# Example: 1D imaging of two spots

Here we present an example of the concept detailed in the previous section. Assume we have two spot-like samples in the plane in the positions and . Let's use a simple FID sequence with constant gradient right after the pulse, during the acquisition. The sequence diagram is shown in Figure 1.

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 :

(1)

Their complex demodulated signal:

(2)

The spatial frequency vector now only has component:

(3)

Hence the signal as a function of spatial frequency:

(4)

The spatial spin density is obtained by the inverse Fourier transform of the signal, now only in one dimension.

(5)

As can be seen, the inverse Fourier transformed signal gives back the original spot-like spin density along the direction.The illustration of the signal and the trajectory in -space are shown in Figure 2.

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.