Band limit

The described approach is suitable for the reconstruction of complicated dielectric profiles of high contrast and demonstrates good stability with respect to the noise in the input data. However, the convergence and the stability of the solution deteriorate if the low-frequency information is lacking. Thus, the method needs to be modified before using in praetiee with real microwave and millimeter wave sourees and antennas, whieh are usually essentially band-limited elements. 

In practice, since x(t) is a frequency band-limited signal, equation (11) shows that H(u) is known only on the finite interval wherein X(u) 0. There are also problems when the input signal is small, reduced to noise. 

Papoulis, A., A new algorithm in speetral analysis and band-limited extrapolation, in IEEE Trans. Circuits Syst., vol. Cas-22, 9, pp. 735-742, (1975). 

As regards the noise spectrum, the different situations can be analyzed ap proximately with NC (noise criterion) and NR (noise rating) curves (Fig. 9.6.3). NC and NR curves define the octave band limits of an acceptable back ground noise each of them is characterized by a number representing the sound pressure level at 1000 Hz. 

The quantity Gy (z) can be easily evaluated, even in the presence of defects, by using a continued fraction technique which does not assume any periodicity of the system 3/4,5 practice n levels of the continued fraction are computed exactly and the remaining part of the continued fraction is replaced by the usual square root terminator which corresponds to using the asymptotic values for the remaining coefficients. Note that these asymptotic values are fixed by the band limits which are known exactly in many cases. The more accurate the calculation of AE is required to be, the more exact... 

Continuous Memoryless Channels.—The coding theorem of the last section will be extended here to the following three types of channel models channels with discrete input and continuous output channels with continuous input and continuous output and channels with band limited time functions for input and output. Although these models are still somewhat crude approximations to most physical communication channels, they still provide considerable insight into the effects of the noise and the relative merits of various transmission and detection schemes. 

Figure 9.7. Noise content of a fiberoptic oxygen sensor signal (a) in the time and (b) in the frequency domains. Time domain signals require broad frequency bandwidths. Frequency domain signals require very limited-frequency bandwidths. Noise is reduced by band limiting the signal, an advantage of frequency domain methods.

In other solids, Pauli paramagnetism is found to be explained in the band limit. In others, still, an intermediate behaviour is encountered. 

In fact, whether to interpret properties in the atomic or in the band limit depends upon the competition between these two quantities the bandwidth W, describing itinerant behaviour, and the Coulombic interaction U, describing atomic behaviour. 

It can be shown that the band treatment (band limit) is valid when the change in is... 

Brief reflection on the sampling theorem (Chapter 1, Section IV.C) with the aid of the Fourier transform directory (Chapter 1, Fig. 2) leads to the conclusion that the Rayleigh distance is precisely two times the Nyquist interval. We may therefore easily specify the sample density required to recover all the information in a spectrum obtained from a band-limiting instrument with a sine-squared spread function evenly spaced samples must be selected so that four data points would cover the interval between the first zeros on either side of the spread function s central maximum. In practice, it is often advantageous to place samples somewhat closer together. 

At this point, we note that there is no mechanism presently built into the relaxation methods to prevent undesirable high-frequency noise from growing with each iteration. Any spurious solution 6(x) satisfies Eq. (1) (see also Chapter 1, Sections V.A and V.B) for co beyond the band limit. If we know that the object 6 is truly band limited, with frequency cutoff co = 2, we can band-limit both data i and first object estimate d(1). The relaxation methods cannot then propagate noise having frequencies greater than Q into an estimate o(k). (One possible exception involves computer roundoff error. Sufficient precision is usually available to avoid this problem.)... 

If we know that the instrument response is band limiting with frequency cutoff Q, we may likewise process the estimate. The resulting solution will then also be band limited. The high-frequency spectral structure beyond cutoff Q that we would wish to restore is forever lost to these linear methods. The data contain no information about the high-frequency content. We must wait until Chapter 4 to see how straightforward and seemingly unimportant... 

Typically, t(co) is small for co large. A spectrometer suppresses high frequencies. If the data i(x) have appreciable noise content at those frequencies, it is certain that the restored object will show the noise in a more-pronounced way. It is clearly not possible to restore frequencies beyond the band limit Q by this method when such a limit exists. (Optical spectrometers having sine or sine-squared response-function components do indeed band-limit the data.) Furthermore, where the frequencies are strongly suppressed, the signal-to-noise ratio is poor, and T(cu) will amplify mainly the noise, thus producing a noisy and unusable object estimate. 

We should, however, be able to carry out the processing within some band limit co < Qp. Frieden (1975) has shown that the processing bandwidth... 

Where t(co) = 0—beyond the band limit Q, for example—I(a>) contains no information about O(co). Clearly it is impossible to restore these lost frequencies based on z(co) and the information in I(co) alone. We see in the next chapter how a simple modification to the various relaxation methods brings about a dramatic improvement. 

It is thus possible to convolve both spread function and data i(x) with s( — x). We may then use the relaxation methods as before. This time, however, we replace i(x) with s( — x) (g) i(x) and s(x) with s( — x) (x) s(x). Not only are we assured convergence, but we have also succeeded in band-limiting the data i(x) in such a way as to guarantee that all noise is removed from i(x) at frequencies where i(x) contains no information about o(x). Furthermore, Ichioka and Nakajima (1981) have shown that reblurring reduces noise in the sense of minimum mean-square error. 

The answer lies in the relatively poor performance of linear methods, especially with band-limited data. Frequently a linear restoration reveals little true structure that could not have been seen in the original data. Even worse, noise-based artifacts often call the result into question. One might even say that the linear methods have helped to give deconvolution a bad reputation in spectroscopy. 

To be sure, linear methods have value where fast computation is necessary. They perform reasonably well when the experimental data are not band limited, and in trials with computer-generated data devoid of noise. Spectroscopic data are often band limited, however, and computation time is becoming less of a problem with advances in computer hardware. The quantity of data required in spectroscopy is far less than that in image processing, for example, another field that has given much attention to deconvolution. Image processing problems are two and sometimes three-dimensional, whereas spectral problems are usually one dimensional. 

Schell (1965) recognized that the major deficiency of the Wiener inverse filter is the nonphysical nature of the partially negative solutions that it is prone to generate. He sought to extrapolate the band-limited transform O(co) by seeking a nonnegative physical solution 6 + (x) through minimization of... 

The method employs a gradual increase in frequency beyond the data band limit. High-frequency components are not sought until the best values of low-frequency components are found. Because frequencies are not sought above the lowest needed to satisfy the data, the method is inherently smooth. Furthermore, Biraud s method appears to be the first to have simultaneously utilized both the constraint of positivity and that of finite extent with specific limits, the latter being inherent in the sampling. These facts are probably responsible for the impressiveness of the restoration in the original publication (Biraud, 1969), which is reproduced in Fig. 4. 

For a basic deconvolution problem involving band-limited data, the trial solution d(0) may be the inverse- or Wiener-filtered estimate y(x) (x) i(x). Application of a typical constraint may involve chopping off the nonphysical parts. Transforming then reveals frequency components beyond the cutoff, which are retained. The new values within the bandpass are discarded and replaced by the previously obtained filtered estimate. The resulting function, comprising the filtered estimate and the new superresolving frequencies, is then inverse transformed, and so forth. 

In previous sections we have seen how constraints of boundedness have enabled recovery of frequencies beyond the band limit of the observing instrument. Starting with the inverse-filter estimate... 

A review of deconvolution methods applied to ESCA (Carley and Joyner, 1979) shows that Van Cittert s method has played a big role. Because the Lorentzian nature of the broadening does not completely obliterate the high Fourier frequencies as does the sine-squared spreading encountered in optical spectroscopy (its transform is the band-limiting rect function), useful restorations are indeed possible through use of such linear methods. Rendina and Larson (1975), for example, have used a multiple filter approach. Additional detail is given in Section IV.E of Chapter 3. 

B. Deconvolving Spectra with Band-Limited White Noise Added 196... 

One procedure for recovering the continuous (band-limited) function exactly is provided by the Whittaker-Shannon sampling theorem, which is expressed by the equation... 

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