Translation Invariant Systems

It is difficult to overstate the power and versatility of the Green's function method. From the viewpoint of mathematics it allows one to generate solutions to any inhomogeneous linear differential equation with boundary conditions. From the viewpoint of radiation physics the Green's function relates a disturbance to its measurable effect or response. From the viewpoint of engineering $G(x;\xi )$ expresses those inner workings of a linear system which relates its input to its output.

Invariant linear systems constitute one of the most ubiquitous of its kind. They are characterized by invariance under space and/or time translations. Their Green's function have the invariance property

$$G(x+a;\xi+a)=G(x;\xi)$$

under arbitrary translations $a$ . Letting $a=-\xi$ , one finds that

$$G(x;\xi)=G(x-\xi;0)\equiv G(x-\xi)$$

Thus Eq.(4.11) becomes

$$u(\xi) =\int^\infty_{-\infty} G(\xi-x)~f(x)~dx \equiv G\star f~(\xi)~.$$ (412)

In other words the response of an invariant linear system is is simply the convolution of the input with the system Green's function.

It is virtually impossible to evade the fact that the essence of any linear translation invariant aspect of nature is best grasped by means of the Fourier representation. The input-output relation of a linear invariant system expressed by means of the convolution integral, Eq.(4.12), is no exception. Take the Fourier transform $\mathcal{F}$ of both sides and find

$$\hat u(k)=\sqrt{2\pi}~\hat G(k) \hat f(k)~,$$

where

$$\hat u(k)\equiv\mathcal{F} [u](k)\equiv \int^\infty _{-\infty} \frac{e^{-ik\xi}}{\sqrt{2\pi}}~u(\xi)~d\xi$$

is the Fourier amplitude corresponding to $u(\xi)$ . Convolution of functions in the given domain has simplified into multiplication of their Fourier amplitudes in the Fourier domain. For each point in this domain the factor $\hat f(k)$ expresses the input of the linear system, $\hat u(k)$ expresses its response. In signal processing and in electromagnetic theory the function $\hat G(k)$ is called the filter function, in acoustics and optics it is called the transfer function of the linear system.