Translational diffusion and transport filters

Here G is a time-dependent, experimental variable, and the space vector r(f) is time dependent because of translational motion of the nuclear spins (cf. Section 1.2). Therefore, the phase 0 is time-dependent as well. For short times, the final position r t) of the spins assumed after the time has elapsed, can be approximated by a Taylor series with a finite number of terms (cf. eqn (5.4.54)). These terms are discriminated by the power of the time lag t and involve initial position r(0), velocity v(0), and acceleration a(0) as coefficients to different moments mk of the time-dependent gradient vector G(t), 

For convenience, but not by necessity, the moments are usually defined with respect to the gradient G, = dB jdXi of the magnetic field. Moments with higher-order spatial derivatives of the magnetic field can be considered as well and may 

The second term in the second line of (7.2.8) provides the familiar definition of the k vector for space encoding (cf eqn (2.2.23)), because 

The higher-order gradient moments encode parameters of motion like velocity v and acceleration a and are related to the Fourier conjugate variables and e of these 

As a consequence, every odd echo is attenuated by molecular diffusion and flow and every even echo is not [Carl]. 

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