Bandwidth Extrapolation

The Fourier frequency bandpass of the spectrometer is determined by the diffraction limit. In view of this fact and the Nyquist criterion, the data in the aforementioned application were oversampled. Although the Nyquist sampling rate is sufficient to represent all information in the data, it is not sufficient to represent the estimates o(k) because of the bandwidth extension that results from information implicit in the physical-realizability constraints. Although it was not shown in the original publication, it is clear from the quality of the restoration, and by analogy with other similarly bounded methods, that Fourier bandwidth extrapolation does indeed occur. This is sometimes called superresolution. The source of the extrapolation should be apparent from the Fourier transform of Eq. (13) with r(x) specified by Eq. (14). 

With suitable scan-line delay circuits, the filter may be applied in real time to live video images. In data communications, bandwidth extrapolation offers the opportunity to make far better use of channel capacity than is now possible. The bounded methods produce their most impressive restorations when o(x) spends a lot of time at or near the bounds. The present method was designed for use with both upper and lower bounds, which makes it ideal for the bilevel signals used in digital transmission. 

Substantial bandwidth extrapolation may be anticipated on the basis of the powerful a priori knowledge that the signal can assume only one of two values. Although the nonlinear hardware filter described here is readily feasible, and perhaps practical with today s standard commercial electronic technology, experimental work along these lines is unknown to the author as of this writing. 

This chapter has attempted to give some flavor of the historical development of nonlinear methods. Early investigators of these methods expended great effort in overcoming the popular notion that bandwidth extrapolation was not possible or practical. It was, for example, believed that the Rayleigh limit of resolution was a limit of the most fundamental kind—unassailable by mathematical means. To be sure, the popular notion was reinforced by a long history of misfortune with linear techniques and hypersensitivity to noise. Anyone who still needs to be convinced of the virtues of the nonlinear methods would benefit from reading the paper by Wells (1980) the nonlinear point of view is nowhere else more clearly stated. 

An amplitude bound has the added virtue of producing solutions having reduced noise sensitivity, fewer artifacts, superior resolution, and possible bandwidth extrapolation. In contrast, methods having an output that is linear in the irradiance data i(x) either produce artifacts or trade off resolution to suppress artifacts. If a bound makes physical sense and can be computationally afforded, use it. Simple clipping of unphysical parts does not always work well, however. Subtle techniques may be more desirable. 

Low and High frequency can be restored by use of a deconvolution algorithm that enhances the resolution. We operate an improvement of the spectral bandwidth by Papoulis deconvolution based essentially on a non-linear adaptive extrapolation of the Fourier domain. 

The NO2 dissociation rate was measured by a two-color picosecond pump-probe method in which the product NO was monitored by LIF. Of particular significance in this study is that the NO2 density of states at the dissociation limit of 25,130.6 cm is relatively well established from an extrapolation of experimentally determined densities at an energy of 18,500 cm . This density (for cold samples where the rotations do not contribute significant densities) is 0.3 states per cm , (Miyawaki et al., 1993) which leads to a minimum rate constant l/h p( ) = 1 x 10 sec . The experimentally measured rate increases from 0 to 1.6 x 10 sec at the dissociation limit. It is interesting that the subpicosecond laser pulses with their transform limited resolution of about 20 cm do not excite individual NO2 resonance states (see section 8.3, p. 284) but, instead, prepare a superposition of those states that are optically accessible within the laser bandwidth. It is thought that all resonance states in this bandwidth are... 

IPs and EAs are overestimated by up to 0.77 eV compared with experimental bulk values. Extrapolated bandwidths agree reasonably well with bandwidths from band structure calculations. The HOMO-LUMO gaps are significantly closer to experimental values with hybrid functionals than with LSDA and vertical excitation energies from UV spectra are reproduced to within 0.01-0.8 eV. 

The problem of acquiring impedances at sufficiently low frequencies is amply demonstrated by the data (Syrett and Macdonald [1979]) for 90 10 Cn Ni alloy corroding in flowing seawater (Figure 4.4.1). Thus, for an exposure time of 22 h, the impedance function can be defined over the entire bandwidth, and an accurate value for Rp may be obtained by probing the interface at frequencies above 0.01 Hz. On the other hand, at much longer exposure times, frequencies as low as 0.0005 Hz are not sufficient to completely define the interfacial impedance, and considerable extrapolation is required to acquire a value for Rp. 

A compromise between the two methods has been recently used to evaluate y in frog nodes (20). Fluctuation spectra, obtained after subtracting TTX insensitive components, were analyzed and shown to fit a simple sum of relaxation spectra. The noise variance was simultaneously and independently measured as the mean square of the fluctuations. It was then corrected for bandwidth limitations according to the extrapolation of the spectra to infinite frequency, and inserted in eqn.(5), to yield i j, after subtraction of the TTX insensitive contribution. estimated in this way, was found... 

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