IERG2051A_T3_Solutions

IERG2051A_T3_Solutions - IERG2051A Exercise Solutions for...

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IERG2051A Exercise Solutions for Tutorial 3 Exercise I 1. 𝑑𝑑 𝑑𝑑𝑑𝑑 [ 𝑒𝑒 −2𝑑𝑑 𝑢𝑢 ( 𝑡𝑡 ) ∗ 𝑢𝑢 ( 𝑡𝑡 )] = 𝑒𝑒 −2𝑑𝑑 𝑢𝑢 ( 𝑡𝑡 ) 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑢𝑢 ( 𝑡𝑡 ) = 𝑒𝑒 −2𝑑𝑑 𝑢𝑢 ( 𝑡𝑡 ) ∗ 𝛿𝛿 ( 𝑡𝑡 ) = 𝑒𝑒 −2𝑑𝑑 𝑢𝑢 ( 𝑡𝑡 ) 2. 𝑒𝑒 −2𝑑𝑑 𝑢𝑢 ( 𝑡𝑡 ) ∗ 𝑢𝑢 ( 𝑡𝑡 ) = 𝑑𝑑 𝑑𝑑𝑑𝑑 [ 𝑒𝑒 −2𝑑𝑑 𝑢𝑢 ( 𝜏𝜏 ) ∗ 𝑢𝑢 ( 𝜏𝜏 )] 𝑑𝑑 −∞ 𝑑𝑑𝜏𝜏 = [ 𝑒𝑒 −2𝑑𝑑 𝑢𝑢 ( 𝜏𝜏 )] 𝑑𝑑𝜏𝜏 𝑑𝑑 −∞ = 1 2 (1 − 𝑒𝑒 −2𝑑𝑑 ) 𝑢𝑢 ( 𝑡𝑡 ) 3. 𝑒𝑒 −2𝑑𝑑 𝑢𝑢 ( 𝑡𝑡 ) ∗ 𝑢𝑢 ( 𝑡𝑡 − 1) = 𝑑𝑑 𝑑𝑑𝑑𝑑 [ 𝑒𝑒 −2𝑑𝑑 𝑢𝑢 ( 𝜏𝜏 ) ∗ 𝑢𝑢 ( 𝜏𝜏 − 1)] 𝑑𝑑 −∞ 𝑑𝑑𝜏𝜏 = ��𝑒𝑒 −2𝑑𝑑 𝑢𝑢 ( 𝜏𝜏 ) 𝑑𝑑 𝑑𝑑𝜏𝜏 𝑢𝑢 ( 𝜏𝜏 − 1) �� 𝑑𝑑 −∞
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