Harmonic detectors

In order to understand the functioning of the FFT mechanism, imagine a filtering process in which a bank of harmonic detectors (e.g. band-pass filters) are tuned to a number of frequencies (refer to Chapter 4 for a brief introduction to filters). The greater the number of harmonic detectors, the more precise the analysis. For instance, if each detector has a bandwidth of 30 Hz, it would be necessary to employ no less than 500 units to cover a band... 

The size of the window defines the frequency resolution and the time resolution of the analysis. This value is normally specified as a power of two e.g. 256, 512,1024, etc. Longer windows have better frequency resolution than smaller ones, but the latter have better time resolution than the former. For example, whilst a window of 1024 samples at a rate of 44100 Hz, allowing for a time resolution of approximately 23 milliseconds (1024/44100 = 0.023), a window of 256 samples gives a much better resolution of approximately 6 milliseconds (256/44100 = 0.0058). Conversely, the harmonic detectors will be tuned to scan frequencies spaced by a bandwidth of approximately 43 Hz (44100/1024 = 43) in the former case and to approximately 172 Hz (44100/256 = 172) in the latter. This means that a window of 256 samples is not suitable for the analysis of sounds lower than 172 Hz (approximately an F3 note), but it may suit the analysis of a sound that is likely to present important fluctuation within less than 23 milliseconds. 

The size of the STFT window is constant. Hence all harmonic detectors have the same bandwidth and are placed linearly across the audio range. The wavelets method improves this situation by introducing a mechanism whereby the size of the window varies according to the frequency being analysed. That is, the bandwidths of the harmonic detectors vary with frequency and they are placed logarithmically across the audio range. 

The size of the window defines the number of input samples to be analysed at a time. The larger the window, the greater the number of channels, but the lower the time resolution, and vice versa. This should be set large enough to capture four periods of the lowest frequency of interest. The sampling rate divided by the number of channels should be less than the lowest pitch in the input sound. This may be set in terms of the lowest frequency of interest or in terms of the number of channels required. Note that the term channel is the Phase Vocoder jargon for what we referred earlier to as harmonic detectors (Figure 3.11). 

Figure 3.11 The Phase Vocoder uses channels to act as harmonic detectors . The greater the number of channels, the better the frequency resolution. Note that (b) does not distinguish between the two uppermost partials that are perfectly distinct In (a)...

In a second kind of infrared ellipsometer a dynamic retarder, consisting of a photoelastic modulator (PEM), replaces the static one. The PEM produces a sinusoidal phase shift of approximately 40 kHz and supplies the detector exit with signals of the ground frequency and the second harmonic. From these two frequencies and two settings of the polarizer and PEM the ellipsometric spectra are determined [4.316]. This ellipsometer system is mainly used for rapid and relative measurements. 

In summary, the NFS investigation of FC/DBP reveals three temperature ranges in which the detector molecule FC exhibits different relaxation behavior. Up to 150 K, it follows harmonic Debye relaxation ( exp(—t/x) ). Such a distribution of relaxation times is characteristic of the glassy state. The broader the distribution of relaxation times x, the smaller will be. In the present case, takes values close to 0.5 [31] which is typical of polymers and many molecular glasses. Above the glass-to-liquid transition at = 202 K, the msd of iron becomes so large that the/factor drops practically to zero. 

The development of hydrodynamic techniques which allow the direct measurement of interfacial fluxes and interfacial concentrations is likely to be a key trend of future work in this area. Suitable detectors for local interfacial or near-interfacial measurements include spectroscopic probes, such as total internal reflection fluorometry [88-90], surface second-harmonic generation [91], probe beam deflection [92], and spatially resolved UV-visible absorption spectroscopy [93]. Additionally, building on the ideas in MEMED, submicrometer or nanometer scale electrodes may prove to be relatively noninvasive probes of interfacial concentrations in other hydrodynamic systems. The construction and application of electrodes of this size is now becoming more widespread and general [94-96]. 

As area detectors (other than multiwire systems) are not energy discriminating devices, apotential source of error lies in the contamination of the data with harmonics of the assumed wavelength of the primary beam. The importance of this effect has been estimated for molybdenum Ka radiation using a graphite monochromator [1],... 

The time resolution of a phase fluorometer using the harmonic content of a pulsed laser and a microchannel plate photomultiplier is comparable to that of a single-photon counting instrument using the same kind of laser and detector. 

Figure 12.1 SGX-CAT beamline schematic. The components of the beamline include (1 (not shown), 8) photon shutters (2,4) beam transport tubes (3, 5) collimators and vacuum pumps (6) beam-defining slits (7) monochromator (9,10) focusing and harmonic rejection mirrors and (12) CCD detector, supporting base, and sample robot.

When a component of the modulated absorption wave which synchronizes with the fcth harmonic of the modulation frequency is detected by a phase-sensitive detector, the signal obtained is proportional to the coefficient of the kih term ak (// ). 

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