Fourier Transform - Scaling Property

In summary, the Fourier transform of (1/p)e^{[(-pi*x^2)/p^2]} for any p > 0 is e^{-pi*p^2 * u^2} or p*e^{(-pi*u^2)/p} depending on how the scaling property is applied.
  • #1
snesnerd
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Homework Statement



Find the Fourier transform of (1/p)e^{[(-pi*x^2)/p^2]} for any p > 0

Homework Equations



The Fourier transform of e^{-pi*x^2} is e^{-pi*u^2}.

The scaling property is given to be f(px) ----> (1/p)f(u/p)

The Attempt at a Solution



Using the information above, I got p*e^{(-pi*u^2)/p}.
On another attempt, I got e^{-pi*p^2 * u^2}.
I am not sure if either one of these is correct. I have a hard time following the scaling property.
 
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  • #2
We can't tell where you're going wrong if you don't show your work. Posting your final results is virtually useless.
 

Related to Fourier Transform - Scaling Property

1. What is the scaling property of the Fourier Transform?

The scaling property of the Fourier Transform states that if a function f(x) is stretched or compressed by a factor of c, then its Fourier Transform F(ω) will also be stretched or compressed by the same factor c.

2. How is the scaling property mathematically represented?

The scaling property is represented by the equation: F(c·x) = (1/c)·F(ω/c), where c is the scaling factor.

3. What is the significance of the scaling property in signal processing?

The scaling property allows us to analyze signals at different scales without losing information or altering the signal's frequency components. It also helps in scaling signals to fit within a certain range, making it easier to analyze and process them.

4. Can the scaling property be applied to both continuous and discrete signals?

Yes, the scaling property can be applied to both continuous and discrete signals. In the case of discrete signals, the scaling factor c represents the sampling rate.

5. How does the scaling property affect the Fourier Transform of a signal?

The scaling property alters the amplitude and frequency of the Fourier Transform of a signal. Specifically, increasing the scaling factor will decrease the amplitude and increase the frequency, while decreasing the scaling factor will increase the amplitude and decrease the frequency.

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