Imaginary Fourier coefficient

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... 

In summary, the Fourier transform of a continuous signal digitized in 2A/ + 1 data points returns N real Fourier coefficients, N imaginary Fourier coefficients and the average signal, also called the DC term, i.e. in total 2N + 1 points. The relationship between the scales in both domains is shown in Fig. 40.9. 

In Section 40.3.4 we have shown that the FT of a discrete signal consisting of 2N + 1 data points, comprises N real, N imaginary Fourier coefficients (positive frequencies) and the average value (zero frequency). We also indicated that N real and N imaginary Fourier coefficients can be defined in the negative frequency domain. In Section 40.3.1 we explained that the FT of signals, which are symmetrical about the / = 0 in the time domain contain only real Fourier coefficients. 

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... 

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. 

It should be recognized that the discrete Fourier coefficients G(x, y, co) are represented by complex numbers. The real part Re(G(x, y, to)) of the complex number represents the amplitude of the cosine part of the sinusoidal function and the imaginary part Im(G(x, y, co)) represents the amplitude of the sine wave. 

The complex exponential can be broken down into real and imaginary sinusoidal components. The results of the transform are the Fourier coefficients g[u] (or g[u,v]) in frequency space. Multiplying the coefficients with a sinusoid of frequency yields the constituent sinusoidal components of the original descriptor. 

FIGURE 6.5 Fourier coefficients are stored in a Fourier spectrum typical of the one above. Note that most of the information is below the first 100 coefficients or 50 pairs. Only the right half of the Fourier spectrum is stored. Both the real a and the imaginary fc coefficients are stored alternatively in the same spectrum (see Table 6.1). 

For the reverse transformation the same routines (source codes) can be used in FFT and FHT. However, for the reverse Fourier transformation the real and imaginary arrays of the coefficients (which are now input) should be divided by N (number of coefficients) and the imaginary array must be conjugated (multiplied by -1), while in the case of reverse Hadamard transformation only a division of N real coefficients by N is necessary. 

A central ingredient in the model is the generalized friction coefficient y(co), which is the Fourier transform of the retarded memory kernel y(f). To compute y(oo), following a standard procedure, one first attributes a small imaginary part e > 0 to co. One thus defines... 

In the structure factor equation the coefficients in the summation were all nonzero electron densities fj occurring at Xj, yj, Zj, which is really p(xj, yj, zj), and Fhki were the entities being calculated. Hence the coefficients in the electron density equation yielding p(x, y, z) must be the reciprocal space entities Fhki Finally, to keep units consistent, and the mathematics consistent with Monsieur Fourier, the sign of the imaginary term must be changed to minus, and the constant V must be inverted to 1/V, the volume of the reciprocal unit cell. Thus the electron density equation assumes the form. 

By comparing it with the equivalent one of Eq. (5.158) they will generate the formula of absorption linear coefficient as function of imaginary component of the Fourier s coefficient and atomic susceptibility for the transmitted wave, namely ... 

As is the case with molecular quantities, Fourier components of E and P are accompanied by frequency-dependent, complex susceptibilities % The macroscopic susceptibilities are used in the physical description of NLO effects, such effects typically being analyzed using wave equations in which the nonlinear polarization produced by a given type of interaction constitutes a source term. Quantities other than the susceptibilities are often used for describing specific NLO interactions. The most useful of these are the electro-optic coefficient r related to co ca,0), the nonlinear refractive index ti2, related to the real component of the degenerate third-order susceptibility Re(x —m,ai)), and the two-photon absorption coefficient jSg, related to the corresponding imaginary component Im(x —[Pg.66]

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