 Let us continue our discussion with regard to the practical aspects of Fourier transform NMR. There are so many special features in FTNMR which are not present in the continuous wave or slow passage experiments. So it is important to discuss all of these in some greater detail. We have looked earlier about some of the important aspects of these such as folding of spectra, why does it occur, what are the consequences of digitization of the FID and what are the things one should take care to improve the signal to noise and the resolution in the spectra, how to optimally choose the offset so that you do not have artifacts in the spectra, the folding is avoided. So in the same way let us continue with one other feature which is called as phase correction. You may recall the basic experiment that is when you apply a 90 degree pulse to an initial magnetization which is here, your initial magnetization is here and you apply a 90 degree pulse to this it gets rotated here, if you apply 90 minus x pulse it comes on to the y axis. The magnetization then processes as it recovers back to the z axis. Now if you look at the cosine component of the induced signal because the y magnetization if you detect along the y axis it will produce your cosine omega i t it will produce a time domain signal which is like okay now this is the time domain signal cosine omega i t e to the minus 8, this is the decay of the magnetization due to transverse relaxation. If you look at the sine component of the processing magnetization then it will be sine omega i t e to the minus 8, in other case there is a decay of the signal due to the transverse relaxation characterized by the time constant t2. If I were to Fourier transform this FID which looks like this then it will produce me a pure absorptive signal which has the shape this we have seen before and if I were to Fourier transform this sort of an FID which is the sine function and it will produce a dispersive signal which has the shape like this. This is when your pulses are perfectly applied that the magnetization rotates exactly into the along the y axis. Now what happens if there are some errors instead of applying the pulse exactly along the x of the minus x axis suppose the RF phase is slightly shifted it is not exactly along this but it is somewhere along the axis which is represented here as x prime then what happens to the magnetization when you apply this pulse 90 minus x pulse then this will get rotated not exactly on to the y axis but somewhere here in the transverse plane. So it makes certain angle theta with respect to the y axis. Now what will be the form of the signal if you collect the y component or the x component if you collect let us say the y component of the signal this is in precision magnetization which will induce the time domain signal right time dependent signal. So if I were to collect the y component it will have this form S of t is equal to e to the minus a t cosine omega t plus phi. So this some initial phase is added and that is this angle so this is shifted away from the y axis by this angle theta this will be constant phase which will be added to the processing magnetization. Now this one is if you can expand this function as e to the minus a t and you write cosine omega i t cos theta minus sin omega i t sin theta. So therefore even if you are measuring this y component it has a cosine omega i t component here and a sin omega i t component here. So what will be the consequence when you Fourier transform this there will be a mixture of contribution from the absorptive signal and a dispersive signal. So I write here that as Fourier transformation of the cosine dependent FID will have contributions from both the absorptive component and also the dispersive component because the cosine omega i t leads to absorptive signal, the sin omega i t leads to dispersive signal and these are weighted by cos theta and sin theta and the real part of that will have this sort of a form the real part of the Fourier transformed spectrum will have a cosine theta minus d sin theta what I showed in the previous slide was the real part only. The imaginary part of the same spectrum will be 90 degrees out of phase with respect to this and that leads to again a mixture of the dispersive signal and the absorptive signal but this now is because of the 90 degree phase shift from here this will have d cosine theta plus a sin theta. So if I were to look at the real part which is normally what you do it will have a line shape which is like this, this signal we normally do not observe it is kept in a computer but this is not we do not use this one. However when you have shapes like this you will see that we may require to use this how that we will immediately see because what we want to get is a we want to get a pure a from recording a spectrum of this type. So therefore what we can see from here is that if I were to mix this and this in a suitable manner that is indicated here the pure phase of sort of signal can be recovered by mixing the real and imaginary components. So then let me show you this. So we had the R is equal to a cos theta minus d sin theta and the I was d cos theta plus a sin theta. Now what I do I multiply R by cos theta so I take R cos theta plus i sin theta. So this will give me a cos square theta minus d cos theta sin theta and plus I multiply this by sin theta so this is d cos theta sin theta plus a sin square theta. These two terms will cancel and I have a cos square theta plus a sin square theta this is a cos sin square theta plus a sin square theta this is equal to a. So therefore by suitably mixing in this manner I got the pure absorptive signal this is known as phase correction. So we have the initial spectrum which is not phase corrected because of the error in the way we applied the pulse which looks like this. Now after you do the phase correction we will have a clean signal which is looking like this. So the value of theta is determined by continuously varying it. You do not know how much should be that value of theta but you keep changing that in a continuous manner so that at some place you will find that this is correct and because when you do it you can actually monitor the spectrum also continuously you can monitor in real time how the spectrum is changing as you are changing the value of theta on the computer. And therefore you can easily correct the phase of the spectrum then you got a neatly phased spectrum. There can be another reason why you get distorted spectrum. So now suppose you do not start collecting the data from time t is equal to 0 typically you are supposed to collect the data as soon as your pulse ends and you have to start collecting the data from this red line. However for some reason you do not collect the data from that point but delay it a little bit. Delay it a little bit by quantity which you will call it here is delta and therefore your t is equal to 0 actually starts from here. So if delta is equal to 0 your magnetization is entirely along the y axis all the spins are along the y axis. But suppose you are giving a delay here because we did not start from t is equal to 0 during that time delay the different spins we have processed by the respective frequencies and they would have moved to different extents in the transverse plane. Here you have shown three lines which you have moved to different extents. So this is similar to the previous case except that the different frequencies have different phase shifts. So you see this fellow has a phase shift of this and this one has a phase shift of this much the third one has a phase shift of this much. So therefore all of them have different phases in the to the begin with and this leads to what is called as frequency dependent mixing of the absorptive and dispersive line shapes and this is called as the first order phase error. The previous one was called as zero order phase error because it is the same for all the spins and here it is called first order phase error and this is different for different spins and this is indicated here in this spectrum you can see this spectrum there are so many lines and as you move from here to here the different lines have different phases and this is the first order phase error. Now we have to see how we can correct this. So mathematically I write it like this the signal that we detect is now called S of T plus T naught notice here this T naught is the same as the delta this is the same as delta in the previous slide. So therefore let us not confuse and we keep this T naught here because in the rest of the things also we use the T naught and then we continue this function is now if a Fourier transform this I get E i omega T naught into F of omega. So in the absence of the delay what is F of omega F of omega as you have seen it has a real component R omega plus I i omega this is the imaginary component of the spectrum and this is the real component of the spectrum and it has it can be represented therefore with an amplitude which is the modulus of F of omega with a phase and there is a phase e to the i phi. Now what are these components R omega is modulus of F of omega sin phi and here again you see this phi is the same as this phi let us correct that this is the same as the phi. So therefore and the same thing applies in the next one also and the modulus of F of omega is given by this square root R omega square plus R omega square and the phi is given by the ratio of this imaginary component divided by the real component of the spectrum. This is in the absence of the delay what happens when there is a delay. So now we have a new spectrum which is F dash omega F dash omega is now e to the i omega T naught into F omega. So now to this F omega you add its expansion as we wrote in the previous slide modulus of F omega and e to the phi i phi. So put that together then you have here omega T naught plus phi sorry no there is so this is i omega T naught plus phi. The phi comes from the phase of the F omega and omega T naught is the phase coming from the delay. So and this angle is now given by i dash omega divided by R dash omega. Let us expand that here. So F R dash omega is modulus of F omega into cosine phi plus omega T naught and if I were to expand this here so then I will have here there is there has to be a square bracket here this F omega multiplies the whole thing. Now you recall F omega modulus of F omega into cosine phi was R omega in the absence of the delay and the sin phi into modulus of F omega is the i omega in the absence of the delay. So this is sin omega we write written in this form expanded it will be written in this form. If omega T naught is extremely less than 1 then for the whole range of frequencies because this we can choose T naught you choose your T naught to be much much less. So very very small then it turns out that i omega T naught can be simply expanded into the first order as a linear dependence on T naught. So e to the i omega T naught we simply write as 1 plus i omega T naught into F omega this. So this is the expansion of e to the i omega T naught. So once you do that then you have a little bit simplification because otherwise here you have to do cosine omega T naught which is for every frequency you will have to calculate that but here it is now a linear dependent form and that will be easier to apply across the length of the spectrum. So you write here now with that you write your R dash omega becomes R omega minus omega T naught into i omega i dash omega is i omega minus omega T naught into R omega because that is what multiplication leads to and therefore by making this suitable adjustment now R omega will be R dash omega plus omega T naught into i dash omega divided by 1 plus omega T naught whole square. So therefore we make this simple multiplication here to the observed R dash omega and i dash omega then you have the real part of the spectrum which is pure phase. So now under the condition this is much much less than 1 we can ignore this term in the denominator then it will be simply R dash omega plus omega T naught into i dash omega and this is a very simple linear equation and easy to apply across the whole spectrum. So omega T naught phase constant for all the frequencies can be easily calculated these are called as the phase constants this is what you use to multiply this individual frequencies individual spectra and you will get a pure absorption spectrum from this operation. So this is the first order phase correction. Next issue in Fourier transform NMR is called as dynamic range in F T NMR. What is this all about? Dynamic range is a special feature that limits the range of intensity that can be properly recorded. This is the consequence of the limited ADC resolution. You recall that we have to digitize the spectrum right you have every point has to be digitally stored. So when the this is done by what is called as analog to digital converter or the ADC. So the data is stored in the binary form and we remember the how much signal we can collect will depend upon what is the ADC resolution here. Assume for the sake of understanding that we have a 4 bit ADC. 4 bit ADC meaning there are 4 bits here in the ADC all the data will be converted into this form and assume that we have 1111 because this is all binary data. Binary data means either you have 1 or a 0. So if I have 1111 then this if you add all of these convert this into a decimal form then you have this number is equal to 2 to the power 4 minus 1 and that is equal to 15. This is 2 to the power 0, 2 to the power 1, 2 to the power 2, 2 to the power 3 right. So 2 to the power 3 is 8, 2 to the power 2 is 4. So that is 8 plus 4 12 then this is power 1 is 2, 14 and 2 to the power 0 is 1 and that is 15. So if you have a number which is larger than 15 then it cannot store this in this because there is no place for it. If I have a number 16 it cannot be stored here. Now what is the consequence of this? Now each point in your FID is a sum of all the signals in the spectrum. Suppose you have 10 signals, every signal contributes to every point in the FID and therefore the if all of these signals had to be represented here the total has to be less than 15 then only all of them will get a space they will get a representation here. If there are two signals for example with an intensity ratio which is greater than 15. Suppose there is a large signal one of them is 20 other one is 1 then of course the one which is with 20 will completely fill this ADC and the number 1 will not be represented there at all. Therefore the smaller signal will not be represented in the ADC and then this signal will be lost. Of course one can say that we try to change the receiver again to see the total scale down the entire signal by certain factor but that will indeed will reduce the total intensity of the signal. But if the ratio what we are looking at is the ratio but the entire ratio will the ratio will not change therefore after reducing the signal also if the ratio is greater than 15 that signal will never be represented in this no matter how much signal averaging you do. Typically one has 12 to 16 bit ADCs in modern spectrometers. So if I have a 12 bit ADC the largest number is 2 to the power 12 minus 1 and that is 4095. However sometimes you use one bit to represent the sign because you have positive and negative signs from the FID then you use one bit to indicate the sign then it will be one bit less in which case it becomes 2 to the power I mean 2 to the power 12 minus 1 divided by 2 then you have 2047. Let us consider this situation. Suppose you have a sample a molecule and you prepared a 1 millimolar of that sample in water and water you see is 110 figure out to the proton we are looking at the proton signal water is 110 molar and this one is 1 millimolar. Now what is the ratio here 110 molar divided by 1 millimolar this is 1.1 into 10 to the power 5. See in such a situation since the ratio is so large this can never be represented in this 12 bit ADC or even 16 bit ADC. In such a situation the solute signal will not be registered in the ADC at all. So what we need to do here in this situation we will have to design some techniques for suppression of strong solvent signals. Presence of such a strong signal leads to overroading of the receiver what does that mean if I have an FID which is very strong then ADC is not able to represent this and then the signal will be chopped off here and this will result in various kinds of artifacts near the bottom of the signal near the baseline it will have artifacts and that is the point number 1. Of course after you have scaled it down also this signal will be still very strong and notice here this is your sample signal is here in this case it is represented but there may be several others which are not represented this is a signal in water and this is a very very tiny signal and there are many other signals which are not represented here at all. So it becomes therefore necessary to suppress this water signal so that firstly you have avoid this overloading of the receiver number 1 and then you are able to recover all of these small signals and I will show you an experimental spectrum how does it look. So when you suppress this it will produce a spectrum you will have all these signals coming up here and we will be able to show that in the next class. We will discuss techniques how to suppress the solvent water in various ways there are many ways of doing it and this is the thing which we will take up in the next class solvent suppression. So therefore today we have covered two important artifacts of features not artifacts two special features of the Fourier terms of NMR spectra one thing is the phase correction how do the phase errors occur there are two types of phase errors one is the zero order phase error which happens because of the improper phase of the RF pulse and the second is the first order phase error which happens due to delayed acquisition in the FID and that leads to frequency dependent phase errors and this has to be corrected in a linear pair one has to try and keep this delay in the acquisition to a very small value so that the phasing becomes little bit easier in that case it is linearly dependent on the frequency and then it can be easily corrected and then we discussed about the dynamic range in Fourier terms of NMR and this happens because the limited ADC resolution typically one cannot increase the ADC resolution too much because the larger the number of bits you have in your ADC it will also add to the noise therefore you do not want to increase their digitizer resolution too much because it will come at cost because your signal to noise ratio will be reduced because the digitizer will introduce its own noise so therefore one does not go beyond 16 bit ADC and a better way would be to suppress the solvent signal in your spectrum so that all your signals get appropriately represented by the ADC and they can be signal to noise can be averaged by increase by signal averaging with that we will stop here and next time we will see the methods to suppress the solvent signal thank you