So, if you recollects we have looked at the definition of random processes. These are

the 2 ways of looking at random process 1 is to

look up on it as a collection of waveforms which we are randomly available right. Any

particular waveform is certain probability depending on the probability distribution

of the underlying probability space right. There

is a mapping from omega to the set of functions which are

functions of time. How a ways to think of it have a sequence in this sequence of

we are there is some kind of index which gives out the functional

dependency eventually. It could be a functional dependency on t index could be t the

index could be some special variable or any other kind of variable right.

So, you have a sequence one after another or random variables occurring the certain

probability distributions in that constitute of random process. The later approach is

convenient from the point of view of characterization of the random process in terms of a

probability distribution functions. Only thing is the complete characterization is a very

elaborate affair and we discuss that it is very very difficult for a complete

characterization of an object random process. Because you when need to characterize the

process at every time instant individually at every pair of time instant jointly and

so on so forth for every the points every quadrate

the points that you select. And therefore, 1 and infinitive number

of going to distribution functions distribution function

as well as joined distribution function of various orders you

completely correct raise the process. To simplify the process to simplify the characterization

you made some assumptions above the process. We can make some assumptions

of course, they must be valid if you are going to use this typically many of this

functions are valid why assumption that we made is the stationary assumption. Then we

defined processes as 1 whose turbidity distribution functions or

whose characterization whose statistical characterization is independent of timology

right time shifts. So, these are the things are

we cover last time, we also looked at the definition of certain limits of a elements

process mainly the new value function and the auto

correlation function. So, we will start from .

So, look at the auto correlation function is second order characterization why do we

say size of characterization? Because required

the joint density function of the 2 in the Xt way and Xt. So,

next call it Rx t 1 t 2 Then integral . So, take the random variables

values random process values are time instance t 1 that

will be denoted by the value x 2. Of course, this

could occur anywhere from this value could be anywhere lying between the value of this

minus infinitive plus infinitive x 1 into x 2 multiply these two. Multiply to the join

density function of these 2 random variables. So, this is 1 notation by which we can

denote the join density function of random variables t 1 by samply the random process

x t at the time instance t 1 and t 2 right.

So, this is the notation this t 1 and t 2. This in general there will be

a time dependency. Yes there will be a time dependency of the processes not stationary.

If the processes is stationary the time dependency only being in terms of P 1 minus 2.

This t 1 and t 2 separately right. But yes in general this is evaluate

as this joint density function depends on this dimensions t 1 and t 2. Therefore, the

auto correlation function that dependent the various

of symptoms of random process right. So,

this is the general definition for strictly stationary processors. For stationary processors

this will be a function only or you could say this you could right as Rx t 2 minus t

1 if the be if you are Rx tau the tau is defined to

the variable t 2 minus t. This is I thing we are

we stop last. So, now let us look at a few things.

Let us look at the values of Rx tau for tau equal to 0 that is it mean that we are looking

at what is this equal to? You can think of this

portion in terms of expectation of portion are

equal to expected value of x t 1 into x t 2 right. This integral is nothing for the

expected value or average value of the product of the

p random variables sampling the process of time instance t 1

and t 2 right. So, where is Rx if you choose t 1 equal to t 2 That will be that will give

you tau equal to 0 right. So, it could say this is equal to expected

value of x square t any arbitrary time instance t.

So, this the value of the auto correlative function for a stationary process at all could

0 it going to be equal to the mu square value of

the process it is a constant. Suppose the process was not stationary then with this

be a constant? No. In fact, we will not write Rx

ca in that case. We will write Rx t t or t 1 t 1 right dependent the

value of t. So, this equivalently for a non stationary

process the corresponding relation will be Rx t t

it represent t 1 equal to t 2 equal to t. And this will be equal to expected value of

x square t that is correct. But this is going to be

a function of time it is are to be constant. In this

case it is not going to be a function of time it is going to be constant right is it clear?

So, for a time for a non stationary process this

is the relation for a stationary process ((Refer Time: 08:25)). It is clear from the definition

of the auto correlation function at least for

real value processors that if I interchange t 1 and t 2 if I interchange write x t 2 first

x t 1 later it would make any difference. We just

multiplying the same 2 numbers right the average value will be the same right what

did it mean? The Rx Rx tau will be equal to Rx

minus tau right. So, this is a important property of the auto correlation function for real

value processors right. Student: what is the difference between Rx

They are same for stationary processors. This will become this for stationary process; for

stationary process this will be equal to t minus t minus t. But in

general it is not a stationary process this we have have to work

with. If it is a stationary process work with this place.

Student: . That is right because the value of this auto

correlative function has been discussed we depend only on the separation of the t time

instance have a on value in the time instance. Because if it

is a the basic function .

Student: So for Rx 0 should not be used . No no no please understand if it is what is

the Rx t 1 and t 2? t 2 value x t 1 and x t 2. So,

ordinary choosing t 1 and t 2 will be the same time instance. The

tau 0 any kind of term look nearly these value that is R 0. Rx stop

is a functional stop then tau equal to 0 if it is it becomes it could be a least four

validity function that is the point that you are trying

make sure. Look at the basic definition can place of the .

So, this is true for real value processors. So, sometimes I simply use a word process

when I want to say random process. In general you could also field

that the Rx stop. Of course, we can prove it more formally like it is

with that yourself the magnificent of RX stop cannot be more than the value of tau equal

to 0 is not it? Again for the same reason we can expect next line correlation to occur

when the p random variable of the same right. The way you example of a different time

extend the additional random variables it can have correlation right. But; obviously,

that will be less than the correlation that you

will have of random variable with itself that will

be perfectly right. So, of course, one can with mathematically which are likely to be

yourself. It is very easy to do easy to check that the magnitude of auto correlation function

for a laptop the variable size as sometimes sometimes called the long variable,

because long between 2. Time instance t 1 and t 2 are the time difference t 1 and

t 2. It is always less than or equal to the value

of the auto correlation function the tau equal to

0. Please understand that Rx tau is not a random quantity in case this there is any

kind of confusion in your mind. Rx tau is the highly

deterministic function right it is a deterministic function, because a basically

look here this interview right it is a function of I like x t which is a random process Rx

tau Rx t t or Rx t 1 t 2 They are deterministic functions which are properties which are which

describe the second order of the random process x t right. It tells you the behavior

of the random variables with respect to each other on an average by looking at the average

value of the product. By sampling a random process the different kinds of things

the multiplying this random variables and looking at the average value of this product

it is some kind of a characterization of the process some kind of a characterization of

the process. It is a value it is a fixed value for

again process even though the random process x t random in nature.

It is the value that you will see random in nature just nothing random in a Rx t 1 t 2

or Rx t t or Rx 0 or Rx stop right. Lot of correlation

function is a deterministic function. So, these are some of the properties of a stationary

process. You say it is just I will mention possibly here and we leave it if . If the

process x is a complex value process. For example, if you are working with

in a naroboam process naroboam process when we looking are its complex envelop then

it will be a complex value process right. In that case the definition are slightly modify

that correlation function is defined as x t 1

into x t 2 right we do a correlation that you will use that one in the

. Suppose in a same instinct process could briefly simplify our characterization of the

process right we discussed earlier. In many

many instrances your real life when you are working with random signals or random

processors. We do not really need to worry about distribution

beyond second order right. Typically we are interested in collectors in the process

in terms of individual value that you might see are various time instance or in terms

of join behavior of a sphere of values right. Typically very early need to work work a situation

very need to look at more than 2 values at a time right, which essentially

means that we do not need to worry about characterization beyond certain order of in

many many application in practice right. So, if that with the case

wherever the process executes stationality in the

stripes or not is another key all of them really then want us that the process be second

order stationary right. The second order stationary means that the first order density

function E x t 1 x should be independent of t. And second order joint density function

E x t 1 and x t 2 should be independent of should

over dependent on t 1 minus t 2 right. So, if the process set as files this time

origin you various only the respective these 2

distribution function we say the process is second order stationary ratherstic

sensationally right. In reality you do not to worry about second order sationarity as

engineers. It is sufficient for us if the first 2 elements of the process execute these

properties right. So if that happens we are working with that process which is set to

be stationary in the widest sense or wide sense.

So, we define a nation of wide sense stationarity which is a special case of stripces

rather stripes sentationallity imply of course, this. So, wide sense stationarity

is defined as follows. If the mean value function the various function which of course, is redundant,

because if the auto correlation. And the third thing auto correlation; if the auto

correlation is a lead to various as a special case or if these 2 are independent of time.

And if the automobiles function is a function only of t 2 minus t 1 t 2 minus t 1 is a function

only of the difference variable tau. Then the process x t is set to be void sensation

this, your usual notation WSS. So, what I am saying? We not asking even in the density

functions to be invariants even the first and second order density function you only

say that the new value function. You know what is new value function how will you new

value function define? Mu x t will be simply equal to expected value of x t. Now,

it is a that is the definition of mew looking at

the random variable of time t look in survey value what is this going to be equal to x

E x x it will also dependent t in general dx right.

So if it is a first our stationary process Px t

could not dependent t right therefore, it is obvious that mew x with constant right.

But other definition of wide sense stationary

process does not even require this to be independent of t all as various if this is

independent t that for enough for us right. We

look all the other first of the movement associate with this x t function right that is one.

What is this various? Various is sigma x square t it will be expected value of x t minus x

bar t whole square right. Again this will depend only on the density function p x t

right. So, this will be equal to x minus x bar that

the definition of various is in it that expected value of x minus x bar whole square that is

integral of x minus x bar and the density function associated with x right. Again this

is this is independent of t therefore, this should be in dependent of t, but again in

WSS whereas, in the, this should be independent of t are you say yes this should

be independent of t right. The second order movement also independent of t. And finally,

what we are saying is that auto correlation function Rx t 1 t 2 as defined earlier should

be a function only is tau right. Of course, the various function is related

to this function this will display this property actually this will be automatically implied

this is something that we can check right. So,

if these three conditions satisfy actually 2 conditions then we the mew value function

is independent of time. And the auto correlation

function is dependent only of on the time difference between the 2 dimensions t 1 and

t 2 These are the processors stationary wide

sense stationary. And many times we are quite happy the processors wide sense

stationary be many times you do not you are need to worry about the distribution

functions right. Do not need to worry about the first and second order movement

functions then we the mean value function and the auto correlation function.

The mean value function is a first order function, because the race

power race power of x that we racing 2 is 1 right when we taking the movement is a first

order movement the auto correlation function the various function in a second order

function, second order movement function. And many time it is sufficient to work with

the first movements right they contain most to the physical information’s we are

generally interested in in looking other processes right. So, therefore, introduce the

concept of strict sense stationary process a process which is stationary up to second

order or third order could be a special case of

there. But wide sensational process is something which is all together much more tolerate of

non stationary, process maybe non stationary stationary in the wide sense right.

So, what your is that if a process is strict sense stationary this

stands for strict sense stationary it could; obviously, implied that it is wide sense

stationary is not it? That obvious should be obvious I think this could imply WSS

property the will happen only very very special cases. WSS could

not in general imply strict sense stationary right because this is independence only in

terms of movements there also if the first thing first 2 orders right. And therefore,

we cannot say that all density functions or the

complete statistical characterization could be

independent of time right. But in very very special cases this could imply this for

example, for a class of association which we call casion processors right that is very

very special.

To clear example of a process which could be which is not strict sensationary and get

it is wide sense stationary right I will just

give you one simple example. I leave a file to work out yourself we can do that quickly

where itself suppose, I generate a Raymond process in a following bit. The many reasons

of n m process right that we define a random process which

essentially we can random by virtual of its dependents on it is a finite function of time

right. It is a finite sense function of time, but A that 3 parameters in the cosine functions

the amplitude A is if you and phase theta right. Let me look to one

of them that it the phase random way that is say the value of theta is something that

of course, randomly from 1 function to another

function. If you look upon this as an example of functions

right all every function in this collection should be cosine function. But with different

phase and the value of phase is govern by some distribution right. So, as much as you

have a collection of function it is a random process is it clear and that every function

occurs with a certain probability right. So, becomes random if any one move of its parameters

random variable. So, for this becomes random because we assuming the theta

is random variable let us say with uniform distribution. That is why if I say

its uniformly distributed between 0 and 2 pi

there could be the density function of theta with B equal to 1 by 2 pi 0 0 and 2 pi and

0 elsewhere. This will imply this is the density

function of the random variable theta we have could be.

Next look at the mean value of this next we look at the mean value function we want to

see whether this function is a function of time or not right. Mind you we are not looking

at density function of x t here right now. But sometimes we can work around that even

the doubt completely the base density function of x t you can compute. So, this is the

important point we can compute this quantity even without move density function of x of

t right, because we know how to compute the expected value of or function of a random

variable. So, you can think of you can think of this constant as a function of random

variable theta and complete this every value on that process. So, will be at the it will

be A cosine omega a t plus theta you multiply

with the density function of theta right think of this as a function of theta is entire thing

at any given time right. Multiplied to the density function of theta which is 1 by 2

pi or 0 to 2 pi d theta is there. Your function of

theta multiplying the density function of theta integrating over the range of theta

or the d 1 of theta right that is

That is one way of computing the expecting this and using the

let me recollect file the discussed. Expected value of any function

of x requires simply do this a portion is not it? I am using this solution here. X here

is our theta variable theta; f of x is this function

right. We are multiplying the density function of x or density function of theta the integration

of theta. Now, look upon what will be the value of this integral? 0 right. So, it is

independent of time right. So, value of this is

equal to 0. Just look at the auto correlation function just look at x t 1 into x t 2 or

lets say more conveniently. Just write this as x t

into x t plus tau right p 1 to be some arbitrary time is complete and t 2 T plus tau for the

difference between ls equal to tau right. So,

this will be equal to 0 to 2 pi again now the function is different that is how? The

function is now this kind of function everything else with the same. So, become A square

cosine omega t plus theta into cosine omega 0 t plus tau plus theta multiply this with

the joint if the basic function of theta right

integrate theta right. And a simple evolution of

this intergral we will show you that this is equal to A square by 2 into cosine omega

0 tau.

Student: . No no no no joint density function I am just

looking at this product as a function of theta I am again

using the same definition here, because I am not

going through the path of first joint density function of x t 1 and x t 2 That could be

1 way. That is very complicated in this phase

it is not necessary. In this case mush easier to

look upon this simply as a function of random variable theta this product function right

multiply the basic function of theta and into our theta. So, you can see that after this

integration it is not going to depend on t it depends only tau right.

Student:

Please repeat your question. No no it has do dependent time theta is a parameter of

this function right. Depending on the value of

theta you have a different time function this different time function are all cosine a same

frequency in same but different phase right.

So, therefore, basically this collection if you are looking at is a collection like this

and so on and so forth. It is a infinite collection

you can define a fairly different value of theta

right. So, this shape of article this shape of article this shape of

right. The way document the mew value function yeah say if I pick up some times is

empty what is the average value that I am likely to see here across the example it also

is 0. If I pick up 2 Time instance t 1 and t

2 which are separated by time instance separate by some interval at all. What is the average

cos . It is these 2 random variables answer is cosine variable

top it does not depend on the class of t 1 and t 2 This is what you

have demonstrate right is it clear? So, here is

an example of a process which is; obviously, not strict sense stationary is it obvious

it is not strict sense stationary. It is not obvious

I am assuming that it obvious you think about it. It is not obvious, but it is possible

to argue very easily to see that. Let us looks at let us say 1 time instance

like this. I think it look at little bit of thinking I

think like linear of the timing, because we are going to a square a

bit. It is possible to argue the density function is not constant with I right. You will a first

add density function is not constant this specific time actually it is one of the problems

in the book. Please look at the problem with

very carefully and you will arrive this argument. But even though the density function

itself is a function of time the first elements then will the mean value function

expected value of x of t and the auto correlation function. Set if pi the required

properties of a wide sense stationary process there is not a strict stationary process and

we are taken a wide sense stationary process. It

is. So, only you need to wide pi rates not a strict sense stationary process. So, please

do the as an exercise. Now, what are let us take

talk about what we are discussed so far. We say that if we are working with random

form random process we can characterize it for all technical processes as engineers most

of the time it is sufficient for is to characterize it with

2 kinds of functions. The mean value function that

gives us an idea of what is the average behavior of the time functions. What is the

average behavior the various time functions which constitute the process right that is

that is 1 property. The second is the auto correlation

function which tells us if I look at 2 random variables which are which other bian

interface tau seconds what will be the obvious value of the cross

correlation of the 2 random variables right. So, rather than trying to specify the density

function and the joint density function which has much more information many times as engineers.

We are sufficient you are sufficiently happy with these 2 information’s

namely the mean value function property and the auto correlation function property

right. In some cases we need to going to more detail, but many times this is good enough.

So, all purposes the function may mew x t which is going to be a constant function for

a wide sensation process. And the function R

x t 1 t 2 which is going to be the function only of tau for wide sensation processors

is good enough us. You do not going to worry about the density functions in many many cases

right. Now, as electrical engineers we are used to describing things in the time

delay as well as in the right. When we talked about the time delay

function immediately ask ourselves what its specters domain description? What is it spectrum

live is not it? I you have worked out the elaborate theory for bring that in the

transcription. You could have similar interest here you have occur

whether the signal is deterministic signal; however, is a random signal. You can ask some

properties post in the time domain as well as in the frequency domain right. So,

we like to also see whether it is possible to

characterize a random process in a frequency domain.

Now, let us see look me just discuss a few basic concepts or conceptual difficulties

associated with this and then just give you the important results in this connection.

Let us look at the difficulty just say you are defining

a random process as a collection of waveforms right that is from you are looking

at. Now, when we take a tele transform, what is the tele transform? Tele transform

is let us say you have function x of t. And you multiplied e to power

minus j 2 pi ft and take the intervals from minus infinitive to infinitive that this is

the that is the initial 1 thing that you work with what are the assumptions

in this? The assumption x of t is a

energy signal right it is a its absolutely integral right. Remember the base shape

condition specify with the definition of a whether the system

transform associated with a system approach transform.

So, what are the requirements are that x t should be absolutely integrable that way we

should be a power signal hen you are working a random

processors. We do not know where we are be working a energy signal of process that is

one issue one difficulty with need to work with right. A functions maybe power signals

your function maybe energy signals. But that is the problem that we are also direct with

the in the context of deterministic signals and we typically better around that by

introducing in function in the frequency domain right if you

recollect that is one problem. The second problem is more difficult more conceptually

more difficult. Except you have a very well defined waveform very well defined

mathematical function of time right. It is very transform you are taking for example,

e to the power minus alpha t or cosine omega 0

t right. But the collection when we talk about x of t be a random process you need not know

what waveform you are working with actually.

It is one of infinite number waveforms an every one of them

transform what we exist for that waveform you have a different transform and therefore,

you could have a different kind of spectrum. So, what is we talk about does it make sense

to talk about transform or the spectrum in the normal sense. Are

you ? No it does not make sense right. However, what that make

senses on an average where is a energy distribution as a function of frequency? What is

the power distribution as a function of frequency? Depending on whether we are working

a energy signals of approximate right. So, important frequency domain concept for

random processors is not the usual spectral which is just the, which is just a

transform of a function, but a average kind of function which is known

that the layer of power spectral density function. First just define this process density

function conceptually right let us say we have random process x of t.

So, what we will do is to convert this into an energy alpha can

takes this process. So, lets random waveform just look at this

waveform between that is a minus 1 to plus t right. This waveform

one same sample function in the process

with here one sample function the process. As you know random process is a infinite

collection of such sample functions right its some arbitrary selected sample function

from that collection. So, you pick up 1 and truncated between minus t to plus t you note

the result in process x sub T t. So, x sub T t has been generated from x t by looking

at its interval between minus t to plus t making

it 0 outside this interval. This artificial construction ensures that high converted even

a power signal into an energy signal right is it clear?

Because now clarifying that energy and for the peri transformer this could be define

again sense again this is your random quantity I do not want to take the full transform of

this right where I am tested in what is first of all I scored this function. Because of

I do not interested in the power energy I am not

interested in the individual functions values by themselves right. Squaring is measure of

the energy at various time instance right is

of course, not literally, but some approximations. is no that thing

in beginning long . So, 1 let me define a free transform of X T all

here using the capital letter please remember I have pick a 1 sample function right. I pick

a one sample function and that sample function is low energy signal I am taking pay

transfer right. This pay transfer what exist now whatever the function maybe whichever

it does not matter. Because I have converted this into energy signal

the pay transform will exist all.

So, I take the pay transfer. This will now become a function of frequency right take

the magnesium square of that right. Now, this

will be the magnesium square or the energy as

a function of frequency of one sample function . If I want to look at

the average value average properties what shall I do here is the average value of this

right. And if I want to convert this energy function this a energy function is not it?.

Into a part function what shall I do? Do varied the

2 T, because of my function duration is 2 T

and if I want look at the original function take the limit us theta theta as infinitive.

So, this motivates my definition of the parse

with density function of a random process S x f.

I denote it by S sub x f as limit as T tends to infinity of 1 by 2 T expected value of

magnitudes square of the free transform of X T t. Look at this carefully I have gone

through the argument reading to this that if you have a doubt please ask your questions.

That is a formal definition of the density function. It is a measure

of the average distribution of power for given random process X t in the right. How is the

power distributed among difference frequency components in

the frequency in the net? Are you agree, you have any questions? So, that is you can take

that as a definition. Now, without going through the details of a result I just like to give

a result which is should no off which is a very

important result in the characterization of NM processors. It relates the density function

the density function when we talk about density function as a process

you must ask the where you talking primitive density function we are talking

about density function. density function is a it is what so, say it

is a once again. Do you agree with that?

Because you are taking a random process in this I just not enough scoring in that in

the time domain in the limit. Now, we are any

way squaring it up right and you taking the away value of the x squared in a function

of frequency right. So, it is a second order

movement just like the auto correlation function was the second order movement right. So, if

you look at this way it must be natural to expect the least 2 second order movements

must be some more related is not it? We are saying in the auto correlation function is

a second order movement description of the random process x t. And now we are saying

similarly that the less function at the just define the function is also a second order

movement

characterization in some cases only thing this is the frequency

characterization that was characterization. Therefore, it sounds logic error the, these

2 second order movement description should be related to each other right. And that is

the important result that I am talking about there is a very simple way to not exactly

very simple. But it possible to show that these 2

Things the second order element description in time delay which is auto correlation

function. And the second order movement remaining in the frequency domain which is

the density function or essentially free transform phase right. So,

which is your result to very similar to what you are used to doing for deterministic

signals what you thing is where reject the free transform of the signal directly where

we taking the free transform of the auto correlation

function right. This result is note the name Wiener Khinchin

theorem which essentially states that the

density function of a process x t and the auto correlation function

of a process x t or the auto correlation function of a process x t or previous one determines

the other. So, p s t we put here like that in auto correlation

function of free transfers. And therefore, in as much as the mean value function and

the auto correlation function are complete second

order characterization of a random process. Similarly, the mean value function

that the density function of complete second order characterization

of the random process. Let me now finally, define one more concept and then

next time we will now will essentially concentrate on concept that we are specifically

go to need in our treatment of the communication systems.

The final concept is the concept of ergodicity. I will just mention it here will have to

discussion of this concept, but we need to know it right a twice.

The concept is as follows we say that the process that I need a random process is if

it is statistical average is or equal to or can be replaced with its time

averages. That is statistical averaging of any is equal to the

corresponding time average of a given any any sample function. Now, this is the very

peculiar concept that the very useful concept because without this concept you know

really we particulate we have to work with random process. Particularly when comes to

measurement of what properties of random process just look at the very quickly a

modification for this property before talk about the property itself.

Motivation is as follows I was said our concept of random process is limit of how you

recovery basically it is a collection of infinite number of waveforms. And you not know

which waveform we are going to see suppose perform the

experiment are you look at the random processes as it as it occurs and display waveform

that you see some arbitrary waveform which we cannot predict

right. Now, anytime in typical situations I will see just one such waveform from this

infinite collection. So, for what measures properties average

properties let us us say if you want to measure the mean value. So,

measure the auto correlation function right suppose I do not know a density functions.

How will I have find out this qualities? right very difficult, because I have only 1 sample

function available in front of t right. I have just 1 sample function which I have observed

I do not know what are the other infinite what are the other members of the this infinite

family. In the other hand this property if it is valid for a given random process helps

to with find this properties just from a single

sample function that we might have observed. So, what we are saying is even if I do not

know if I even if I average they cross the whole family because

I done have the whole family with we. If I just look at the average across the time of

one sample function of the family it is good enough I get the same value right. So, processes

with exhibit such properties are called algolic process. Fortunately for us many physical

processes which we work with I got it in nature right.

And therefore, many times we can obtain this the auto correlation function by looking at

the time mean auto correlation function of a given sample function or the mean value

function by just looking at the mean value of a given time to mean function. Just like

you do for any time . So, what is a time mean

average are you simply this right take the limit as the T junction

infinite that the time mean average. This does

not involved the density function of x is taking a sample function right and integrating

minus T to t that is the sum of all the values that you see between minus infinitive the

vary for the time interval that is the time mean average. What is the correspondent

statistical average? It will be x Px t x dx that is the correspondent statistical average.

So, what you are saying is this is equal to this

similarly for the auto correlation function for

any other kind of average. If this kind of relation hold for all time averages

corresponding statistical averages the process is continue step by step I.

Final statement in this connection I know time this one final statement, because overall

this continuous. If I denote by this space as a clause of all stationary clause of all

random processes right clause of white sensationary

processes. So, this is all processes clause of

white sentationary process is a sub clause its satisfy those 2 conditions which I

mentioned linearly function is independent of time auto correlation function is dependent

only on time difference. Strict sense stationary is a further sub set of that right and

it is a smallest sub set that just I want to it make and complete this

discussion. So, if a process I go to it will also be strict sentationary it will also be

wide sentationary etcetera.

Thank you.