Fourier sine function

Mathematical theory shows that any periodic function of time, /(f), can be represented as a series of sine functions having frequencies a>, 2a>, 3ft), 4ft), etc. Function /(f) is represented by the following equation, which is referred to as a Fourier series ... 

Figure 4.11. Left Simulated EXAFS spectrum of a dimer such as Cu2, showing that the EXAFS signal is the product of a sine function and a backscattering amplitude F(k) divided by k, as expressed by Eq. (6). Note that F k)/k remains visible as the envelope around the EXAFS signal xW- Right The Cu EXAFS spectrum of a cluster such as CU2O is the sum of a Cu-Cu and a Cu-O contribution. Fourier analysis is the mathematical tool used to...

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... 

Summarizing, two complementary representations of a signal have been derived f(0 in the time domain and [A - jB ] in the frequency domain. The imaginary Fourier coefficients, represent the frequencies of the sine functions and the real... 

This gives the following expression for the Fourier transform of a sine function ... 

The phase spectrum 0(n) is defined as 0(n) = arctan(A(n)/B(n)). One can prove that for a symmetrical peak the ratio of the real and imaginary coefficients is constant, which means that all cosine and sine functions are in phase. It is important to note that the Fourier coefficients A(n) and B(n) can be regenerated from the power spectrum P(n) using the phase information. Phase information can be applied to distinguish frequencies corresponding to the signal and noise, because the phases of the noise frequencies randomly oscillate. 

Fig. 1.10 Soft rf pulses (left) in the shape of a sine (sin x/x) function, and their Fourier transforms (right), being equivalent to the excited slice in the presence of a constant magnetic field gradient. The well defined sine function (top) produces an excitation that is a slice...

Sometimes the FID doesn t behave as we would like. If we have a truncated FID, Fourier transformation (see Section 4.4) will give rise to some artefacts in the spectrum. This is because the truncation will appear to have some square wave character to it and the Fourier transform of this gives rise to a Sine function (as described previously). This exhibits itself as nasty oscillations around the peaks. We can tweak the data to make these go away by multiplying the FID with an exponential function (Figure 4.1). 

Fig. 1. Top Scheme of an inversion recovery experiment 5rielding the longitudinal relaxation time (inversion is achieved by mean of the (re) radiofrequency (rf) pulse, schematized by a filled vertical rectangle). Free induction decays (fid represented by a damped sine function) resulting from the (x/2) read pulse are subjected to a Fourier transform and lead to a series of spectra corresponding to the different t values (evolution period). Spectra are generally displayed with a shift between two consecutive values of t. The analysis of the amplitude evaluation of each peak from — Mq to Mq provides an accurate evaluation of T. Bottom the example concerns carbon-13 Tl of irans-crotonaldehyde with the following values (from left to right) 20.5 s, 19.8 s, 23.3 s, and 19.3 s.

Fig. 13 Simulation of a monochromatic source with a finite arm displacement of the interferometer, (a) Truncated cosine function (30 discrete data points), (b) Its Fourier transform, the sine function, which simulates the infrared spectral line.

Fourier analysis is used to find the velocity and attenuation of surface waves. Let the range in z over which data is available be (. If there were no attenuation, then by the convolution theorem the Fourier transform F ( ) would be a sine function centred at a spatial frequency... 

One of the fundamental theorems of Fourier transforms states that multiplying two functions in one Fourier domain is equivalent to convoluting the two functions in the other domain [60], The FT spectrum thus has a lineshape corresponding to the Fourier transformation of >(<5), which is the sine function... 

According to Fourier theory, any complicated periodic function can be approximated by this series, by putting the proper values of h, Fh, and ah in each term. Think of the cosine terms as basic wave forms that can be used to build any other waveform. Also according to Fourier theory, we can use the sine function or, for that matter, any periodic function in the same way as the basic wave for building any other periodic function. 

Sine function sinc(x) = sin(x)/x. The sine function is most often encountered as the (frequency-domain) Fourier transform of a (time-domain) rectangular pulse. 

Thus, each coordination shell contributes a sine function multiplied by an amplitude, as illustrated in Figure 6.11 for the simple case of a dimer. EXAFS analysis boils down to recognizing all sine contributions in /(fe). The obvious mathematical tool to achieve this is Fourier analysis. 

Fig. 2. (a) The free induction decay, G(t) for 19F in a single crystal of CaFi for B0 along [1,0,0]. The experimental points are given by circles and crosses from the CW and pulse measurements, respectively, and the theoretical curve is that of Eq. (14), corresponding to an exponential decay multiplied by a sine function. Note that F(t) is equivalent to G(t) in the present notation. Reproduced with permission from A. Abragam, The Principles of Nuclear Magnetism, p. 121, Oxford University Press, London, 1961. (b) The lineshape in the frequency domain corresponding to the Fourier transform of the theoretical curve. 

After you Fourier transform your FID, you get a frequency-domain spectrum with peaks, but the shape of the peaks may not be what you expected. Some peaks may be upside down, whereas others may have a dispersive (half up-half down) lineshape (Fig. 3.36). The shape of the peak in the spectrum (+ or — absorptive, + or — dispersive) depends on the starting point of the sine function in the time-domain FID (0° or 180°, 90° or —90°). The starting point of a sinusoidal function is called its phase. Phase errors come in all possible angles, including those intermediate between absorptive and dispersive (Fig. 3.37). The spectrum has to be phase corrected ( phased ) after the Fourier transform to obtain the... 

Now let us consider the Fourier transform of the applied AC voltage, U(t), and the current response, 7(t). The transform of a sine function is a positive complex delta function at the appropriate positive frequency and a negative complex delta at the appropriate negative frequency ... 

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