Transformation, convolution, and correlation

The terms transformation, convolution, and correlation are used over and over again in NMR spectroscopy and imaging in different contexts and sometimes with different meanings. The transformation best known in NMR is the Fourier transformation in one and in more dimensions [Bral]. It is used to generate one- and multi-dimensional spectra from experimental data as well as ID, 2D, and 3D images. Furthermore, different types of multi-dimensional spectra are explicitly called correlation spectra [Eml]. It is shown below how these are related to nonlinear correlation functions of excitation and response. 

Experimental NMR data are typically measured in response to one or more excitation pulses as a function of the time following the last pulse. From a general point of view, spectroscopy can be treated as a particular application of nonlinear system analysis [Bogl, Deul, Marl, Schl]. One-, two-, and multi-dimensional impulse-response functions are defined within this framework. They characterize the linear and nonlinear properties of the sample (and the measurement apparatus), which is simply referred to as the system. The impulse-response functions determine how the excitation signal is transformed into the response signal. A nonlinear system executes a nonlinear transformation of the input function to produce the output function. Here the parameter of the function, for instance the time, is preserved. In comparison to this, the Fourier transformation is a linear transformation of a function, where the parameter itself is changed. For instance, time is converted to frequency. The Fourier transforms of the impulse-response functions are known to the spectroscopist as spectra, to the system analyst as transfer functions, and to the physicist as dynamic susceptibilities. 

Impulse-response and transfer functions can be measured not only by pulse excitation, but also by excitation with monochromatic, continuous waves (CW), and with continuous noise or stochastic excitation. In general, the transformation executed by the system can be described by an expansion of the acquiired response signal in a series of convolutions of the impulse-response functions with different powers of the excitation [Marl, Schl]. Given the excitation and response functions, the impulse-response functions can be retrieved by deconvolution of the signals. For white noise excitation, deconvolution is equivalent to cross-correlation [Leel]. 

Some basic knowledge about convolution, correlation, and transformation is required for a more general understanding of measurement procedures [Angl]. In NMR 

By Fourier transformation, a signal is decomposed into its sine and cosine components [Angl]. In this way, it is analysed in terms of the amplitude and the phase of harmonic waves. Sine and cosine functions are conveniently combined to form a complex exponential, coscot 4- i sinwt = exp i yt). The complex amplitudes of these exponentials constitute the spectrum F((o) of the signal f(t), where co = In IT is the frequency in units of 2n of an oscillation with time period T. The Fourier transformation and its inverse are defined as 

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As already noted, the properties of convolution and correlation are the same, whether or not a continuous or discrete transformation is used, but because of the cyclic nature of sampled sequences discussed previously, the mechanics of calculating correlation and convolution of functions are somewhat different. The discrete convolution property is applied to a periodic signal 5 and a finite, but periodic, sequence r. The period of 5 is N, so that 5 is completely determined by the N samples s0, Sj,. .., %. The duration of the finite sequence r is assumed to be the same as the period of the data N samples. Then, the convolution of 5 and r is... 

In addition to its applications as an integral component of the measurement process, the Fourier transformation can be extremely useful for various signal-conditioning operations, such as smoothing, convolution, and correlation. Smith et al. have discussed the possibilities in this area at some length, and the interested reader is urged to pursue their discussions (36, 38). 

Fourier transform to provide integration, differentiation, convolution, and correlation. These operations are straightforward and normally require little user intervention. 

Nonetheless, by using the Fourier transform and two other mathematical operations, convolution and correlation, we can obtain several important structural parameters fiom the experimentally measured scattered intensity. The usual assumption is that the system can be represented by two-phases, with either sharp or difiuse boundaries between the phases. Some structural parametas of interest include average separation distance between the phases specific surface, Og average thicknesses cf the two phases and width of the boundary between phases, if diffose. These parameters are obtained from the "auto-correlation" function of p(x), which is a specific type of convolution. 

The discrete Fourier transform and its inverse are implemented as fft and ifft. In N dimensions, the routines are fftn and ifftn. In two dimensions, fft2 and ifft2 should be used. The examples in this chapter demonstrate the use of these functions. To compute convolutions and correlations, multiply the Fourier transforms appropriately. 

Simple product operations in the z transform domain apply for calculating the system response and the impulse response analogous to the expressions for convolution (4.2.14) and correlation (4.3.5) in the Fourier domain. The z transform of the result is readily found by expansion into a polynomial in z , and the coefficients determine the result on the equidistant time grid [Bral]. 

An alternative method for calculating the time correlation function, especially useful when its spectrum is also required, involves the fast Fourier transformation (FFT) algorithm and is based on the convolution theorem, which is a general property of the Fourier transformation. According to the convolution theorem, the Fourier transform of the correlation function C equals the product of the Fourier transforms of the correlated functions ... 

Starting with AT/T = kit (A x Ip), where and x are the correlation and convolution operators, respectively, we apply the Fourier transform, keeping in mind the fundamental identities ... 

Relation (C.ll) may be viewed as an implicit definition of the direct correlation c(R) in terms of the total correlation h(R). It can be made explicit by taking the Fourier transform of both sides of equation (C.ll), and noting that the integral on the rhs of (C.l 1) is a convolution (for spherical particles) hence, applying the convolution theorem, we obtain... 

One of the most useful properties of Fourier transformation is that it converts a convolution into a multiplication. (Convolution is one of many correlations, i.e., mathematical operations between functions, that can be greatly simplified by Fourier transformation.) Since convolution is a rather involved mathematical operation, whereas multiplication is simple, convolutions are often performed with the help of Fourier transformation. We will explore this property in more detail in sections 7.5 and 7.6. 

Methods for evaluation of analytical signals are as follows transformation, smoothing, correlation, convolution, deconvolution, derivation, and integration. 

In the equation, the boldface letter signifies a matrix consisting of the site-site correlation functions, haj r) and Cay(r), and the asterisk denotes the convolution integrals as well as the matrix products, coa-y is the Fourier inverse transform of u>ay k) defined by Eq. (1.10), which can be written as... 

Much as for the ID solvent-solvent correlations, the renormalization of the 3D-RISM/HNC equations is not necessary in respect to convergence. Nor is it required for the 3D fast Fourier transform (3D-FFT) employed to evaluate the convolution in Eq. (4.A.41). For a periodic solute neutralized by a compensating background charge, the Coulomb potential of the solute charge is screened at a supercell length. Therefore the 3D site direct, and hence total correlation functions, are free from the Coulomb singularity at fc = 0 and can be transformed directly by the 3D-FFT. 

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