Exact Kramers-Kronig derivation and applications

Tissue attenuation is generally modelled as a power law in frequency. The exact Kramers-Kronig (K-K) relation has been derived for power law frequency variation using infinite series representations of operators in certain function spaces and little known infinite series expansions of functions. We give a simple derivation using only contour integrals and show some associated results from bovine femur and phantoms. The method is extended to power laws higher than unity using a subtraction technique which is required for applications. The exact variation of the speed vs frequency is important to determine the frequency independent part of the pseudo wave number that results from a standard transformation of the variable density wave equation to the Helmholtz equation. This frequency independent part may be helpful in the estimation of density. This simple derivation is similar to standard methods in wave analysis to derive Green’s functions and integral representations and indeed the K-K integral relation itself. The extension to a>1 is required to be physically relevant and is also carried out very simply. The singular case at a=1 is established as a limiting case.