Frequency domain, Fourier series

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. 

Before discussing the Fourier transform, we will first look in some more detail at the time and frequency domain. As we will see later on, a FT consists of the decomposition of a signal in a series of sines and cosines. We consider first a signal which varies with time according to a sum of two sine functions (Fig. 40.3). Each sine function is characterized by its amplitude A and its period T, which corresponds to the time required to run through one cycle (2ti radials) of the sine function. In this example the frequencies are 1 and 3 Hz. The frequency of a sine function can be expressed in two ways the radial frequency to (radians per second), which is... 

The coefficients of the sines and cosines will be real for real data. Restoring a high-frequency band of c (unique complex) discrete spectral components to a low-frequency band of b (unique complex) spectral components will be the same (when transformed) as forming the discrete Fourier series from the high-frequency band and adding this function to the series formed from the low-frequency band. When applying the constraints in the spatial domain, the Fourier series representation will be used. 

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-. 

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... 

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.

FIGURE 5.6 Fourier transformation of a series of FIDs like the ones in Figure 5.5 (C) to give the frequency-domain spectrum as both a peak and as contours. The contour plot also shows a projection parallel to F2. 

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 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. 

The process of Fourier transformation converts the raw data (e.g. a time series) to two frequency domain spectra, one which is called a real spectrum and die other imaginary (diis terminology conies from complex numbers). The true spectrum is represented only by half the transformed data as indicated in Figure 3.18. Hence if there are 1000 datapoints in the original time series, 500 will correspond to the real transform and 500 to the imaginary transform. 

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). 

Often in an experiment it is possible to eliminate the contributions from the two power spectra leaving only the interference term. It is only this interference term that is dependent on phase and phase fluctuations. Note that for two identical pulses the signal is simply proportional to 2 cos [cox /2], which is a series of peaks in the frequency domain separated by 2/cx cm. Thus a x = 1 ps delay yields a peak separation of 67 cm In general the peak separations in the frequency domain are not independent of frequency and instead depend on the spectral phase difference at each frequency. Therefore spectral interferometry presents a method by which to determine the phase differences of two pulses. When the pulses are the same, we can use spectral interferometry to determine their time separations. The inverse Fourier transforms of the first two contributions to the spectrogram in Eq. (18) peak at f = 0 whereas the cross term peaks at t = x. Therefore Fourier transformation of S (a) can permit a separation of the cross term from the power spectra of the signal and reference fields [72]. 

According to the Fourier method, the measured line integral p r,4>) in a sinogram is related to the count density distribution A(x,y) in the object obtained by the Fourier transformation. The projection data obtained in the spatial domain (Fig. 4.2a) can be expressed in terms of a Fourier series in the frequency domain as the sum of a series of sinusoidal waves of different amplitudes, spatial frequencies, and phase shifts running across the image (Fig. 4.2b). This is equivalent to sound waves that are composed of many sound frequencies. The data in each row of an acquisition matrix can be considered to be composed of sinusoidal waves of varying amplitudes and frequencies in the frequency domain. This conversion of data from spatial domain to frequency domain is called the Fourier transformation (Fig. 4.3). Similarly the reverse operation of converting the data from frequency domain to spatial domain is termed the inverse Fourier transformation. 

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. 

The existence of the Fourier series makes it possible to represent a signal either in the time domain, as signal level v>y. time, or in Xht frequency domain, as the set of amplitudes and phase angles of the component sinusoids. Sometimes it is useful to dissect a time-do-main signal into its components or to synthesize the time-domain signal from its components. Section 10.8 provides excellent illustrations of both cases. Our concern here is with the mechanism of interdomain conversions. 

Figure 5.10 Radio frequency variation of My(t) transverse magnetisation observed, acquired and stored digitally with time is known as a Free Induction Decay (FID). Stored FID either singly or averaged, are processed by fourier series transformation (FT) from time domain signal information, SnoCti), into frequency domain (spectral) information, /nmr( i)- Only chemically equivalent nuclei without spin-spin coupling and with an equivalent Lamor frequency, V, are being observed here hence only a single signal will result of frequency Vi.

Fourier series transformation of time domain data, Snoffi), into frequency domain signal intensity data Inmr(Ei) then yields a characteristic classical ID NMR spectrum. 

Figure 5.11 Alternative diagrammatic representation of general FT ID NMR experiment. There is a single 90° pulse then signal observation and acquisition in time domain tj prior to fourier series transformation of time domain signal information SFiD(ti) into frequency domain (spectral intensity) information, 7nmr(Fi).

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,...

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