Time domain, Fourier series

Figure 5.17 Cartoon diagram to represent general structure of 4D correlation experiments. This is the same as for 3D correlation experiments (Fig. 5.14) except that an extra resonant population of heteroatom nuclei are involved in generation of transverse magnetisation (in time ts) and magnetisation transfer (during M3). Final pulse sequence generates transverse magnetisation in the Destination Nuclei S that is observed, acquired and digitised in time t/,. Fourier series transformation is used to transform time domain signal information Sfid (ti, ta, ts, 4) into frequency domain (spectral intensity) information, /NMR(fi, F2,...

Both methods for time domain (Fourier) and frequency domain (direct) filters are equivalent and are related by the convolution theorem, and both have similar aims, to improve the quality of spectroscopic or chromatographic or time series data. Two functions, /"and g, are said to be convoluted to give h, if... 

Time domains and frequeney domains are related through Fourier series and Fourier transforms. By Fourier analysis, a variable expressed as a funetion of time may be deeomposed into a series of oseillatory funetions (eaeh with a eharaeteristie frequeney), whieh when superpositioned or summed at eaeh time, will equal the original expression of the variable. This... 

The frequency-domain format eliminates the manual effort required to isolate the components that make up a time trace. Frequency-domain techniques convert time-domain data into discrete frequency components using a mathematical process called Fast Fourier Transform (FFT). Simply stated, FFT mathematically converts a time-based trace into a series of discrete frequency components (see Figure 43.19). In a frequency-domain plot, the X-axis is frequency and the Y-axis is the amplitude of displacement, velocity, or acceleration. 

Eq. (2.11) can be solved by developing it as a complex series of sines and cosines according to relation (2.9). This is a Fourier series [14]. Thus, an exponential in the time domain, F ((), and a Lorentzian in the frequency domain, f (op, are Fourier transforms of each other [15-17],... 

As any time domain function F (r), a square wave rf pulse of width tp can be approximated by a Fourier series of sines and cosines with frequencies w/2 tp (n — 1, 2, 3, 4, 5,...) [14, 7]. An rf pulse of width t thus simulates a multifrequency transmitter of frequency range A = 1/4 (p (eq. (2.14)). Accordingly, an rf pulse of 250 ps simultaneously rotates the M0 vectors of all Larmor frequencies within a range of at least A = 1 kHz. It simulates at least 1000 simultaneously stimulating transmitters, the resolution in the Fourier transform depending on the number of FID data points (eq. (2.16)), not the stimulation time t-. 

The data flux of a two-dimensional carbon-proton shift correlation is similiar to that described in Fig. 2.50(a) for a. /-resolved 2D CMR experiment, with one difference Instead of carbon-proton couplings JCH, proton chemical shifts (iH are stored in the evolution time tl. Fourier transformation in the (2 domain thus yields a series of NMR spectra with carbon-13 signals modulated by the attached proton Larmor frequencies. A second Fourier transformation in the domain generates the dH, Sc matrix of a two-dimensional carbon-proton correlation. 

In an ordinary Fourier transform NMR experiment the time-domain signal (the FID) is converted into a frequency-domain representation (the spectrum) thus a function of time, S(t2), is converted into a function of frequency, S(f2). The very simple basic idea of 2D NMR is to treat the period preceding the recording of the FID (known as the evolution period ) as the second time variable. During this period, tu the nuclear spins are manipulated in various ways. In the 2D experiment a series of S(t2) FID s are recorded, each for a different t u and the result is considered a function of both time variables, S(tu t2). A twofold application of the Fourier transformation (see Fig. 82) then yields a 2D spectrum, S(fi,f2), which has two frequency... 

Fig. 4.4. (a) Excitation-ionization spectrum of the H atom Balmer series around the ionization limit in a static homogenous magnetic field, (b) Fourier-transformed time domain spectrum of the spectrum shown in (a). The square of the absolute value is plotted. The time scale is given in units of the cyclotron period Tc = 2 k/u c. Reprinted from Main, Holle, Wiebusch, and Welge (1987). 

Figure 4.2b is a presentation of the FID of the decoupled 13C NMR spectrum of cholesterol. Figure 4.2c is an expanded, small section of the FID from Figure 4.2b. The complex FID is the result of a number of overlapping sine-waves and interfering (beat) patterns. A series of repetitive pulses, signal acquisitions, and relaxation delays builds the signal. Fourier transform by the computer converts the accumulated FID (a time domain spectrum) to the decoupled, frequency-domain spectrum of cholesterol (at 150.9 MHz in CDC13). See Figure 4.1b.

Any continuous sequence of data h(t) in the time domain can also be described as a continuous sequence in the frequency domain, where the sequence is specified by giving its amplitude as a function of frequency, H(f). For a real sequence h(t) (the case for any physical process), H(f) is series of complex numbers. It is useful to regard h(t) and 11(f) as two representations of the same sequence, with h(t) representing the sequence in the time domain and H( f) representing the sequence in the frequency domain. These two representations are called transform pairs. The frequency and time domains are related through the Fourier transform equations... 

The raw data or FID is a series of intensity values collected as a function of time time-domain data. A single proton signal, for example, would give a simple sine wave in time with a particular frequency corresponding to the chemical shift of that proton. This signal dies out gradually as the protons recover from the pulse and relax. To convert this time-domain data into a spectrum, we perform a mathematical calculation called the Fourier transform (FT), which essentially looks at the sine wave and analyzes it to determine the frequency. This frequency then appears as a peak in the spectrum, which is a plot in frequency domain of the same data (Fig. 3.27). If there are many different types of protons with different chemical shifts, the FID will be a complex sum of a number of decaying sine waves with different frequencies and amplitudes. The FT extracts the information about each of the frequencies ... 

Lastly it should be noted that the time or scan rate issue equally plagues time as well as frequency domain methods for obtaining Rf, since in the time domain measurement, the triangle waveform is simply the Fourier synthesis of a series of sinusoidal signal functions. However, voltage sweep, potential step, and impedance methods should all yield the same value of Rf when all the scan... 

The Fourier transform (FT) has revolutionised spectroscopy such as NMR and IR over the past two decades. The raw data are not obtained as a comprehensible spectrum but as a time series, where all spectroscopic information is muddled up and a mathematical transformation is required to obtain a comprehensible spectrum. One reason for performing FT spectroscopy is that a spectrum of acceptable signal to noise ratio is recorded much more rapidly then via conventional spectrometers, often 100 times more rapidly. This has allowed the development of, for example, 13C NMR as a routine analytical tool, because the low abundance of 13 C is compensated by faster data acquisition. However, special methods are required to convert this time domain ... 

The time domain consists of a sum of time series, each corresponding to a peak in the spectrum. Superimposed on this time series is noise. Fourier transforms convert the time series to a recognisable spectrum as indicated in Figure 3.16. Each parameter in the time domain corresponds to a parameter in the frequency domain as indicated in Table 3.9. 

Filtering a time series, using Fourier time domain filters, however, involves multiplying die entire time series by a single function, so diat... 

The convolution theorem states diat /, g and h are Fourier transforms of F, G and H. Hence linear filters as applied directly to spectroscopic data have their equivalence as Fourier filters in die time domain in other words, convolution in one domain is equivalent to multiplication in die other domain. Which approach is best depends largely on computational complexity and convenience. For example, bodi moving averages and exponential Fourier filters are easy to apply, and so are simple approaches, one applied direct to die frequency spectrum and die other to die raw time series. Convoluting a spectrum widi die Fourier transform of an exponential decay is a difficult procedure and so die choice of domain is made according to how easy the calculations are. 

The integration converts the time-domain quantities into the respective frequency-domain quantities. Integration is carried out over a period T comprising an integer number of cycles. This serves to filter errors in the measurement. The complex current Ir + jlj and potential Vr + jVj are the coefficients Ci of the Fourier series expressed as equation (7.30). 

Figure 6-3 The solid line slanting upwards at frequency on the horizontal axis serves as the baseline for a series of H spectra of chloroform, according to the COSY pulse sequence for a series of values of t. Each peak results from one cycle of 90°-/i -90° followed by Fourier transformation during ti of Figure 6-1 to give frequency on the axis labeled V2 (corresponding to the time domain tj). The period is ramped up after each cycle. Fourier transformation in the fi dimension has not been carried out.

All modern ACC radar sensors make use of the Fourier transformation for signal processing. Viewed in a simple manner, the Fourier transformation is a calculation-intensive transformation from the time domain to the frequency domain and reverse. A series of measured values defined in frequency steps, the frequency spectrum, is derived from a series of defined time steps. Modern signal... 

A time domain function can be expressed as a Fourier series, an infinite series of sines and cosines. However in practise integrals related to the FOURIER series, rather than the series themselves are used to perform the Fourier transformation. Linear response theory shows that in addition to NMR time domain data and frequency domain data, pulse shape and its associated excitation profile are also a FOURIER pair. Although a more detailed study [3.5] has indicated that this is only a first order approximation, this approach can form the basis of an introductory discussion. 

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