Integral transforms Fourier cosine transform

In the previous example the transform integral was separated into one part containing a cosine function and one containing a sine function. If the function f x) is an even function, its Fourier transform is a Fourier cosine transform ... 

Integral transforms were discussed, including Fourier and Laplace transforms. Fourier transforms are the result of allowing the period of the function to be represented by a Fourier series to become larger and larger, so that the series approaches an integral in the limit. Fourier transforms are usually written with complex exponential basis functions, but sine and cosine transforms also occur. Laplace transforms are related to Fourier transforms, with real exponential basis functions. We presented several theorems that allow the determination of some kinds of inverse Laplace transforms and that allow later applications to the solution of differential equations. 

The quantities Aik) and Bik) are interpreted as the amplitudes of the sine and cosine contributions in the range of angular spatial frequency between k and k + dk, and are referred to as the Fourier cosine and sine transforms. If we consolidate the sine and cosine transforms into a single complex exponential expression, we arrive at the complex form of the Fourier integral. This is the integral in Eq. (26.32), known as the Fourier transform, which for the one-dimensional function fix) is... 

The inverse of the transformation Ca / is given by the Fourier cosine integral formula... 

Because of the form of as a convolution integral, the governing equation is readily solved by means of Fourier transforms. If attention is restricted to surface shapes with the reflective symmetry prescribed by h(x,—z,t) = h(—x,z,t) = h(x,z,t), then the real Fourier cosine transform can be used this feature accounts for the relative transparency of the steps that follow. Suppose that H(a,f3,t) is the time-dependent double Fourier cosine transform of h(x,z,t) in both x and 2 , that is,... 

Let us assume that f s) is an even function and note the relation that exp( 2jii50 = cos 2%st i sin 2%st. Then, exp( 23ti5 t) in Equations (Dla) and (Dlb) can be replaced with cos 2%st, because the part of the integrand containing sin(2)Wt) is an odd function with respect to either s or t, and the integration of this part from -oo to oo tends to zero. As a result, the following two equations are obtained, and the Fourier transform in this expression is called the Fourier cosine transform. In this expression, no distinction exists between the Fourier transform and its inverse transform. 

Another view of this theme was our analysis of spectral densities. A comparison of LN spectral densities, as computed for BPTI and lysozyme from cosine Fourier transforms of the velocity autocorrelation functions, revealed excellent agreement between LN and the explicit Langevin trajectories (see Fig, 5 in [88]). Here we only compare the spectral densities for different 7 Fig. 8 shows that the Langevin patterns become closer to the Verlet densities (7 = 0) as 7 in the Langevin integrator (be it BBK or LN) is decreased. 

The Fourier sine transform F, is obtainable by replacing the cosine by the sine in these integrals. 

Data analysis in phase fluorometry requires knowledge of the sine and cosine of the Fourier transforms of the b-pulse response. This of course is not a problem for the most common case of multi-exponential decays (see above), but in some cases the Fourier transforms may not have analytical expressions, and numerical calculations of the relevant integrals are then necessary. 

The results of the integrations depend on the spectral density, which is defined as the cosine Fourier transform of the dynamical friction Eq. (8) ... 

In this Figure we have chosen a cenirosymmetric function, g y), that is a cosine function with a periodicity of 1. The integration has been approximated by a summation with small increments in y in order to demonstrate how a Fourier transform can be calculated. 

The electrical output signal from a conventional scanning spectrometer usually takes the form of an amplitude-time response, e.g. absorbance vs. wavelength. All such signals, no matter how complex, may be represented as a sum of sine and cosine waves. The continuous fimction of composite frequencies is called a Fourier integral. The conversion of amplitude-time, t, information into amplitude-frequency, w, information is known as a Fourier transformation. The relation between the two forms is given by... 

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. 

Two further aspects of Fourier transformation with respect to NMR data must be mentioned. With quadrature detection a complex Fourier transformation must be performed, there is a 90° phase shift between the two detectors and the sine and cosine dependence of the sequential or simultaneous detected data points are different. In addition because the FID is a finite number of data points, the integral of the continuous Fourier transform pair must be replaced by a summation. 

The Fourier transform allows us to take any well-behaved function and rewrite it as an integral over sine and cosine functions. If we have a function ff(f) that varies with (as an example) the time t, then we can rewrite this as a new function where o) is a coordinate with units of 1/t. The function f (co) is not really a new function but a splitting of the original f t) into many pieces, where each piece is a sine or cosine function with frequency co. 

FIGURE 19.10 Simulated free induction decay of a pure cosine component of the field after the 90° pulse from the side. The My component is 90° different as a sine wave while the magnetization rotates in the X-Y plane. For a real sample these would not be clean waves but would contain the information from many different chemical shift environments. The time scale could also be as much as 10 times longer in MRI. The two components are available for Fourier transformation with M as the real part and My as the imaginary part of the complex transform integral. 

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