So, let us continue the discussion with Fourier transform NMR spectroscopy which we started in the last class. While we do so, it is a good idea to have a recap on what was done last time in a brief manner. So, therefore I have here a general introduction and general coverage of what we did last time, slow passage versus pulsar acceleration. This is an integral element of the Fourier transform NMR spectroscopy and this was the conventional method which was used before the discovery of the Fourier transform NMR. To give a recap of what this means, let us just have a look at the NMR spectrum once more. You have an NMR spectrum which looks generally like this with lots of signals at various places in the NMR spectrum. Now we said in the slow passage experiment, so one has to satisfy the resonance condition for each one of these and so therefore when the resonance condition is satisfied for this peak you will see absorption signal there, when it is satisfied this peak you will see the absorption signal there and so on and so forth. And we said this has to go on slowly because the system has to follow the magnetic field. When you change the magnetic field to satisfy resonance condition one by one the system has to be in equilibrium all the time therefore this passage has to be slow and we do a sequential excitation of the individual spins. Sequential excitation of spins is what is done in the slow passage experiment whereas in the pulsar excitation we apply an RF pulse, the RF is applied for a short period of time and the response of this is generation of a large number of frequencies and therefore we have a simultaneous excitation of all the spins by the RF pulse. The RF pulse is applied for a short time tau and then we also saw what is the response of that which is the kind of a sync function and we have a certain bandwidth which is excited by the RF pulse. Now this is a slow and time consuming process this can take minutes to hours whereas this is a rapid excitation the pulse is applied for a few microseconds and we detect the signal as a free induction decay so and therefore this is a much faster way of recording the NMR spectrum. As a result of this we can do rapid signal averaging here and we will see more of that in the present today's class. And this slow experiment in the slow passage puts lots of constraint on spectrometer stability and thus limits the practical spectral weights because if we have to scan through a 10 ppm range which is about 1000 hertz in the case of proton for 100 megahertz spectrometer we said it will take 15 to 16 hours and then if we have to signal average with a large number of scans then it can take 1500, 1600 minutes that puts a heavy constraint on the spectrometer stability it has to be stable producing the same kind of a magnetic field and there should be no fluctuations in the temperature around and things like that therefore this puts a serious limits on the practical numbers for the spectral weights whereas here spectrometer stability is not an issue because you are collecting the data in about a few milliseconds spectral bandwidths are determined by the pulse weights you remember we had the pulse weights which is typically of order of 5000 to 6000 hertz and that can be readily excited and we when we applied a pulse like this which is here we have the RF going on a short period and the response of this was something like this. So then we had this is the main frequency omega naught of the RF and which is this is the area which is quite substantial and this one is roughly approximately equal to 1 by tau. So if the tau is the length of the pulse if that is of the order of 1 microsecond then here you have almost 1 megahertz here this range is 1 megahertz this range is 1 megahertz therefore a few kilohertz here will have the uniform excitation efficiency and therefore we are able to select these with the application of the RF pulse. So therefore excitation of all the spins which happens a few microseconds and we can collect the data quite rapidly and in this case the spins are in equilibrium as I mentioned throughout the process of sweep it follows the magnetic field and that is the reason why we had to do it slowly. In this case magnetization is rotated into the transverse plane flip angle notice that magnetization which is otherwise on the Z axis when you apply the RF pulse it gets tilted into the XY plane therefore we said we can apply a 90 degree pulse or a 180 pulse or a 270 pulse we described all of these things and 90 degree pulse puts the magnetization in the transverse plane so also 270 degree pulse but we can apply any kind of flip angle it may not be a 90 degree it can be 20, 30, 40 whatever depending upon your choice and that discussion we can do later. And then the signal is detected in the absence of the RF and therefore and it is a it is called as a free induction decay because it is free precession in the absence of any perturbation the signal is induced in the detector is considered a voltage because the precessing magnetization induced is the voltage in your detector and that is your signal and the signal decays because of the relaxation behavior of the spins transverse magnetization decays and therefore this whole signal is called as free induction decay. Signal detection in the slow passage happens in the frequency domain directly whereas in the pulse excitation signal detection is in the time domain and frequency decoding is achieved by Fourier transformation. We said all these signals are excited at the same time and the different magnetization components precess in the transverse plane as the precess they all individually induce signal in the detectors therefore what we will observe with the superposition of all of these signals components from the various precessing magnetization therefore this can be decoded by doing the process of Fourier transformation. So a consequence of this is the following excitation and detection are separated in time in pulse excitation whereas here excitation and detection happen together. What I mean excitation and detection are separated in time we apply an RF and following the RF we detect the signal as an FID. So therefore signal averaging is much better here compared to this here for one scan it takes about 15 minutes for here one scan it takes about few milliseconds few hundreds of milliseconds therefore as a result of signal averaging you have better signal to noise ratio per unit time in this case as compared to what it is in the slow passage. This was actually a major breakthrough and that led to the application of NMR to various kinds of systems. So here one required very high concentrations of materials, some molar concentrations whereas here one could work with low concentrations of materials submillimolar materials you may be often be limited by the solubility of the molecules in your solution therefore you can work with small concentrations of materials you can do better signal averaging and get higher signal to noise ratio. In the case of nuclei such as carbon 13 and nitrogen 15 and many others which have very low natural abundance it would be almost impossible to detect signals by the slow passage experiments because this signal to noise will be very very low remember C13 is 1.1 percent and nitrogen 15 is 0.37 percent to collect signals from such kind of low abundant nuclear slow passage experiment would be almost impossible to perform whereas here you can do it because you signal averaging several times you can co-add the FIDs and Fourier transform at the end. So these are the important differences between the pulse excitation and Fourier transform and MR and compared to the slow passage experiments. So now we will go forward from here on and that is to make some formal definitions of the Fourier transform. Some theorems on Fourier transform is what we are going to consider now. First of all the definition the Fourier transform definition this is a mathematical operation. So time domain function is if F of t is given by 1 by 2 pi minus integral minus infinity to infinity of a frequency domain spectrum F omega e to the i omega t d omega. So this is the definition and this actually represents our FID. FID is superposition of all the frequencies processing in the transverse plane and that is mathematically represented in this manner. Conversely you can write the frequency domain spectrum as the Fourier transform of the FID this is F of t and 1 by 2 pi integral minus infinity to infinity F of t e to the minus i omega t dt. So this is the operation one performs we consider the signal we collect the signal F of t as FID and then do a Fourier transformation in this manner and we get in the frequency domain spectrum. Now what are the properties of these functions and these will follow from these definitions without going to the actual detail calculations here we can list some of these properties. Now you can see if F of t is even that is if the function is cosine omega t cosine omega t is even function what does that mean if you change t to minus t it still has the same value cosine omega t then F of omega is also even that is if F of t is equal to F of minus t then your frequency domain spectrum F of omega will also be equal to F of minus omega. So if you have an FID which is going like this this is time t is equal to 0 and on this side it is going like this this is time on the minus side this is time t positive if t of F of t is equal to F of minus t that means if I take a particular value of t here and a similar value of t if it is symmetrically written here like this and if these values are the same then it would mean F of t is equal to F of minus t that is the evenness. So if I have a cosine function this is what I get this is the cosine function is symmetrical with respect to t is equal to 0 therefore if you have t F of t is equal to F of minus t then your frequency domain spectrum draw the frequency domain spectrum this is omega is equal to 0 you have the positive frequencies here and you have the other frequencies here if this is omega this is minus omega and these two are equal this is what we said if F of omega is equal to F of minus omega the intensity of the signal is the same and it has the same sign. So that is what we get when the from the first theorem similarly if F of t is odd that is when we consider a sign function for example if F of t is equal to sin omega t then F of minus t is equal to F of is minus F of t if suppose this were sin omega t sin omega t if I change t to minus t then it becomes minus sin omega t and that is minus F of t in that case your frequency domain spectrum F of minus omega will be minus F of omega. So what was positive in the on both sides earlier then on one side it will be positive and the other side it will be negative. So this half of the spectrum if it is positive like this the other side of the spectrum will be negative like this that is minus F of omega. So that is the that is what you get if F of t is odd how do we get such kind of situations this depends upon what we actually collect if you collect only one of these components mx or my depending upon what you collect this can be collected as a cosine function or a sin function and depending on that you will have this sort of behavior. Now there are other properties which will come as a result of the definition of Fourier transform itself suppose F of t is e1 remember in the earlier case we had these limits going from minus infinity to infinity. If it is e1 if the function is e1 you also notice that F of t is equal to F of minus t therefore I can simplify this as going the time domain going from 0 to infinity or the frequency domain also going from 0 to infinity. I eliminate the negative components in the time domain or in the frequency domain. So when I do this that definition will simplify to this particular definition F of t will now become equal to 1 by pi 0 to infinity integral fc omega cosine omega t d omega where fc omega is defined here as is equal to 2 0 to infinity f of t cosine omega t dt. Now this function fc omega is called as the cosine transform. Now it will be easy to prove that F of omega as a whole is equal to 2 times fc omega you simply have to rewrite your basic definition in this and then you can it will be easy to show that f of omega is equal to 2 times fc omega. Likewise if f of t is odd f of t is odd that means f of t f of minus t is equal to minus f of t and that happens when you have f of t as a sine function for example here as a considered here as a sine function in this situation f of t can be written as 1 by pi 0 to infinity fs omega sine omega t d omega these things will simply follow from the basic definition which I gave in the very first slide and here fs omega is equal to 2 times integral 0 to infinity f of t sine omega t dt. Now this function is called as the sine transform and here one can also prove then following the main definition that f of omega is equal to minus 2i fs omega. The next theorem we would like to consider which will follow from again from the definitions. So this one can work it out explicitly suppose I have two functions f of t and g of t they may represent for example two different FIDs on the same sample once you collect it and then you collect it once more you essentially like signal averaging. So you collect two times the function f of t and then you can add or subtract g of t that is the both your two FIDs you added here then you take the Fourier transform of this total time domain function what you get here this is equivalent to taking the Fourier transform of this function plus or minus the Fourier transform of the second function. So therefore it is the sum or the difference of the Fourier transforms of the individual function. Now f omega is the Fourier transform of f of t and g omega is the Fourier transform of g of t therefore if I add these two time domain function the Fourier transform of the whole thing is equal to the sum or the difference of the Fourier transform of the individual functions and here notice that f plus represents the Fourier transform operator that is going from time domain to frequency domain this operates in the time domain function and the result what we get is the frequency domain function. This theorem is important implications for signal averaging you see the question is when you want to add the signals and the spectra do you add them in the frequency domain or in the time domain. If you were to do it in the frequency domain every time you will have to do Fourier transform store it and then you add it. Now which is a more time consuming process the Fourier transformation takes several seconds whereas the data acquisition takes few hundred milliseconds. If you have to Fourier transform every time with the FID and then add in the frequency domain it will be very very time consuming of course not as time consuming a slow pace but nonetheless it will take much more time than if you simply add in the time domain itself you add the various FIDs one after the other add them hundred times each one of them is few hundred milliseconds you collect all the FIDs together and add them and then you do Fourier transformation once at the end. So that means it will be a great saving in the time and therefore the signal to noise per unit time will be further enhanced as a consequence of this therefore this is an important theorem from the point of view of signal averaging. Now if you modify the time variable for example you have a function f of t earlier what we considered we are taking a Fourier transformation now instead of t I multiply this by a constant a what will be its consequence in the frequency domain if I do this I multiply the time domain variable by constant here then I ask when you do Fourier transformation what will be this consequence you can see here it will become 1 divided by 2 pi times modulus of a into f of omega by a you can easily verify using this equation the properties which we described with respect to the evenness of the time domain function or the evenness of the frequency domain function suppose I put a is equal to minus 1 here then f of minus t what will happen here f of minus t and this is the modulus here therefore it does not matter and here it is omega by a is minus 1 this becomes equal to minus 1 therefore it is equal to the f of f of omega is equal to f of minus omega if f of minus t is the same as f of t then f of omega will be the same as f of minus omega I mean this easily proves the earlier theorem multiplication of f of t and if I were to multiply this f of t by constant here a is a constant then it will amount to multiplication of the whole frequency domain spectrum by the same constant so this is like when you are doing an experiment you enhance the gain this is so called gain so if you multiply the entire function by a constant then your entire frequency domain spectrum will be multiplied by the same constant a delayed acquisition so here is a new property what we have and this notice here I would like say something here this is one one term and delta t this is delta t is one unfortunately when we made this t has come in a different color and delta has come in a different color but that is not what it is supposed to be it is delta t this delta t which is present here is the same one which should be there it is not a product of delta and t it is delta t is one entity so delta t represents the delay in the sort of data acquisition what does that mean I have the FID which is going like this right so this was my time t is equal to 0 suppose I do not start the acquisition from his time t is equal to 0 but I start from somewhere here and this period is delta t I give a delay I give a delay in the data acquisition although I should have collected from here because when I apply the pulse if I apply the pulse immediately after the pulse I should start collecting the data as soon as they remove the RF that is the time t is equal to 0 and then I should start collecting the data from there but however if I wait for some time and insert a delay there that means I do not start acquisition from here but I start acquisition from there so this time is delta t what is the consequence of that the consequence of that is your frequency domain spectrum f of omega is multiplied by this phase factor e to the i omega delta t and notice this depends upon the frequency here and depending upon what delta t is at a different frequency we will have different phase factor therefore we have a frequency dependent introduction of the phase of the frequency spectrum and this of course one has to get rid of it these are the issues which come as a result of the Fourier transform Fourier transformation so therefore they are very specific to Fourier transform NMR and these things we will discuss at a later stage as to how to correct these for various purposes delta t represents the delay in the start of data acquisition as I explained to you the frequency domain spectrum is phase modulated by delayed acquisition now from the definition of the Fourier transform we also will realize that f of 0 that is if time t is equal to 0 what we get here 1 by 2 pi this the definition was minus infinity to infinity f of omega e to the i omega t d omega so if t is equal to 0 that e to the i omega t term vanishes therefore I have this integral f omega d omega now what this integral means this integral means this is the total area under the curve this is the total area under the spectrum the integral of the spectrum is therefore equal to the value of the first point in the f i d the f i d as you remember is is going like this the first point in the f i d is your f of 0 and that is that value is equal to the total integral of the frequency domain spectrum this is important implication since again for various purposes okay now multiplication in the time domain translates into a convolution in the frequency domain this is another mathematical operation called convolution now we looked at what happens if we do addition of 2 f of 2 f i d's with therefore then we said it is the result will be the sum of the two Fourier transforms in this suppose we were to consider a multiplication here f of t multiplied by g of t there is another function this is like modifying the f i d in some manner so in that case what happens is this the 2 individual Fourier transforms we calculate and then you take a convolution of these 2 of these functions the convolution is an operation which is defined in this manner you take f of omega prime into g of omega minus omega prime take the integral over d omega prime so this is often used to simplify your spectra remove some of the artifacts but it also has an implication for actual data collection as we will see in the next theorem and that is called as digitization digitization means that your f i d is an analog signal it is a continuous signal which is coming as in this manner but do we collect the signal in a continuous manner suppose we were to collect the data as a series of Dirac delta functions that in other words we collect the points at certain time intervals then its Fourier transform is also a series of Dirac delta functions 1 by tau holds apart mathematically it is represented in this manner this is delta function delta t minus n tau f t this is sum over all of these represents the f i d so each term here represents one data point so as n increases you get a different data points a summation of all of these essentially is a arranging all of these points at different time intervals and that is your f i d so if I take the Fourier transform of that then from the previous theorem I get the convolution of this Fourier transform of this function and the Fourier transform of this function so that I get a convolution of these two frequency domain spectra and eventually that is represented in this manner 1 by tau summation over f of omega minus n by tau so this is a series of spectra what it implies is there is a series of spectra which are separated by 1 by tau and this has important implications later as we will see for the folding of signals and so on. So this is schematically indicated here for your easy understanding this f i d was like this we are not collecting the f i d in this manner we collected the f i d a series of points and the time between two consecutive points is tau this is called as the dwell time and this also this is also represented as sampling so how you sample your f i d you collect the data here a point here and a point here and a point here and a point here and a point there so this is called as sampling so you collect the data points systematically over the entire f i d and then what you get after the Fourier transform such a discrete data set you will get a spectrum which also is a discrete set of points so you do not have an analog signal which is the which is this curve here but you get points like this which are equally spaced and then you connect those and extrapolate in between to generate your frequency domain spectrum therefore how well you represent your data your spectrum here depends upon how well these points are separated how close they are closer they are the better is the representation of your spectrum therefore all these things becomes important with regard to the spectral resolution in the data sets. So therefore with all of these we now look at the FTNMR spectrometer what are the integral important elements of this of course the magnet and all of those things remain as they were before but we have certain essential improvements or essential elements which are present here because now the whole thing has to be very highly precisely controlled so therefore the new element comes into the picture here we have the transmitter controller as before the shim system the flip frequency lock the sample probe this lies in the magnet and today we can have magnets of a different type we have superconducting magnets which can produce very high magnetic fields which are very high stabilities and then you have the RF transmitter where produces the pulse and we have the this RF frequency is also applied to the receiver here which means that we actually detect the signal in the rotating frame this basic transmitter frequency omega naught is subtracted here from the signal what you are collecting from the sample and therefore it amounts to saying that we are collecting the signal in the rotating frame and then we do do the processing data processing means you collect the FID here therefore you will have to do Fourier transformation and that becomes the data processing all this requires computer why do you need the computer because you need your precise control you have to have the pulse you have pulse programmer here which generates the pulse RF pulse and sends it here the pulse programmer means you apply a pulse which is of the order of microseconds 10 microseconds 10.2 microsecond 1.2 microsecond so that precision is required timing control is required and therefore the time integral element of the spectrometer will be the computer the computer can manage all of this and then finally after data processing you have the spectrum. So, if I were to put summarize this in words you have the computer control is essential precise timing controls of pulses delays because you need to know how much delay you have given because we see the delayed acquisition leads to phase modulation of the spectrum and therefore you need to know how much delay you have put and you should be precise and we need to know exactly how much is that data processing data storage etc. Now if I want to collect the data apply RF pulse with so that I accept all the spins at the same time you remember we have to apply high power transmitter why do we need high power and this is because our gamma H1 which is our effective field has to be much larger than all the frequencies in the rotating frame. So, let me put that back to you here so recall that once more. So, we had the rotating frame frequency omega r is equal to omega i minus omega r and this can be of the order of several kilohertz omega i minus omega naught this is the RF frequency this is the frequency of your precision and we said in the effective field gamma H1 should be much larger than omega r. So, if this is of the order of 5000 hertz that is if you collect a proton spectral range of 10 ppm for 100 megahertz it is 1000 hertz for a 500 megahertz NMR spectrometer this is 5000 hertz. Now, if I have to have the same effective field for the entire set of spins we said that gamma H1 should be much much larger than omega r. So, if this were 5 kilohertz then I should have this one at least 20 kilohertz or 25 kilohertz that kind of a range I must have. So, therefore high power transmitter is required and that is the pulse generator. Then we need a digitizer digitizer means the signal that comes out from the NMR sample is a analog signal but we have to consider points because you are now collecting data at certain intervals as systematically placed time intervals and therefore we need the digitizer. The digitizer converts the analog signal into a digital signal that means the signal will be a set of points as indicated earlier. Then we need to have filters the filters are required to remove unwanted frequencies and that we will we already mentioned that when you have the excitation in the sink pulse we take the selective central RF excitation going like this and we had a small portion if you remember here we were to filter out that portion only and that requires filters and then you have the receiver the receiver collects a signal signal they are finding this is also called as a detector to this we also apply the main RF to bow in the rotating frame. So, this also is a time domain signal which has to be collected at regular intervals of time and this will be of the order of few 100 microseconds at the time points are collected intervals of several 100 microseconds depending upon the spectral width what we have and how much would should be the how is the relation with respect to the spectral width that we will consider in the next classes. So, these are the essential elements which have become important as a result of the Fourier transform process of the spins excitation and data collection. We will stop here and we will continue with the further aspects of Fourier transform NMR and what are the features which are very specific to Fourier transform NMR in the next class. Thank you.