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Design >> Analog Design >> Fourier Transform of Integration
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Message started by qiushidaren on Sep 14^th, 2008, 1:12am

Title: Fourier Transform of Integration
Post by qiushidaren on Sep 14^th, 2008, 1:12am

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

Title: Re: Fourier Transform of Integration
Post by pancho_hideboo on Sep 14^th, 2008, 2:32am

qiushidaren wrote on Sep 14^th, 2008, 1:12am:

πX(0)δ(ω) come from in the formulation for Fourier Transform of Integration.
I can’t find any details from Oppenhim’s classical book Signals and Systems,
can anybody help me about this confused question?

As far as my memory is correct, such issue is descripted also in a famous classical text book you refer.
Or see text books written by "A.Papoulis".

From direct thought, if X(0) is not zero, it is very reasonable there is an impulse spectrum around ω=0 for integral of x(t).
Here X(ω) is fourier transform of x(t).

[Mathematical Hint]
Fourier transform of step function u(t) is πδ(ω)+1/jω.
Here you have to learn a nature of delta function δ(ω).
If ω*F1(ω)=ω*F2(ω), it must be F1(ω)=F2(ω)+k*δ(ω), here k is any constant.

Title: Re: Fourier Transform of Integration
Post by thechopper on Sep 18^th, 2008, 7:28pm

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

Title: Re: Fourier Transform of Integration
Post by qiushidaren on Oct 16^th, 2008, 6:48pm

Thank you, I understand it now. ;)

-Terry