Discrete Hilbert transforms in crystallography

Abstract

Under the assumption that the structure amplitude of X-ray diffraction from a crystal satisfies the causal Fourier transform condition and appears to be a function with a band-limited spectrum, discrete Hilbert transforms (DHT) linking structure amplitudes having half-integral-valued Miller indices with structure amplitudes having integral-valued indices are obtained. DHT are then used to derive an interpolation formula that permits structure-amplitude reconstruction from samples with half the sampling frequency of the Nyquist rate. Some one-dimensional test calculations are also given.

The joint probability distribution function of structure factors with rational indices. IV. The P1 case

“…The problem was partially overcome by Mishnev (1993), who applied the Shannon sampling theorem (Shannon, 1949) to reconstruct F(x à ) from the values sampled at the reciprocal-lattice points. It may be of use to the reader to recall such a basic theorem:…”

“…Recently, the discrete Hilbert transform has been evoked (Ramachandran, 1969;Mishnev, 1993Mishnev, , 1996Zanotti et al, 1996) to link structure amplitudes having half-integral Miller indices with structure amplitudes of standard re¯ections. The relation between such techniques and our probabilistic method is discussed.…”

The Use of Discrete Hilbert Transforms in Phase Extension and Improvement

“…However, they are valid in the case of scattering by amorphous samples and can be used to constrain the phase values (Makowski, 1981). Only recently, the problem was partially overcome by Mishnev (1993), who, applying Shannon's sampling theorem (Shannon, 1949), derived the following expression for the structure factors:…”

“…the condition of causal Fourier transform, would not be fulfilled. The use of twice the crystallographic sampling rate becomes a necessary condition for the application of the Hilbert transforms to structure factors [more complete discussion of the validity conditions of (3) can be found in Mishnev (1993)]. …”

The joint probability distribution function of structure factors with rational indices. V. The estimates

“…In papers III and IV, the joint probability distribution functions of structure factors with rational indices were derived. Probabilistic relationships relating the real and imaginary parts of the structure factors were obtained, which encompass those derived via the Hilbert-transform method (Mishnev, 1993(Mishnev, , 1996Zanotti et al, 1996). The ®rst part of this paper is devoted to the applicative aspects of our theory.…”