Short-time Fourier transform

Reconstruct the time-scaled short-time Fourier transform at time 11 according to Eq. (7.15). 

When localisation is an issue, the intuitive solution still making use of the Fourier transform would be to cut up the signal and to transform the pieces. This approach is called the short-time Fourier transform, it adds a dimension to the Fourier transform, namely time, as it allows following frequencies over time. Where the Fourier transform is a frequency analysis, the short-time Fourier transform is a time-frequency analysis. Instead of describing the signal in either the time or the frequency domain, we describe it in both, a joint time-frequency domain. When we do this, we are faced with a fundamental limitation we cannot localise in the time domain and the frequency domain at the same time. 

In this case the researchers purchased carbon-13 enriched [l- C]glucose and [6- C]mannose and at time zero mixed each with cell suspensions of the bacteria. The sample was placed directly in an NMR probe and the signals accumulated over 256 repeated scans and the resulting FID were Fourier-transformed. The time series spectra in Figure 7.3 show that the glucose was consumed and converted to [3- C]lacetate (21.1 ppm), [2- C ]-ethanol (17.8 ppm) and lower concentration products not yet observable in the short time span shown. (Note also that the two... 

Figure 6. The Fourier transformed signal AS[r, i] of CH2I2/CH3OH. The pump-probe time delays vary between i = —250ps and 1 ps. The pair distribution function gi i peaks in the 3 A region. If i < 50 ns, the 1—1 bond corresponds to the short-lived intermediate (CH2TI), and if I > 100 ns it belongs to the (I3") ion. Red curves indicate the theory, and black curves describe the experiment. (See color insert.)...

Greatly enhanced sensitivity with very short measuring time is the major advantage of PFT (pulse Fourier transform) experiments. In the CW (continuous wave) experiment, the radiofrequency sweep excites nuclei of different Larmor frequencies, one by one. For example, 500 s may be required for excitation over a 1-KHz range, while in a PFT experiment a single pulse can simultaneously excite the nuclei over 1-KHz range in only 250 jits. The PFT experiment therefore requires much less time than the CW NMR experiment, due to the short time required for acquisition of FID signals. Short-lived unstable molecules can only be studied by PFT NMR. 

The 13C NMR sensitivity can sometimes be a problem, but for the kind of samples studied here the effective concentration of monomer units is several molar which does not place excessive demands on present Fourier transform NMR spectrometers. In addition to the sensitivity of the chemical shift to structure (9), the relaxation of protonated carbons is dominated by dipole-dipole interaction with the attached proton (9). The dependence of the relaxation parameters T, or spin-lattice, and Tor spin-spin, on isotropic motional correlation time for a C-H unit is shown schematically in Figure 1. The T1 can be determined by standard pulse techniques (9), while the linewidth at half-height is often related to the T2. Another parameter which is related to the correlation time is the nuclear Overhauser enhancement factor, q. The value of this factor for 13C coupled to protons, varies from about 2 at short correlation times to 0.1 at long correlation... 

Second, the fluctuation is delayed by a time 5t which is a function of the residence time t, of the element in the reservoir. For an infinite residence time the argument of the tangent tends towards n/2 and the delay 5f towards T/4, while for a short residence time, the delay tends towards zero. As expected, reactive elements respond more rapidly than inert elements. The phase shift and the damping factor relating input to output concentrations represent the angular phase and argument of a complex function known as the transfer function of the reservoir. Such a function, however, is most conveniently introduced via Laplace and Fourier transforms. Applications of these geochemical concepts to the dynamics of volcanic sequences can be found in Albarede (1993). 

Fourier transform infrared spectroscopy spectrometers can cover wide spectral ranges with a single scan in a relatively short scan time, thereby permitting the possibility of kinetic time-resolved measurements. 

Pulse fluorometry uses a short exciting pulse of light and gives the d-pulse response of the sample, convoluted by the instrument response. Phase-modulation fluorometry uses modulated light at variable frequency and gives the harmonic response of the sample, which is the Fourier transform of the d-pulse response. The first technique works in the time domain, and the second in the frequency domain. Pulse fluorometry and phase-modulation fluorometry are theoretically equivalent, but the principles of the instruments are different. Each technique will now be presented and then compared. 

Fourier transform spectroscopy spect A spectroscopic technique in which all pertinent wavelengths simultaneously Irradiate the sample for a short period of time, and the absorption spectrum is found by mathematical manipulation of the Fourier transform so obtained. fur e a tranz,form spek tras-ko-pe fp See freezing point. 

The advent of computers and Fourier transform completely revolutionized the detection and identification of organic compounds. Modern automated instruments allow very small samples in the nanogram (10 g) range to be characterized in a very short time. The application of Fourier transform nuclear magnetic resonance (FTNMR) and Fourier transform infrared (FTIR) allows recovery of the sample in contrast to mass spec-trometric (MS) determination which is a destructive but quite often a necessary technique. 

Fig. 4 Stacked Fourier transforms of the NMR spin echoes in D-RADP-25 versus echo delay time at T = 65 K. In contrast to D-RADP-20 (Fig. 2) there are still two rims present at this low temperature indicating the coexistence of two different phase states. The rim at the Larmor frequency vl originates from Rb spins localized in short range ordered glass clusters, whereas the rim at vl + 8 kHz is produced by spins sitting in FE clusters [17]...

Recently, Darowicki [29, 30] has presented a new mode of electrochemical impedance measurements. This method employed a short time Fourier transformation to impedance evaluation. The digital harmonic analysis of cadmium-ion reduction on mercury electrode was presented [31]. A modern concept in nonstationary electrochemical impedance spectroscopy theory and experimental approach was described [32]. The new investigation method allows determination of the dependence of complex impedance versus potential [32] and time [33]. The reduction of cadmium on DM E was chosen to present the possibility of these techniques. Figure 2 illustrates the change of impedance for the Cd(II) reduction on the hanging drop mercury electrode obtained for the scan rate 10 mV s k... 

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