What Happens to the Fourier Transform When Functions Are Scaled and Shifted?

In summary, to find the Fourier transform of g(t) = f(3t) - f(4t + 7), you can use the shift property and time scaling property of the Fourier transform.
  • #1
gomes.
58
0
If f(t) has the Fourier transform F(ω), what is the Fourier transform of the function g(t) =
f(3t) − f(4t + 7)? Use the shift property and time scaling property of the Fourier transform
to obtain your answer.


I have no idea how to start, please help, thanks..
 
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  • #2
Well, there is a reason why the homework help posting template (the thing you deleted when you typed your post) asks you to list some of the relevant equations -- they often give you an idea of where to start. So let's do that:

What is the shift property of the Fourier transform? How might it be applied to the functions you have been given?

What is the time scaling property of the Fourier transform? How might it be applied to the functions you have been given?
 

FAQ: What Happens to the Fourier Transform When Functions Are Scaled and Shifted?

What is a Fourier transform?

A Fourier transform is a mathematical tool used to represent a function as a sum of sine and cosine functions. It is used to convert a signal from the time or spatial domain to the frequency domain.

What is the significance of a Fourier transform?

A Fourier transform allows us to analyze the frequency components of a signal and understand the underlying patterns or structures. It is used in a variety of fields such as signal processing, image processing, and physics, to name a few.

What is a shift in a Fourier transform?

A shift in a Fourier transform refers to the translation of a function in the time or spatial domain. It is represented by multiplying the original function with a complex exponential term, which results in a phase change in the frequency domain.

How does a Fourier transform handle shifts in a signal?

A Fourier transform handles shifts in a signal by applying a phase shift in the frequency domain. This allows us to analyze the frequency components of a shifted signal and understand the changes that occur due to the shift.

What are some applications of a Fourier transform with shifts?

Some applications of a Fourier transform with shifts include image registration, which involves aligning two images by applying a shift to one of them. It is also used in digital communication systems to correct for phase shifts in signals caused by noise or interference.

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