Continuous-time Fourier transforms

The term Fourier transform usually refers to the continuous integration of any square-integrable function to re-express the function as a sum of complex exponentials. Due to the different types of functions to be transformed, many variations of this transform exist. Accordingly, Fourier transforms have scientific applications in many areas, including physics, chemical analysis, signal processing, and statistics. The continuous-time Fourier transforms are defined as follows [1-3] ... 

The continuous, infinite Fourier transform defined in Equation 10.9, unfortunately, is not convenient for signal detection and estimation. Most physically significant data are recorded only at a fixed set of evenly spaced intervals in time and not between these times. In such situations, the continuous sequence h(t) is approximated by the discrete sequence hn... 

The discrete-time Fourier transform converts a digital waveform into a continuous spectrum ... 

Discrete-time waveform A waveform, herein represented by x( ), that takes on values at a countable, discrete set of sample times or sample numbers , the assumed independent variable. The discrete Fourier transform, the discrete-time Fourier series, and discrete-time Fourier transform apply to discrete-time waveforms. Compare with continuous-time waveform. 

Many DSP concepts can be demonstrated by examples which involve a great deal of computation. A list of some of the concepts is as follows convolution, filtering, quantization effects, etc. The curriculum begins with discrete Fourier transform (DFT). DFT is derived from discrete-time Fourier transform expression. The continuous and discrete Fourier transform are covered in Signals and Systans. The flow of the topics is as follows DFT, properties of DFT, Fast Fourier Transform, Infinite Impulse Response filter and Finite Impulse Response fillers and filter structures. If the topics are linked to a project with each block of the project demonstrating the various topics of the curriculum, it is easier for the student to comprehend what is being taught. 

In practical applications, x(t) is not a continuous function, and the data to be transformed are usually discrete values obtained by sampling at intervals. Under such circumstances, I hi discrete Fourier transform (DFT) is used to obtain the frequency function. Let us. suppose that the time-dependent data values are obtained by sampling at regular intervals separated by [Pg.43]

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. 

As mentioned before, the smallest observable frequency (v ,in) in a continuous signal is the reciprocal of the measurement time ( I2T ). Because only those frequencies are considered which exactly fit in the measurement time, all frequencies should be a multiple of namely n/2T with n = -< to -l-oo. As a result the Fourier transform of a continuous signal is discrete in the frequency domain,... 

DGE a AC AMS APCI API AP-MALDI APPI ASAP BIRD c CAD CE CF CF-FAB Cl CID cw CZE Da DAPCI DART DC DE DESI DIOS DTIMS EC ECD El ELDI EM ESI ETD eV f FAB FAIMS FD FI FT FTICR two-dimensional gel electrophoresis atto, 10 18 alternating current accelerator mass spectrometry atmospheric pressure chemical ionization atmospheric pressure ionization atmospheric pressure matrix-assisted laser desorption/ionization atmospheric pressure photoionization atmospheric-pressure solids analysis probe blackbody infrared radiative dissociation centi, 10-2 collision-activated dissociation capillary electrophoresis continuous flow continuous flow fast atom bombardment chemical ionization collision-induced dissociation continuous wave capillary zone electrophoresis dalton desorption atmospheric pressure chemical ionization direct analysis in real time direct current delayed extraction desorption electrospray ionization desorption/ionization on silicon drift tube ion mobility spectrometry electrochromatography electron capture dissociation electron ionization electrospray-assisted laser desorption/ionization electron multiplier electrospray ionization electron transfer dissociation electron volt femto, 1CT15 fast atom bombardment field asymmetric waveform ion mobility spectrometry field desorption field ionization Fourier transform Fourier transform ion cyclotron resonance... 

Fourier transform spectroscopy technology is widely used in infrared spectroscopy. A spectrum that formerly required 15 min to obtain on a continuous wave instrument can be obtained in a few seconds on an FT-IR. This greatly increases research and analytical productivity. In addition to increased productivity, the FT-IR instrument can use a concept called Fleggetts Advantage where the entire spectrum is determined in the same time it takes a continuous wave (CW) device to measure a small fraction of the spectrum. Therefore many spectra can be obtained in the same time as one CW spectrum. If these spectra are summed, the signal-to-noise ratio, S/N can be greatly increased. Finally, because of the inherent computer-based nature of the FT-IR system, databases of infrared spectra are easily searched for matching or similar compounds. 

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