made in this manner are usually rhodium-plated (tips) and then platinized. They serve well in registering fast transients and exhibit a noise figure close to the theoretical limit for their equivalent resistance. 

The theoretical limit of sensitivity in electrical measurements is determined by the noise generated in the resistors of the actual circuit (Johnson noise). In any resistance, the charge carriers carry out a random motion which is due to their thermal energy. At the terminals of the resistor, R, a power, P, is available which is given by... 

Discussions of relaxation kinetics (see section 6.2) and of transient kinetics, often contain the following general statements. In principle the relaxation spectrum of a reaction contains the necessary information to evaluate all the rate constants of the elementary steps of the reaction. Similarly one can state that in principle the time profile, and its concentration dependence, of the appearance of products during the transient approach to the steady state, contains all the information for the evaluation of the individual rate constants of the formation and interconversion of intermediates. However, in both cases there are important limitations. The theoretical limitations are that the degeneracy of the sequential time constants and the position of the rate limiting step within the sequence of events can reduce the information contents, even if the record of the reaction has an unlimited signal to noise ratio. In real life, noise and restricted time resolution further reduce the number of steps which can be resolved in any particular experiment. The time resolution of different... 

The detection limit metric, as used here, is different from the sensitivity metric. Unlike sensitivity, which deals only with absolute signal, detection limits consider signal-to-noise ratio (S/N). Detection limits also consider the entire analytical method including sample extraction efficiencies and injection volumes in addition to the response characteristics and noise levels of the mass spectrometer signal. Detection limits are therefore analytically more relevant than sensitivity. Sensitivity measurements allow one to estimate the theoretical limits for the quantity of a compound that can be detected, whereas detection limits define what can actually be achieved in a practical setting to provide a meaningful result toward the solution of some application of mass spectrometry. 

This is a theoretical limit - the best we can do with a thermal detector. Since the derivation expresses the noise in terms of incident power. Equation 3.6a is the ideal or best possible noise-equivalent power (NEP) that can be achieved with a thermal detector. We do not need to do any further work to derive this composite figure of merit. 

It is important to understand that this material will not be presented in a theoretical vacuum. Instead, it will be presented in a particular context, consistent with the majority of the author s experience, namely the development of calibrations in an industrial setting. We will focus on working with the types of data, noise, nonlinearities, and other sources of error, as well as the requirements for accuracy, reliability, and robustness typically encountered in industrial analytical laboratories and process analyzers. Since some of the advantages, tradeoffs, and limitations of these methods can be data and/or application dependent, the guidance In this book may sometimes differ from the guidance offered in the general literature. 

The fitted and calculated vibrational frequencies and normal mode composition factors corresponding to the 17 most important NIS bands are presented in Table 5.9. It is evident that the vibrational peaks in the calculated NIS spectrum are typically 0-30 cm lower than to the experimental values. In the calculated NIS spectra, there are two small peaks at 635 and 716 cm (Fig. 5.14b) that are not visible in the experimental spectrum. According to the normal mode calculations these are Fe-N-N and Fe-O-C deformation vibrations. Small admixtures of Fe-N and Fe-O stretching modes account for the calculated nonzero normal mode composition factors. Although the calculated relative intensities are slightly above detection limit dictated by the signal-to-noise ratio, they are determined by values of pea which are very small (0.028 and 0.026 for the peaks at 635 and 716 cm ). They must be considered to be within the uncertainties of the theoretical... 

Theoretically, 8/lp in the resonant wavelength shift scheme is independent of resonance shape or resonant bandwidth, and should be determined merely by instrument resolution, typically less than 10 pm. However, in reality, noise can perturb resonance spectra such that accurate determination of resonant wavelength shift becomes difficult for a broad resonance curve. To enhance accuracy in detecting wavelength shift, narrower resonance is required. This is equivalent to obtaining higher-g resonance behavior. To take into account noise-included detectability of 8/lp, 8/lp can be simply described as a fraction (p) of the full width at half maximum (FWHM) bandwidth of resonance, A7.. WnM. In this fashion, optical detection limit becomes pA/.. WnMAS or p/-vl(QS). In practice, p can be chosen as a reasonable value of 0.1. In the intensity variation scheme, 87 is determined by noise from environment and photodetectors. It can reach as low as several nanowatts with care. 

Theoretical considerations shown in the above equation also indicate that peak area precision is inversely proportional to the peak sig-nal/noise ratio, and to the number of sampling points across the peak width. For very noisy peaks, the peak area precision is limited by random noise fluctuations (Figure 6). Figure 7 shows that the precision of the peak area degrades rapidly when the signal-to-noise ratio is less than 100. Statistical considerations also stipulate a minimum data sampling... 

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