Anti-symmetrization of a PIP

Similar to the PIP, the Hamiltonian [Eq. (52a)] of a periodic pulse shows an infinite number of effective RF fields with both x and y components of the scaling factors X a and the phases 0na. The periodic pulse, however, acquires a different symmetry as that of the PIP. From Eq. (52c) and = ana, it follows that the scaling factor Xm, is symmetric in respect to the sideband number n, while the phase 6na is anti-symmetric according to Eq. (51c). These symmetries seem to be a coincidence arising from the mathematical derivations. As a matter of fact, they are the intrinsic natures of the periodic pulse. Considering the term f x i)Ix for instance, any Iy component created by the rotating field denoted by a> must be compensated at any time t by its counter-component oj n in order to reserve the amplitude modulated RF field. 

Fig. 8. The procedure for anti-symmetrizing a PIP. A PIP8(0°, 144°, 50 is, 1.3214 kHz, 10) is shown on top and it is anti-symmetrized (bottom) by splitting the first step into two equal parts and moving the first part to the end of the PIP. The total number of steps (A=ll) of the anti-symmetrized PIP increases by one but the pulsewidth remains the same. The phase satisfies the condition of (pk = k-...

Fig. 9. The BSOS as a function of the relative phase 0l(t( = 0t — 0O) between the RF fields /10 and fn where the solid line is a theoretical curve calculated with Eq. (74), while the open circles are computer simulated directly from the two PIPs./iO = 0.5 kHz and o = 0°. /u is created by a PIP10 (0°, 3.6°, 1.0 is, 0.5 kHz, 500) that is anti-symmetrized according to the procedure discussed in Section 4.5. Reprinted from Ref. 33 with permission from the American Institute of Physics publications.

A periodic pulse also introduces multiple effective RF fields that are symmetric in amplitude and anti-symmetric in phase. For the centre band RF field, the phase is not shifted and the strength is equal to the average value of the periodic pulse. Similar to the case of the PIP, all the phases and scaling factors of the RF fields can be obtained analytically and to fulfill the energy conservation law in periodic pulses, the scaling factors obey the equation ofEr=-oo = (/l2nns//l2)-... 

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