 This lecture is on frequency domain signal analysis like as I told you before in condition based monitoring signal analysis is very very important and that too every machinery has a distinct frequency of operation. So, through the signal which has been obtained through the transducers we need to analyze them to find out the characteristics frequencies of the signal or in other words the signature of this machinery because every machinery has a distinct frequency. So, how do we find out the frequency of that machine in a particular signal which we have acquired. So, in this class we will be going briefly into the mathematics of such signals how through Fourier analysis you know people do determine the frequency components of a signal. For example, you all would recall in your under gadget maths class on Fourier transform the teacher would have given you many waveforms it could have been saw tooth and square wave etcetera triangular and told you find out the a b and c coefficients and then expand this waveform as a series of sums of sines and cosines and in fact, this is the very starting point of frequency domain signal analysis and in this class we will be focusing our attention into such analysis and later on we will see how such analysis can be used to analyze real world mechanical signals. Well, you as you all know the need for frequency analysis is because of the fact that every mechanical component has a characteristic defect a characteristic frequency or frequencies which could be related to the defect in the machinery. So, we will be having a distinct signature out of this machinery this signature is its characteristics frequency. For example, if a machine is rotating at say n r p m its rotational frequency is nothing, but n by 60 hertz. So, if I have a say just a machine may be a soft supported on a bearing and this was rotating at n r p m and if I had put an transducer here. You all can appreciate the fact that in the time domain the signal obtained by such a transducer would somehow look like this and the inverse of its time period is a frequency f is equal to nothing, but n by 60 is equal to 1 by t where t is in seconds. So, this is a simple signal coming out of a machinery. Well, if the signal is of this form I can very easily tell you know the frequency is the inverse of the time period and such a signal having just one frequency is usually known as a pure tone signal, but you will realize later on you will see actual signals coming out of machinery no such signal from an actual machine is of such a single pure tone. In fact, if you are to measure or record the signals out of real machineries this could be of this nature signal out of an actual machine. So, if I asked you the frequency of the signal you all will be lost as to well how do I find out the frequency of such signals because I do not see a distinct time period they are randomly varying. So, this is the problem we have in front of us how do we estimate frequencies of the signal and obviously, this is not a pure tone signal, this is not a single frequency signal there are many frequencies. The problem before us is how do I find out the frequency contents in the signal. So, that is what we are going to look into in this frequency domain signal analysis in machineries and we will see how we can do that. So, because of the fact why again frequency domain signal analysis is important because signature of a machine component is unique when I am talking about say in a machinery ball bearing gear pulley impeller etcetera rotating shaft rotating shaft itself can manifest in many ways as an unbalance as a crack shaft as a misaligned shaft as a loose shaft as a board shaft. So, there are many ways. So, each of these mechanical conditions be it a defective bearing defective gear defective pulley unbalanced impeller impeller with a blade not present. So, they all will have different characteristic frequencies. So, if I got such a signal from a machinery where all these components were there and if I can find out the characteristic frequencies of the defects I can pretty well say well that defect has occurred because I see that frequency in the signature obtained from this machinery. Very simple to know you have the students role list and you ask everybody to sign those who are present they will sign. So, you will know in this class these the students are present similar to that if I know everybody's signature beforehand and if I do a signature analysis and if I see the frequency components showing up I can pretty well say well perhaps this this this defects occur because their characteristic frequencies have come up in the analysis which I have done. Now, let us see how we do such analysis. For example, I have just explained to you before that this is a 10 hertz pure tone because the time taken from 1, 0, 1 minimum to the next minimum is about 0.1 seconds exactly 0.1 seconds. And if I take the inverse of this I will get its frequency 10 hertz pure tone. In an oscilloscope if I get such a signal I can just take the inverse of the time period and then find out the frequency and another signal. But and if I do the spectrum of this pure tone I will get what is known and this is in frequency. So, I will get a peak at 10 hertz. If I notice this this had an amplitude of 10 you know whatever the unit 10 volts or 10 whatever mechanical units. So, this is its amplitude I know its time period. So, in the frequency domain once the analysis has been done I also see a 10 amplitude and also frequency of 10 because this is 50 and if I can divide it by 5 times this comes to 10 hertz. Now, we will take it a little forward in the sense if I have just added two sinusoids it will be very difficult if I asked you what are the two frequencies present in this signal. If some of you may estimate and do a correct estimation, but now imagine today it is 2 sinusoids tomorrow it could be 30 sinusoids 50 sinusoids. If I sum them up together you will have a signal which are just described like the signal or of a real world machinery signal and this is a problem we have in hand. So, a real world signal looks in fact even worse than this it is not even as uniform and simple like this. Those of you have done experiments in the lab must have seen this in the on the oscilloscope or on your analyzer. So, I have just done the analysis of this signal and I will show you this is what in fact it is it is the summation of a sine wave of 10 hertz with an amplitude 10 and another with an amplitude of 5 and a frequency of 50 hertz. So, if I have to write this signal. So, the composite signal x t is nothing but 10 sin 2 pi times 10 t plus 5 sin 2 pi 50 t. So, if I if I just replace this as a sin omega t. So, omega is equal to 2 pi f. So, in this problem we have f in one case f 1 is 10 hertz other case f 2 is 50 hertz a 1 is 10 sorry 10 and a 2 is 5. Just by comparison I have such a signal now I could mathematically add many functions and then get up this signal. But in the real world signal I have signal like this if I can break it up into such components I will know yes this frequency as f 1 f 2 f 3 and that is what we will be using the concept of Fourier analysis to find out the frequency components. We will first see for the case of signals which are well defined mathematically that means they can be described by a mathematical expression or a function. Now, I had shown you the case when one was 10 hertz another was another was 50 hertz. But if I add 2 signals 10 hertz plus a signal having 10.5 hertz that means these frequencies are very close very close. Then what happens when the frequencies are close a phenomena the resultant amplitude looks something like this that is the amplitude certainly increase decrease and this is known as the signals are beating or beat phenomena has occurred where the signals are independent each other. Now, I will give you an example there are ways you know Fourier series is one method of estimating the frequencies of the signal. But there are other methods to find out the frequency of the signals one such method is what is known as by signal heterodyning. I will pose you a problem in the case in a sense suppose in this signal one is 10 hertz and another is 10.5 hertz. I slowly bring by an frequency where I have a provision that I will have a frequency oscillator or a generator wherein I can change the frequencies and I have a meter which is observing the amplitude of this meter of this signal resultant signal. So, when the frequencies of this matches with the 10 hertz this amplitude will is going to be steady is not it. So, then I will say that the frequency of the unknown signal is equal to the frequency of my known signal. Let me explain to you in this method suppose I have a signal s 1 suppose I have a signal s 2 where is the s 2 frequency is known to me and I have a well this is the unknown signal. Now, my problem here is to know the frequency of this unknown signal. I have another signal which is my reference signal whose frequency is known and whose frequency is set by me right. So, if I was to these two frequencies were well apart I will be having certain frequencies here, but once they come close to the unknown signal is close to this frequency what is going to happen? I just told you this beating is going to happen right signal as is going to reduce in amplitude with time. Now, once the unknown signals frequency matches with the known signals frequency this is going to be a steady amplitude. So, this will be a steady thing. I will relate this to you know in the earlier days you know people use transistor radios when they tune the transistor radios and when they change the frequency setting of the station when they came close to the frequency of the station you will hear kind of a noise like waning and wax. And then finally, it will catch on to a a cause one here because it has last on to the because the frequency which you are setting is equal to the frequency of the received signal and then they will match and then you will hear the clear voice, but once you are very close to the station while tuning it manually you will hear this signal which is waning waxing because of the beating. And earlier days you know when you know people were not having Fourier systems where they are not having fast computers they were tuning the frequency of the because one frequency is known to you. So, the unknown frequency if it matches there will be a steady amplitude. And this is what is known as signal heterodyning and I will show you the mass behind this and this is what happens. For example, when I have the signal 2 pi f naught t and it is very close to f naught plus delta f and once this 2 frequencies match delta f becomes 0 it is and once this becomes 0 I will have because delta f is nothing but f 1 minus f naught. So, this will become 0 and whole things will disappear and then I will have a steady amplitude this is known as signal heterodyning. To know the signal of an unknown signal by matching with the signal of a known signal I hope this is clear to all of you. And this is just from the trigonometric functions, but mind you the signals are independent of each other if they are dependent they will modulate and that is something we will discuss later on in the classes, but beating signals because you know the known signal and unknown signal they are independent of each other. I have a set of known signal some unknown signal is coming I am summing it up and then seeing this resulting signal. So, if I just add those signals this is the from trigonometry this is the sin a plus sin b formula has been used here and I will get this relationship. Now, another way of finding out the frequency because I am I am telling you certain tricks which are used without even using Fourier analysis and one such is heterodyning another one is comparing the signals. If signals you know if you do this if the signal is the two signals are the same frequency this is if one is in the x axis other in the y axis I will get a plot of like this. If one is the twice of the other I will get such figures and these are known as Lesageau's figures and this are mostly used in orbit analysis by this we can find out the relative ratios of the frequency of the signals. Now, before I come to Fourier series there is another method by which we can do is set up filters this could be analog in nature in the sense what is a filter? Filter means it will allow signals of certain frequencies to pass through and not pass through. For example, if I do the frequency response in the frequency domain this is how typically the response of a filter looks like and this is known as 0 dB this means logarithm of output by input it could be log 10. When output is equal to input what happens it becomes 1 and log of 1 to the base 10 is 0. So, this is known as 0 decibel and this is the typical frequency response of a filter this is a band pass filter wherein it has a lower limit f of l and this is an upper limit f of c or f of u and this is known as the lower cut off frequency and this is known as the higher cut off frequency. Now, an ideal filter what would happen in an ideal filter they should be very very sharp ok, but this does not happen it takes certain other frequencies because this is known as the filter roll off and this depends on the order of the filter I mean I can have a sharp filter sharp band or a steep depending on the order of the filter we will not discuss this in details, but such filters are available in the market wherein I know the lower frequency and the higher frequency. So, if there is an unknown signal which frequency is not known I will pass it through a filter set wherein I know the f lower and f upper and if I get some signal I know that this signal is devoid of or this signal has only this signals in or I can look into them in also in time domain this is all in time domain this means that the frequency in this is between f upper minus f lower and this is the frequency band width. Here all frequencies are present I do not know what they are in stage one I pass them through a filter in stage two in stage three if I get certain signal I know the frequency of this signal is only between f upper and f lower right no problem all in time domain I need not know what the exact frequencies are well question is now if I want to know what the frequencies are I can make this bandwidth very very small I can put a series of filters with very narrow bandwidths is not it. So, I can put a series of narrow bandwidth filters and if I know each one's bandwidth I will see what are the frequencies which are present or not just by monitoring the amplitude coming out of this frequency of this filter set. So, I can have filter set wherein this bandwidths could be tuned they could be digitally programmed they could be operated through a software they could be set manually. So, there are many ways by which we can do that traditionally when computers were not available people use such analog filters very broadly analog filters were used they would manually tune the frequency of this filter set see the amplitude at the output of the filter set, but nowadays there are computers which we where we can vary in fact this bandwidth could be as low as 0.01 hertz very fine resolution of 0.01 hertz next is 0.01 plus 2 into 2 into 3 into 4 into 5 and then I can have series of such digital filters ok and this could be done very serially one after the other, but this as you will see will be very very time consuming and it cannot be done in real time and there are problems associated with it ok. Nevertheless filters can be used analog filters digital filters to do such analysis. While I am on the subject of filters I should tell you about few other filters and how their responses looks like in the frequency domain. Today we will be using about 4 types of signal of signal of filter of this sort this is a response of the filter another could be another we just talked about and another is or I am sorry this one is the other way in fact please make a correction. So, this is as you will see is known as a low pass that means this allows the low pass frequencies signal to go up and then after certain frequencies they are cut down. Because by using such a low pass filter I can know that no frequencies beyond this cut off frequencies are present in the result signal low pass filter and opposite to this this is known as the high pass filter. That means every frequency beyond the high pass frequency are available are allowed to pass through and this is as you know the band pass and this is it will kill a particular frequency and this is known as a notch filter. Notch filter I will just give an example for example in many of the electrical circuits we have a predominant frequency of 50 hertz. So, if I put a notch filter of 50 hertz it is going to remove that 50 hertz signal from the actual signal. So, this is where a notch filter is used. So, if I have such filters I can also have an estimate of the frequency response of the signals. Now, we will look into the another theory behind frequency domain analysis and that is what is known as this Fourier analysis and I am sure this is a very very important analysis which is used in engineering. In fact, mathematicians say that out of 10 great inventions or discoveries in mathematics Fourier series is one such analysis. Fourier Fourier series has lot of engineering applications be it engineering, be it mechanical, civil, electrical, be it in medicine. Everywhere Fourier analysis is useful in the sense that the most important thing is anything I will say what this you anything can be broken up into its frequency components. Be it signal, be it physical dimensions, be it the surface roughness they can all be broken up into different sums of sines and cosines and that is why this signal is this is very important. So, let us go back to how to determine the frequency spectrum of periodic signals. Now, as you all know what is a periodic signal? Signals which repeat with time and Fourier series says that the signal has to occur from plus infinite to minus infinite and periodic in nature. So, such a signal y t where y is in time domain can be broken up into components as a naught by 2 plus n is equal from a n cosine n omega t plus b n sin n omega t where n goes from 1 to infinite. There is another way of representing this, but we cannot find out the components a n and b n by this integration. I will not go in the details of this, but I will just tell you the fact that y t to determine a n and b n you will see from this equation, y t has to be mathematically known to me and then we have a serious problem here. The real world mathematical signal which I have the real world machinery signal which I measure I do not know it is y y t, I cannot represented it in a mathematical expression. If I was able to do it represented mathematically I could very easily find out its a n and b n by doing this integration and that is a problem. We will see how we can overcome this. So, this capital T is given by 2 pi omega is the period of the signal where omega is the fundamental frequency or known as the first harmonic that the fundamental it is always known as the first harmonic and twice of that is the second harmonic and so on. And this you will find in any mathematics textbook in your third year Fourier series or second year Fourier series and this is the generic form of such an analysis. I will not go into the details of such an analysis here, but I must tell you because of this Fourier you can all make a note of it and because I always find it now should know about the great mathematician who has given us this Fourier series to us John Baptist Joseph Fourier. You must have heard of Fourier law of heat conduction it is the same person. And we will see in this work he showed that any function of a variable may be expanded in a series of sine functions result which is frequently used in mathematics and science today and which we are going to use in this class. So, to begin with if I have such an signal wherein its time period is T and amplitude is A I can pretty well say well this signal has frequency of 1 by T. So, if the signal is pure tone I will get I would be thinking that yes very easily I can construct back the signal well if I was to construct I will only get a sine wave back is not it. If I just take the fundamental, but then I am not getting my original signal my original this is my original square wave, but then if I expand the Fourier series there will be the other terms and other terms are very important. So, if I was to take the definition of this Fourier apply this on this square wave and I was to estimate A n. So, A n comes out to be 0 because the frequency is divided from 0 to T by 2 if I go T by 2 it is this is there is a typo here this is not 1 this is T by 2. So, 0 to 2 by 2 this is A sorry excuse me this is A and T by 2 to T it is minus A the second half. So, this one is A and this one is minus A. So, I have broken up the signal into 2 halves and then similarly if I do this I will land up with an component of this form of A n is 0 and B n is equal to 4 A by n pi. So, square wave can be broken up into different components and thus if I was to represent a square wave in the frequency domain this is the square wave which we originally had. If I did the Fourier series expansion by this expression here I will land up with such an expression wherein this is at the fundamental is at 4 A by pi the second harmonic in this case it is 3 omega because the second one does not exist. So, it has omega 3 omega 5 omega 7 omega and so on. So, a square wave X T can be broken up into 4 A by pi sin omega t plus 4 A by 3 pi sin 3 omega t plus 4 A by 5 pi plus sorry sin 5 omega t and so on. And if I take this term still infinite if I add them up together I will get back my original square wave. So, then with confidence I can say well I have broken up in this Fourier this square wave by Fourier series into its individual frequency components. So, a square wave is just not a sin wave, but a summation of such sin waves omega 3 omega 5 omega 7 omega so on. And I am sure in your maths classes you must have done such exercises for triangular waves you know in a first year circuit the teacher would have given you break up such a wave periodic wave form into such Fourier components. Now you notice interesting thing here I have now put all of them together all the harmonics individually they have been plotted and eventually they have been added up this is the sum of the first three harmonics with an amplitude of A is equal to 1. And you see this dark blue line we have quite not got back the original square wave had I added more terms of course we cannot go up till infinite, but with large number of terms you can you will see that you will get back your original square wave. What this means is this square wave consists of frequencies of omega 3 omega 5 omega 7 omega etcetera. So, if I get such a signal I will know this is its frequency content, but the problem still rises that I do not know y t and that is the something we have to live with it. So, we have to do what is known as a mathematical or a numerical estimation of the signal and try to go from it. So, you will see the limitations of the traditional Fourier analysis difficult to implement numerically for a measure signal because one obviously has no obvious mathematical form and which we short. If I knew y t I would have done this integration very easily easily got back and another limitation of this Fourier analysis is it is limited to periodic signals it cannot handle transient waveforms or random signals. For example, a period signal which just occurs like this I possibly cannot find out its periodic components because it is not periodic and these are the limitations of Fourier series analysis. So, well how do you do go about it there are the Fourier series there is an integral method to do it wherein instead of summations I do an integration from minus infinite to plus infinite and in such a way I can transform a signal from the time domain to the frequency domain by doing this forward transform and we will see how this is done. And the vice versa is also true if I have a signal in the frequency domain I can get back its individual time domain components and this is known as the inverse Fourier transform when we have it in the integration form we call it as integral or in the digital form it will be known as the Fourier transform or digital Fourier transform and this can be all done digitally. But in order to do this Fourier transform I need to have representatives of the signal and the Fourier transform or integral this y t could be anything may be transient could be random could be periodic and the Fourier transform is a complex quantity. And then we always have a Fourier transform pair wherein the signal can be related from time to frequency or frequency to time. So, we have this Fourier series which is for periodic function we have the Fourier integral transform which can be for non periodic random signals. And then if you look back here this integration could be done numerically once I do numerically this integration I need to have an estimate of the signal y t at every numerical steps. So, to digitally implement the Fourier transform I can call it as digitally implement Fourier transform or sometimes it is known as discrete Fourier discrete Fourier transform or d f t I need to have digitally acquired time data then this integration can be done I will. So, the problem is now how do we get this digitally acquired data I have a signal which is occurring like this I want to do is d f t. So, that I can get the frequency spectrum, but then question is how do I do d f t well there are many algorithms to do d f t we will discuss that in the subsequent classes, but most important is I need to digitally acquire the time data and that is known what is known as the computer aided data acquisition data has to be acquired by a computer converted from analog to digital sample data. This is an real analog data I have to digitally sample them so that this digital data y k is can be put in the d f t algorithm to get the Fourier spectrum once such Fourier spectrum is obtained by us we can see how this analysis can be done. So, as a homework I could ask you to try out the Fourier series of find out the Fourier series representation of this signals you can take them as a this is you know t t by 2 and this is going to infinite this is a triangular wave another is you could have t a a positive sided square wave where the negatives are chopped you can find out the Fourier series expansion of such waves this could be a homework for all of you to do and we will either put the results in the website or we will discuss that in the subsequent tutorial classes this lecture. So, in the next class this would conclude this lecture on the frequency domain signal analysis, but we still do not know how to implement this with real world signal, but the problem I have posed to you is we need to acquire this analog data into a digital form once this data is there there are many algorithms which will be putting in to find out the frequency estimate of the signal and we will discuss that in the subsequent classes. Thank you.