Fourier transform scaling property

The radius of the pole, r controls both the amplitude and the bandwidth of the resonance. As the value of oi decreases, its pole moves towards along the real axis towards the origin, and hence the frequency response has a lower amplitude at fi) = 0. From the difference equations, we know that a first order HR filter with oi < 1 is simply a decaying exponential. Small values of ai correspond to slow decay and narrow bandwidth, large values (still < 1) a steeper decay and wider bandwidth. This is exactly what we would e q)ect from the Fourier transform scaling property The standard measure of bandwidth is the 3db down point which is the width of the resonance at 1/ /2 down from the peak. For larger values of r = a, two useful approximations of the bandwidth (in normalised frequency) are ... 

Bioactive macromolecules like peptides, proteins, and nucleic acids have been successfully embedded in planar LbL films. An important question is the retention of the bioactivity of the film-embedded biomolecules. The structural properties and stability of the LbL films formed from synthesized polypeptides of various amino acid sequences were recently reported [50]. The authors showed that control over the amino acid sequence enables control over non-covalent interpolypeptide interaction in the film, which determines the film properties. Haynie and coworkers showed by circular dichroism spectroscopy that the extent of adsorption of poly(L-glutamic acid) (PGA) and poly(L-lysine) (PLL) in the LbL films scales with the extent of secondary structure of the polypeptides in solution [51]. Boulmedais demonstrated that the secondary structure of the film composed of these polypeptides is the same as the peptide structure in the complex formed in solution [52], as found by Fourier transform IR spectroscopy (FUR). 

A number of reports on phthalocyanines and porphyrins have been published. Spectral diffusion and thermal recovery of spectral holes burnt into phthalocyanine doped Shpol skii systems has been examined . An absorption, emission, and thermal lensing research on carboxylated zinc phthalocyanine shows the influence of dimerization on these properties. Fourier transformation of fluorescence and phosphorescence spectra of porphine in rare gas matrices has yielded much structural and electronic state data on this compound . Exciton splitting is an effect which is seen in the spectra of covalently linked porphyrins . A ps fluorescence study of the semirigid zinc porphyrin-viologen dyad has provided evidence for two dyad conformers . Spectral diffusion in organic glasses has been measured by observing the hole recovery kinetics over the time scale of 1 to 500 ms for zinc tetrabenzoporphyrin in PMMA . 

One of the most important developments in pharmacentical process research of the past several decades has been the introdnction of bench-scale experimental tools for efficient and accnrate in situ monitoring of chemical reactions. Snch tools inclnde reaction calorimetry, which measnres a property directly proportional to reaction rate (a differentiaf method), and spectroscopic methods such as Fourier transform infrared (FTIR) spectroscopy, which measnres a property proportional... 

Any function of the form 5/ (x) jx is known as a sine function. The real spectrum of this is shown in Figure 10.10b, and we can see that this has the exact shape suggested by the envelope of the harmonics in Figure 10.10. Figures 10.10c and lO.lOd show the waveform and Fourier transform for a pulse of shorter duration. The Fourier transform is now more spread-out , and this demonstrates an important of the Fourier transform known as the scaling property. 

We have already described the Fourier transform of a pulse of different durations in section 10.1.6. The shorter duration pulse has a spectrum that is more spread out, and the longer duration pulse has a more contracted spectrum. This is known as the scaling property, which states that compressing a signal will stretch its Fourier transform and vice versa. This can be formally stated as ... 

As expected from the scaling property, the Fourier transform of an impulse is a function that is infinitely stretched , that is, the Fourier Transform is 1 at all frequencies. Using the duality principle, a signal x(t) = 1 for all t will have a Fourier transform of 6( ), that is, an impulse at time 00 = 0. This is to be expected - a constant signal (a d.c. signal in electrical terms) has no variation and hence no information at frequencies other than 0. 

---