Frequency noise spectral density

The generation of photons obeys Poisson statistics where the variance is N and the deviation or noise is. The noise spectral density, N/, is obtained by a Fourier transform of the deviation yielding the following at sampling frequency,... 

For 8=1, the noise spectral density is a constant (white noise), at least in the angular frequency range co oo, the Langevin force F(t) is delta-correlated, and the Langevin equation is nonretarded. The white noise case corresponds to Ohmic friction. The cases 0 < 8 < 1 and 8 > 1 are known respectively as the sub-Ohmic and super-Ohmic models. Here we will assume that 0 < 8 < 2, for reasons to be developed below [28,49-51]. 

Noise spectral density (Eq. (2)) can be expressed through special functions. Since there is no problem in numerical calculation of S(co) using Eqs. (11) and (2), we will not set it out here. We present, however, simple results for the low-frequency noise. In the case of comparable and... 

We can observe that in this case, the cell measured noise emission spectral density is not flat versus frequency. Therefore we may consider that is composed of two components a constant noise spectral density due to the electrolyte itself an excess noise spectral density... 

The noise spectral density is l/f type in the frequency band 10 mHz to 300 Hz in normal and reverse operation mode. Noise spectral density is a quadratic function of the current, when the electric field strength in isolating layer is so low that avalanche process caimot occur. Measurement performed at very low frequency band 10 mHz to 1 Hz reveals that for some samples noise is l/f type, but it was observed some time instability, which is probably related to the self-healing process. 

The most important sources of fluctuation in tan-taliun capacitors consist in regenerative micro breaks, fluctuation of polarization and mechanical strain. The frequency dependence of the noise spectral density in mHz region gives the information on slow irreversible processes of tantalum pentoxide crystallizations and oxide reduction. The self-healing process can improve... 

Qualitative characterization of the excess noise is possible with the use of noise spectral density at given frequency (Sikula et al. 1994). In our investigation, the measurable quantity is an indicator of sample quality and reliability Cq given as... 

Where Su is noise spectral density,/ - frequency and U - apphed voltage. The lower value of Cq and its dispersion, the better is the technology. This hypothesis was proved by classical ageing methods. 

Where 5/ is the current noise spectral density,/ is the frequency, N is the total number of carriers in the sample active region, I is the device DC current and a is an empirical constant, which is now extensively used to characterize the device structure perfectness. 

Figure 18. The voltage noise spectral density vs. frequency for sample N51 at T = 298K and 260K.

In lasers, luminescent diodes, power diodes and solar cells it was found that in the low injection region the current noise spectral density is a quadratic function of the forward current. Typically, the excess current is a dominant current component in this region. The current noise spectral density vs. frequency for PN junction is shown in Fig. 19. Curve 1 denotes the current noise spectral density for the low injection range, curve 2 is the current noise spectral density for the... 

This noise power spectral density expressed in Equation (1.5) and Equation (1.6) increases linearly with absolute temperature, and is independent of frequency. By solving Equation (1.6) for the temperature T, it is possible to determine it from the available noise spectral density. This can be extended to define a noise temperature for any source of white noise, even nonthermal noise, in order to allow convenient... 

Noise power density represents the average noise power over a 1-Hz bandwidth as a function of frequency. A companion quantity is noise spectral density denoted by e (f) with units given by volts/ hertz. 

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. ... 

At the resonant frequency the noise spectral density has a maximum thus resonant frequencies should lie outside the bandwidth of the detector or be narrow enough to be filtered in the data analysis. Far from resonance, the spectral density of thermal noise is lower for low loss factors

test masses and the last stage of isolation systems are made of low loss materials. There is experimental evidence that for most materials the loss factor is almost independent of frequency thus below resonance x ( >) a T and above resonance x (isolation systems contribute with several modes, mainly the pendulum mode of the test mass and the violin modes of the suspension wires. Since test masses are suspended as pendulums, the effective loss factor is... 

By frequency response we normally mean the detector response to modulated IR input as a function of the modulation frequency. One way to measure the frequency response is to directly measure the output as the modulation frequency is changed. Another method relies on the response sampled quickly after viewing a quick change in the IR input. Still a third infers the frequency response of the DC output from the frequency dependence of the noise spectral density. 

Frequency Response by Noise Spectral density Still another way to determine the frequency response is to measure the noise spectral density (noise in a small, constant bandwidth) versus frequency. At frequencies above the 1 // noise region, the frequency response of the noise and the signal are (theoretically) the same, so the measurement of the noise spectral density provides the desired signal response. This method eliminates the need for a high-speed chopper, but one must still compensate for any amplifier rolloff, and use Fourier processes. 

We noted that for narrow electrical bandwidths the noise voltage is proportional to the bandwidth. The noise in a given bandpass A/ divided by the square root of the A/is the noise spectral density n(f), with units of V/Hz (volts per root Hz, or V/rHz). We will use the symbol n(f) or NSD for the noise spectral density, but there is no standard symbol. Within small frequency regions, the noise spectral density is independent of frequency. If, however, we look at a wide range of frequencies, n(f) has significant frequency dependence, as shown in Figure 10.16. 

Figure 10.16 Ideal noise spectral density versus frequency. Note the increase at low frequencies, the flat region ( white noise ), and the eventual falloff at very high frequencies. Measured data will not be as smooth, and will have spikes at some frequencies due to external source (60 Hz power, fluorescent lights, etc.).

Noise is characterized by the time dependence of noise amplitude A. The measured value of A (the instantaneous value of potential or current) depends to some extent on the time resolution of the measuring device (its frequency bandwidth A/). Since noise always is a signal of alternating sign, its intensity is characterized in terms of the mean square of amplitude, A, over the frequency range A/, and is called (somewhat unfortunately) noise power. The Fourier transform of the experimental time dependence of noise intensity leads to the frequency dependence of noise intensity. In the literature these curves became known as PSD (power spectral density) plots. 

Both Ti and T2 relaxations of water protons are mainly due to fluctuating dipole-dipole interactions between intra- and inter-molecular protons [62]. The fluctuating magnetic noise from all the magnetic moments in the sample (these moments are collectively tamed the lattice) includes a specific range of frequency components which depends on the rate of molecular motion. The molecular motion is usually represented by the correlation time, xc, i.e., the average lifetime staying in a certain state. A reciprocal of the correlation time corresponds to the relative frequency (or rate) of the molecular motion. The distribution of the motional frequencies is known as the spectral density function. 

This function (Fig. 3.3) has its inflection point at spectral density function. The nucleus picks up the needed to frequency for its relaxation. The probability of this to occur depends on the spectral density (i.e. on the value of function (3.2)) at that frequency. 

In contrast to the diffusion processes, the fluctuating force 3F t) can no longer be derived from the Wiener process, and its spectral density defined as 2kBT 0 times the Fourier transform of the memory function C(f) is frequency limited and has no more the white noise characteristics. 

Cancelling of the signal. For a sinusoidal signal of frequency 0 (which is supposed to correspond to the center frequency of one of the filters of the filter-bank), it is easily checked (assuming that the additional noise power spectral density is sufficiently smooth) that Eq. 4.18 becomes... 

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]. 

We shall do that with the same kind of the operational definition as in Section IV.B.5. Specifically, we assume that each signal-to-noise value emerges as a result of a three-step procedure (1) one decides on the frequency il at which the test would be performed then (2), at double this frequency, the noise spectral power density in the state with no driving field is evaluated, thus yielding... 

White noise is by definition a random signal with a flat power spectral density (i.e., the noise intensity is the same for all frequencies or all times, of course, within a finite range of frequencies or times). A time-random process u>w(t) is white in the time range a < t < b if and only if its mean value is zero ... 

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). 

More detailed information can be obtained from noise data analyzed in the frequency domain. Both -> Fourier transformation (FFT) and the Maximum Entropy Method (MEM) have been used to obtain the power spectral density (PSD) of the current and potential noise data [iv]. An advantage of the MEM is that it gives smooth curves, rather than the noisy spectra obtained with the Fourier transform. Taking the square root of the ratio of the PSD of the potential noise to that of the current noise generates the noise impedance spectrum, ZN(f), equivalent to the impedance spectrum obtained by conventional - electrochemical impedance spectroscopy (EIS) for the same frequency bandwidth. The noise impedance can be interpreted using methods common to EIS. A critical comparison of the FFT and MEM methods has been published [iv]. 

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