Derivation of Broadband and Narrowband Excitation Pulses Using the Floquet Formalism

Amplitude-modulated broadband and narrowband excitation pulses for spin systems with I = 12 are derived using Floquet theory. These pulses are defined in terms of the coefficients of their finite Fourier transforms. A methodology for the design of band-selective pulses is presented and examples are given. Two perturbation-theory approaches are exploited. In the first, normal perturbation theory about the resonance condition is used to evaluate the eigenvalues and eigenvectors of the Floquet Hamiltonian of the irradiated spins. ln the second, a simultaneous perturbation-theory calculation is performed at off-resonance positions equal to multiples of the principal frequency of the Floquet matrix. With these methods, amplitude-modulated 180° broadband excitation pulses and narrowband inversion and 90° pulses were obtained and compared with similar pulses introduced by others. In addition, the conversion of 180° broadband pulses to pulses of arbitrary flip angles is discussed by representing the spin system in its interaction representation.