 Alright, today let us look at some properties of noise in general, in particular I want to look at a phenomenon called short noise and we would like to study the statistics of this because of the very large number of applications that such a process has. Now to recall a few things to you, we looked at what a Poisson sequence did, Poisson pulse sequence did, namely we said if you have a completely uncorrelated set of epochs or instance of time distributed according to a Poisson distribution, they form a Poisson sequence and then some properties of this Poisson sequence we studied in connection with statistics. But now let us look at a more general version of this and that is the following. So as a function of time, let us suppose that at random instance of time you have an event occurring, a pulse occurring like neuronal spikes from the brain activity or radio signals from outer space or whatever, some kind of random pulses appearing at random instance of time. The only proviso being for the moment that the times at which these pulses occur are completely uncorrelated with each other. So in general you might have a signal that perhaps looks like this, there is a sudden spike and then it decays and then a little later there is another one etc in this fashion. And these pulses occur at random instance of time, let us call it ti, ti plus 1, ti plus 2 and so on and it is assumed that this has been an ongoing process for a long time. And the question one would like to ask is what is the statistics? What kind of statistics does the stochastic process have corresponding to these signals that you detect? And let us suppose that these pulses occur at times t sub i and let me call the shape of each pulse, let me call it f of t minus ti that is the shape of the ith pulse which starts at time ti and then goes on decays could be finite duration could go on forever could exponentially decay etc we do not care. So the pulses the shapes could actually overlap with each other we do not care about that but the starting instances, instance are completely uncorrelated with each other. So the probability so this sequence ti is a Poisson pulse sequence we will assume that each pulse has a shape given by this function t minus ti. So this f of t is not is 0 could be taken to be 0 for t less than ti okay starts at ti and then it goes off in some fashion compact or otherwise typical examples would be for instance if this is an exponentially decaying pulse with time it could be of the form some e to the minus lambda t minus ti multiplied by some height or whatever but it is also multiplied by a step function t minus ti that is one typical shape or it could be something which is a set of pulses it looks like this and so rectangular pulses of some kind. So perhaps it is of the form theta of t minus ti minus theta of t minus ti minus tau if tau is the duration of the pulse this quantity is 0 everywhere except between this and that is given by this difference of theta functions step functions. So we have in mind a signal psi of t this is what is detected and what you have is all these pulses getting detected so this is equal to a summation from i which could be from minus infinity to infinity might have an infinite train of these pulses times f of t minus ti and they could be of different heights we do not care. So let us put a height variable there let us put h sub i and the h sub i is could themselves be random drawn from some independent distribution okay so these are random pulse heights you could generalize it even further and say even the shapes of these pulses could change over the pulses we are assuming here for simplicity that the pulse is exactly the same shape each pulse is of the same shape we have allowed for a little variation by changing its height for instance in the case of these rectangular pulses for example you could say even the duration of the ith pulse could be different from that of the jth pulse so perhaps you have this and so on. So a lot of general possibilities occur and the question is what is the statistics of this Xi this process Xi what is the first order statistics what is the power spectrum what is the correlation function and so on this is what we would like to find out okay. Now if you go back a few steps you realize that the way to get this at this process is to say alright let us first look at simply the Poisson process itself let us just look at something which is a every time there is a ti you add one to it that is it. So that process let me call it u of t equal to a summation from i equal to minus infinity to infinity theta function of t minus ti now what does this process look like well it is just a step every time there is a it crosses one of the ti is it adds one to it and keeps going so it is a staircase function that is increasing so the first thing to do is to find the statistics of this and then we find the statistics of successively of its derivative and so on and so forth. Now what is its derivative look like let us let us call it x so let us call x of t equal to u dot of t and what does this look like this is equal to d over dt of these theta functions over i t minus ti that is equal to summation over i delta of t minus ti so you have a spike of unit strength a delta function of unit strength at every instant ti so what does this look like this function this thing simply looks like this so every time there is a ti there is a spike etc. So this is ti ti plus 1 ti plus 2 and so on you could ask what is the number of pulses up to time t this number would be n of t would be the integral of this from minus infinity up to time t because every time there is a delta function it counts one so this is simply equal to integral minus infinity up to time t dt prime this guy so summation over i delta of t minus ti and that counts precisely this what is the rate at which these pulses are occurring well that is the rate at which the original Poisson process is occurring this pulse sequence so let us give it a name let us say this is a Poisson pulse sequence with mean rate new so the rate at which the pulses are occurring that itself then the number of pulses per unit time is itself a random variable the mean rate of course is given to be new so that is clear from here that dn over dt the derivative is precisely delta of t minus ti this is precisely your x of t because if I differentiate under the integral sign this gives me a 1 and I put t prime equal to t and I get just x of t what is the average value of x of t that is the rate at which the pulses are occurring so the average value is new that is immediately clear right so it is right then down so it says x of t equal to new is x of t a stationary stochastic process is the stationary because remember the original pulse sequence is a Poisson process and new is independent of time that rate does not change with time so the question the way you decide whether a random variable is stationary stochastic process or not is to ask whether x of t and x of t plus constant have exactly the same statistics or not and it is clear if I just add a constant to x of t the statistics does not change at all at any order right so x of t is a stationary random process strictly stationary random process because we have random processes which are strictly stationary which means the statistical properties simply do not change none of the distributions changes when you have an add constant added to the random variable or wide sense stationary which means that the mean value is independent of time and the two point correlation is a function of the time difference alone but higher order correlations might not be so this is strictly stationary this process is strictly stationary here and this quantity x of 0 x of t this guy here can be computed I am going to leave this as an exercise to you and tell you how to do this give you a hint and tell you how to do this you can actually compute the autocorrelation of this process and this turns out to be new times delta function at t plus new square now a word about how you actually compute autocorrelations of these of stationary processes first of all if a process is stationary its derivatives are stationary and so on and so forth now the way to do this is if you give me a call let me just call the random variable Xi for the moment so that so Xi of t has a correlation function let us call this correlation function I use a symbol for it right I use C for it see but you got to be careful there could be a mean value if this Xi does not have a non-zero if it does not have a zero mean but as a non-zero mean then the correct way to define the covariance of this object is delta Xi of 0 delta Xi of t as you know right so let me let me write this down if the average value of Xi equal to mu is not equal to 0 then the correlation a covariance it is called the covariance equal to delta Xi of 0 delta Xi of t which is equal to this quantity Xi of 0 Xi of t minus new squared the average value we know that over and over again we have gone through this and we define the power spectrum as a Fourier transform of this fellow here that is not the same as the this is the correlation function and that is the covariance function so I need a slight change of notation or to be little careful so let us call this C of t and let us call this R of t that is the standard notation in books on random processes so C of t and R of t are slightly different objects they just differ by this guy here now the way to do this is the following if you have a stationary process then R of Xi well I should let us be even more careful this is R Xi Xi of t that means it is the correlation of Xi at time 0 with Xi at time t then the first step is Xi Xi dot so the correlation of Xi of 0 with Xi dot of t this guy here is equal to minus d over d t R Xi Xi and R Xi dot Xi dot minus d 2 over d t 2 R Xi Xi so you got to differentiate each time which is why when you do the power spectrum you multiply by omega by i omega each day so let us not get into this thing here but let me just point out that to find this result for the delta function sum sum of delta functions you start with the theta functions and find its auto correlation and that is easier to do and once you do that twice differentiating it will give you this okay so I am going to leave this to you as an exercise you must remember that the derivative of a delta of a theta function is a delta function you need to remember that that is where this comes will turn out to come from okay. So whatever let us take this result it is easy enough to establish so let us take this as our basic result and then what do we add want to add on to this you see now I am going to simply motivate it heuristically we can go through the derivations but it is not particularly enlightening the result is very clear we want the auto correlation we want of Xi of 0 Xi of t this guy here what this implies but before that let us write the power spectrum of this fellow out so this is S X again let us use proper notation both are X this guy here is a Fourier transform it is simply 1 over 2 pi integral minus infinity to infinity dt e to the i omega t of this so this fellow will give me nu over 2 pi plus what does this give me yeah it gives me a delta function of omega right because it is just 1 over 2 pi e to the i omega t minus infinity to infinity and that is a delta function that is the power spectrum we have the auto correlation now the question is what is the auto correlation of Xi given that of X okay and one can sort of guess what it is going to be in fact let us guess what the let us guess what the average value of Xi of t is going to be I defines i of t in this fashion and we want to know what is its average going to be this is an independent random variable so this average is going to come out when you take averages and then you need to take the average over the random variable ti so it is clear that somewhere along the line you are going to get an integral of this because we want the average value of this f but now we appeal to a godicity namely you take the time average of f and that is going to be the ensemble average okay so there is definitely going to be and then it is multiplied by the rate at which these fellows have happened because the average for the process X of t is itself nu so there is definitely going to be a nu and then there is going to be an average of H in this multiplied by minus infinity to infinity dt f of t now of course remember that f of t might be 0 up to t equal to some given value some say 0 and then it drops down in this fashion so this integral is formal it is not going to run up to minus infinity because a pulse starts at some point and I said you could define f to be 0 for negative values of its argument and that is it that is the average value of xi okay what is the variance of xi going to be this is xi squared and it is squared and this is equal to and now this is a non-trivial statement it is definitely going to involve the rate nu because it is all controlled by this guy by this process and then we want the square of this fellow here and then you want this function squared inside okay so this is equal to and this is a non-trivial result it is nu times H squared times integral minus infinity to infinity dt f of t and it squared inside okay this requires proving this is not a state trivial statement and together these statements are called a Campbell's but physically it is very clear where these terms come from and so on in particular the fact that nu appears here multiplicatively outside in all of them in fact there is an extension of Campbell's theorem to the case when this function f itself has to be ensemble diverged the shape itself might be different and so on and as a further there is another extension of this even in this case to the higher cumulants it turns out that if KN is the nth cumulant of Xi then KN turns out to be nu times H to the n inside there and then minus infinity to infinity dt f of t to the power n it is quite a remarkable thing that that is all it is you can write down the full statistics of this I we still not come to the point of what is its correlation it is a stationary process so all these averages are time independent explicitly they are all single time averages this is independent of time so we have complete first order statistics here by Campbell's theorem and its extension on the other hand the fact that you know what this fellow is what the power spectrum is for X also tells you what the power spectrum is for Xi itself and this implies that the power spectrum of Xi and let me write it as Xi Xi of Omega this is equal to now depends on the Fourier transform conventions but let me be careful here this is there is a 2 f tilde of Omega mod square you would expect that because this f is going to appear everywhere and it is got to be quadratic because it is a power spectrum going to be 2 of these fellows size multiplying each other times 2 nu average value of H that is what this is going to give me and it comes from this term plus 4 pi nu squared sorry this is average value of H squared and then this is H squared delta of Omega that comes from here and this goes by another theorem which is familiar to electrical engineers what is this called well the one for the mean and the variance are called Campbell's theorem and this one is called Carson's this term as you can explicitly see arises there is a weight here at Delta there is a Delta function at Omega equal to 0 there is a DC contribution coming entirely from the fact that the average value of H may not be 0 that is 0 this is gone by the way what is the power spectrum for the process which is again controlled by a pulse process pass on pulse process but where the variable takes just 2 values say plus 1 or minus 1 or plus C and minus C the dichotomous Markov process what is the power spectrum going to be for that. So in that case you have a process I of t equal to plus or minus C and it goes like this with the rate which is got to be prescribed to you in some fashion the rate we use a symbol lambda for it but let us use nu right now because these things here are controlled by a pass on process assuming that both the down to up and up to down the transitions are controlled by the same pass on process okay. So we know that Xi of 0 Xi of t equal to C squared e to the minus 2 nu t 2 nu modulus t. So for t greater than 0 it is exponentially correlated what is the power spectrum therefore C squared over 2 pi and then the Fourier transform e to the i omega t etc. So there was a twice you could do the cos omega t by integrating 0 to infinity so you have 0 to infinity d t e to the minus 2 nu t cos omega t and what does that give you this is equal to 2 nu C squared it is Lorentzian this is Lorentzian in this case. So we can plug in all kinds of functional forms here and see what this looks like the power spectrum of this noise looks like now once you have this theorem in place that is it I mean let us look at the simplest instance these are short noise the original short noise right. Now the whole thing I should have titled the whole thing short noise because it is general short noise in general that means a set of pulses of some kind uncorrelated with each other that was our assumption but the original short noise was the noise which was seen when you had these electrons coming out of a cathode in some electronic tube you collected the current and you discovered the current went up in spikes every time an electron hit the collector you got a spike right it was just pure delta function spikes in the simplest case so it is essentially our process X of t except that each time you had a charge and therefore there was a current and what is the power spectrum of this going to be in this case so if your rate of at which things were happening was your X of t process S X X of t of omega was equal to we wrote this down somewhere there was a new apart from the 2 pi factor etc etc and then there was plus a new square delta of omega right so let us look at it for omega greater than 0 it was essentially this and then if you put in there was a factor 2 this in the engineering convention of doing things and H was 1 in that case so it was basically 2 new because this fellow here was 1 shape was 1 inside so it became 2 new and then therefore it was 2 e squared new that is it but this is equal to twice e times e new but that is the average current because the rate at which these electrons are coming in is new and e times new is the rate is a current average current this was the original short noise proportional to e and not proportional to the temperature whereas we saw that for white noise you had a temperature factor so this was the way in which they could we could distinguish the 2 kinds of noise this was noise was independent of temperature what happens if you have a square pulse so let us do that incidentally one can write this formula a little more simply in the following way so let us write that down a little more simply we will use the fact that Xi squared average Xi sorry Xi average equal to new times average of H times an integral minus infinity to infinity dt if dt f of t but we know that 2 pi f tilde of omega was the integral e to the i omega t times f of t so f tilde of 0 is this integral so we have Xi equal to new average H Xi 2 pi new average H f tilde of 0 so this squared is let us write this as 2 pi f tilde of 0 whole squared new squared H so this thing here becomes 2 pi f tilde of omega whole squared and then there is a 2 new H squared plus 2 pi by the way once I multiply by delta of omega I could write this as f tilde of 0 right so we have 2 pi f tilde of 0 whole squared and then 4 pi squared so the new squared is there H squared is there and then 2 pi this guy is there right so this is equal to 4 pi average Xi whole squared little more convenient to write it like this in terms of the average of Xi itself look at what happens for a square pulse this is the very very common situation so f of f of t minus ti is this guy and let us take the pulses to have the same so what is f tilde of omega is 1 over 2 pi integral minus infinity to infinity blah blah let us find 2 pi f 2 pi this guy equal to an integral from 0 to tau dt e to the i omega times 1 pulse of unit height a theta function right so this is equal to e to the i omega tau minus 1 over I so we need this fellow here mod squared mod squared we need mod squared of this this is equal to I pull out an e to the i omega tau over 2 and then I get mod squared that goes away and then I am going to get 2 sin omega tau over 2 over omega so this becomes 4 sin squared omega it is the sinc function squared and you plug that in here so for the case of square pulses s Xi Xi of omega for square pulses rectangular pulses is equal to apart from various constants and so on for omega greater than 0 let us forget the DC contribution it is proportional to nu times sin squared omega tau over 2 over omega squared so that is a very very typical result what happens if the tows themselves varied randomly oh by the way in general of course it is nu times H squared here what happens if the tows themselves varied randomly and independently the only place tau appears is here so it just get averaged I mean there is further average over whatever distribution you care to prescribe generally with cutoff between some lower and upper range some window of tows continuous window distributed in some fashion and you can find what it is averages so let us look at another example a very simple example of physical one if you take a piece of iron and you magnetize it you apply a magnetic field and you jack up the field as you go along then you discover that domains inside this piece of iron start orienting in the direction of the field and they do so at random so every time it does so the magnetization changes by a small amount so in the simplest instance let us assume the magnets were pointing in one direction and under the field and the upward direction they orient upwards right so the change in magnetization is going to be of the form twice so if it runs from if there is a little domain with magnetization m pointing down and then it points up the same domain points up then the change in the flux magnetic magnetization is 2 m and this change in flux will die down exponentially okay so there is some 2 m e to the minus t over tau this is what the flux will look like and now if you detect this this flux will induce a current in a coil and then that current can be used to drive a loudspeaker and you can actually hear it it is called Barkhausen noise so what is the rate of change of flux and that is what the voltage is going to be the random voltage so v of t is going to be 2 m the derivative of this which is going to be tau e to the minus t over tau with a minus sign that is what Faraday's law of induction says and if you got a coil with n turns there is going to be an n so that is going to be the random voltage that occurs if there is a change of domain at time t equal to 0 this is your f of t the pulse shape right so what is the correlation going to be now what is the power spectrum going to be it is essentially like you have to take this quantity and take its transform right that is it f tilde of omega so what is f tilde going to be for this guy this is for t greater than 0 so what is f tilde of omega going to be and what is its modulus going to look like Lorentzian again it is just going to be Lorentzian okay so this will imply that s v v s psi psi of omega is proportional to once again nu is going to sit there and then there is going to be 1 plus omega square power square apart from some constants just going to be Lorentzian shape okay and you can measure it explicitly so this formalism these two theorems Campbell's theorem and Carson's theorem actually very very powerful as long as the pulse sequence is uncorrelated you can get fairly simple closed form answers for the power spectrum assuming the process is stationary and of course the original uncorrelated assumption now it gets more and more intricate when things get correlated to each other and some results are available for the case where the height is say the correlated with the width of the pulse and so on but that is a second secondary problem that is a separate problem altogether so I thought I had mentioned this simply to round off the fact that in practical situations you measure the power spectrum directly and there are these two theorems which help you understand what a no pulse sequence would do okay the interesting problems arise in correlated pulses when you have pulses which are not independent of each other in other words you do not have a Poisson process then the you have a much more complicated situation and I will make some comments about that little later okay so let us now move on to a class of processes which go beyond the Markov process well let us look at some non Markovian processes and what is the simplest such process we should look at well there are several ways to approach this but since we have been doing a lot of diffusion let us take a problem which involves diffusion itself and ask what happens if I have a position variable for a diffusing particle which will obey a non Markov process which will be a non Markovian process okay and here is the way one does it you know that the simplest model of diffusion for a particle on a line such that its probability density obeys the diffusion equation was in fact as the integral of white noise so we said x dot equal to square root of 2d times this is Gaussian white noise and then immediately we got a process which was Markovian but not stationary we discovered that x of t the displacement starting from 0 say is 2d integral 0 to t dt prime eta of t prime and this was the Wiener process out here we call that the Wiener process okay and we found out what the correlation of this was the average value we found x of t x of t prime average 2d and so on. So this diffusion process the position was the integral of white noise now I say alright and we also saw that white noise came about I mean we saw that this process came about as the continuum limit of a particle undergoing a random walk on a line in which the step length went to 0 and the time between steps went to the time step also went to 0 such that a squared over 2 tau went to the finite limit d okay. So it was a limit of a simple random walk where the walker tossed a coin and with equal probability took a step to the right or a step to the left and we took a limit we said that the step length so this is a continuum limit of a simple random walk in which a squared over tau went to twice d we also accommodated bias diffusion here by saying that there is a probability of moving to the right or left for different p and q etc that led to a diffusion equation with a drift term present which took into account the fact that you had a constant force such as gravity right. So that should be clear that the simple random walk in the unbiased case gave you unbiased diffusion gave you this unbiased diffusion limit and the biased random walk gave you a diffusion with drift as if you had a constant force present right. But in both cases the X process was the integral of white noise Gaussian white noise now you could say well suppose this particle should also have some memory of its momentum we are now looking at the process saying that well the position alone is not enough to describe the state of the particle you should know its position and velocity at any instant of time or position and momentum. Now what would be the discrete way of looking at of implementing this is to say well the walker moves to the right or left with different probabilities depending on the direction in which he was moving in the previous step. So with probability p he continues in the same direction that he was moving either to the right or left and with probability q he reverses direction okay now what is the effect of this it means there is memory in the process the walker is remembering where he came from in the previous step. So it immediately tells you this cannot be a Markov process at once there is some memory right earlier the person was just tossing a coin and saying p with probability p I move to the right probability q I move to the left hand you do this at each side and now you say alright I now move to the right or left with I now continue in the same direction or reverse direction with probabilities p and q respectively. So I remember where I came from in the previous step I can write difference equations for this process also and what would happen in the continuum limit it would be correspond to saying that the walker remembers the direction of motion and now if you say that the drift velocity is constant in a suitable limit we can show immediately that drift the velocity of the particle is constant then it is equivalent to saying that the particle is diffusing with velocity either plus c or minus c and there is a certain average rate at which the particle changes direction right then let us write down what the mastery equations could be for this process. So I am going to skip going through the discrete and going to the continuum we will write down the continuum immediately. So now instead of saying p of x, t is the probability density of this particle at any given time t right I should also tell you the state of the particle by saying what is the direction of the velocity is it moving towards the right or is it moving towards the left right. So there is a p r and a p l of x and I need to write down rate equations for p r and p l together and that is done very simply there are only two states r and l for each x. So this is like saying I am writing down probability densities for p x v t except that I am saying v is plus or minus c that is all I am doing right and what do you call a process in which you have a plus c and a minus c and that is a Markov process it is a dichotomous Markov process. So it is essentially saying that this velocity process is a dichotomous Markov process so this is dmp and the position is the integral of the velocity. So we are saying x dot equal to v and this fellow here is a dmp with some correlation time. So if the reversals are occurring with a mean rate nu for instance then this process we are guaranteed that v of 0 v of t in this problem this can only have the values plus or minus c and it is a dichotomous Markov process the mean value of the velocity is 0 so this must be equal to c squared e to the minus 2 nu mod t and now we are asking a hard question we are saying what about the integral of a dichotomous Markov process just as we looked at the integral of white noise and we got a vener process now we are asking what is the integral of a dichotomous Markov process which has a finite correlation time. So what kind of correlation time does this have it turns out the problem is now non Markovian okay because x this process x is not a Markov process on the other hand the combined process x together with v is a two-dimensional Markov process okay but let us write the equations down for this and you immediately see where this is taking us what is dp r over dt of x t equal to what can this be equal to where there is a gain term and a loss term right. So if at rate nu you change from l to r so there is going to be a nu times p l of x t but you are also going to lose the fact that you are going to switch out to the other state so this is minus p r of x you should actually write this as p r of x t plus delta t and put the delta t is outside there and so on so with probability nu delta t you are going to actually reverse direction and that is going to if you go from l to r that is a gain for r if you go from r to l that is a loss for r okay and this is p l of x t over dt equal to nu times p l sorry p r of x t and that is it right but these are derivatives with respect to time total derivatives right. So what is the partial derivative equal to x t plus the convective derivative v dot del but v is plus c in this state and that is equal to so that is your equation and similarly this equation now becomes delta p l delta t x t minus c delta p l x these are the equations for this problem and this is called persistent diffusion or correlated diffusion so our task now is to solve these equations what would you expect will happen if I eliminate p r or p l in favor of the other probability and write an equation for p r alone you get a second order equation in time which means that your initial condition has to be specified for both p r and p l that is equivalent to saying you are specifying the initial position as well as the velocity whether it is plus c or minus c. So it is non Markov the x process alone is non Markov but the combined processes Markov the x comma v process is Markov and they satisfy these two equations so our next task is to solve these two guys which we will do write down the solution they look reasonably trivial but it is not all that trivial but they got many interesting properties we will see what this process does in detail but this is the physical motivation the original physical motivation this was introduced by G. I. Taylor and this original motivation was to introduce momentum into the problem because he had the hope of understanding turbulence using this of course that is not true but this process persistent diffusion itself is very important in something called dispersion in fluid layers and we will talk about that. Now these two are also called the telegraph was the equation for a reason I will explain the first thing you observe is that if you add these two guys the right hand side vanishes so that is one simplification but then you have to be careful there is a minus sign here so it does not immediately give you the solution on the other hand you have a little bit of simplification their first order equations so we could perhaps use a matrix method to try to solve this on the other hand you eliminate one of the variables you are going to get a second order equation and then you should have to deal with that but it will have the advantage that there will be a second derivative in space and a second derivative in time so it looks like you know you can use a little bit of homogeneity property here would you expect this to have diffusive behavior would you expect the mean square displacement to increase like time the second order in both space and time so it is not immediately clear what is going to happen it does you do see that the mean square velocity is c square so it looks like there might be ballistic motion so the mean square displacement may actually increase like the square of the time on the other hand it is not immediately obvious what will happen as t becomes very very large because you do expect the central limit theorem to start operating right so we will see what happens.