Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w))

In summary, the w-derivative of Fourier transform,d/dw (F{x(t)}), is calculated using the regular Fourier transform,X(w)=F{x(t)}, and the differentiation and duality properties of the Fourier transform.
  • #1
Alexei_Nomazov
3
0
TL;DR Summary
Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w))
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 w-derivative 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
25905b02d53ae042b113c038d50ac131fcf020d0


I can find that w-derivative 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:
tkBL5.png


is it possible to express the derivative, d/dw (F{x(t)}), in frequency domain using terms of X(w) ?

Many Thanks,
Desperate Engineer.
 
Physics news on Phys.org
  • #2
First I want to clarify your question, since I find your notation to be complicated. If ##X(\omega)## is the Fourier transform of ##x(t)##, are you asking if there is a way to write ##\frac{d}{d\omega} X(\omega)## in terms of ##X(\omega)##?

If you are looking for an actual formula, then the answer is no. Consider a couple of examples. If ##X(\omega)=1##, then ##\frac{d}{d\omega} X(\omega) = 0 = 0 \, X(\omega)##; on the other hand if ##X(\omega) = e^{j \omega \tau}## then ##\frac{d}{d\omega} X(\omega) = j\tau\,e^{j \omega \tau} = j\tau\, X(\omega)##. Fourier transforms are just functions - and when you learned calculus you had to learn a bunch of different examples of derivatives (polynomials, sinusoids, exponentials, etc) - so it shouldn't be a surprise that there isn't a simple expression like you seem to be looking for.

On the other hand, if you just want a symbolic expression that allows you to write it a slightly different way, then you can just write ##\frac{d}{d\omega} X(\omega) = \delta^\prime(\omega) \ast X(\omega)##, where ##\delta^\prime(\omega)## is the derivative of the delta function, and ##\ast## represents convolution. But this is just using the definition of the delta function and convolution, and doesn't actually help you compute anything.

Good luck,

jason
 
  • Like
Likes Alexei_Nomazov
  • #3
Well, when you learned calculus did you learn one single formula that gives you the derivative of any arbitrary differentiable function in terms of the original function? Yes, but it is just the definition of derivative, which probably isn't what you are looking for.

Perhaps if you tell us what you are actually trying to do we could help.

jason
 
  • #4
Thank you Jason.
I think i was misleading myself a little bit reducing my problem to strict technical one.
Your answer clarifies it.

Thanks again,
Alexei.
 

1. What is the Fourier Transform and why is it important in scientific calculations?

The Fourier Transform is a mathematical tool that allows us to break down a complex signal or function into its individual frequency components. It is used in a wide range of scientific fields, including signal processing, image analysis, and quantum mechanics, to name a few.

2. How is the Fourier Transform derivative calculated?

The derivative of the Fourier Transform is calculated by taking the derivative of the function in the time domain and then applying the Fourier Transform to the resulting function. This can also be expressed as the derivative of the Fourier Transform being equal to the Fourier Transform of the derivative of the function.

3. What does the derivative of the Fourier Transform represent?

The derivative of the Fourier Transform represents the rate of change of the frequency components of a signal. It can be used to analyze the changes in a signal over time or to identify specific frequencies that are changing rapidly.

4. How is the Fourier Transform derivative used in practical applications?

The Fourier Transform derivative is used in a variety of practical applications, such as image processing, audio signal analysis, and data compression. It can also be used in scientific research to analyze and understand complex systems and phenomena.

5. Are there any limitations or assumptions when calculating the Fourier Transform derivative?

Yes, there are some limitations and assumptions when calculating the Fourier Transform derivative. These include the assumption that the function being transformed is continuous and has a finite energy, and the limitation that the function must be differentiable at all points.

Similar threads

Replies
3
Views
990
Replies
11
Views
847
  • Differential Equations
Replies
4
Views
2K
Replies
17
Views
1K
Replies
4
Views
272
  • Calculus and Beyond Homework Help
Replies
1
Views
775
  • Calculus
Replies
8
Views
4K
Replies
18
Views
3K
Back
Top