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!