Fourier transform of an integral
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From: fisico32 on 30 Apr 2010 08:26 Hello Forum, I am trying to find out what is the FT of the integral of a generic function f(t) with arbitrary and finite limits of integration a and b. Most textbooks give the FT of the integral of a function with lower limit=-infinity and upper limit equal to t. Why? Why not use general limits of integration? Clearly, the FT of integral of a function is the FT of a function (which is the result of the integral)...that function need to satisfy the Dirichlet conditions to be Fourier integrables of course... thanks fisico32
From: Rune Allnor on 30 Apr 2010 09:03 On 30 apr, 14:26, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com> wrote: > Hello Forum, > > I am trying to find out what is the FT of the integral of a generic > function f(t) with arbitrary and finite limits of integration a and b. > > Most textbooks give the FT of the integral of a function with lower > limit=-infinity and upper limit equal to t. > > Why? > > Why not use general limits of integration? Because of convenience. Somebody - your guess of 'who' is as good as mine - have decided that the half-open integral is more useful than the closed integral between two finite limits. Of course, the closed integral is an intermediate result in the derivation of the half-open integral, so if you know how to derive the half-open integrals you already know how to handle the closed integral. Rune
From: dvsarwate on 30 Apr 2010 11:08 On Apr 30, 7:26 am, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com> wrote: > Hello Forum, > > I am trying to find out what is the FT of the integral of a generic > function f(t) with arbitrary and finite limits of integration a and b. > Assuming that a and b are fixed, the integral of a function f(t) from a to b is a fixed real number, not a function of time, and so the concept of Fourier transform does not apply. Can you rephrase your question?
From: fisico32 on 30 Apr 2010 11:19 >Assuming that a and b are fixed, the integral of a function >f(t) from a to b is a fixed real number, not a function of >time, and so the concept of Fourier transform does not apply. >Can you rephrase your question? > Hello dvsarwate, thanks for your input. You are right, we get just a number if the limits are finite numbers or both infinity, + and -. I guess I am considering the case where the two limits are finite numbers but still variables. So the final result I am looking for is an expression for the FT of a integral of a function with upper and lower limit of integration that are real (or complex)variables..... The common result in the book has the upper limit of integration equal to t and the lower limit equal to -infinity. I guess that is because we would eventually get a 1-dimensional function out of that integral say g(t), to be then Fourier transformed. If the two limits of integration were both variables we could only do a partial fourier transform or need to do a double Fourier transform...correct? thanks for any validation and further suggestion..... fisico32
From: dvsarwate on 30 Apr 2010 12:36
On Apr 30, 10:19 am, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com> wrote: > > So the final result I am looking for is an expression for the FT of a > integral of a function with upper and lower limit of integration that are > real (or complex)variables..... > > The common result in the book has the upper limit of integration equal to t > and the lower limit equal to -infinity. I guess that is because we would > eventually get a 1-dimensional function out of that integral say g(t), to > be then Fourier transformed. > If the two limits of integration were both variables we could only do a > partial fourier transform or need to do a double Fourier > transform...correct? Let f(t) be a function with Fourier transform F(w). Let g(t) be another function whose value equals the area under the curve f to the left of t, that is, area from -infinity to t. Thus, g(0) = area under the curve f to the left of 0, g(3.1415926) = area under the curve to the left of 3.1415926, g(2010) = area under the curve f to the left of 2010, etc. The standard result in Fourier theory gives the Fourier transform G(w) of g(t) in terms of F(w). So, tell us. What is your definition of the function h(t) whose Fourier transform you want to find? Just saying a and b are variables is not enough, you need to define the function just a tad more. How about telling us the values of h(0), h(3.1415926) and h(2010) for starters? What areas are we talking about since your function h, being an integral of f(t), is obviously an area? --Dilip Sarwate |