Cosine and Sine Transforms

The cosine transform Ffs) and the sine transform Fs(s) of a function/ (x) are defined as 

It should be noted that both the cosine and sine transforms take no account of /( ) to the left of the origin. To see the relationship of the cosine and sine transforms to the Fourier transform, let us now write (B.l) as 

Iff(x) is an even function, i.e., /(— ) = f(x), then the second integral in (B.7) is zero, and we see that the Fourier transform F(s) is the same as the cosine transform Fc(s). Since Fc(s) is also an even function,/( ) can be recovered by taking the cosine transform of Fc(s)  

F(s) is then an odd function, and therefore its inverse transformation becomes 

On comparing (B.6) with (B.9) we note that F(s) is equal to —iFs(s). From (B.10) we therefore find 

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It is seen that c and s are the Fourier cosine and sine transforms of/(f, d, d) at the angular frequency of intersysfem crossing 2Q/5 and p is the total probability of reencounter. 

Figures 24b,c are, respectively, the cosine and sine transforms of this signal. Note that, as expected, both of these transforms yield valid line shapes with fractional amplitudes, and that the modulus transform. Figure 24e, yields the correct line shape.

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 derivation of the D matrix for a given contour is based on first deriving the adiabatic-to-diabatic transformation matrix, A, as a function of s and then obtaining its value at the end of the arbitrary closed contours (when s becomes io). Since A is a real unitary matrix it can be expressed in terms of cosine and sine functions of given angles. First, we shall consider briefly the two special cases with M = 2 and 3. 

Fig. 5.1 Comparison of the FORTRAN source codes of FFT and FHT algorithms, respectively. Besides the fact that FHT does not need the evaluation of cosine and sine functions (or the use of complex arithmetics), it saves time and computer memory by applying the transformation only on real coefficients.

To calculate Fourier coefficients, each row of the Free-Wilson matrix, denoted by FW(n, p), where n is the number of molecules and p the number of site/substituent indicator variables, is transformed into cosine and sine terms according to the following equation ... 

In the case of reflection measurements, the sample replaces one of the mirrors in the Michelson interferometer (see Fig. 32). The reference mirror is assumed to be 100% reflecting in the far-infrared, and in the sample interferogram the power reflectance R of the sample and the phase shift y> at the reflection (usually n for nonabsorbing media with w > 1) take over the role of T and q> in transmission measurements. The interferogram obtained in this case is also somewhat shifted and as3nmnetiic (see Fig. 33, KBr sample). By means of the cosine and sine Fourier transforms, R and y>, and finally n and x, are evaluated from the experimental data. 

The details of the Fourier transform itself are usually of little consequence to anyone using NMR, although there is one notable feature to be aware of. The term e " can equally be written cos cot + i sin cot and in this form it is apparent that the transformation actually results in two frequency domain spectra that differ only in their signal phases. The two are cosine and sine functions so are 90° out-of-phase relative to one another and are termed the real and imaginary parts of the spectrum (because the function contains complex... 

The cosine and sine data sets are transformed with respect to t2 and the real parts of each are taken. Then a new complex data set is formed using the cosine data for the real part and the sine data for the imaginary part ... 

With amplitude modulation the cosine and sine components may be handled in two ways to achieve quadrature detection in fl. They may be acquired in subsequent scans by either incrementing the pulse or receiver phase and the data co-added in the computer memory or they may be acquired sequentially and stored separately. With the first approach direct Fourier transformation yields frequency discrimination in fl but no absorptive lineshapes whilst with the second approach additional processing steps are necessary to achieve both, frequency discrimination and absorptive lineshapes. 

A Fourier transform enables one to convert the variation of some quantity as a function of time into a function of frequency, and vice versa. Thus, if we represent the quantity that varies in time as x(f), then Fourier analysis enables us to also represent that quantity as a function X i>), where i/ is the frequency (—oo < i/ < oo). Fourier analysis is usually introduced by considering functions that vary in a periodic manner with time which can be written as a superposition of sine and cosine functions (a Fourier series see Section 110.8). If the period of the fvmction x f) is r then the cosine and sine terms in the Fourier series are functions of frequencies 27m/r, where n can take integer values 1, 2, 3, ... 

Fig. 1 Idea of complex FT. Two signals of the same frequency and amplitude, shifted in phase by 7i/2, ScosC) and Snm(t) are transformed with cosine and sine FT and added. This may be described as one complex operation on one complex signal s(t) = ScosQ) + Wsin(t)...

According to Equation 52, the phase of the Fourier transformed signal in the frequency domain changes over the bandwidth of the instrument. This is, however, not the only phase problem that occurs. In addition, a frequency-dependent phase shift is introduced because the data collection can normally begin only a certain time (ca. 100 nsec) after the pulse is switched off. This and other phase shifts due to the use of band limiting filters are well-known in NMR and can easily be corrected with a digital computer. The true absorption spectrum, Sg(w), is obtained as a linear combination of the cosine and.sine Fourier transforms of the signal in the time domain Scos( ) SsinM respectively. 

If a function does not have either even or odd symmetry, its Fourier transform is complex. The real and imaginary parts of the transform consist of the cosine and sine Fourier transforms discussed in Appendix 4. 

The cosine and sine terms in this expression can be viewed as, respectively, normalized sine and cosine Fourier transforms of the fluorescence decay function. If we define the normalized sine and cosine Fourier transforms of F(t) as... 

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