Study the autocorrelation function

[T.Engineer]

I would like to know how to find the autocorrelation function.
If there are any links for free e-books that can I downloaded to study the autocorrelation function and its properties in details.
Thanks!

 

[EnumaElish]

http://en.wikipedia.org/wiki/Autocorrelation
http://en.wikipedia.org/wiki/Autocovariance

 

[T.Engineer]

I already know about http://en.wikipedia.org/wiki/Autocorrelation.
I meant another links that will provide any free download E-books if you know.
Thanks alot!

 

[EnumaElish]

http://www.google.com/search?hl=en&q=autocorrelation+finance&btnG=Search
www.springerlink.com/index/M106653187216806.pdf
www.ier.hit-u.ac.jp/Common/publication/DP/DP448.pdf
http://www.stat.tamu.edu/~ljin/Finance/stat689-R.htm

 

[T.Engineer]

I am going to find the autocorrelation function for the following equation
Hn(t)= (-1)^n * e^(t^2)* d^n/dt^n * e^(-t^2)

So, I didnt find the previous links can help me to solve this problem.

 

[EnumaElish]

You need to solve

Lim_s-->0∫_-s^s Hn(t)Hn(t+s)ds / (2s).

Note,
∫_-s^s Hn(t)Hn(t+s)ds = ∫_t-z^z-t Hn(t)Hn(z)dz

 

[T.Engineer]

You need to solve

Lims-->0∫-ss Hn(t)Hn(t+s)ds / (2s).

Note,
∫-ss Hn(t)Hn(t+s)ds = ∫t-zz-t Hn(t)Hn(z)dz


but I want to understand why you suggest this. I want to understad exactly by steps how can I find a solution.
Thanks alot.

 

[EnumaElish]

The reason I suggest is because this is the definition of autocorrelation (AC) function. See http://mathworld.wolfram.com/Autocorrelation.html

As for step-by-step solution, I'll suggest to think about how the fundamental theorems of calculus might apply to this AC function. See http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html

 

[T.Engineer]

Thanks a lot EnumaElish!
I will try to read these and I will reply you.
Thanks alot.

 

[T.Engineer]

You need to solve

Lim_s-->0∫_-s^s Hn(t)Hn(t+s)ds / (2s).

Note,
∫_-s^s Hn(t)Hn(t+s)ds = ∫_t-z^z-t Hn(t)Hn(z)dz

I couldn't solve the equation
∫_t-z^z-t Hn(t)Hn(z)dz

may be I should to follow such courses that explain Autocorrelation function or may be I must make reveiw for the integrals.
I want to advise me, please.

 

[EnumaElish]

For starters,

∫Hn(t)Hn(z)dz = Hn(t)∫Hn(z)dz.

Then, insert Hn(z) = (-1)^n * e^(z^2)* d^n/dz^n * e^(-z^2). Now you have:

Hn(t)∫(-1)^n * e^(z^2)* d^n/dz^n * e^(-z^2)dz = Hn(t)(-1)^n ∫e^(z^2)* fn(z) dz

where fn(z) = d^n/dz^n * e^(-z^2).

You need to work on solving ∫e^(z^2)* fn(z) dz.
___________________
CORRECTION: Lim_s-->0∫_-s^s Hn(t)Hn(t+s)ds should have been Lim_T-->0∫_-T^T Hn(t)Hn(t+s)ds.

Then, defining z = t+s, we have ∫_-T^T Hn(t)Hn(t+s)ds = ∫_-T^T Hn(t)Hn(z)dz.

This does not change the substance of the advice I have given until now.

 

[T.Engineer]

For starters,

∫Hn(t)Hn(z)dz = Hn(t)∫Hn(z)dz.

Then, insert Hn(z) = (-1)^n * e^(z^2)* d^n/dz^n * e^(-z^2). Now you have:

Hn(t)∫(-1)^n * e^(z^2)* d^n/dz^n * e^(-z^2)dz = Hn(t)(-1)^n ∫e^(z^2)* fn(z) dz

where fn(z) = d^n/dz^n * e^(-z^2).

You need to work on solving ∫e^(z^2)* fn(z) dz.
___________________
CORRECTION: Lim_s-->0∫_-s^s Hn(t)Hn(t+s)ds should have been Lim_T-->0∫_-T^T Hn(t)Hn(t+s)ds.

Then, defining z = t+s, we have ∫_-T^T Hn(t)Hn(t+s)ds = ∫_-T^T Hn(t)Hn(z)dz.

This does not change the substance of the advice I have given until now.

let g(z)=∫e^(z^2)* fn(z) dz
for n = 1

fn(z)= d/dz e^-(z^2)= -2 z e-(z^2)
So, g(z) = ∫-2 z * e^(z^2)* e-(z^2) dz
= ∫-2 z *e^[(z^2)-(z^2)] dz
= ∫-2 z dz
=-z^2

is not that true?

 

[EnumaElish]

Your calculation looks right.

Now you need to put in the integration limits (-T, T), multiply with Hn(t)(-1)^n, then take the limit T --> 0.

 

[T.Engineer]

Now you need to put in the integration limits (-T, T), multiply with Hn(t)(-1)^n, then take the limit T --> 0.[/QUOTE]

we have fn(z) = -Z^2 for n =1
by multiplying it by (-1)^n
we will have Z^2
by multiplying it by Hn(t),
where Hn(t)= (-1)^n * e^(t^2)* d^n/dt^n [e^-(t^2)]
Then, we will have:
Z^2 * (-1)^n * e^(t^2)* d^n/dt^n [e^-(t^2)], where z = t+s
So, how I will find the integral and I have two variables (t and Z)??
Can you help me, Please.

 

[EnumaElish]

Remember, the integral is a definite integral. It has limits "from -T to T." You need to evaluate the integral according to the limits.

Before you multiply the integral with Hn(t), you need to apply the integ. limits to -z^2. This will make z "disappear." The z^2 will be replaced with a quadratic function of T, say q(T).

Do you understand what q(T) is? Can you derive it explicitly?

 

[T.Engineer]

Remember, the integral is a definite integral. It has limits "from -T to T." You need to evaluate the integral according to the limits.

Before you multiply the integral with Hn(t), you need to apply the integ. limits to -z^2. This will make z "disappear." The z^2 will be replaced with a quadratic function of T, say q(T).

Do you understand what q(T) is? Can you derive it explicitly?

OK, I will have:
-z^2$$\left|^{T}_{-T}$$
then I will get
-(T)^2 - [- (-T)^2]= 0
So, the autocorelation for
Hn(t)Hn(t+s)ds = 0

is that true!

 

[EnumaElish]

It seems to be true for n=1.

Can you solve the general case for arbitrary n?

 

[T.Engineer]

It seems to be true for n=1.

Can you solve the general case for arbitrary n?

Actually, this is exactly what I am going to ask you about.
So, can you just help me a little bit?
And I have another question:
What about the autocorrelation of
Hn(t)Hm(t)dt , where m not equal to n

 

[EnumaElish]

I cannot see the LaTex equation for m. There seems to be a typesetting error.

EDIT: I can see it now.

For the Hn(t)Hn(t+s) case, you need to work on solving ∫e^(z^2)* fn(z) dz for arbitrary n.

The Hn(t)Hm(t)dt case is not a regular AC function. Go back to the MathWorld link (http://mathworld.wolfram.com/Autocorrelation.html) and review the definition. The AC function has a single argument (a single function), for example, ac(t) = AC(Hn(t)) or equivalently ac = AC(Hn). What you are proposing has two arguments, Hn and Hm. So it doesn't fit the definition of AC.

 

[T.Engineer]

when I written:the autocorrelation of Hn(t)Hm(t)dt
I meant that what about the autocorrelation function between the first derivative of H(t) and the second derivative of H(t)
Such as :
the autocorrelation function of H1(t)H2(t).
Thanks alot!

 

[T.Engineer]

I cannot see the LaTex equation for m. There seems to be a typesetting error.

EDIT: I can see it now.

For the Hn(t)Hn(t+s) case, you need to work on solving ∫e^(z^2)* fn(z) dz for arbitrary n.

The Hn(t)Hm(t)dt case is not a regular AC function. Go back to the MathWorld link (http://mathworld.wolfram.com/Autocorrelation.html) and review the definition. The AC function has a single argument (a single function), for example, ac(t) = AC(Hn(t)) or equivalently ac = AC(Hn). What you are proposing has two arguments, Hn and Hm. So it doesn't fit the definition of AC.

for n=1
∫e^(z^2)* fn(z) dz = -z^2
for n=2
∫e^(z^2)* fn(z) dz = $$\frac{3}{4}$$ z^3

So, should I continue for n=3,4,5,...

 

[EnumaElish]

when I written:the autocorrelation of Hn(t)Hm(t)dt
I meant that what about the autocorrelation function between the first derivative of H(t) and the second derivative of H(t)
Such as :
the autocorrelation function of H1(t)H2(t).
Thanks alot!

The AC function has a very specific definition; and it is defined for a single parent function.

I am going to look at your question that way. You can write the AC function of H1*H2 by following the general definition of AC for function F:

AC_F(t) = Lim_T--->0 ∫_-T^T F(t) F(t+s) ds / (2T)

and making the substitution F = H1*H2.

 

[EnumaElish]

for n=1
∫e^(z^2)* fn(z) dz = -z^2
for n=2
∫e^(z^2)* fn(z) dz = $$\frac{3}{4}$$ z^3

So, should I continue for n=3,4,5,...

You can continue and hope that you will "see" the general solution eventually by induction and guesswork. But the more direct approach is to try to solve the general case algebraically.

 

[T.Engineer]

The correlation is equal to the average of the product of two random variables and is defined as
cor(X,Y) = E[XY]
= ∫∞ −∞∫∞ −∞ xyf(x,y)dxdy

So, if the above equation seems to be correct, then I can define H1(t) as X and H2(t) as Y.
Can I work like this depending on the above?


And for a given set of variables, we use the correlation coefficient to give us the linear relationship between our variables. The correlation coefficient of two variables is defined in terms of their covariance and standard deviations, as seen below
ρ= cor(X,Y)/σχ σΥ

If there is no relationship between the variables then the correlation coefficient will be zero and if there is a perfect positive match it will be one.
If there is a perfect inverse relationship, where one set of variables increases while the other decreases, then the correlation coefficient will be negative one.

So, my second equastion is how to find the mean and variance for H1(t) and H2(t)
Hint: if Hn(t)= (-1)^n cos(2π fc t)* e^[(t^2)/4] *d^n/dt^n *e^[(t^2)/4]

Thanks alot!

 

[EnumaElish]

The correlation is equal to the average of the product of two random variables and is defined as
cor(X,Y) = E[XY]
= ∫∞ −∞∫∞ −∞ xyf(x,y)dxdy

This is the definition of covariance, not correlation.

So, if the above equation seems to be correct, then I can define H1(t) as X and H2(t) as Y.
Can I work like this depending on the above?

To obtain Cov(X,Y), yes.

And for a given set of variables, we use the correlation coefficient to give us the linear relationship between our variables. The correlation coefficient of two variables is defined in terms of their covariance and standard deviations, as seen below
ρ= cor(X,Y)/σχ σΥ

The correct formula is ρ = Cov(X,Y)/(σχ σΥ) where ρ is the correlation coefficient between X and Y.

If there is no relationship between the variables then the correlation coefficient will be zero and if there is a perfect positive match it will be one.
If there is a perfect inverse relationship, where one set of variables increases while the other decreases, then the correlation coefficient will be negative one.

Correct.

So, my second equastion is how to find the mean and variance for H1(t) and H2(t)
Hint: if Hn(t)= (-1)^n cos(2π fc t)* e^[(t^2)/4] *d^n/dt^n *e^[(t^2)/4]

Thanks alot!

You should follow the approach in this post: https://www.physicsforums.com/showthread.php?p=1387935#post1387935

 

[T.Engineer]

You should follow the approach in this post: https://www.physicsforums.com/showthread.php?p=1387935#post1387935[/QUOTE]

Thank you very much for being patient with me!
Do you know what my exact problem is?
I am going to transmit a signal, which it is described by Hn(t).
I want to prove the orthogonality of this function by using different derivative for this signal, If they are orthogonal to each other or not.
So, I suggest to find the autocorrelation function for them, I mean between H1(t) and H2(t). After that I have to find the mean and the variance for the result of autocorrelation function.

Am I on the correct way?

 

[EnumaElish]

I am not sure I understand completely. Can you fill in the blanks in the following statement:

"I'd like to show function ___ is orthogonal to function ___."

If your answers are H1 and H2, then why does this have anything to do with the AC function? In this case, you need to prove Cov(H1,H2) = 0, which is different than the AC function.

On the other hand, if your statement is "I'd like to show function H[n] is orthogonal with itself (over time)" (i.e. AC=0), then you are correct to study the AC function.

 

[T.Engineer]

I am not sure I understand completely. Can you fill in the blanks in the following statement:

"I'd like to show function _H1_ is orthogonal to function _H2_."

If your answers are H1 and H2, then why does this have anything to do with the AC function? In this case, you need to prove Cov(H1,H2) = 0, which is different than the AC function.

On the other hand, if your statement is "I'd like to show function H[n] is orthogonal with itself (over time)" (i.e. AC=0), then you are correct to study the AC function.

"I'd like to show function _H1_ is orthogonal to function _H2_."

 

[T.Engineer]

when we want to receive these signals, a correlation receiver will be used
So, the autocorrelation function for the transmitted signal should be determined and from the result of the autocorrelation function we will determine the Mean and Variance

 

[EnumaElish]

"I'd like to show function _H1_ is orthogonal to function _H2_."

The way to show H1 is orthogonal to H2 is to show Cov(H1,H2) = 0. This does not have anything with the AC function, which is a very specific transformation of a single parent function (for example, H1) with itself, over time (over a very short time interval).

So, the autocorrelation function for the transmitted signal should be determined and from the result of the autocorrelation function we will determine the Mean and Variance

I am not a signal processing engineer. I don't know what a correlation receiver is.

My best understanding is this:
You are trying to find the autocorrelation coefficient of a signal. http://en.wikipedia.org/wiki/Autocorrelation#Signal_processing

You need to first determine the R_ff(T) function.

Then you'd like to convert it into an AC coefficient as: r(T) = [R_ff(T) - Mean_f(T)^2]/Var_f(T). http://en.wikipedia.org/wiki/Autocovariance (look under "Normalization")

Am I close?

You can determine R_ff(T) analytically. But a simpler method is computer simulation. If you simulate N signals from your H1 function, then you can calculate the autocorrelation coefficient numerically (for n lags, or t - n), as in this link: http://w3eos.whoi.edu/12.747/notes/lect06/l06s02.html

 

[T.Engineer]

My best understanding is this:
You are trying to find the autocorrelation coefficient of a signal. http://en.wikipedia.org/wiki/Autocorrelation#Signal_processing

You need to first determine the R_ff(T) function.

Then you'd like to convert it into an AC coefficient as: r(T) = [R_ff(T) - Mean_f(T)^2]/Var_f(T). http://en.wikipedia.org/wiki/Autocovariance (look under "Normalization")

Am I close?

yes, exactly! My points are to determine the mean and variance.
So, as I understand from your comments that if I will find the correlation coefficient then I can determine the mean and variance.

 

[EnumaElish]

No. Typically you need to:
1. Calculate the mean
2. Calculate the variance
3. Calculate Rff
4. Put 1, 2, 3 together to calculate r (the corr. coeff.).

Is this helpful? If not, why not?

 

[T.Engineer]

No. Typically you need to:
1. Calculate the mean
2. Calculate the variance
3. Calculate Rff
4. Put 1, 2, 3 together to calculate r (the corr. coeff.).

Is this helpful? If not, why not?

Yes, it’s fine. I agree.
But I think I should start with determining Rff(T) and after that the mean and variance.
Is that’s right?

 

[EnumaElish]

Any order would be fine.