Subject : Signals & Systems

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

Difficulty: Medium

Duality:

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 …