Magnetic Resonance Imaging

Phase Encoding

Consider a measured signal consisting of the sum of two unknown sinusoidal signals: $S_1(t)=s_{11}(t)+s_{21}(t)$ , where $S_1(t)=A_1\sin (\omega t+\theta)$ and $s_{11}(t)=a_{11}\sin (\omega t +\theta)$ and $s_{21}(t)=a_{21}\sin(\omega t+\theta)$ . This received signal is measured at $t_1$ . Since $s_{11}$ and $s_{21}$ are both sinusoidal signals with the same frequency, we can combine them both and conclude that

$A_1=a_{11}+a_{21}$ .

Next, we apply a gradient along the y direction. This has the effect of modifying the frequency of precession of nuclei based on the location of their y-coordinate. Let’s assume that the gradient field is linear and can be modeled as

$B_y=my$ .

When the gradient is turned on, the signals are now

If the gradient is turned on for a duration of $t=\Delta t$ , and then turned off, subsequently, then the signals will have experienced an increase in phase given by