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Question: Hilbert transform

Question:Hilbert transform

hugocoolens 5
cauchyprincipalvalue transform
March 06 2015
1

I'm trying to use Maple to show that the Hilbert transform of the natural log of |H(jw)| is -arctan(H(jw))

for a minimum phase network; The network I choose is the simplest filter there exists i.e. a low pass RC-filter

with transfer function H(jw)=1/(1+tau*j*omega) therefore |H(jw)|=1/sqrt(1+(tau*omega)^2))

Here is what I did:

assume(omega > 0); assume(tau > 0); interface(showassumed = 0);

result5 := (int((1/2)*log(1+(tau*nu)^2)/(nu-omega), nu = -infinity .. infinity, CauchyPrincipalValue))/Pi

simplify(result5, symbolic)

 

unfortunately this does not give me the expected result: -arctan(w*tau*omega)

can anyone here tell me what the right way to do it is?

 

thanks in advance

hugo
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