Fourier theorem

If one bears in mind that the Fourier theorem states that a periodic non-sinusoidal response can be described by a series of sinusoidal... 

QEEG makes use of the Fourier theorem to analyze the electrical power in the different frequency bands of the EEG (Itil 1981). Animals, with electrodes fixed to the skull (cortical leads) and sometimes also implanted stereotaxically in selected structures (e.g. hippocampus, striatum), are exposed to brief measurement periods during which they are either left free to move spontaneously, or activated by means of a treadmill to ensure a stable heightened level of vigilance. 

As another example, consider the diffraction by a grating consisting of a large number N of parallel slits of width a, separated by opaque strips that are also of width a, as shown in Figure 1.8(a). The fundamental spatial period is d = 2a. We now make use of the Fourier theorem, which... 

The Fourier theorem states that any periodic function may be resolved into cosine and sine terms involving known constants. Since a crystal has a periodically repeating internal structure, this can be represented, in a mathematically useful way, by a three-dimensional Fourier series, to give a three-dimensional Fourier or electron density map. In X-ray diffraction studies the magnitudes of the coefficients may be derived from... 

Recently, a new technique has been developed [1323] that allows the direct comparison of widely different reference frequencies and thus considerably simplifies the frequency chain from the cesium clock to optical frequencies by reducing it to a single step. Its basic principle can be understood as follows (Fig. 9.91) The frequency spectrum of a mode-locked continuous laser emitting a regular train of short pulses with repetition rate 1/AT consists of a comb of equally spaced frequency components (the modes of the laser resonator). The spectral width Aw = 2jt/T of this comb spectrum depends on the temporal width T/Ar of the laser pulses (Fourier theorem). Using femtosecond pulses from a Tusapphire Kerr lens mode-locked laser, the comb spectrum extends over more than 30 THz. 

Many texts do not distinguish between the four possible data types . The four methods are all variations of one Fourier Theorem, and a good mathematician would... 

Unfortunately, the basic Fourier theorems are not directly applicable to most real data - and that is the data type we most often want to analyze. Real data is usually an arbitrarily truncated subset of an infinite dataset, so it is not a perfectly accurate representation of the whole dataset, and because of random noise, it is never exactly periodic. We can use Fourier methods to handle this data type only by being creative , or cheating - and we need to understand the consequences of that approach. We discuss these consequences in Sections 15.4 and 15.5. 

Typical tomographic 2D-reconstruction, like the filtered backprojection teelinique in Fan-Beam geometry, are based on the Radon transform and the Fourier slice theorem [6]. 

The last relation in equation (Al.6.107) follows from the Fourier convolution theorem and tlie property of the Fourier transfonn of a derivative we have also assumed that E(a) = (-w). The absorption spectmm is defined as the total energy absorbed at frequency to, nonnalized by the energy of the incident field at that frequency. Identifying the integrand on the right-hand side of equation (Al.6.107) with the total energy absorbed at frequency oi, we have... 

In this Fourier representation the Hamiltonian is quadratic and the equipartition theorem yields for the thennal... 

Showing that T(p) is the proper fourier transform of T(x) suggests that the fourier integral theorem should hold for the two wavefunetions T(x) and T(p) we have obtained, e.g. 

Now let us eonsider a funetion that is periodie in time with period T. Fourier s theorem states that any periodie funetion ean be expressed in a Fourier series as a linear eombination (infinite series) of Sines and Cosines whose frequeneies are multiples of a... 

It is a well known fact, called the Wiener-Khintchine Theorem [gardi85], that the correlation function and power spectrum are Fourier Transforms of one another ... 

This is the autocorrelation and by the Wiener-Khintchine theorem the power spectrum of the disturbance is given by its Fourier transform,... 

Thus the mutual intensity at the observer is the Fourier transform of the source. This is a special case of the van Cittert-Zernike theorem. The mutual intensity is translation invariant or homogeneous, i.e., it depends only on the separation of Pi and P2. The intensity at the observer is simply / = J. Measuring the mutual intensity will give Fourier components of the object. 

We are now ready to derive an expression for the intensity pattern observed with the Young s interferometer. The correlation term is replaced by the complex coherence factor transported to the interferometer from the source, and which contains the baseline B = xi — X2. Exactly this term quantifies the contrast of the interference fringes. Upon closer inspection it becomes apparent that the complex coherence factor contains the two-dimensional Fourier transform of the apparent source distribution I(1 ) taken at a spatial frequency s = B/A (with units line pairs per radian ). The notion that the fringe contrast in an interferometer is determined by the Fourier transform of the source intensity distribution is the essence of the theorem of van Cittert - Zemike. 

The fundamental quantity for interferometry is the source s visibility function. The spatial coherence properties of the source is connected with the two-dimensional Fourier transform of the spatial intensity distribution on the ce-setial sphere by virtue of the van Cittert - Zemike theorem. The measured fringe contrast is given by the source s visibility at a spatial frequency B/X, measured in units line pairs per radian. The temporal coherence properties is determined by the spectral distribution of the detected radiation. The measured fringe contrast therefore also depends on the spectral properties of the source and the instrument. 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

From the convolution theorem it follows that the convolution of the two triangles in our example can also be calculated in the Fourier domain, according to the following scheme ... 

Since the function A k) is the Fourier transform of (x, t), the two functions obey Parseval s theorem as given by equation (B.28) in Appendix B... 

Finally, integration over the variable k yields ParsevaTs theorem for the Fourier integral,... 

Favorable properties of the Fourier transform itself provide general means either to split the general problem of data analysis into sub-problems or even to obtain structure parameters without much modeling work. In this respect the Fourier slice theorem must be pointed out because of its superior impact on scattering (Bonart [16] ... 

In combination with other theorems of Fourier transformation theory many of the fundamental structural parameters in the field of scattering are readily established. Because the corresponding relations are not easily accessible in textbooks, a synopsis of the most important tools is presented in the sequel. 

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