 about modulation and side bands, well by now I understand you must have got a feel of what the different types of signals are whether they are stationary, non-stationary and then how we can distinguish these two just by looking at it in the time domain, but more importantly we can look at them in the frequency domain know their frequency contents and then try to characterize the signal. One thing you will realize when we are going to do FFT, we always assume that the signal is stationary, but you know that the process of FFT takes some time, what would happen if the signal was varying the instance you are doing the FFT. So, if so for example you are doing a number of FFT's you have taken the first block of data, first block and the second block is also being digitized and ready for FFT, but if the you are trying to do and similarly you will be doing an average FFT, you will be doing summation of all the FFT's and then dividing by the number of FFT's, so that you get your averaged FFT. The reason we do this averaging is because to reduce the random noise and this random noise error will actually reduce by this equation where n is the number of averages. This equation comes from the probability theory and we will not go in the details of this, but it is suffices to say that once we are doing the FFT of a stationary signal, if I do increase the number of averages the random noise will be reduced. So, averaging helps us to improve the signal characteristics or signal features while doing an FFT by removing the random noise. Otherwise, let me just give an example here that if you did an FFT of a signal you will perhaps get lot of low frequency high frequency and then and this this this could be because of the random noise present in the signal, this is in frequency, this is the amplitude. So, my objective is to reduce this random noise by having more number of averages of this stationary signal, then I will have a sharp peak and this is the averaged FFT, but this is very realistic, very ideal, but what would happen if the machines machines rotating speed undergoes a change. By change I mean not that you know you are running a machine at 1200 rpm you want to run it at 1600 rpm and so on. No I am not talking about that I am talking about say for example, if you are running at 1200 rpm because of power supply variation because of load variation which could be intermitted on and off. So, this 1200 rpm will then vary by an amount of delta rpm very small that could be 1 or 2 rpm, but what happens if you are taking this data continuously you had assumed it to be stationary. So, you are violating the definition of stationarity. So, what happens one average you know may be I will draw them by different lines. These frequencies once it can be here another time it will be here another time it could be here. So, once you have an average of this you may have a signal something like this and this is not desired this is something. So, I cannot identify this frequency of my machinery because of the this is because of what is known as signal smearing. Signal smearing has occurred because the signals frequency has changed while you are doing the FFT or while you are doing the average. But if it is a nice constant steady state constant speed machinery if such average things are done then we will not have the problem of signals smearing. To avoid this sometimes we do what is known as synchronous time averaging or frequency averaging. Once we do that the special variation is reduced and another method is by having a triggered acquisition. So, some of the features will be prominent once you do such time synchronous time averaging and frequency averaging. But many a times even sometimes just a single acquisition is good enough for example, once you know the signal is fluctuating quite often. So, we should be able to capture the signal just by taking one block of data particularly while we are doing with transients kind of signal this happens only once. So, the next block there may be no signal previous block there could be no signal. So, in such a case may be just one block of data processing is sufficient particularly once we have such transient signals or for non stationary signals. Now, here I would like to draw your attention to signal processing module which we have developed at IIT Kharagpur under the virtual labs program of the Ministry of Human Resource and Development MHRD. So, I will there is a lab on mechanical systems signal processing which you can access through my website sorry www.iitnoise.com it is linked to this website and once you go to the student resources you will click there and I will take you there then because of course on machinery condition and signal processing cannot be just 40 lectures it has to be also done through some bit of labs. Now, we possibly will cannot go to the lab. So, I have tried to bring the lab to this classroom. So, there are few experiments in this portal where you can do the virtual lab at home at a leisure and then try to generate signals try to do their time domain analysis frequency domain analysis I will not take it to the detailed exercises in this class, but I will just show you how to access it. So, what you need to do is you can go to this website on www.iitnoise.com and this is the address. So, once you come to the student resources and virtual laboratory and mechanical systems and signal processing if you click here it will take to the NPT MHRD website on virtual labs and this is the lab for mechanical systems and signal processing. In fact, throughout the course of this course when we are going through the other examples other lectures we will come to this experiments one by one. Right now I can you can access basics of dynamic signals basics of frequency domain signal I mean the first four you could perhaps do right now. I will just show you how to invoke basics of dynamic signals all you do is click there and then there is an introduction of understanding about the signals there is a bit of theory on this topic and then how do you how you measure all the different time domain parameters and then you can go to the different modules. There are three modules by which you can generate the signal one is by user defined waveforms other is by mathematical function and the last one is your actual measured data. So, then in the user defined waveforms you could generate a square wave, sine wave, triangular wave, sawtooth wave you could write your own mathematical expression of further signals you can add signals see the beating phenomenon etcetera or you may have your own real world data which you could perhaps bring into this files. So, the module one the simulation I will just show you the screen shot of this signal what happens here is you know this is the screen shot you will be having. So, you can generate your signal right now it is sawtooth you can increase the change the amplitude given offset to the amplitude change the frequency see the waveform and then this program itself will calculate the mean the maximum the standard deviation of the cutos is crest factor form factor skewness variation. So, you got a feel of the signal feature of the signal just generating yourself as if you do not need a signal generator or an oscilloscope or a computer to do the analysis you go to this website. And next module of course, is you could add your own sorry you can generate your own signal and then you could also see the same effect. And the second experiment is on the responses of first order and second order systems you need not go through this right now for this course, but the most important for the frequency analysis is you can find out in the third experiment basics of frequency domain signal analysis you could be measuring the finding out the amplitude spectrum the real imaginative spectrum power spectrum power spectral density of any type of signal which you want to give. Similarly here it could be user defined functions mathematical function a measured data and then in fact this is how the screen shot of the frequency analyzer looks like again here you generate the signal you can decide on the sampling frequency you can decide on the averaging mode on the type of window and then you can see the amplitude spectrum you can see the real and imaginary spectrum here. So, you could use this at home to understand more about the signal analysis and the link through this is this website of IITNoise.com ok. Now, let us come back to this frequency domain analysis regarding modulation and side bands which I will focus mostly in this course here rather in this lecture here. We had discussed about beats in the last class. Beating occurs when we have well before I come to beats I will tell you something about the harmonics and side bands. For example, I have a frequency spectrum I have certain amplitude. So, I have got a certain frequency the first frequency this is the fundamental. So, this is at a function of say X I am writing this as X and usually people write it as 1 X in the literature of condition monitoring. It may so happen this signal and this is one type of signal there could be a spectrums which are equally spaced and this is an exact multiple of the value here. So, these are known as the harmonics this helps us to know whether these harmonics are multiple of the fundamental. Suppose this value is for example, if this corresponds to 20 hertz. So, if I see a value at a peak at 40, a peak at 60, a peak at 80, a peak at 100 I know that these frequencies are the harmonics. What happens this is a very ideal situation in the real one once you go to the FFT analysis you will see that there will be lot of noise like this buried and these harmonics are perhaps buried in the FFT. So, to our naked eyes sometimes it becomes very difficult to determine what the harmonics are. So, sometimes there are what is known as the options are available which are known as harmonic cursors. So, once you set at the fundamental a cursor which happens to be the harmonic cursor automatically it will calculate because it knows this distance or it knows this value. So, it will calculate and put the harmonic cursors at all the places in the spectrum and then you can very easily identify if there is a peak you know well these are the harmonics and so on. So, this is what is known as the harmonics and harmonics occur because of particularly in unbalance and misalignment you will see lot of such things occur they are all multiple survivors things. Then we have what is known as side bands side bands occur in the frequency spectrum around a particular frequency. For example, I have a peak here I will also notice that there are 2 very prominent peaks almost equally spaced about this is usually known as the carrier frequency and this is the left side band and this is the right side band and this value is the value of the what is known as the modulating frequency. So, the modulating frequency was value was f m and this is f c. So, the value here left side band will be f c minus f m and here it will be f c plus f m. Now, there could be sometimes you know f c minus twice of f m f c plus twice of f m. So, basically there will be a group of side bands around a carrier frequency. It would suffice right now to say particularly this kind of signals occur in speech signals occur in gear boxes. Now, these are very ideally represented signals if you go to the real wall machines again there will be lot of noise and on top of it there will be many carrier frequencies. On top of it the frequencies could be smearing because of the effect of the loads or the speed fluctuations. So, and then once you do an FFT you may not be getting a sharp frequency because of the frequency sparing. So, because our objective is to identify frequencies I mean we should why are we studying signal processing. Our primary objective is to identify frequencies so that we can do fault detection ok. This is our primary objective we should never forget that no matter what amount of signal processing we do it may be a mathematical exercise altogether, but end of the day unless I detect the faults my signal processing is no good ok, but so what are the tools available excuse me excuse me. So, what are the tools available to do this kind of signal processing? We will come to this in the later classes regarding substrum analysis wherein we can identify the side bands etcetera, but it would suffice to say that in a frequency spectrum I will be seeing side bands as well as harmonics or I may see none of them also at times. So, now let us see why and how this side bands are generated in the first place and harmonics are generated in the first place ok. To begin with let us talk about signal beating for example, I have a machine a running at say 1200 rpm that corresponds to 1200 by 60 that is 20 hertz ok. Machine is there it is operating I have another machine B which is for some reason it had a defective supply or something it is running at 21 hertz ok. Of course, I am standing here now because it is running at 20 hertz and this is heavy mechanical machinery it is also going to generate noise at these frequencies. To my ear I am getting 20 hertz and also I am getting 21 hertz ok and then you know if I have two signals say a 1 is equal to sin omega 1 t plus a 2 equal to sin omega 2 t where omega 1 is equal to 2 pi f 1. So, in one case f 1 is equal to 20 hertz in another case f 2 is equal to 21 hertz ok. So, once I see this or hear this signal in fact, if you mathematically sum this signal they are going to look like this of course, I am not being accurate while drawing in my by my hand. This is how it will look in the time domain and if I can draw a waveform this is the envelope it is known as and this is the resultant signal A. So, you see this resultant signal which I hear is no longer constant in amplitude it is changing with frequency and this resultant frequency will be because f f 1 minus f 2 and f 1 plus f 2. So, two independent frequencies because the relationship between machine A and machine B they are totally independent but the net effect produces a resultant signal something like this wherein the frequency then this signal will sound like as if you must have gone to a factory where in two machines are running and you will get this kind of howling noise you know suddenly noise increasing decreasing increasing decreasing and that is because two machines are running side by side at very similar very close by frequencies. So, frequency of this beating is nothing but f 1 minus f 2 and this is a very small quantity and the time period of this beating is the inverse of this. So, in the frequency domain all I will have is two frequencies close by one is may be f 2 one is f 1 the signal amplitudes I will not know whether they are beating unless I see the resultant waveform in the time domain. But I will see two distinct frequencies and I will also see frequencies of f 1 minus f 2 f 1 plus f 2 if I see this I will know the signals are beating another very important characteristics of signal or types of signal with the signal modulation. Now, suppose I have a signal x t is equal to A sin omega t now this A itself in this case is a constant but suppose A was also varying with a frequency is given by A m sin omega m t and this I will write as omega c. So, my x t will be A m sin omega m t sin omega c t. So, product of two sin functions will be a summation you will have components in the resultant as two frequencies one with omega c plus omega m and another with omega c minus omega m. So, the product of two sin waves one is modulating the other like I give an example of a very common example is ceiling fan. The hub rotates at a particular r p m say n r p m the rotation of the hub and then I have say number of blades equal to 3 I will have the blade pass frequency as 3 n by 60. So, you see the rotational speed of the ceiling the blade pass frequency and the rotational frequency they are related one is affecting the other and then we can have modulations to begin with. But, modulations you will see in details when we talk about defects in ball bearings in gears and so on. So, but the fact that x t is equal to A m sin omega m t tan sin omega c t this gives rise to omega c plus minus omega m and this is what is the side bands. If you look at this f f t of this signal because modulation has occurred the resultant wave form will have signals which will have frequencies of omega c plus minus f m omega c plus minus omega m or f c plus minus omega f m and this is known as the carrier frequency and this is known as the modulating frequency. So, what are the features of this distinguishing features of amplitude modulation suppose I have taken an amplitude modulated signal and I have done a fast Fourier transform of such a signal one is I will see the carrier frequency f c I will see the modulating frequency. But, most important is I will see the sum of the carrier and modulating frequencies difference of the carrier and modulating frequencies. In the case of the beating these frequencies may not be visible carrier and modulating frequencies there will be two independent frequencies showing up it could be f 1 and f 2 very close by and usually in amplitude modulation one is pretty high compared to the other and then this sum and difference frequencies plus these two frequencies will be seen when you can know for sure the amplitude modulation has occurred. And once we go to gear but because our objective again is you know how do I find out my carrier frequency and modulated frequencies because once you do the say for example, this is a typical modulating signal I will show you how this becomes modulating because if you look at this signal this in time domain this is signal a. So, this is my high frequency carrier signal. So, the transducer has collected the signal into sending to the receiver or the analyzer and the information is actually this one here this is the low frequency the machines defect or machines features are actually there in the blue signal machines characteristic. If I do so I have to find out a mathematical technique by which I can only obtain this low frequency modulated signal because otherwise what happens once you go to the frequency domain there will be so many frequencies you will be kind of getting lost which one is your modulating frequency which one is your carrier frequency which are the multiples of carrier frequency etcetera there will be side bands. So, there are mathematical techniques which will cover subsequently to how do we demodulate demodulate demodulation or sometimes known as envelope. There are many ways to do demodulation and envelope analysis like using Hilbert transform that is something we will discuss later on and then this can be used to obtain this blue signal envelope of this modulated signal and thus know more about the machinitic. I should again tell you one thing other than this virtual lamps I would also recommend that all of you use the MATLAB signal processing tool box to determine what will be the generate waveform signal feature extraction in time frequency domain. And MATLAB is a software which is freely available in educational institutes as a floating open license students can use them or you can buy them and then use them. But some of the examples which I will be subsequently showing you could be from the our virtual labs website on signal processing and also some examples done on MATLAB. So, in MATLAB if you use the help menu you can know how to do signal amplitude modulation frequency modulation how to feature envelope out find out the side bands etcetera all these functions are available in the MATLAB. Now, another type of signal modulation is what is known as the frequency modulation. In frequency modulation what happens FM I am writing FM A naught sin omega m t plus cosine omega c t. Now, in such a case what happens if I was to draw the signal in time domain the distinguishing features of frequency domain signal in a time domain is the amplitudes are constant. But the frequencies are vary. So, there will be an increase and decrease of the frequency about the same amplitudes because you can see these are the high frequency signals these are the low frequency signal because the time period has changed and then amplitudes are constant. Such signals also we will come across in machinery and then we will see some of the features of this in the signal processing. Now, what are the distinguishing features of the frequency modulated signals carrier frequency at the center of component array you will see lot of side bands and the presence of a multiple side bands with a spacing of f c f c minus f m or f c plus f m some x times f m this is plus and this is again f c plus minus some multiple of f m. And this actually occur in rotating shafts and particularly when we are monitoring torsional vibrations as a function of the angular displacement. So, there will be multiple side bands at carrier frequency minus orders of the modulator and multiple side bands at the carrier frequency plus orders of the modulator and constant signal amplitude when viewed in the time domain which would have seen. But there is no distinct component occurring specifically at the modulating frequencies which was present in the case of amplitude modulated signals. So, you will not see this happening in the case of the frequency domain f m signals. Now, this a m and f m signals have traditionally been used in electronics mostly for radar communication wherein the because of the frequency of the modulator you know the carrier frequencies are usually high frequency signals and high frequency signals are very directive. For example, the radio signal the TV signal they are very very high frequency signal in the guards and very directional. So, if you have an antenna you must have observed that when an antenna particularly with your short wave radios or m radios you have to point the antenna towards the source and because their high frequencies they are very directional. And on this carrier frequency carrier high frequency carrier signals my information signal is this is the information signal it could be a radio station signal TV it could be machinery. These people talk in the order of gigahertz I am talking in the order of only hertz or kilohertz. Our signals are of much much lower bandwidth than the radio signals but nevertheless the physics of data communication is the same laws are the same. But objective again from a machinery condition monitoring point of use how do I get this information whether it is frequency modulated or amplitude modulated is immaterial to me as long as some signal processing algorithm I can find out this information green information signal. And that is what we will discuss in the subsequent classes regarding how do I find out the harmonics how do I find out the side bands most important importantly how do I find out the modulated signals from such composite signals. And mind you once you go to the frequency spectrum of a signal because see my machine is there my machine is an inanimate object. And many things are happening in it simultaneously all I have is put this poor transducer. So everything depends on what my transducer is capturing. So the phenomena of resonance could be occurring the phenomena of modulation could be occurring everything and of course our good friend noise is always present. So everything is happening together in this system and I only capture this by a transducer. And transducer is no matter what you see in an oscilloscope it will always give me signal which is you know I can make nothing out of it. Our starting point in CBM is this I will not get a signal a transducer will never tell this is your modulated signal this is your resonance signal this is your noise signal no everything is there built in summation some effect of all signals are there. So we have to now use our skills mathematical power how do I analyze such signals. First fall is whether it is a stationary signal whether it is a non-stationary signal whether I can find out features in the time domain which are repeating repeatability of the signals or whether it is one of a kind. Then we have to go into the frequency domain see if my FFT I can see the frequencies as sharp peaks or there are no frequencies smearing this is to be avoided. Now on top of it if signals are getting modulated there are presence of lot of side bands and end of the day I will land up with a signal wherein a lot of these components are there. So then we will see how we can analyze the signals to get the different frequency content of the signal. So I think with this I will stop here. Thank you.