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State duality property of Fourier Transform.

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Subject : Signals & Systems

Topic : Continuous Time Fourier Transform (CTFT) and Discrete Time Fourier Transform (DTFT).

Difficulty: Medium

1 Answer
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Duality:

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Proof:

Inverse Fourier transform is given as $x(t)=12π \int_{-∞}^∞ X(ω)e^{jωt} \, dω $

Interchanging t by ω we get, $x(ω)=12π \int_{-∞}^∞ X(t)e^{jωt} \, dω $

Interchanging ω by –ω we get, $x(-ω)=12π \int_{-∞}^∞ X(t)e^{-jωt} \, dt $ $i.e. 2π x(-ω) =12π \int_{-∞}^∞ X(t)e^{-jωt} \, dt $

Right hand side of above equation is Fourier transform of X(t) i.e.

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To find: Fourier transform of 11 + t

Solution:

Duality property which states

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