The concept and calculations of convolution occur in many fields that involve data/signal analyses. I found that many people in astronomy were confused about its physical meaning BECAUSE a conceptual mistake was made in deriving a widely used formula in astronomy (Voigt line profile) many years ago. I made some comments to clarify the issue in the following link:
http://www.astronomy.com.cn/bbs/thread-62027-1-11.html
The following comments taken from the above link are worth reading and carefully digesting to those who are interested in astronomy theory or radiative transfer.
^^^^^^^^^^^^^^^^^^^^^^^^
If F(s) and G(s) are Fourier/Laplace transforms of f(t) and g(t), respectively, then the Fourier/Laplace transform of the convolution between f(t) and g(t) is the product of F(s) and G(s).
In most cases, only one side of the above identity has a (clear) physical significance. For example, the frequency-dependent coefficients of phenomenological conductivity and permeability define product relations, which carry a clear physical meaning. On the other hand, its counterpart that the electric polarization and magnetization of the electromagnetic waves will be related to the electric and magnetic fields through convolution integrals of Fourier transforms is not very clear/easy to understand.
The most interesting example about convolution (and product) in astronomy is the formation of line profiles. It is often said that the Voigt line profile is the convolution of the Lorentz line profile and the Doppler line profile whereas in fact its physical meaning has nothing to do with the concept of convolution!
^^^^^^^^^^^^^^^^^^^^^^^^
A correct and logical derivation of the Voigt profile
----------------------------------------------------------
I made the following statements in the above discussions:
“The most interesting example about convolution (and product) in astronomy is the formation of line profiles. It is often said that the Voigt line profile is the convolution of the Lorentz line profile and the Doppler line profile whereas in fact its physical meaning has nothing to do with the concept of convolution!”
I was trying to answer a common question like the following ones that I also encountered when I was learning this subject:
比邻星yue:“为什么是两个函数的卷积?”
快乐中微子:“我就问卷积是怎么得来的……为什么有卷积？”
My point is that there is no 卷积 involved in a correct and logical derivation of the Voigt profile. The final form of the Voigt profile happens to look like a 卷积 is solely by coincidence. The central idea of this problem is that the measured line profile represents a bulk/ensemble average of many atoms/molecules that have different wave forms and/or line profiles. This is very similar to the situation that many people have different heights and weights. You average their heights to get their average height. You take an ensemble mean of people’s weights to get the mean weight.
A correct and logical derivation is as follows:
(-4) Emission or absorption has a finite lifetime (due to either or both natural broadening or/and pressure broadening) so you do not have an infinitely long sinusoidal wave in time that will give you a sharp Dirac-delta function line profile in frequency.
(-3) The line profile you measured is an average of many atoms/molecules that have different lengths of finite lifetimes.
(-2) Different lifetimes of those atoms/molecules satisfy a Poisson distribution of P~exp(-t/tau).
(-1) You take the bulk average of the individual line profiles of different atoms/molecules by integrating over the Poisson distribution function to get the measured Lorentz line profile (or average over “dvdt相空间”).
(0) The Lorentz line profile can be expressed as a simple closed form so nobody has ever talked about convolution when taking the average over the Poisson distribution function to get the Lorentz line profile.
(1) Finite lifetime gives you a Lorentz line profile (due to either or both natural broadening or/and pressure broadening) with a half-width alpha_L
(2) The line profile approaches a Dirac-delta function as alpha_L -- > 0
(3) The random/thermal motion gives you a frequency shift in Lorentz profiles for different atoms/molecules related to the Boltzmann distribution function of velocity.
(4) You take the bulk average by integrating over the velocity distribution function to get the measured line profile (or average over “dvdVr相空间”).
(4.5) The bulk/ensemble average over the velocity distribution function here is similar to taking the bulk/ensemble average over the Poisson distribution before: the measured line profile represents all those atoms/molecules that have different lifetimes AND different velocities, i.e., people have different heights AND different weights.
(5) There is a nonlinear relationship between velocity v and the Doppler shifted frequency. However, if v << c, the relationship becomes linear.
(6) Changing integral variable from dVr to d[nu] gives you the Voigt line profile.
(7) Letting alpha_L-->0 in the Voigt line profile and using the Dirac-delta function relation give you the Doppler line profile.
(8) You now check again and find that the Voigt line profile happens to be the convolution between Lorentz profile and Doppler profile.
(9) It also happens that there is no closed expression for the Voigt line profile so people have always been taking that expression as a physical convolution between two line profiles.
Note: You do not have such a convolution relation if v<<c does not hold. Doppler line profile has been derived after you get the Voigt line profile. In other words, you really cannot have a pure Doppler line profile in theory.