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T.Engineer
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Homework Statement
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]
Homework Equations
The Attempt at a Solution
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