Bandwidth inversion

One should note that the phase shift becomes time-independent and maximal for a = 1, i.e., at the resonance condition v = vG. The frequency spectrum 4>(a) bears a sine shape with a bandwidth inversely proportional to the number of oscillations of the gradient field (Fig. 4). Such a behaviour was also predicted in Ref. 15. Recording in a systematic way the phase shift as a function of vG without space encoding would be a very fast and efficient method to scan in a whole object the possible frequencies of spin motions. 

To improve URT, we used a deconvolution technique. Our enhancement procedure is based on Papoulis deconvolution i.e. on an extension of the generalized inversion in the complementary bandwidth of the electro-acoustic set-up. 

Thus the settling time is inversely proportional to the bandwidth. Comparing equation (6.70) with equation (3.68) gives... 

In the hrst case, the degree of self coherence depends on the spectral characteristics of the source. The coherence time Tc represents the time scale over which a held remains correlated this hme is inversely proportional to the spectral bandwidth Au) of the detected light. A more quantitative dehnition of quasi-monochromatic conditions is based on the coherence time all relevant delays within the interferometer should be much shorter than the coherence length CTc. A practical way to measure temporal coherence is to use a Michel-son interferometer. As we shall see, in the second case the spatial coherence depends on the apparent extent of a source. 

The bandwidth is inversely proportional to the pulse duration. Following is a computer simulation of a short pulse kft) and its calculated... 

Experimentally, dvz is directly proportional to the laser bandwidth and is a constant dvx is determined by the slit width and is inversely proportional to the ion arrival time (i.e. a ID solid angle factor) and dvy/dt denotes the time-to-speed transformation in the ion TOF measurement, which can readily be derived from the equation of motion. It was found... 

For the Normal distribution, the exponential term has become a constant, and we see that the maximum magnitude of the derivative is inversely proportional to <72 (for the constant area expression) or inversely as a (for the constant height expression). This confirms our observation from figure 54-1. For the Lorentzian distribution, we see that the derivative decreases with the second power of the bandwidth. 

The negative sign in equations 54-19 and 54-20 reflect the fact that the maximum second derivative is a negative value, which also agrees with Figure 54-1, and it also tells us that the magnitude of the second derivative decreases inversely as the cube of a (for the Normal band shape) and inversely as the fifth power of a (for the Lorentzian band shape), that is as the bandwidth of the absorbance band increases. This explains why the derivatives of the broad absorbance band decrease with respect to the narrow absorbance band as we see in Figure 54-1, and more so as the derivative order increases. 

D(co)VV (cn)/ii m equal to the expression (l/rr) y y co2). Integrating from frequency zero up to infinite, one gets the empirical formula K(t-x)= (X/ft) y exp(-y t-x ). Here, 1/y represents the memory time of the dissipation and is essentially the inverse of the phonon bandwidth of the heat bath excitations that can be coupled to the oscillator. It reduces to a delta function when y->infinite. The correlation function (t-t), in this model is [133]... 

For pulse radar, the receiver s bandwidth B is inversely proportional to the pulse width tp. Substituting the receiver s bandwidth by pulse time in (7), one obtains the equation... 

Due to the BSOS discussed in Section 4.3, the inversion profiles are shifted by 1.38 kHz to the left and right side, respectively (Fig. 13b). The amount of the shift compared to the bandwidth of the inversion profile is significant. To have a right position of inversion, the effect of the BSOS needs to be taken... 

Figure 22a shows the inversion and null profile of this scheme. It has a narrow inversion and null bandwidths and in addition, the null point moves when the RF field strength /j varies. 

For simplicity, we will let W = 1/(2M), i.e the bandwidth is inversely proportional to the number of grid cells. Another alternative is to use the M grid cells to define the kernel function ... 

The Bandwidth is essentially a normalized half confidence band. The confidence interval bandwidths for 9 data sets using inverse transformed data are given in Table X. The bandwidths are approximately the vertical widths of response from the line to either band. The best band was found for chlorpyrifos, 1.5%, at the minimum width (located at the mean value of the response) and 4.9% at the minimum or lowest point on the graph. Values for fenvalerate and chlorothalonil were slightly higher, 2.1-2.2% at the mean level. The width at the lowest amount for the former was smaller due to a lower scatter of its points. The same reason explains the difference between fenvalerate and Dataset B. Similarly, the lack of points in Dataset A produced a band that was twice as wide when compared to Dataset B. Dataset C gave a much wider band when compared to Dataset B. 

Table X. Confidence Interval Bandwidths from the Regression of Transformed Data Sets. Inverse Transformed Data.

Table XII. Response Error Bound Bandwidths of Inverse Transformed Data. a sQ.025 where 95% of Response Unknowns Will Lie within the Response Error Bounds of the True...

Table XIV. Estimated Amount Interval Bandwidth from Inverse...

The linewidth Afl of such a single mode laser is determined by the bandwidth A of the laser cavity (which is inversely proportional to its g-factor), the laser frequency v and the output power P at this frequency. 

The stimulated Raman emission exhibits a threshold and only a few vibrational lines are observed, corresponding to the lines of highest intensity in the normal Raman spectrum. For observation of the inverse Raman spectrum, both the continuum and the monochromatic stimulating radiation are supplied. There is no threshold and therefore the complete Raman active spectrum can be obtained if the continuum light source has a sufficient spectral bandwidth. 

As a hrst experiment, we wanted to return to the fundamental premise of TPM, which states that the efficiency of the two-photon excitation should be inversely proportional to the pulse duration. For TL pulses it is equivalent to having two-photon excitation efficiency proportional to the spectral bandwidth of the laser pulses. 

Another limiting factor is the bandwidth of the op-amp. On the factory specifications, the commonly used indicator is the gain-bandwidth product. The nominal dc gain is valid up to a cutoff frequency / which is typically 10 Hz. Above that frequency, the gain g is inversely proportional to the frequency. The product of gain and frequency, the gain-bandwidth product/a is typically 1 MHz. The input impedance of the amplifier increases with frequency ... 

Turning to the mostly ionic oxides, the localized character of the 5 f emission in the valence band spectrum is illuminated by a series of experimental results. Direct and inverse photoemission for UO2 gives the Coulomb energy Uh, which governs together with the bandwidth the localization vs. itineracy double aspects of the open shells. 

The build-up of the laser pulse from the statistical fluctuation is beautifully demonstrated in 31), where a short period of statistical fluctuations was registered with a streak camera while they were traveling back and forth in the resonator. A detailed theoretical treatment of this process is given in 32> for the case of solid-state lasers and in 33> for the case of dye lasers in which the saturation of the active medium plays an important role. The halfwidth of the pulse which is finally reached in this mode-locking process is theoretically determined by the inverse spectral bandwidth of the active medium provided that the dye relaxation time rp is sufficiently short. 

Figure 7.12 compares the theoretical predictions with the experimental values across the 4d series, assuming one valence s electron per atom and taking x = 12 corresponding to close-packed lattices. The experimental values of the bandwidth are taken from the first principles LDA calculations in Table 7.1. The ratio b2 a is obtained by fitting a bandwidth of 10 eV for Mo with Nd = 5, so that from eqn (7.42) b2/a = eV. The skewed parabolic behaviour of the observed equilibrium nearest-neighbour distance is found to be fitted by values of the inverse decay length that vary linearly across the series as... 

Several parameters, for example focal length, determine linear dispersion, which is expressed in millimetres per nanometre (or its inverse, which is called reciprocal dispersion). Linear dispersion represents the spread, in the focal plane, of two wavelengths differing by l nm. Bandwidth, which must not be confused with the width of the slit, is the interval of the spectrum that corresponds to the width in picometres exiting the slit. This width is generally greater than the natural width of the line being transmitted. 

In WMS, phase-sensitive (i.e. lock-in) amplification demodulates the periodic signal with a very narrow electrical bandwidth to precisely measure the RMS amplitudes of the fundamental sinusoid and its second harmonic component, averaged over a period of time equal to the inverse of the electrical bandwidth. These are called the If and 2/ signals and each amplitude is expressed by the following equations, respectively,... 

For the STFT case, the bandwidth of the filter-bank is inversely proportional to the duration of the short-time frame and it is shown in [Cappe and Laroche, 1995], using standard results concerning the simultaneous frequency masking phenomenon, that the processing can suppress audible signal components (ie. components that were not masked by the additive noise) if the short-time duration is well below 40 ms. 

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