Cosine function Fourier transform

BPTI spectral densities Cosine Fourier transforms of the velocity autocorrelation function... 

Fig. 8. Spectral densities for BPTI as computed by cosine Fourier transforms of the velocity autocorrelation function by Verlet (7 = 0) and LN (7 = 5 and 20 ps ). Data are from [88].

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 first Fourier transformation of the FID yields a complex function of frequency with real (cosine) and imaginary (sine) coefficients. Each FID therefore has a real half and an imaginary half, and when subjected to the first Fourier transformation the resulting spectrum will also have real and imaginary data points. When these real and imaginary data points are arranged behind one another, vertical columns result. This transposed data... 

Ifourth(fd, 2 Q) was multiplied with a window function and then converted to a frequency-domain spectrum via Fourier transformation. The window function determined the wavenumber resolution of the transformed spectrum. Figure 6.3c presents the spectrum transformed with a resolution of 6cm as the fwhm. Negative, symmetrically shaped bands are present at 534, 558, 594, 620, and 683 cm in the real part, together with dispersive shaped bands in the imaginary part at the corresponding wavenumbers. The band shapes indicate the phase of the fourth-order field c() to be n. Cosine-like coherence was generated in the five vibrational modes by an impulsive stimulated Raman transition resonant to an electronic excitation. 

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... 

A relationship, known as Euler s formula, exists between a complex number [x + jy] (x is the real part, y is the imaginary part of the complex number (j = P )) and a sine and cosine function. Many authors and textbooks prefer the complex number notation for its compactness and convenience. By substituting the Euler equations cos(r) = d + e -")/2 and sin(r) = (d - e t )l2j in eq. (40.1), a compact complex number notation for the Fourier transform is obtained as follows ... 

It is easy to show that the eigenspectrum of H is simply a cosine Fourier transform of the Chebyshev autocorrelation function ... 

In practice, the phase shift and the modulation ratio M are measured as a function of co. Curve fitting of the relevant plots (Figure 6.6) is performed using the theoretical expressions of the sine and cosine Fourier transforms of the b-pulse response and Eqs (6.23) and (6.24). In contrast to pulse Jluorometry, no deconvolution is required. 

S( at) and G oS) are the sine and cosine Fourier transforms of the luminescence response to r5-function excitation, respectively. N yields the total number of photons of the response to the (5-function excitation. Equation (9.59) can be rewritten in the following form... 

Platinum and palladium porphyrins in silicon rubber resins are typical oxygen sensors and carriers, respectively. An analysis of the characteristics of these types of polymer films to sense oxygen is given in Ref. 34. For the sake of simplicity the luminescence decay of most phosphorescence sensors may be fitted to a double exponential function. The first component gives the excited state lifetime of the sensor phosphorescence while the second component, with a zero lifetime, yields the excitation backscatter seen by the detector. The excitation backscatter is usually about three orders of magnitude more intense in small optical fibers (100 than the sensor luminescence. The use of interference filters reduce the excitation substantially but does not eliminate it. The sine and cosine Fourier transforms of/(f) yield the following results ... 

Conversely, a rf field is totally correlated because it is represented by a sine (or cosine) function and, as a consequence, its value at any time t can be predicted from its value at time zero. The efficiency of a random field at a given frequency co can be appreciated by the Fourier transform of the above correlation function... 

ESEEM is a pulsed EPR technique which is complementary to both conventional EPR and ENDOR spectroscopy(74.75). In the ESEEM experiment, one selects a field (effective g value) in the EPR spectrum and through a sequence of microwave pulses generates a spin echo whose intensity is monitored as a function of the delay time between the pulses. This resulting echo envelope decay pattern is amplitude modulated due to the magnetic interaction of nuclear spins that are coupled to the electron spin. Cosine Fourier transformation of this envelope yields an ENDOR-like spectrum from which nuclear hyperfine and quadrupole splittings can be determined. 

The t2g density is zero where two of the direction cosines x, y, and z are zero, that is, in the planes of the coordinate axes, but peaks along the eight cube diagonals. The density functions are Fourier-transform invariant, as discussed in chapter 3, and expressed by the equation... 

The friction function (Eq. 4) is then the cosine Fourier transform of the spectral density. 

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.

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-. 

Now let s look briefly at just enough of the mathematics of fiber diffraction to explain the origin of the X patterns. Whereas each reflection in the diffraction pattern of a crystal is described by a Fourier series of sine and cosine waves, each layer line in the diffraction pattern of a noncrystalline fiber is described by one or more Bessel functions, graphs that look like sine or cosine waves that damp out as they travel away from the origin (Fig. 9.3). Bessel functions appear when you apply the Fourier transform to helical objects. A Bessel function is of the form... 

Spectral Manipulation Techniques. Many sophisticated software packages are now available for the manipulation of digitized spectra with both dedicated spectrometer minicomputers, as well as larger main - frame machines. Application of various mathematical techniques to FT-IR spectra is usually driven by the large widths of many bands of interest. Fourier self - deconvolution of bands, sometimes referred to as "resolution enhancement", has been found to be a valuable aid in the determination of peak location, at the expense of exact peak shape, in FT-IR spectra. This technique involves the application of a suitable apodization weighting function to the cosine Fourier transform of an absorption spectrum, and then recomputing the "deconvolved" spectrum, in which the widths of the individual bands are now narrowed to an extent which depends on the nature of the apodization function applied. Such manipulation does not truly change the "resolution" of the spectrum, which is a consequence of instrumental parameters, but can provide improved visual presentations of the spectra for study. 

Equation 8 mathematically describes the interferogram (intensity versus optical retardation, which is a function of time) and that which is physically measured by the spectrometer. It represents one half of a cosine Fourier transform pair, the other being... 

Therefore these expressions permit us to transform the compliance function from the time domain to the frequency domain. The relationships of Eqs. (6.24) can also be written in terms of sine and cosine Fourier transforms ... 

FIGURE 6.13. Graphical examples of the Fourier transforms of (a) a cosine and (b) a sine function. Note that the Fourier transform contains information on phase, but that this information is lost when intensities (which involve the square of the displacement) are measured. The designation real and imaginary derives from the presence of i = in the Fourier transform Equation 6.14.1. 

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. 

In other words, the Chebyshev polynomials are essentially a cosine function in disguise. This duality underscores the effectiveness of the Chebyshev polynomials in numerical analysis, which has been recognized long ago by many,[7] including the great Hungarian applied mathematician C. Lanczos.[9] In particular, Fourier transform (and FFT) can be readily implemented in the spectral method involving the Chebyshev polynomials. 

After the application of weighting functions (primarily in NMR), the next step in data processing is to zero fill the data to at least a factor of two (called one level of zero filling). The reason for this step is that the complex Fourier transform of np data points consists of a real part (from the cosine part of the FT) and an imaginary part (from the sine part of the FT), each containing np/2 points in the frequency domain. Therefore, the actual spectrum displayed is described by only half of the original number of points. The technique of zero... 

Classical force-force correlation functions and their (cosine) Fourier transformations are shown in the left and middle column of Fig. 5 for five different trajectories. Here we have defined a ( function as... 

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