Basic Physics of MR Signal and Image Generation - Advance Image Processing in Magnetic Resonance Imaging - page 26

90-degree

RF

Time

Gz

FIGURE 1.13 Example of a pulse timing diagram relevant to the slice selection operation.

after an excitation is the homogeneous B 0 field plus another linear gradient field
(xG x ), the Larmor frequency at position x is

ω (x) = ω 0 + γ xG x (1.34)

Correspondingly, the FID signal ds(x,t), generated locally from spins in an
infinitesimal interval dx at point x without considering the transverse relaxation
effect, is

ds(x,t) = c ρ (x)dxe

− i ( BxG

0

+ ) t

(1.35)

x

where the constant of proportionality c is dependent on the flip angle, the main
field strength, and so on. The signal in Equation 1.35 is said to be frequency
encoded because its oscillation frequency is linearly related to the spatial location;
consequently, G x is called a frequency-encoding gradient . The signal received
from the entire sample in the presence of this gradient is:

∞ ∞

&
(

∞

)
+

∫ ∫ ∫

st () = dsxt ( , ) = c ρ ( ) xe

− iBxG x γ ( + )

dx = c ρ (( x e

− ixGt γ

dx e

− i t ω 0

0 x

−∞ −∞ −∞

(1.36)

In the general case, the received frequency-encoded FID signal after demod-
ulation (i.e., after removal of the carrier signal exp( − i ω 0 t )), is given by

∞
∫ ρ

− iG t

freq

γ r

st () = c () r e dr (1.37)

−∞

where G freq is the frequency-encoding gradient defined by G freq = (G x , G y , G z ).