Let's talk about the concept of frequency encoding. Again, please keep in mind that with respect to spatial encoding, the coordinate in case space at a certain time is determined by the integral area of gradient as a function of time along each axis. So integral area of gradient determines location on the frequency domain or called K-space. So k_x corresponds to gamma hat g_x t_x and k_y correspond to gamma hat g_y t_y. So this is linear multiplication of time and gradient. It's actually corresponding to the integral area of the gradient. Okay, so let me give you an example of pulse sequence shown here. So this is g_x and g_y. So this is right after slide let's assume like that then the location one, at the location time 0.1, there is no gradient is applied. So there is a gradient g_x but there's no time so the integral area is going to be zero. So that is going to start from the center of the frequency domain. So it's zero. But at the time point of two, so there is integral area along x direction, so it's going to move to the location two. So if we sample a data at number one and that can be used to fill the case-based center as shown here, and the if we sample another point at time point number two, then it can be used to sample a point here. Okay, it's going to move along x direction. For instance, and then number three case, there's time integration along y direction but the polarities minus so it's going to move downward along y direction and it can be used to sample a point here if we sample a data as number three. Then number four and then there's a narrative gradient along g_x and then the area is a little bit bigger so it's going to be moved or folder. So it can be used to sample a point number four as shown here. Therefore, the number of five case, both g_x and g_y, they are applied all together and then it's going to move in this direction because both x and y accumulate as a function of time. So it's going to move all the way along there. Then for the number five, it'll be used to sample here and the number six and so x direction and it's with a negative polarity it going to move left. Then number six, from number six number to number seven. So both gradient, x and the y direction, is turned on. So it's going to move along this direction, minus x, minus y. Then folder number seven to eight, at the number eight positive gradient along x direction so it's going to move along this direction and negative gradient along y direction is going to move along this direction. So it's going to move like that and return back to original case-based center. So as shown in this example, summation of integral area of gradient determines the location on the frequency domain for the sampled data. There are infinitely many ways of filling the whole K-space as you can imagine. There are infinitely many ways to filling the K-space but the one simple, very simple and also very common way of filling the K-space which is still used for most of the clinical scans in the actual clinical scanners is to fix gradient strengths and the duration along one direction, for instance, y direction in this case and then apply a constant gradient along the other direction. For instance, x direction. For a time period during which data sampling is performed. So as shown in this video, this is slice or selection gradient then we can turn on gradient and then we can keep up client data samples at multiple points following this gradient along x direction, for instance. Then let's see what happens on the frequency domain. So at the first data points, it is going to be case-based center. Okay, and there is gradient which is turned on continuously and then the position is going to keep moving along x direction. At the end of this data location, is going to be here. But data is now continuously sampled during this application of gradient and then so data point is going to keep sampled on the frequency domain following this trajectory. Okay, so if we fill data like that, then we can fill only the right-hand side of the K-space. Well, we want to feel both side of K-space along one complete line, along this k_x direction. This is K-space and k_x(y,u) and k_y(y,v). How to fill both sides of the K-space and then to do that we have to slightly modify the pulse sequence diagram. So as shown here, this is a pulse gradient. So it is a slice selection. There's FID signal. This is transmission of energy and that is going to be detectable signal. That is eligible site. It could be confusing but on the same axis but they are two independent events. So now the gradient along x direction is slightly modified so there is a gradient but we apply for a negative gradient first and then positive gradient later with longer duration. Okay, and then we are sampling data during this positive gradient. If we change the sequence diagram, pulse sequencing as shown here then what happens is this prephase gradients. We call this gradient as a prephase gradient because this makes the K-space location from center so this number one point is going to be K-space center and at the number two point, it's going to move to the left-hand side of the K-space. Then, applying for a positive gradient up to number three and then it's going to move, the trajectory move from number two as shown in this figure and they move all the way to right to the number three. If we applying for sampling during this trajectory continuously, then we can fill K-space from left-hand to the right-hand completely. So we can cover all the way to the right during those data sampling. So this prephase gradient is we make the phase gradient for data sampling, so it's called the prephase gradient. During this period, we don't acquire data. Okay, it just moves the K-space frequency location on the frequency domain to be all the way left. So in the viewpoint of signal, there is FID signal, but it may interfere the data sampling but applying for this prephase gradient make free induction decay signal decay faster. Then, what we acquire is higher signal intensity in the middle of the location. This procedure is called frequency encoding. This procedure itself is called frequency encoding. Combination of prephase gradient and negative polarity and also positive polarity gradient is applied which is called readout gradient. This is called readout gradient because we are acquiring data during this period. This is called readout gradient. This whole procedure is called frequency encoding. Also, this application of prephase gradient along this direction and also applying for the readout gradient. This combination of gradient negative polarity and positive polarity back-to-back. This combination generate a high-signal intensity in the middle which is called echo and this echo signal is called gradient echo because it is echo signal formed by gradient polarity changes, and the signal is called echo because originally FID signal is high but the FID signal has high intensity in the beginning. But if this signal has high intensity in the middle, so which is desired to fill the K-space and this is called, because of that this shape is called echo because it's kind of echo signal of the free induction decay. Let me give you another viewpoint of frequency encoding. To give spatial information, the gradient of course change the frequency of the precession spin along that direction. So if we consider, if there is no gradient and we have two different locations, for both of them have proton spins with different locations, if there is no gradient and both of them will have the same precession frequency and if they are acquired induced signal on the RF coil and both of them will have the same frequency and then there will be no spatial information. But if the location is combined with readout gradient. So with this readout gradient, we will make the precession frequency of the spins spatially different along that direction, in this case, x direction. This location and this location will have different precession frequency and then the detected signal will have two different frequency components. For instance, in this case. Then Fourier transform will distinguish the spatial location of these two proton spins. That is concept of frequency encoding because during the data readout, the gradient is turned on and then we can separate the spatial location by applying for Fourier transform. So that is the concept of frequency encoding. So again, the frequency encoding gradient is called readout gradient, and because the gradient is turned on during data sampling. The signals from two object at different locations as shown here can be detected as different frequency component which can be converted into original location which is image, that this can be performed by offline for Fourier transform to the acquired data. This is concept of frequency encoding.
