An estimator algorithm for the rotation time of magnetization vector in nuclear magnetic resonance for imaging (NMRI)

The purpose of this paper is to propose a useful method to investigate the rotation time of the magnetization vector in the nuclear magnetic resonance for imaging (NMRI) system. The ninety degrees rotation of the magnetization vector is the first step in order to establish the free induction decay that radiates electromagnetic energy inside the NMRI chamber. The estimator involved in this research is called Luenberger's observer which is a state estimator of a dynamical system. The Bloch's equation is a dynamical system characterized by a radio frequency (RF) impulse located inside the dynamic matrix, which means the system is not linear. The observer algorithm involved in this paper estimates each vector's component of the Bloch's dynamic model characterized by the magnetizations along the x, y and z direction which are axles located inside the NMRI chamber where the z axis has the same direction of the uniform magnetic field. The result is compared with one shown in the literature which results coincident with estimation. The estimator has been calculated in a closed form except in some cases where the symbolic expression makes the mathematical characterized by a high computational burden. The expression of the solutions is calculated by using the Heaviside expansion once the poles of the dynamical systems characterizing the Bloch's differential equations system are known. A set of simulations is carried out by using different configurations of the observer that have been calculated formerly without considering the RF pulse and subsequently with its introduction showing how the Bloch's dynamical system is affected by the skew symmetric matrix which is typical of a gyroscopic dynamical system. This likelihood produces a time estimation of the rotation vector which is slightly higher than the estimated value offered in the literature.