Efficient design of pulses with trapezoidal magnitude and linear phase response profiles

A variation of the Shinnar-Le Roux (SLR) method of pulse envelope design that allows for control of the phase of the frequency response profile has been developed. The method makes use of the fact that a knowledge of one of the SLR polynomials in combination with a root inversion pattern for the other polynomial is sufficient to fully define the second polynomial. Optimization of the first polynomial, when cast in this form, remains nonlinear. However, it was demonstrated that the relationship between the SLR polynomials and the frequency response profile may be used to generate an initial guess for the SLR polynomials that is sufficiently accurate to allow for the application of linear optimization techniques in most cases. In practice several pulse envelopes having different root inversion patterns are investigated for each target profile. The resulting collection of pulses allows the user to trade off pulse power for profile accuracy. The proposed technique was used to design a large number of amplitude modulated excitation pulses having trapezoidal magnitude and linear phase frequency response profiles. A few examples of the resulting pulses and their response profiles are presented.