Fourier Transform Derivation: Calculating X(t)

First Page 𝑋(𝑡) = 𝑋(𝑡) = 1 2𝜋 𝑋(𝜔)𝑒 𝑑𝜔 1 ∫ 2𝜋 (−2)𝑒 𝑑 𝜔 +∫ (2 )𝑒 𝐼 = (−2)𝑒 𝑑𝜔 = −2 𝑒 𝑑𝜔 𝑑𝜔 - The Rule 𝑒 𝑑𝜔 = −2 𝑒 𝑒 j𝑡 𝑑𝜔 = −2 𝑒 - 𝐼 = −2 𝑒 𝑑𝜔 = −2 j𝑡 - 𝑒 ( ) − 𝑒 (- ) = −2 j𝑡 (2)𝑒 𝑑𝜔 = 2 𝑒 𝑒 ( ) -𝑒 ( ) = −2 j𝑡 𝑖𝑡 𝑒 -2𝑒 = j𝑡 𝑑𝜔 = -2𝑒 -1 j𝑡 Second Page 𝑋(𝑡) = 𝑋(𝑡) = 1 -2 (1-𝑒 j𝑡 2𝜋 1 1 × [-1 + 2e2𝜋 jt ) 2(𝑒 + + 2e -2] = − 1) j𝑡 1 1 × [2(e 2𝜋 jt Rule 𝑒 +𝑒 = 2cos(2t) 𝑋(𝑡) = 1 1 ⋅ ⋅ [2⋅2cos (2t)−4] 2𝜋 𝑗t 𝑋(𝑡) = 1 1 ⋅ ⋅ (4cos (2t) − 4) 2𝜋 𝑗𝑡 𝑋(𝑡) = 1 4(cos (2t) − 1) ⋅ 2𝜋 jt 𝑋(𝑡) = 2 ⋅ cos (2t) − 1 𝜋jt + e- )-4]

Fourier Transform Derivation: Calculating X(t)

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Fourier Transform Derivation: Calculating X(t)

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