A general form of Parseval's Theorem says that if two functions are expanded in a Fourier Series
f(x) =1/2 ao + Sigma [(an cos(nx)) + bn sin(nx)]
g(x) 1/2 ao' + Sigma [(an' cos(nx)) + bn' (sin(nx)]
Then the average value, < f(x)g(x)>, is:
1/4 ao = sigma[an an' + bn bn'] prove this and using any two functions
Please give an example.© BrainMass Inc. brainmass.com October 16, 2018, 8:51 pm ad1c9bdddf
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The Fourier eigenfunctions and are orthonormal for any integers n, m in the interval , which means that:
Of course, for any integers n, m
So assume we can write the function as a series expansion of the eigenfnctions:
To find the coefficients we simply multiply both sides of the equation by an eigenfunction.
For we get:
Since the summation is over n and not m we can bring the eigenfunction into the sum:
The Parseval's Theorem is analyzed. The average value is provided. Two functions are analyzed.
Fourier transform and energy
I am trying to get a feel for Fourier transforms and would like some help.
Please show all work/explanations.
1. Let g(t) = x(2t) - x*(2t- 1/2T0), assume that X(jw) is known
Find G(jw) in terms of X(jw).
2. Let x(t) = 10sin(200t)/t
a) Find X(jw)
b) Let x(t) be the input to a continuous LTI system with impulse response h(t) = sin(100t)/t.
Find the output y(t)
c) Find the energy in x(t) and the energy in y(t).