Hi all,

I have a question recently which I couldn’t find an answer to, when I was an undergraduate in univercity, that is where dose the second term πX(0)δ(ω) come from in the formulation for Fourier Transform of Integration which is showed below, I can’t find any details from Oppenhim’s classical book Signals and Systems, can anybody help me about this confused question?

-Terry

Hi

I could find it finally....

The Fourier transform of the  ∫x(t) dt between -∞ and t is 1/ω*F(ω) provided that ∫x(t) dt between -∞ and +∞ is zero (such integral is the "DC component of x(t) )

If that is not the case, then ∫x(t) dt between -∞ and t can be expressed as ∫x(t)u(k-t)dt between -∞ and +∞, which is the convolution integral. Applying the Fourier transform to the convolution you get the product of the two functions spectra and therefore the composite fourier transform will be:

1/jω*F(ω)+ π*δ(ω)*F(ω) and δ(ω)*F(ω) = δ(ω)*F(0)


Hope this helps

Tosei