How to Use Duality Property for Finding Fourier Transform of sin x / x?

In summary, the person is seeking help in finding the Fourier transform of the function sin x / x using the duality property of Fourier transforms. They mention using Jordan's Lemma but it does not apply in this case. They also mention the use of a square wave in the time domain with duty cycle d as a possible solution.
  • #1
eckiller
44
0
Hi, how do I find the Fourier transform of this function sin x / x, i.e.,

f* = Integral( sin x / x * exp( i*w*x) dx from -infinity to +infinity ).

I've been using Jordan's Lemma up to this point, but it doesn't seem to
apply here as a way to evaluate the integral.

Thanks for any help.
 
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  • #2
Hint: Use the duality property of Fourier transforms. Remember that d*sinc(w) = d*sin(pi*w*d) / (n*pi*w*d) is the Fourier transform of a square wave in the time domain with duty cycle d.

Edit: Fixed some things. If the time domain part confuses you, ignore it; I learned this stuff primarily from a signals & systems perspective.
 
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What is the Fourier Transform?

The Fourier Transform is a mathematical tool used to decompose a function into its constituent frequencies. It allows us to represent a function in the frequency domain instead of the time domain.

What is the significance of "sin x / x" in the Fourier Transform?

"Sin x / x" is a commonly used function in the Fourier Transform because it is a basic building block for creating more complex functions. It has a Fourier Transform that is well-defined and can be used to represent a wide range of signals.

What is the mathematical formula for the Fourier Transform of "sin x / x"?

The Fourier Transform of "sin x / x" is given by F(ω) = π/2 for |ω| ≤ 1 and F(ω) = 0 for |ω| > 1.

How is the Fourier Transform of "sin x / x" used in signal processing?

The Fourier Transform of "sin x / x" is used in signal processing to analyze and manipulate signals in the frequency domain. It allows us to filter out specific frequencies, extract useful information from a signal, and perform operations such as convolution and correlation.

Can the Fourier Transform of "sin x / x" be applied to non-periodic functions?

Yes, the Fourier Transform of "sin x / x" can be applied to non-periodic functions. The function can be extended and used to represent non-periodic functions in the frequency domain. However, the Fourier Transform of a non-periodic function may not have a nice closed form like it does for "sin x / x".

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