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August 11th, 2019, 02:14 AM  #1 
Newbie Joined: Jun 2017 From: Israel Posts: 2 Thanks: 0  Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w))
Hello to my Math Fellows, Problem: I am looking for a way to calculate wderivative of Fourier transform, d/dw (F{x(t)}), in terms of regular Fourier transform, X(w)=F{x(t)}. Definition Based Solution (not good enough): from I can find that wderivative of Fourier transform for x(t) is Fourier transform of t*x(t) multiplied by j: d/dw (F{x(t)})=d/dw(X(w))=j*F{t*x(t)} Question: But, taking into account the differentiation and duality properties of Fourier transform: is it possible to express the derivative, d/dw (F{x(t)}), in frequency domain using terms of X(w) ??? Many Thanks, Desperate Engineer. Last edited by skipjack; August 11th, 2019 at 10:00 AM. 
August 11th, 2019, 09:52 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,529 Thanks: 1389 
$X(\omega) = \displaystyle \int_{\infty}^\infty~x(t) e^{j\omega t}~dt$ $\dfrac{d}{d\omega} X(\omega) = \displaystyle \int_{\infty}^\infty~x(t) \dfrac {d}{d\omega}e^{j\omega t}~dt = $ $ \displaystyle \int_{\infty}^\infty~(jt) x(t) e^{j\omega t}~dt= \mathscr{F}\{(jt)x(t)\}$ repeated derivatives add another power of $(jt)$ obtaining the full property $\dfrac{d^n}{d\omega^n} X(\omega) = \mathscr{F}\{(jt)^n x(t)\} $ This is property 11 on your sheet. 

Tags 
calculation, d or dw, derivative, fourier, fourier transform, fxtd or dwxw, transform 
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