In this chapter and the next, we demonstrate how the problem of Fourier-transform estimation from sampled data arises in the processing of measurements obtained by sampling electromagnetic- or acoustic-field fluctuations, as in radar or sonar.

In many areas of remote sensing, what we measure are the fluctuations in time of an electromagnetic or acoustic field. Such fields are described mathematically as solutions of certain partial differential equations, such as the wave equation. A function u( x, y, z, t) is said to satisfy the three- dimensional wave equation if

where u _tt denotes the second partial derivative of u with respect to the time variable t twice and c > 0 is the (constant) speed of propagation. More complicated versions of the wave equation permit the speed of propagation c to vary with the spatial variables x, y, z, but we shall not consider that here.

We use the method of separation of variables at this point, to get some idea about the nature of solutions of the wave equation. Assume, for the moment, that the solution u( t, x, y, z) has the simple form

Inserting this separated form into the wave equation, we get

or

The function...