Spectral density of noise

It is clear that the nature of the electromagnetic phenomena is the same for optics and radio wave, the only experimental differences being that radiowave photons are far below the spectral density of noise of actual detectors and that the temperature of the source is such that each mode is statistically populated by many photons in the radio wave domain whereas the probability of presence of photons is very small in the optical domain. 

In optical domain, preamplifier is no more an utopia and is in actual use in fiber communication. However quantum physics prohibits the noiseless cloning of photons an amplifier must have a spectral density of noise greater than 1 photon/spatial mode (a "spatial mode" corresponds to a geometrical extent of A /4). Most likely, an optical heterodyne detector will be limited by the photon noise of the local oscillator and optical preamplifier will not increase the detectivity of the system. 

The main source of noise of such a heterodyne detector is the photon noise that takes place at the splitting of the local oscillator. Quantum physicists see this noise as originating from vacuum fluctuation on the input arm. This gives directly the spectral density of noise at input hv/2. 

Wiener inverse-filter however yields, possibly, unphysical solution with negative values and ripples around sharp features (e.g. bright stars) as can be seen in Fig. 3b. Another drawback of Wiener inverse-filter is that spectral densities of noise and signal are usually unknown and must be guessed from the data. For instance, for white noise and assuming that the spectral density of object brightness distribution follows a simple parametric law, e.g. a power law, then ... 

K. Tachibana, K. Miya, K. Furuya, G. Okamoto, Changes in the power spectral density of noise current on type 304 stainless steels during the long time passivation in sulfuric acid solutions, Corros. Sci. 31 (1990) 527-532. 

Describing SR in terms of a susceptibility is particularly advantageous for systems that are in thermal equilibrium, or in quasiequilibrium. In such cases the fluctuation-dissipation relations [9] can be used to express the susceptibility in terms of the spectral density of fluctuations in the absence of the periodic driving. This was used explicitly in the case of noise-protected heterodyning. It is true in general that the analysis of fluctuations is greatly facilitated by the presence of thermal equilibrium when the conditions of detailed balance and of the time reversal symmetry are satisfied [44]. 

For weak noise the spectral density of fluctuations (SDF) at the output of the... 

More precisely, die quantity displayed is the signal power estimated from 10ms frames. As die power spectral densities of die two types of noise exhibit a strong peak at the null frequency, the two noises were pre-whitened by use of an all-pole filter [Cappe, 1991]. This pre-processing guarantees that the noise autocorrelation functions decay sufficiently fast to obtain a robust power estimate even with short frame durations [Kay, 1993]. 

Flicker-noise spectroscopy — The spectral density of - flicker noise (also known as 1// noise, excess noise, semiconductor noise, low-frequency noise, contact noise, and pink noise) increases with frequency. Flicker noise spectroscopy (FNS) is a relatively new method based on the representation of a nonstationary chaotic signal as a sequence of irregularities (such as spikes, jumps, and discontinuities of derivatives of various orders) that conveys information about the time dynamics of the signal [i—iii]. This is accomplished by analysis of the power spectra and the moments of different orders of the signal. The FNS approach is based on the ideas of deterministic chaos and maybe used to identify any chaotic nonstationary signal. Thus, FNS has application to electrochemical systems (-> noise analysis). 

One d i sad vantage ot the iiseof randont nttisc excitation in. spectro.scopy and imaging isthe unavoidable presence of systematic noise as a consequence of the variance of the power spectral density of the excitation It is reduced but not eliminated by formation of... 

Initial measurements on valinomycin-doped membranes (13, 14) showed that lipid bilayers, which provide cells with an effective permeability barrier, are equilibrium objects by this criterion. Within an accuracy of several percent, the experimentally obtained values of the spectral density of voltage noise showed agreement with those calculated from relation 1. Figure 1 illustrates this agreement for three valinomycin concentrations. For the valinomycin-K+ system chosen for these experiments and for the frequency range used in measurements, the dispersion in membrane impedance was caused only by geometrical capacitance of the bilayer the characteristic times of the transport process itself were too small to influence impedance in this range. 

A noise that has a clearly distinct origin from noise discussed in previous sections is the electric noise that originates in modulation of ion transport by fluctuations in system conductance. These temporal fluctuations can be measured, at least in principle, even in systems at equilibrium. Such a measurement was conducted by Voss and Clark in continuous metal films (44). The idea of the Voss and Clark experiment was to measure low-frequency fluctuations of the mean-square Johnson noise of the object. In accordance with the Nyquist formula, fluctuations in the system conductance result in fluctuations in the spectral density of its equilibrium noise. Measurement of these fluctuations (that is, measurement of the noise of noise) yields information on conductance fluctuations of the system without the application of any external perturbations. The samples used in these experiments require rather large amplitude conductance fluctuations to be distinguished from Johnson noise fluctuations because of the intrinsic limitation of statistics. Voss and... 

Figure 5. Spectral density of polymer-induced current noise in the open alamethicin channel vs. polymer molecular weight (46). Data represent noise in different channel-conducting levels at 150 mV in the presence of polyethylene glycols of different sizes added to 1-M NaCl aqueous solutions to obtain 15% weight-to-weight concentration. The vertical scale is given in 10 27 A2/Hz...

Studies of single channels formed in lipid bilayers by Staphylococcus aureus alpha toxin showed that fluctuations in the open-channel current are pH-dependent (47). The phenomenon was attributed to conductance noise that arises from reversible ionization of residues in the channel-forming molecule. The pH-dependent spectral density of the noise, shown in Figure 6, is well described by a simple model based on a first-order ionization reaction that permits evaluation of the reaction parameters. This study demonstrates the use of noise analysis to measure the rate constants of rapid and reversible reactions that occur within the lumen of an ion channel. 

A sharp drop, by several orders of magnitude, of (f)llP with the change in concentration of CO in the air suggests the possibility of making a precise estimation of the gas content in air - that is, we can offer a new method of estimation of the concentration of gases in an environment and the noise spectroscopy (Figs 12.3 and 12.4). Su is the spectral density of the noise voltage. 

Flicker noise has long been a nuisance in measurements of the spectral density of conductance fluctuations in experiments with biological membranes. In 1973, Fishman measured the differential voltage noise spectra of a native membrane and a membrane whose potassium conductance has been blocked with tetraethylammonium. The differential spectrum he obtained fitted well... 

A more general process known as least-squares filtering or Wiener filtering can be used when noise is present, provided the statistical properties of the noise are known. In this approach, g is deblurred by convolving it with a filter m, chosen to minimize the expected squared difference between / and m g. It can be shown that the Fourier transform M of m is of the form (1///)[1/(1 - - j], where S is related to the spectral density of the noise note that in the absence of noise this reduces to the inverse filter M = /H. A. number of other restoration criteria lead to similar filter designs. 

The spectral density of the process is constant at constant temperature, i.e., 6 co) is independent of the noise frequency the case of the so-called thermal noise. 

To conclude our discussion of noise measurements in electrolytes, in the absence of an applied electrical field, we can see that spectral density of electrolyte itself is constant versus frequency and corresponds to thermal white noise the electrode-electrolyte interface spectral density is not a characteristic of concentrated electrolytes it cannot be detected in the case of dilute electrolytes because... 

Zhigalskii (1993) found the correlation between the occurrence of 1/fnoise andnon-hnearity ofVA characteristics of Mo and Cr thin films. It is supposed that this kind of noise in thin films is related to the vacancy concentration and mobile impurities. Noise with spectral density of the l/f type is observed in many different monocrystals as well as in polycrystals. 

Figure 15. Noise spectral density of two 100 resistors. Curve 0 is the apparatus noise, curve 1 is noise spectral density of a low noise resistor and curve 2 is noise spectral density of a resistor with burst noise source.

If we know the exact forms of these correlation functions, we can use the Fourier transform to obtain the power spectral density of the output which will, in turn, enable us to evaluate the final output signal-to-noise power ratio for the three-frequency system. 

The frequency noise power spectral density of a SL typically exhibits a 1/f dependence below 100 kHz and is flat from 1 MHz to well above 100 MHz. Relaxation oscillations will induce a pronounced peak in the spectrum above 1 GHz. The "white" spectral component represents the phase fluctuations that are responsible for the Lorentzian linewidth and its intensity is equal to IT times the Lorentzian FWHM.20 xhe 1/f component represents a random walk of the center frequency of the field. This phase noise is responsible for a slight Gaussian rounding at the peak of the laser field spectrum and results in a power independent component in the linewidth. Figure 3 shows typical frequency noise spectra for a TJS laser at two power levels. 

Measured input-noise spectral densities of VLSI nanoclamp and backgroimd in the nanopore sensing. 

The membrane component of the background noise consists mainly of shot noise [8-11], that is the expected electrical noise created by the ions that cross the membrane by leakage or ionic pumps. The spectral density of shot noise is directly proportional to the unidirectional membrane current. Thus, the spectral density of the noise will increase by increasing the surface of the membrane patch, and consequently the total current (leakage and pumps). This implies that the noise conditions will be improved when current is recorded from a small piece of membrane (a patch). 

Johnson noise [11-13], which occurs on the seal, i.e., in the junction between the pipette and the membrane. The spectral density of this noise is inversely proportional to the seal resistance. Therefore, it is important to have a very tight seal, the so-called gigaseal, which should significantly improve the noise characteristics of the patch-clamp record. 

Two common varieties of noise power densities are white noise and 1 // noise. The spectral density of white noise is a constant e w, and its rms value, from Eq. (7.141), is... 

The noise power density of 1 // noise has the form e (/) = K /f where Ky is a constant representing the extrapolated value of the noise spectral density at / = 1 Hz. The spectral density of l/f noise is Kyl f indicating that on a log-log plot the spectral density has a slope of—0.5 decades/decade. The rms value of 1 // noise, obtained from Eq. (7.141), is... 

The noise sources in the noise model for an op-amp are composed of a mixture of white and 1 // noise as shown for a voltage noise source in Fig. 7.98. At low frequencies, l/f noise dominates, and at high frequencies, white noise dominates. The boundary is called the corner frequency and is denoted as fey in Fig. 7.98. A similar plot appHes to the noise spectral density of current noise sources in the op-amp model. The corner frequency for a noise current source is denoted as f. ... 

Figure 7.99 shows the op-amp noise model. A noisy op-amp is modeled by three noise generators and a noiseless op-amp. Polarities and reference directions are not shown since noise generators do not have true polarity or reference direction assignments. The spectral densities of the current sources... 

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