Thompson - Computing for Scientists and Engineers (523188), страница 73
Текст из файла (страница 73)
The proof is easier done than said, so why not try it?Exercise 10.25(a) Use (10.46) to express the Fourier transform of the Gaussian convolution,G12(k), in terms of the product of the two transforms, G1 and G2, where(10.24) is used for each of these. Thus show directly that(10.48)where the convoluted width is obtained from(10.49)in terms of the standard deviationsof the two Gaussians in x space.(b) From (10.48) show immediately, by using the inverse Fourier transform result (10.47), that(10.50)where g is the Gaussian defined by (10.21). nTherefore, two Gaussians convolute to a single Gaussian whose standard deviation10.3CONVOLUTIONS AND FOURIER TRANSFORMS403is given by the sum-of-squares formula (10.49).
This result can readily be generalized, as follows:Exercise 10.26Use the method of mathematical induction to show that successive convolutionsof K Gaussians always produces Gaussians with total standard deviation,obtained from the sum-of-squares fomula(10.51)a result which holds for widths in either x space or in k space, depending onwhich space is used to perform the convolution. nThis formula is the same as that used in statistical error analysis for K independentand normally-distributed variables.
The Gaussian (normal) distribution was introduced in Section 6.1 in the context of maximum likelihood and least squares. Butwhat is the connection of error analysis with convolutions? If the error distributionfor the i th source of errors is Gaussian with standard deviationand if the errorsare all independent, then each point on a given error distribution has to be convoluted with those due to all the other sources of error in order to estimate the total error distribution.
The standard deviation of this distribution is just given by (10.5 1).We derived the convolution theorem and its application to convolution of Gaussians by formal analysis. How can you be convinced that the result (10.49) is plausible? Just use the program Convolute Arrays that we developed in the preceding subsection.Exercise 10.27(a) Modify program Convolute Arrays, given above, so that CONARRAY1and CONARRAY2 are filled with Gaussians extending well into their tails for input choices of the two standard deviations. Both Gaussians should have thesame stepsize, chosen small compared with the smaller of the two standard deviations. Then run the program to estimate the discretized convolution (10.44).(b) Use the trapezoidal rule to estimate the area under the resulting distribution,then divide each point of the distribution by this area, in order to produce a function with unit area.(c) Add some code to compare how well the discretized convolution approximates the Gaussian with standard deviation given by (10.49).
Discuss the reasons for discrepancies as a function of position along the distribution and thestepsize. nConvolution of two Lorentzians is also most easily accomplished by using theconvolution theorem (10.46) applied to the exponentials, (10.33), that result from404FOURIER INTEGRAL TRANSFORMSFourier transforming Lorentzians. Several changes of notation are required. Wewrite the convolution theorem as(10.52)Now there are several straightforward analytical steps to obtain the convolution formulas.Exercise 10.28Substitute the expressions for the Fourier transforms P1 and P2, obtained using(10.33), into the integral (10.46), then regroup real and imaginary exponents inthe two exponentials. Next, identify the integral as the Fourier transform of aLorentzian whose parameters are additive in the parameters of the original Lorentzians, namely(10.53)where the convolved Lorentzian is(10.54)in which the total FWHM is additive in the input FWHM values(10.55)as is the resonance frequency(10.56)Thus, Lorentzians convolve into Lorentzians with additive parameters.
nThe additivity of Lorentzian widths expressed by (10.55) has a consequence forobserved spectral line widths of radiating systems, such as atoms or molecules emitting (or absorbing) light or other electromagnetic radiation, as discussed at the end ofSection 10.4 and in Dodd’s book. The width that one observes is the sum of thewidths of the initial and final states between which the transition occurs. Therefore,the lifetime for decay (which is proportional to reciprocal of the width) depends onthe widths of both initial and final states.
This is discussed in the context of stellarspectroscopy in Section 9-l of the book by Mihalas.Note in the convolution result (10.54) that extra factors are obtained because wechose to be conventional and scaled the initial Lorentzians to unity peak values.10.3CONVOLUTIONS AND FOURIER TRANSFORMS405Since convolutions preserve areas, as shown by (10.43), one then cannot force thepeak value of the convolved Lorentzian, L12, to be unity.
In order to understandthis normalization problem for Lorentzians, it is instructive to define the normalizedLorentzian by(10.57)where the normalization factor, N, is to be chosen so that the integral overtive as well as positive values, is unity.Exercise 10.29Apply a tangent transformation to the variable of integrationintegrateto show that N must be given bynega-in (10.57), then(10.58)if the integral is to be unity. nThus, the normalized Lorentzian is given by(10.59)The convolution result for the normalized Lorentzian now reads(10.60)where the FWHM and resonance frequency relations are given by (10.55) and(10.56). Convolution of such normalized Lorentzians conserves the area under eachLorentzian.Exercise 10.30Use mathematical induction to prove that successive convolutions of any numberof normalized Lorentzians produces a normalized Lorentzian whose FWHM isthe sum of its component Lorentzians and whose resonance frequency is the sumof its component resonance frequencies.
nFor Lorentzians, as for the Gaussians, it is interesting to see convolution resultsoccurring in practice, and therefore to try a numerical example with a discretizedconvolution (10.44).Exercise 10.3 1(a) Modify program Convolute Arrays so that the files CONARRAYl andCONARRAY2 are filled with Lorentzians extending well into their tails for input406FOURIER INTEGRAL TRANSFORMSchoices of the two standard deviations. As we discussed in Section 10.2 whencomparing Gaussians and Lorentzians (Figure 10.6), this will require a widerrange of x values than for a Gaussian of the same FWHM. Both Lorentziansshould use the same stepsize, which should be chosen small compared with thesmaller FWHM.
Run the program to estimate the discretized convolution(10.44).(b) Add some code to compare how well the discretized convolution approximates the Lorentzian with FWHM given by (10.55). Discuss the reasons fordiscrepancies as a function of position along the distribution and of stepsize.
nWe now have a good appreciation of the analysis, numerics, and applications ofconvolving two Gaussians or two Lorentzians. It might appear that it will be assimple to convolute a Gaussian with a Lorentzian, which is a very common examplein applications. This is not so, however, and extensive analysis and numerics arerequired, as we now develop.Convoluting Gaussians with Lorentzians: Voigt profileAs we derived in Section 8.1, a Lorentzian as a function of the frequency oftenarises in the description of resonance processes in mechanical, optical, or quantalsystems. The Gaussian usually arises from random processes, such as random vibration of a mechanical system or of molecules in a gas.
Therefore, convolution ofGaussians with Lorentzians will be required whenever a resonance process occurs ina system which is also subject to random processes, such as when atoms in a veryhot gas emit light. The emission spectra of light from stars is an example we develop in Section 10.4.We now derive formulas for the convolution of a Gaussian with a normalizedLorentzian, which is called the Voigt profile. (Woldemar Voigt, 1850 - 1919, was aGerman physicist who developed the science of crystallography, discovered the Lorentz transformations, and coined the term “tensor.“) In Project 10 (Section 10.4)we present the numerics and applications of the formulas for the Voigt profile.
Themethod we develop is most suitable for preparation of tables (in computer memoryor on paper) that have to be frequently used. If only a few values of the Gaussianand Lorentzian parameters are to be used, then direct numerical integration of theconvolution integral may be more appropriate. The derivations that follow also require several mathematical analysis skills that are commonly needed in numerical andscientific applications.To perform the convolution, it is most convenient (and also instructive) to usethe convolution theorem, since we know the Fourier transforms of Gaussians,and of Lorentzians, P (k).
If their formulas, (10.24) and (10.33), are inserted in the convolution formula (10.34), and if is made the variable of the convolution, then some redefinition of variables leads to a simple-looking expressionfor the Voigt profile.10.3CONVOLUTIONS AND FOURIER TRANSFORMS407Exercise 10.32Substitute the Gaussian and normalized Lorentzian Fourier transforms in theconvolution formula (10.34) as indicated, then make the following substitutionof variable(10.61)and introduce the scaled parameters(10.62)(10.63)to show that the convolution of a Gaussian with a normalized Lorentzian is givenbywhere H (a,v ) is the Voigt function, defined by(10.65)in which all variables are dimensionless.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
