Noise limits

Bialkowski, S. E., Data Analysis in the Shot Noise Limit 1. Single Parameter Estimation with Poisson and Normal Probability Density Functions, Anal. Chem. 61, 1989, 2479-2483. 

Stability may not be as much of a problem as with a diode source. However, there are problems with this method as well. The range of tunability is limited by the absorption properties of the nonlinear crystal which generates the difference frequency. At present, tunability is limited to wavenumbers >2500 cm-1 and conversion efficiencies are low. Typical laser powers in the CH2 experiments (82) were 20 n W (compared to the power of the CO lasers, 10 mW-1 W). This produces a situation where IR detectors, particularly fast ones, may be close to or background noise limited. However, it is clear that more applications of this technique will appear in the future. 

Nir, E., Michalet, X., Hamadani, K. M., Laurence, T. A., Neuhauser, D., Kovchegov, Y. and Weiss, S. (2006). Shot-noise limited single-molecule FRET histograms Comparison between theory and experiments. J. Phys. Chem. B 110, 22103-24. 

Some detectors for the visible and UV spectral regions can detect individual photons. These detectors are shot-noise limited. X-ray and gamma-ray spectroscopy also detects... 

Our first chapter in this set [4] was an overview the next six examined the effects of noise when the noise was due to constant detector noise, and the last one on the list is the first of the chapters dealing with the effects of noise when the noise is due to detectors, such as photomultipliers, that are shot-noise-limited, so that the detector noise is Poisson-distributed and therefore the standard deviation of the noise equals the square root of the signal level. We continue along this line in the same manner we did previously by finding the proper expression to describe the relative error of the absorbance, which by virtue of Beer s law also describes the relative error of the concentration as determined by the spectrometric readings, and from that determine the... 

F(x), here, is (Es + AEs)/ Er + A )), as we just noted. In the previous case, the weighting function was the Normal distribution. Our current interest is the Poisson distribution, and this is the distribution we need to use for the weighting factor. The interest in our current development is to find out what happens when the noise is Poisson-distributed, rather than Normally distributed, since that is the distribution that applies to data whose noise is shot-noise-limited. Using P to represent the Poisson distribution, equation 49-59 now becomes... 

The detailed investigation of superconducting (Al) bolometers is due to Clarke and Richards [64,65], This bolometer had a very low heat capacity and reached the thermal fluctuation noise limits. This bolometer needed a much more complex electronics than those using Ge thermistors. 

The sensitivity of this system is 430 pg/0.0044 absolute. The standard deviation of the baseline noise is about 0.0007 absolute, resulting in a noise-limited detection limit of 140 pg of germanium at the 95% confidence level. 

In detector noise limited spectroscopies such as PAS it is advantageous to enhance the throughput of energy (Jacquinot s advantage) by utilizing a Michel son interferometer. One then Fourier transforms (FTs) the resulting interferogram to yield a PA spectrum that qualitatively resembles an absorption spectrum. 

Even if these problems with overlapping bands can be avoided, at high enough pressures the gas phase absorption will be so strong that the sensitivity will be detector noise limited. This problem can only be handled by keeping the optical path length in the pressure cell as small as possible. 

The Rayleigh resolution criterion can be satisfied within noise limits if spatial frequencies are recorded to the Scherzer point resolution ... 

The Stratonovich interpretation of Eq. (2.238) may also be obtained [31,32] from the white-noise limit of a sequence of stochastic ordinary differential equations (ODEs) of the form... 

Stratonovich SDEs, unlike Ito SDEs, may thus be manipulated using the familiar calculus of differentiable functions, rather than the Ito calculus. This property of a Stratonovich SDE may be shown to follow from the Ito transformation rule for the equivalent Ito SDE. It also follows immediately from the definition of the Stratonovich SDE as the white-noise limit of an ordinary differential equation, since the coefficients in the underlying ODE may be legitimately manipulated by the usual rules of calculus. 

Juma, M., Korterik, J. R, Otto, C., and Offerhaus, H. L. 2007. Shot noise limited heterodyne detection of CARS signals. Opt. Express 15 15207-13. 

Deconvolution of spectra, such as infrared absorption spectra, provides researchers with a tool that they can use to carry out a particular experiment. It provides an extra measure of flexibility in the design of experiments and in the observation process. In dispersive infrared spectroscopic systems, Blass and Halsey (1981) have shown that effective resolution-acquisitiontime trade-offs may be made, owing to the fact that dispersive infrared spectroscopy is usually detector noise limited. Acquisition rates are therefore optical throughput dependent, which is equivalent to saying that acquisition rates are resolution dependent. Blass and Halsey (1981) show that, for a constant signal-to-noise ratio,... 

Because infrared spectrophotometry is detector noise limited, a profitable trade-off is possible between signal-to-noise ratio and resolution. Such a trade-off may be useful when deconvolution is used to recover resolution lost by opening monochromator slits to increase acquisition rates. In any system where degrading the resolution improves the signal-to-noise ratio nonlinearly, a potentially useful trade-off is possible. 

To determine how noise limits pixel dimension, arrange Equation 5 so that normalized exposure time depends on pixel dimension note that 2 3ir = 4 ... 

Remark. The white noise limit is not sufficiently defined by just saying rc 0. We have to construct a sequence of processes which in this limit reduce to Gaussian white noise. For that purpose take a long time interval (0, T) and a Poisson distribution of time points Ta in it with density v. To each Ta attach a random number ca they are independent and identically distributed, with zero mean. Consider the process... 

Exercise. Formulate the white noise limit for a random matrix process. 

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