Time-reversal acoustics can be studied either in the time domain or in the frequency domain. The two domains have complementary advantages and disadvantages.
Pluses and minuses for the time domain include: Acoustical experiments are generally performed in the time domain. Self-focusing in any acoustic medium can be achieved using time-reversal processing in the time domain. Delay-time windowing easily distinguishes arrivals from well-separated scatterers in the time domain. But, if we want to understand the eigenvector/eigenvalue structure of the acoustic scattering operator in the time domain, we can only obtain one eigenvector at a time, and to obtain more eigenvectors (after the first one) a rigorous orthogonalization procedure must be followed.
Advantages of the frequency domain include the fact that eigenvector/eigenvalue or singular value decomposition (SVD) is quite easily done there, although it is done then one frequency at a time. Thus, all eigenvalues/eigenvectors or singular values/singular vectors are obtained simultaneously. Linear subspace imaging methods are then also most easily applied in the frequency domain. The main disadvantage of the frequency domain is that it is not simple to take advantage of target separation in space (and therefore in time) in this domain.
We will return to discuss more about time domain processing later in the paper, but for now we assume that all measured time traces have been Fourier transformed into the frequency domain, and that the pertinent equation to study is therefore the Helmholtz equation.