1) (15 pts) a. Derive the encoding efficiency for the below scenario, and the difference A...

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1) (15 pts) a. Derive the encoding efficiency for the below scenario, and the difference A between the initial phase of the monofrequency vibration u(t), u(t)=us )=u sin(@t +), and the initial phase of the MR signal phase ((s). s corresponds to the start time of the motion encoding gradient (MEG) K(t)=K, sin(@t+k). Perform the derivation for the special case of matched frequencies of the vibration and of the MEG. Hints: Assumption: MEG is applied in same direction as vibration. Integrate the product over the duration of the MEG. Note that the MEG duration always corresponds to an integer number q of MEG cycles. Assume, that the MEG has got zero phase at its start time t = s. The following equation is useful: sin(a) sin(B) === [cos(a - -B) = cos(a + B)] b. Calculate the encoding efficiency of a trapezoidal motion encoding gradient G(t) for a voxel oscillating (in the same direction as G) with frequency 1/T. T Go forts and t s + 4 + and t < < s + - 4 3T Go fort > s ++ and t 1) (15 pts) a. Derive the encoding efficiency for the below scenario, and the difference A between the initial phase of the monofrequency vibration u(t), u(t)=us )=u sin(@t +), and the initial phase of the MR signal phase ((s). s corresponds to the start time of the motion encoding gradient (MEG) K(t)=K, sin(@t+k). Perform the derivation for the special case of matched frequencies of the vibration and of the MEG. Hints: Assumption: MEG is applied in same direction as vibration. Integrate the product over the duration of the MEG. Note that the MEG duration always corresponds to an integer number q of MEG cycles. Assume, that the MEG has got zero phase at its start time t = s. The following equation is useful: sin(a) sin(B) === [cos(a - -B) = cos(a + B)] b. Calculate the encoding efficiency of a trapezoidal motion encoding gradient G(t) for a voxel oscillating (in the same direction as G) with frequency 1/T. T Go forts and t s + 4 + and t < < s + - 4 3T Go fort > s ++ and t