Posting Guidelines for T.Engineer - Graduate/Post-Graduate Level

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As per PF posting guidelines:

if your problem is related to undergrad material, please post it under the relevant homework section,

if your problem can be considered graduate or post-graduate level, then:
-- please identify the problem as completely as possible as best as you can.
-- please show any work you have done to solve the problem
-- please report all of your important results
-- please post specific questions. Describe your problem as narrowly as possible.

Thank you.
 
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EnumaElish said:
As per PF posting guidelines:

if your problem can be considered graduate or post-graduate level, then:
-- please identify the problem as completely as possible as best as you can.
-- please show any work you have done to solve the problem
-- please report all of your important results
-- please post specific questions. Describe your problem as narrowly as possible.

Thank you.

I am going to study the properties of Hermite polynomials mathematically and by simulation.
Hermite Polynomial has the following formula:
Hn(t) = (-1)^n * e^(t^2) * d^n/dt^n e-^(t^2) … (1)

Hermite polynomials had been modulated and modified to be as follows

Hn(t) = (-1)^n * cos(2π*fc*t)e^(t^2) * d^n/dt^n e-^(t^2)…. (2)

Where fc= 6.85GHz, and n= 1,2,3,…,N

The modulated and modified Hermite polynomials are orthogonal to each other for different numbers of n.

In order to transmit a signal which represented by eq(2) with k users active in the system, multiple access interference is a factor limiting for large number of users.
Let each user use different value of n. For example: user1 uses eq(2) with n=1, user2 uses eq(2) with n=2,…., etc.

The medium access to the system is achieved through the assignment of a unique Time Hoping code sequence per user, to reduce the multi-User interference.
Also, the transmitted signal is modulated by pulse position modulation.

So, the transmitted signal will be represented by:

S(t)= \sum^{\infty}_{j=-\infty} Hn(t- jTf - cj Tc - d^kj)

My question is:
I’d like to find an expression for the probability density function, the mean, and variance using eq(2).

My results:
1. The mean and variance depend on pulse shape which is represented by
eq(2) .
2. From research I had found the following equations that I couldn’t
understand what it refer to exactly.
For example:
\int^{\infty}_{-\infty}e^-(t^2) Hn(t)Hm(t) dt =
0 if n \neq m
or
2^n*n! \sqrt{π} if n =m

And I have a paper and it contain all the related equation of Hermite polynomial but I couldn't understand it very well .
if you want I can send it to you to explain it to me.
Thanks alot!
 
1. What is each user doing? Is each user transmitting a signal (e.g. at random times)? Or are they listening to signals?
2. Where does the signal originate from? One of the users? Or not?
3. Who receives the signal? All users? Some of them?
4. What is the objective? Can you state it in non-technical terms? (E.g., "to send a signal from user A to user B, but other users C to Z create a noise.")
 
EnumaElish said:
1. What is each user doing? Is each user transmitting a signal (e.g. at random times)? Or are they listening to signals?

If we will have k users, where every user has a transmitted signal represented by eq(2) with different values of n.
The time of transmitting a signal for each user will be specified according to time hoping techniques which suggest:
For each user it specified a pseudorandom cj taking integer range 0<cj<Nh.
Where Nh is the number of hops.
Let Tf is the time duration of a frame. Tc is the hop width in Time hopping systems satisfying Tf = Tc*Nh
d is the delay associated with PPM.
as the following equation
S(t)= \sum^{\infty}_{j=-\infty} Hn(t- jTf - cj Tc - d^kj) ...(3)


EnumaElish said:
2. Where does the signal originate from? One of the users? Or not?

If you mean how many users will be activated in a specific time, If this is right, let say 4 users in a specific time.

EnumaElish said:
3. Who receives the signal? All users? Some of them?

Let assume we will have a single receiver. The receiver will behave just like a differentiated device.
If perfect synchronization between transmitter and receiver is assumed, given the orthogonality property of the waveforms, the transmit code word can be detected in a
symbol-by-symbol maximum likely hood fashion by correlating the received signal with each of the N possible waveforms.

EnumaElish said:
4. What is the objective? Can you state it in non-technical terms? (E.g., "to send a signal from user A to user B, but other users C to Z create a noise.")

We suggest a system with 4 users (where k=4).
User1 will transmit the following sequence (0110), where at t=t0, bit0 is transmitted.
User2 will transmit the following sequence (1010), where at t=t0, bit1 is transmitted.
User3 will transmit the following sequence (1110), where at t=t0, bit1 is transmitted.
User4 will transmit the following sequence (0010), where at t=t0, bit0 is transmitted.

So, at time t=to , the receiver should detect the following (0110).

The transmitted bit from any user will be considered as a noise for the others.
 
Since cj is random, # = t - jTf - cjTc - dkj is random.

Suppose cj is distributed (discrete) uniformly over the interval [0,Nh] for each user. If nothing else is random in the expression # = t - jTf - cjTc - dkj, then # is distributed (discrete) uniformly over interval [A,B] where A = t - jTf - dkj and B = t - jTf - NhTc - dkj.

See http://en.wikipedia.org/wiki/Uniform_distribution_(discrete)

For given k and n, you can find the mean and the variance of Hn(#) either analytically or by simulation.

I understand k indexes the user. What does j index? Does it index bit sequence (first bit, second bit)?

So, at time t=to , the receiver should detect the following (0110).
At t0, does the receiver detect 0110, or just 0?
 
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EnumaElish said:
Since cj is random, # = t - jTf - cjTc - dkj is random.

Suppose cj is distributed (discrete) uniformly over the interval [0,Nh] for each user. If nothing else is random in the expression # = t - jTf - cjTc - dkj, then # is distributed (discrete) uniformly over interval [A,B] where A = t - jTf - dkj and B = t - jTf - NhTc - dkj.

See http://en.wikipedia.org/wiki/Uniform_distribution_(discrete)
For given k and n, you can find the mean and the variance of Hn(#) either analytically or by simulation.

how? where should I applied my function? I couldn't understand?
please, can you show me an example?
Thanks alot!

EnumaElish said:
At t0, does the receiver detect 0110, or just 0?

I think it should detect just 0.
 
I understand k indexes the user. What does j index? Does it index bit sequence (first bit, second bit)?

j is the index for pseudorandom time hopping sequence.
 
ok I will try to do somthing just give me little time?
I think I find it "I hope so".
Thanks alot!
 
  • #10
E(Hn(#)) = \sum_{q=1}^{Nh} P(cjq) Hn(#(cjq))

where P(cjq) = 1/Nh for q = 1, ..., Nh; cjq = q and #(cjq) is the value of Hn's generic argument # evaluated at cjq, that is # = t - jTf - cjqTc - dkj = t - jTf - q Tc - dkj.

This mean is conditional on the (given) values of all parameters other than cj. For example, t, jTf, Tc, dkj and n are all assumed given (constant).
 
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  • #11
Actually they find the autocorrelation function for just Hn(t)

\int^{\infty}_{-\infty} Hn(t)Hm(t) dt

where m not equal to n.
also Hm(t) is the first derivative function of Hn(t)
And then they try to find the mean and variance.
So, I believe we should get the autocorrelation function and then use the result to find the mean and variance
 
  • #12
can you for example explain to me what it means each equation in the attached file, please.
Thank you!
 

Attachments

  • #13
The text says (15) is the orthogonality condition.

(16) implies 1/N^2 = I/\delta where I is the integral in Equation (15).

(17) is a re-statement of (15) with \psi=H/N.

(18) is a linear approximation to \psi_n(t-\tau) with the c coefficients to be determined.

(19) derives the c coefficients from (17) and (18). Each c coefficient is a correlation function between \psi_n and \psi_m.
 
  • #14
(16) implies 1/N^2 = I/\delta where I is the integral in Equation (15).

Why it is important to find the Normalization coefficient Nn which is represented by equation (16)
 
  • #15
My guess is, in order to simplify things. To use (18) as a linear approximation to H/N, and to derive (19) as a correlation function between H_n/N_n and H_m/N_m.
 
  • #16
EnumaElish said:
My guess is, in order to simplify things. To use (18) as a linear approximation to H/N, and to derive (19) as a correlation function between H_n/N_n and H_m/N_m.

this is if :
Hn(t) = (-1)^n * e^(t^2)* d^n/dt^n * e^(-t^2)

what about if :
Hn(t)= (-1)^n * e^(t^2)* d^n/dt^n * e^(-t^2) * cos (2\pi fc t)

where fc is a constant.
will we get the same result?
 
  • #17
This will change both sides of (15), so either of two things will have to change:

EITHER the definition of N will change,

OR the definition of \psi ("psi") will change. Specifically, new psi = old psi * cos(...). I don't know whether (19) will still be valid.
 
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  • #18
EnumaElish said:
(16) implies 1/N^2 = I/\delta where I is the integral in Equation (15).

Accroding to what, they implies 1/N^2 = I/\delta
If I will apply it to my new function with cos (...)
then how would I work.
May be you can help me to make it more cleare to me,because till now I don't know how they get the result in eq(16) and according to what?
Thanks alot!
 
  • #19
If you take (15) and divide both sides with \delta you will get 1/N^2.

(16) is not a result; it is a definition.

Let's say that you introduce the cos(...) term but N remains the same. In that case, new psi = old psi * cos(...).
 
  • #20
and what about equations 17, 18, and 19.
will they still the same.
when the new psi= old psi* cos (2 pi fc t)
will eqations 17, 18 , and 19 give the same results?
 
  • #21
(17) will apply.

(18) is a definition conditional upon the c's, so my guess is it will apply.

(19) relies on (18) and (17), so my guess is (19) will still apply.

But you should verify these.
 
  • #22
but I don't know how?
 
  • #23
Study your textbook and seek help from your teachers.

Do you know how to integrate?

Can you find antiderivatives?
 
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  • #24
I don't think it will be so easy?
May be I should use a formula that it will help me to find the result of the integrals, or may be I should use a mathimatical program to find it out.
 
  • #25
Do you know how to integrate? Yes or no?

What is the integral of "t dt" from 0 to 1?
 
  • #26
it is
1/2
 
  • #27
Can you spend some time to understand how your textbook derives (19) from (18) and (17)?
 
  • #28
what about
\int^{\infty}_{-\infty} Hn(t) Hm(t-d) dt

where H(t)= (-1)^n * e^(t^2) * d^n/dt^n * e^(-t^2)* cos (2 pi fc t)
 
  • #29
Why are you changing the subject?

Are you, or are you not willing to figure out how you can get to (19) from (18) and (17)?
 
  • #30
for Hn(t)= (-1)^n * e^(t^2) * d^n/dt^n * e^(-t^2)* cos (2 pi fc t)

where fc will be chosen such as to still keeping the orthogonality of Hn(t) for different integer values for n.

so,
Psi_n(t)= N_n * Hn(t)

where N_n = \frac{1}{\sqrt{2^n*n!*\sqrt{pi}}}
according to the definition as you said.

consequently I believe we will get the same results as in equation 17, 18, and 19.

yes I still willing to know how to get 17, 18, and 19.
 
  • #31
That's a good guess, but you should try to derive (19) from (18) and (17) to understand the mechanics.
 
  • #32
let assume that our function g(t) instead of H(t)
So that,

g(t)= (-1)^n * e^(t^2) * d^n/dt^n * e^(-t^2)* cos (2 pi fc t)

1. we can say that g(t) can satisfied equations 15 and 17 because of its orthogonality properties.
Is that true?

2. in equation 18, I didnt understand what C_n,m(t) is reffered to?
 
  • #33
according to the text that I have and I couldn't attached it here, the coeffecient C_m,n(t) is the the cross correlation function between nth and mth order pulses at point t.
 
  • #35
EnumaElish said:
What is equation [9]? Or is it source [9]?

It is a reference which is : A. D. Poularikas, The Transforms and Applications Hand book CRC Press, Boca Raton, Fla, USA, 2nd edition, 2000.
 
  • #36
Do you have it, or can you get it?
 
  • #37
EnumaElish said:
Do you have it, or can you get it?

No, I don't have it.
 
  • #38
Is there a library near you?
 
  • #39
yes, why?
I don't think that they have it.
 
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  • #40
Have you worked on deriving (19)?
 
  • #41
As I said to you:
let assume that our function g(t) instead of H(t)
So that,

g(t)= (-1)^n * e^(t^2) * d^n/dt^n * e^(-t^2)* cos (2 pi fc t)

we can say that g(t) can satisfied equations 15 and 17 because of its orthogonality properties.
Is that true?
 
  • #42
Is cos (2 pi fc t) positive over the relevant range?

You may want to post it in the homework section as a separate question.
 
  • #43
OK, I will.
Thank you very much!
 
  • #44
If the value of fc will be chosen in such a way that still keep the orthogonality property of Hermite polynomial, then orthogonality is preserved. (15) will apply, although with a different normalization constant and different psi functions.

New psi = old psi * cos (2 pi fc t).
 
  • #45
(15) will apply, although with a different normalization constant

do you mean with different N_n which is represented by eq(16)
 
  • #46
Correct; that's because the integral in (15) will evaluate to a different output.

Another possibility is N_n will remain the same, but the delta will be different. Or both might change.

But the "qualitative" result will not change, as long as fc is chosen to preserve orthogonality. That is, you will get to (17) with the new psi functions.
 
  • #47
1. please, can tell me how to find the normalization coeffecient N_n?
2. you said different δ _n,m.
I know thet δ _n,m is Kronecker delta function, how it can be changed?
Thanks a lot!
 
  • #48
1. N is determined by the output of the integral in (15). If the integral evaluated to δ*K for arbitrary K, then the norm. constant would have been N = 1/sqrt(K).

2. The δ itself won't change; but you may have something like Integral = z(δ 2nn!\sqrt{\pi}) for some function z.
 
  • #49
1. you mean for arbitrary n,m.
2. what do you mean by z.
3. can you tell me how to evalute eq(15) to get this result: δ_n,m 2^n n! sqr(pi).
if I you will know how they get this result for Hn, Hm, so I can also evaluted for my equation with Hn * cos (...)
but this is my problem I don't know how they get this general formula.
 
  • #50
1. yes

2. arbitrary function that results from including the cos term in the integrand (I haven't tried to integrate (15) with or without the cos term, so I don't know what z actually "looks like," even if we assume that a closed-form solution exists with the cos term)

3. I don't know; I think [9] might have the answer. Someone has suggested to look it up from an integration table (under another thread in the homework section).
 
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