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Find the solution for the following equation

  1. Aug 27, 2007 #1
    1. The problem statement, all variables and given/known data

    if H_n(t)= (-1)^n * e^(t^2) * d^n/dt^n * e^(-t^2)

    where n=1,2,...,N

    from the orthogonality property of Hermite polynomials we will have:

    [tex]\int^{\infty}_{-\infty} e^{-t^2} H_n(t) H_m(t)dt [/tex] = [tex]\delta_n,m[/tex] 2^n n! [tex]\sqrt{}pi[/tex]

    this gives


    N_n= [tex]\frac{1}{\sqrt{}2^n n! \sqrt{}pi}[/tex]


    consequently

    [tex]\int^{\infty}_{-\infty}\varphi_n(t)\varphi_m(t) dt [/tex] = [tex]\delta_n,m[/tex]

    let g_n(t) = H_n(t) cos (2 pi fc t)

    the value of fc will be chosen in such a way that still keep the orthogonality property of Hermite polynomial.

    find out if cos (2 pi fc t) is positive over the relevant range as follow


    [tex]\int^{\infty}_{-\infty}\varphi_n(t)\varphi_m(t) dt [/tex] = [tex]\delta_n,m[/tex]



    2. Relevant equations




    3. The attempt at a solution
     
    Last edited: Aug 27, 2007
  2. jcsd
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