Autocorrelation convolution

In the field of scattering the autocorrelation is also known by the name convolution square . 

The signal Scoh represents a convolution integral of the intensity of the probing pulse oc EP(t — to) 2 with the molecular response the latter is governed by the autocorrelation functions 0Vib and 0r- Numerical solutions of Equations (2)-(5) are readily computed and will be discussed in the context of experimental results. 

The integral over t is the convolution of the intensities of the two laser pulses at time separation r. This function is identical to the noncollinear second-harmonic-generation intensity autocorrelation of the laser pulses,... 

Thus, the OHD-RIKES signal can be written as a convolution of the autocorrelation function with the response function ... 

Due to the finite propagation time T of the wavepackets, the Fourier transformation causes artifacts known as the Gibbs phenomenon [122]. In order to reduce this effect, the autocorrelation function is first multiplied by a damping function cos jtt/IT) [81,123]. Furthermore, to simulate the experimental line broadening, the autocorrelation functions will be damped by an additional multiplication with a Gaussian function exp — t/xd)% where zj is the damping parameter. This multiplication is equivalent to a convolution of the spectrum with a Gaussian with a full width at half maximum (FWHM) of /xd- The convolution thus simulates... 

If we take the scalar product of an atom s velocity with its velocity a short time later, t, and take an average over all the atoms of the same t)q)e then we may represent it as a convolution v(0) v(i), also commonly written (v(O).v(t)). The Fourier transform of a convolution is equivalent to product of the Fourier transforms of the two functions taken separately. Self convolution, autocorrelation, yields the square of the functions imder Fourier transformation and only the real part of the transformation is available, the power spectrum, p (a). 

Fig. la-d. Small-angle scattering from a dilute, random dispersion of membranes (vesicles), a corrected intensities or thickness factor obtained from the experimental intensity distribution 1(h) by multiplication with h. b Structure factor (amplitude function) with arbitrarily chosen signs (-k,—, +, —,).c Autocorrelation function of the electron density q(x) profile across the membrane obtained by cosine transformation of I,(h) (Eq. 5a) the insert shows the profile obtained by de-convolution. d Centrosymmetric electron density profile obtained by cosine transformation (Eq. 5b) of F,(h). From a study on lipoprotein X, an assembly of unilamellar vesicles (Ref. 84, with permission)... 

This autocorrelation function is the convolution product (symbolized by ) of g(r) and is defined by... 

Figure 3.16 Plot of the autocorrelation function (3.51). The curve pm ( ) centered around md is the m-fold convolution of p (.x) and is accordingly broader in width and shorter in height with increasing m. ...

Another method of estimating the pitch of a signal uses a special form of convolution called autocorrelation defined as ... 

This is a time (lag) domain function that expresses the similarity of a signal to lagged versions of itself. It can also be viewed as the convolution of a signal with the time-reversed version of itself. Pure periodic signals exhibit periodic peaks in the autocorrelation fimction. Autocorrelations of white noise sequences exhibit only one clear peak at the zero lag position, and are small (zero for infinite length sequences) for all other lags. 

The real space function p(r) is directly linked to the spatial autocorrelation function (convolution square) of electron-density fluctuations inside the particle. 

The first integral is independent of x and gives a constant background when the delay time x is varied. The second integral, however, does depend on t. It gives information on the intensity profile 7(r) of the pulse because it represents the convolution of the intensity profile 7(r) with the time-delayed profile I t + x) of fhe same pulse (intensify autocorrelation). 

FIGURE 1. Anisotropic photobleaching transients for PSI-200 particles at 665 and 675 nm. Continuous curves are optimized convolutions of Eqs. 1 with the laser pulse autocorrelation functions. 

As a result, we obtain the convolution of the density functirai p(r) with the same function inverted with respect of the origin of the reference frame p( r). Note that the minus sign appears due to different signs in the exponents for two complex conjugates in (5.28). The P(r) function is known as density autocorrelation function or the Paterson function when used in structural analysis. Thus, we may write the inverse and direct Fourier transforms as follows ... 

Mathematically, this relation follows from the convolution theorem, which relates the Fourier transform of a function to its autocorrelation function. 

An important theorem relates the Fourier transform and convolution operations. The Convolution Theorem (8,9) states that the Fourier transform of a convolution is the product of the Fourier transfomis, or F (f g) = F(u)G(u). Applying this to the autocorrelation yields F Kx) f(-x)] = F(u)F(-u). If f(x) is real, F(u)F(-u)=F(u)F (u)= F(u)p. Thus, "the Fourier transform ofthe autocorrelation of a function frx) is the squared modulus of its transform" (Ref. 9, p. 81). Application to scattering replaces frx) with the electron density profile, p(x). We have then the important result that the Fourier transform of the autocorrelation of the electron density profile is exactly equal to the intensity in reciprocal space, F(u). The autocomelation function cf the electron density has a special name it is called the generalized Patterson fimction(8), P(x), given by ... 

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