 So, let us now look at the practical aspects of FTNMR, first offset or it is also called as the carrier frequency. So, what is the meaning of this? Now, we are exciting all the frequencies in one go here, right? So, let us say I have a spectral region from here to here, this is my spectral region and my spectrometer frequency is somewhere here, let us say this is 500 megahertz. If I am monitoring the protons, actual region of the spectra is here which is little away from here, somewhat away, we do not know how much away is because we do not know how much after you apply the current where the magnetic field has come and what is the exact frequency of the spectrometer, some arbitrariness will be there in that one. So, therefore, nonetheless what we have to do is we want to excite the region of our interest, region of our interest means you remember with respect to the spectrometer frequencies, we are choosing a few kilohertz into the left and right of that one. Therefore, we would like to have that spectrometer frequency here, we would like to have that spectrometer frequency here, right? So, what we have to do is we have to shift this to this place. In other words, we have to add a certain amount of offset to this frequency and that the amount of shifting what we do, this is called as the offset. So, from here to here what we are shifting, this is called as the offset. This is in kilohertz, some kilohertz it will be, it will not be megahertz, it will be some kilohertz. So, 5000 kilohertz, I mean 5000 hertz or 6000 hertz and things like that, that much of range we will occur and this of course is few kilohertz. So, when we do this, we achieve a uniform excitation of all your frequencies, therefore it is necessary to move the spectrometer frequency to this point and then this frequency what I put here, this frequency, this is called as the carrier frequency. Spectrometer frequency is here, main frequency, but when I shift it, then I bring it to the center of my region or one end of the region or whatever, one end of the region if I bring it, then it is called as the carrier frequency, the amount of which I shift is called as the offset. Then the next one is RF pulse, RF pulse. We said we have to apply 90 degree pulse, so this may be let us say 10 microseconds, so you must control it very well, it has to be exactly 10 microseconds, not 11, not 10.5, so therefore perfect control is required in this RF pulse, pulse width. So, 90 degree pulse means 90 degree pulse and that has to be perfectly controlled. And then, so then I have the third part is the free induction decay, free induction decay which we already saw this, what, how does it look like? So, we said there are two ways, I said if I monitor the Y component of the magnetization, so magnetization has come here after the 90 degree pulse, this is, let us say this is Y, this is after 90 degree pulse. Now depending upon where I put the detector, if I put the detector along the Y axis or along the X axis, suppose I have one detector, I put it along the Y axis, so then my FID will look like this. So, this is detector along Y axis. Now, if I put the detector along the X axis, so where it will start? Because at X axis the time t is equal to 0, X magnetization is 0. So, as it starts rotating here, then if it starts, X magnet starts building, therefore it is like a sine wave, this one is like a cosine wave, this is M0 cosine omega i t e to the minus t by t to i and this one will start from 0 and then it will go like this. This is detector along X axis and this will be, this signal will be is equal to M0 sine omega i t e to the minus t by t to i. This is for the individual one particular frequency, it is similarly for the other frequencies as well. So, what is the consequence of this? So, if I were to Fourier transform this, if a Fourier transform an FID like this, this f t will produce me a signal which is like this. And if a Fourier transform an FID which is like this, it will produce me a signal like this. So, this is my absorptive signal and this is a dispersive signal. So, correspondingly you can also have the negative signs of these ones also. In other words, if my FID to show you one, if I collect an FID which is like this, starts from the minus value, then I will get a signal which has a negative sign. So, this is absorptive negative and what is dispersive negative? So, let me also draw that one here. Dispersive negative will be, it will go like this. So, this will produce if I draw like that and this will be, this is dispersive negative. So, depending upon how we actually collect your signal, you will have different kinds of line shapes in your spectrum. These are the important consequences as we will see in the practical aspects later. So, this is with regard to the FID and then the signal and now let us look at the detection system. Single channel versus single channel versus there are two kinds of detection systems and that is called as single channel versus quadrature detection. So, in all of us, let me see the single channel. So, this is my spectral range. I bring my carrier here. This is my carrier. This is the spectral range here. This whole region is my, this is the spectral range. So, therefore, with respect to the carrier, all frequencies are, let us say, this is positive. All frequencies are positive. Suppose I do a experiment where this is again my same spectral width here. If I bring my carrier here, what that means here? I have positive frequencies here and negative frequencies here. So, we have both positive and negative frequencies. So, when we have this sort of a situation, all frequencies are positive, we need to have one detector. So, one detector because all frequencies have the same sign. So, there will be no difficulty. There will only one kinds of signals which are present there. Now, this is called as single channel because there is one detector. This is called as single channel. Now, here if I have to discriminate between positive and negative frequencies because it is necessary to discriminate between positive and negative frequencies. So, how do we do that? So, if you have to do this, I will have to discriminate positive and negative frequencies. I need to collect both X and Y components. How is, why is it so? So, let me draw that here. What is the meaning of positive and negative components? So, positive components meaning they go in this direction and negative components mean they come in this direction. So, therefore, so this one, this will go like this, this will go like this. So, these are positive and negative frequencies. So, if I want to discriminate these two, I have to collect both the Y component and the X component. Let me draw here the X and Y. So, if I collect both the X and the Y components, then I know exactly whether it is a positive frequency or a negative frequencies. At the same time, I should collect both the components at the same time. So, if I have to do that, I will need, so this will need two detectors. And therefore, this is called as quadrature detection. Both the detectors have to be identical and they will collect the signal at the same time. The FID as it is growing, you collect both the X and the Y components simultaneously, you process them and then you will be able to see the single frequency as it happens. So, that is the positive and negative components. We will be able to distinguish between those two. The next parameter which is signal digitization and sampling. So, this is because data is collected in a digital manner. So, in other words, if my FID is like this, what I will be doing? I will be collecting data here, here, here, here, here and so on. So, digitally it is collected with time intervals in between. So, this is called as a signal digitization. Times between two consecutive points has to be the same and this time, if I went to represent like this here, let me call this as dwell time. So, this is the time between two consecutive points. Now, of course, we ask this question, how do we know this? What should be the time between two points? I have the FID. FID has various frequencies. What should be the time in between? So, this is where a particular theorem called as NICS Theorem comes into the picture, NICS Theorem. That is, it says you need at least two points per wave to represent the frequency correctly. This determines what is called as the sampling. Sampling meaning how you collect the data individually and that is called as sampling. So, the NICS Theorem tells me how many data points I should collect. So, if I have a larger number of frequencies in my spectrum, so I have my carrier here, let us say and I have this largest frequency here, omega max. This is the largest frequency. I have various other frequencies in between here. So, if I adjust my sampling so that I have at least two points for the W max. So, this largest frequency meaning what? Let us let me draw the largest frequency here. If this is my largest frequency, how will the frequency is smaller than that will go? So, let me draw that also here. So, a slower frequency will go like this and even slower one will be even slower than that. Therefore, if I have at least two data points to represent the largest frequency, automatically there will be more data points for the lowest frequency. If DW is adjusted for the largest frequency omega max, then all others will be represented. So, this is the NICS Theorem and therefore, this is the signal digitization and the sampling. So, these are all very specific to FTNMR as you can see. So, now a consequence of this will be suppose you make certain mistakes that you do not choose this dwell time properly and then you are not covering all the frequencies correctly. So, what will be the consequence of this? So, this leads to what is called as folding, folding of signals. And that is the following. Suppose I choose a carrier at this point and my spectral width up to here, I choose a spectral width because I choose a dwell time according to that. So, this is my spectral width, this is what I assumed it as the largest search number of frequencies because up here you do not know, if you do not know what is the range of your frequencies, you arbitrarily choose some numbers that okay, my frequencies are supposed to be present within this area. Suppose I have a signal, you chose your dwell time according to this. If you chose that, then suppose I have a frequency which is present here, you did not realize it. You did not realize it, there is a frequency which is outside the region which you have chosen. So, what will be the consequence of this? If you are doing single channel detection, then in your spectrum, this was the region which you chose. And this last fellow which you have missed it, that will appear inside here with some distorted phase. It will appear with a distorted phase meaning which will be neither purely absorptive nor purely dispersive, it will have some sort of a distortion there. So, this is a folded signal, this is called as folding. On the other hand, if you are doing a quadrature detection, now where is my carrier? The spectral range is this much, right? Let me draw the spectral range same as here. But now my carrier is in the middle, my carrier is in the middle that I can detect both positive and negative frequencies, right? I can discriminate between positive and negative frequencies. And in this situation, if I have made a mistake in choosing the offset and I have a signal which is present here, then where will this appear? This will also fold, this is quadrature, then it will appear, this was present at this point. So, which was outside here in this area and now it will appear on the other side here. So, this is the effect of quadrature detection and this is the folded signal. And therefore, this is an important factor one has to take care. How do we know this? That when you detect a folded signal, so what one should do? So, what one should do is to, by trial and error, choose a larger spectral width in the beginning, optimize later, okay? The next point, so what is signal averaging? Let me draw that schematically here. So, you have the first RF pulse, you collected FID, wait for the time certain time, again RF pulse, collect FID, wait for the time, third RF pulse, collect FID. So, this is FID 1, FID 2, FID 3 and you are going to take the sum, sum of all, okay? Let us say the time between these two is TP, time between two pulses is the same everywhere. So, this has to be the time, should be the optimum time and let us say this is the, I have the flip angle beta here, this is also beta, this is not necessarily always be 90 degree, I have to keep the same, okay? So, when I apply in the pulse, what happens? I collect the FID, alright? That is the decay of the transverse magnetization. But the longitudinal magnetization may not have recovered completely because this is dependent on the T1, T1 the spin lattice relaxation time. The longitudinal magnetization may not have recovered completely. So, therefore, when I apply the next pulse, the starting point for the next pulse is not the same as the starting point for the first pulse. Similarly, for the third pulse, it may not be the same as the second and so on and so forth. But we would like it to have the same, every time it should be the same. Therefore, what we should do? We should have a steady state at steady state, it should be the same for every one of those, after that you start collecting the signal. I mean, you collect the signal nevertheless, so you have to reach a steady state as early as possible, so that every time it is there the same. So, therefore, this requires optimization of the flip angle beta and Tp. So, let me write here the expression for that, your x component of the magnetization after the pulse at the after the steady state is reached and the steady state is reached is equal to M naught sin beta 1 minus E 1 divided by 1 minus E 1 cosine beta. And what is E 1? E 1 is equal to E to the minus Tp by T 1. So, you get maximum amplitude will be obtained for a flip angle which is it is called as beta optimum, for a beta optimum. And what is beta optimum? We can see this will be given by which is given by cosine beta optimum is equal to E 1. And E 1 is dependent on the Tp and the T 1. So, if you optimize your beta such that cosine beta optimum is equal to E 1, then you get the maximum Mx plus. So, maximum Mx plus means you will get the maximum signal to noise. Beta optimum for Tp by T 1. So, notice that when I plot this curve it is for everyone every point on this curve you have the maximum signal, every point here you will get the maximum signal. So, therefore if my T 1 is very large compared to Tp what I should do? Now, this one is 90 degree this is a 90 degree. So, all other ones are lower than that. So, if the Tp is 5 times T 1 let us say this is 1, 2, 3, 4, 5 and so on so forth 1, 2, 3, 4, 5 etcetera Tp by T 1. If T 1 is very large then I will have to use a very large Tp. Suppose T 1 is 5 seconds then I will have to use a Tp of 25 seconds which is a very large time. Therefore, the advantage of Fourier transform number will be last when if you are doing that. But if T 1 is very short then of course you can use if T 1 is 0.1 second that I can easily use 0.5 seconds for the Tp therefore there is no difficulty for that. Therefore, if T 1 is large then best signal to noise per unit time is obtained for small flip angles. This is typically so for carbon thirties. So, carbon thirties relaxation times are very long ok. So, carbonyl for example has a very long relaxation time because it has no proton attached to it and therefore carbonyl relaxation times are very long. Similarly, whenever there is no proton attached then you will have or quaternary carbons they have a very long relaxation times. So, these are for example these are carbonyls or quaternary carbons they are very long relaxation times. So, in such a situation what one should do we use a very small flip angle. See the one more point we will have to consider here and that is the digital filtration data processing aspects. So, the next is 7 data processing, data processing is the Fourier transformation basically. But typically when you do the Fourier transformation what we observe suppose I have an FID which is going like this and I do not wait for the entire FID to decay I do not wait until it goes down to 0 ok. Suppose I truncate the FID here I stop here because I do not know what are the T2 values. Therefore, I do not know where to stop typically I collect let us say 1024 points or 2048 points and things like that. So, depending upon the number of data points what I have here n points and the acquisition time will be acquisition time will be will be equal to n times tau, tau is my dual time ok. Let me write it as dual time n times tau ok. Now if I do a Fourier transformation of this of such an FID there will be distortions at the baseline. Suppose normal signal has to be like this then I will get distortions here and here. So, these are distortions due to truncation of the FID. So, therefore what we do here to eliminate this problems. So, we do what is called as digital filtering digital filtering. So, what is the meaning of this? So, I will do the same thing again here. So, then I have an FID which is going like that ok suppose I truncated the FID at this point. So, what I will do is I will multiply this by a function let us say which goes like this and brings it down to 0 exactly at that point multiply by a function which goes to 0 at the last data point. So, this results in elimination of those wiggles in your in the spectrum ok. So, and then you will get a clean absorb your line shapes when we do this then if you do after that if you do a truncation through FD then you get clean signals like this with no truncations here. So, this is digital filtering there are many functions which one can use you can use exponential functions the functions can be cosine functions cosine functions or shifted sine functions we will not discuss all of these things, but depending upon the optimization one can choose different kinds of functions depending upon the number of data points you have. So, how many data points you will have here accordingly one has to choose because of course you have the data points here like that ok. So, the last data point is here you collected data point. So, it has to be 0 at the last data point you can use cosine function there is also what is called as Lorentz Gauss functions. Now, this we need not discuss in greater detail, but this can be these are routinely available on the spectrometers. So, one can optimize which function to use and how much to use ok. Then of course there is one another last one which I will mention that is what is called as a zero filling zero filling this is. So, you have a FID you are collected up till this point all right ok. Now, what you do is to increase the number of points number of points in your spectrum. So, that the each signal is represented properly you fill in zeros I collected n more add n more points here. So, therefore total number of points becomes 2n therefore in your spectrum which is also in a digital form your spectrum which will be looking like this right. So, you get a better resolution in the spectrum and of course when we do this the filtering functions had to be adjusted filtering functions had to be adjusted remove the Wiegels ok. So, I think we will stop here and continue with the next the next class.