Signal Gaussian

Application to cw Radar with Gaussian Input Signals (Gaussian Spectra)... 

The dependence of the in-phase and quadrature lock-in detected signals on the modulation frequency is considerably more complicated than for the case of monomolecular recombination. The steady state solution to this equation is straightforward, dN/dt = 0 Nss — fG/R, but there is not a general solution N(l) to the inhomogeneous differential equation. Furthermore, the generation rate will vary throughout the sample due to the Gaussian distribution of the pump intensity and absorption by the sample... 

We now apply Eqs. (4-194) to (4-201) to the frequency limited, power limited, additive white gaussian noise channel. If N is the block length of a code in samples, then T = N/2W is the block length in time. Furthermore if is the available signal power and if N0 is the noise power per unit bandwidth, then the signal to noise ratio, A, is 8/N0W. Finally we let JRT> the rate in nats per second, be 2 WB. Substituting these relations into Eqs. (4-194) and (4-197), we get... 

Heteronuclear-shift-correlation spectra, which are usually presented in the absolute-value mode, normally contain long dispersive tails that are suppressed by applying a Gaussian or sine-bell function in the F domain. In the El dimension, the choice of a weighting function is less critical. If a better signal-to-noise ratio is wanted, then an exponential broadening multiplication may be employed. If better resolution is needed, then a resolution-enhancing function can be used. 

Matched filter The multiplication of the free induction decay with a sensitivity enhancement function that matches exactly the decay of the raw signal. This results in enhancement of resolution, but broadens the Lorentzian line by a factor of 2 and a Gaussian line by a factor of 2.5. 

The procedure described above is a pictorial approximation of a process called scale-space filtering of a function, proposed by Witkin (1983). The surface (e.g., Fig. 6) swept out by a filtered signal as the Gaussian filter s standard deviation is varied, is called scale-space image of the signal and is given by... 

The conformation of polymer chains in an ultra-thin film has been an attractive subject in the field of polymer physics. The chain conformation has been extensively discussed theoretically and experimentally [6-11] however, the experimental technique to study an ultra-thin film is limited because it is difficult to obtain a signal from a specimen due to the low sample volume. The conformation of polymer chains in an ultra-thin film has been examined by small angle neutron scattering (SANS), and contradictory results have been reported. With decreasing film thickness, the radius of gyration, Rg, parallel to the film plane increases when the thickness is less than the unperturbed chain dimension in the bulk state [12-14]. On the other hand, Jones et al. reported that a polystyrene chain in an ultra-thin film takes a Gaussian conformation with a similar in-plane Rg to that in the bulk state [15, 16]. 

Figure 5.11 shotvs the temporal profile of the intensity change in the SFG signal at the peak of the Vco mode (2055 cm ) at OmV induced by visible pump pulse irradiation. The solid line is the least-squares fit using a convolution of a Gaussian function for the laser profile (FWFJ M = 20 ps) and a single exponential function for the recovery profile. The SFG signal fell to a minimum within about 100 ps and recovered... 

Figure 8.1 (a) Block diagram of the femtosecond near-infrared laser microscope system, (b) Spectrum ofthe light pulse from the Cr F laser, (c) Interferometric autocorrelation trace of SHG signal with envelope curve calculated assuming a chirp-free Gaussian pulse with 35 fs fwhm. 

Table 40.3. As one can see, the filter introduces a slower response to stepwise changes of the signal, as if it were measured with an instrument with a large response time. Because fluctuations are smoothed, the standard deviation of the signal is decreased, in this example from 2.58 to 1.95. A Gaussian peak is broadened and becomes asymmetric by exponential smoothing (Fig. 40.26).

Fig. 40.28. Effect of a low-pass filter, (a) original Gaussian signal, (b) FT of (a), (c) Signal (a) filtered with V() = 10. (d) Signal (a) filtered with Vo = 20.

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