A Simple Relationship between Total RF Pulse Energy and Magnetization Response - The Nonlinear Generalization of Parseval′s Relation

A simple expression for the total energy of an RF pulse is given in terms of the magnetization response of the pulse. It is shown to be a nonlinear generalization of Parseval′s relation and is one of an infinite number of conserved quantities, or integrals of the motion, found in the theory of nonlinear evolution equations. The simplicity of this expression enables the statement of a straightforward criterion for pulses that must have their energy reduced. It also allows the proof of some important theorems in pulse theory, including the known result that minimum-phase pulses are minimum energy pulses, and the previously unknown result that B 1-insensitive pulses must induce (possibly improper) bound states. A comparison of the energy requirements of minimum-phase and self-refocused pulses is given. Further, "Butterworth" pulses are compared to "Chebyshev I" pulses and found to require more energy in general.