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Here we present an example of the concept detailed in the previous section. Assume we have two spot-like samples in the {EQUATION()}$x=0, y=0${EQUATION} plane in the positions {EQUATION()}$z=z_0${EQUATION} and {EQUATION()}$z=-z_0${EQUATION}. Let's use a simple FID sequence with constant {EQUATION()}$z${EQUATION} gradient right after the pulse, during the acquisition. The sequence diagram is shown in Figure 1.
{img fileId="3176" thumb="y" rel="box[g]" width="720" imalign="center" align="center" desc=Figure 1. Left: two spot samples to be imaged. Right: Sequence diagram of a simple 1D imaging FID with constant readout gradient.}
The phase evolution of the spins in the two samples with a gradient strength {EQUATION()}$G_z${EQUATION}:
::{EQUATION()}
\label{1D_imaging1}
\phi_1 = \gamma G_z z_0 t, \hspace{10pt}
\phi_2 = -\gamma G_z z_0 t
{EQUATION}(1)::
Their complex demodulated signal:
::{EQUATION()}
\label{1D_imaging2}
S(t) = S_0 \left(\mathrm{e}^{\mathrm{i} \gamma G_z z_0 t} + \mathrm{e}^{- \mathrm{i} \gamma G_z z_0 t} \right) = 2 S_0 \mathrm{cos} \left( \gamma G_z z_0 t \right)
{EQUATION}(2)::
The spatial frequency vector {EQUATION()}$\mathbf k${EQUATION} now only has {EQUATION()}$z${EQUATION} component:
::{EQUATION()}
\label{1D_imaging3}
k_z = \gammabar \int \limits_0^t G_z(t') \mathrm d t' = \gammabar G_z (t-t_1), \hspace{10pt} 0 < k_z < k_{max}= \gamma G_z (t_2-t_1)
{EQUATION}(3)::
Hence the signal as a function of spatial frequency:
::{EQUATION()}
\label{1D_imaging4}
S(k_z) = S_0 \mathrm{cos} \left(2 \pi k_z z_0 \right)
{EQUATION}(4)::
The spatial spin density is obtained by the inverse Fourier transform of the signal, now only in one dimension.
::{EQUATION()}
\label{1D_imaging5}
\rho(z) = \int \mathrm d k_z 2 S_0 \mathrm{cos} \left( 2 \pi k_z z_0 \right ) \mathrm{e}^{-\mathrm i 2 \pi k_z z} = S_0 \int \limits_{- \infty}^{\infty} \mathrm d k_z \left(\mathrm{e}^{\mathrm i 2 \pi k_z (z+z_0)} + \mathrm{e}^{\mathrm i 2 \pi k_z (z-z_0)} \right)= \\
= S_0 \left [ \delta (z+z0) + \delta (z-z_0) \right]
{EQUATION}(5)::
As can be seen, the inverse Fourier transformed signal gives back the original spot-like spin density along the {EQUATION()}$z${EQUATION} direction.The illustration of the signal and the trajectory in {EQUATION()}$k${EQUATION}-space are shown in Figure 2.
{img fileId="3177" thumb="y" rel="box[g]" width="800" imalign="center" align="center" desc=Figure 2. Upper left: Laboratory signal of the two spot-like samples with its envelope. Lower left: Real part of the demodulated signal forming a cosine. Right: trajectory in k-space. Note that the maximal reached k value is proportional to the acquisition (or sampling) time Ts.}