Solving Bloch Equation with ADvsincðDvtÞ Envelope - Digital Signal Processing for Medical Imaging Using Matlab - page 98

Appendix A
Solving Bloch Equation with ADvsinc ð Dvt Þ
Envelope

Rewriting ( 2.21 ) and ( 2.23 ) with ADvsinc ð Dvt Þ envelope as follows with the
condition Dv\\ B 0 c

2p .
dM xy ð t Þ
dt

¼ jcADvsinc ð Dvt Þ M 0 cos ð a Þ e j ð cB 0 t þ h Þ

ð A : 1 Þ

dM z ð t Þ
dt

ð A : 2 Þ

¼ cADvsinc ð Dvt Þ M 0 sin ð a Þ E 0 sin ð h / Þ

Since sinc ð Dvt Þ is almost one for all t with Dv\\ B 0 c

2p , we treat sinc ð Dvt Þ as the
constant in both the equations and solution is obtained as follows

M xy ð t Þ¼ 2M 0 sin ð a Þ e j ð cB 0 t þ / Þ

ð A : 3 Þ

M z ð t Þ¼ cM 0 sin ð a Þ ADvsinc ð Dvt Þ sin ð h / Þ t þ 2M 0 cos ð a Þ ð A : 4 Þ