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EE 341 HW 4S Spring 2017.pdf - EE 341 Discrete Time Linear...

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EE 341 Discrete Time Linear Signals and Systems Problem Set #4 Solution Problem 1 (a) when 𝑛 ≥ 0 ?[𝑛] = ?[𝑛] ∗ ℎ[𝑛] ?[𝑛] = ∑ ?[𝑘]ℎ[𝑛 − 𝑘] 𝑘=−∞ = ∑ ? 𝑘 ? 𝑛−𝑘 𝑢[𝑛 − 𝑘] 𝑘=−∞ = ∑ ? 𝑘 ? 𝑛−𝑘 𝑘=0 = ? 𝑛 ∑ ? 𝑘 ? −𝑘 𝑘=0 = ? 𝑛 ∑ ( ? ? ) 𝑘 𝑘=0 = ? 𝑛 1 − ( ? ? ) 𝑛+1 1 − ( ? ? ) = ? 𝑛+1 − ? 𝑛+1 ? − ? 𝑢[𝑛] (c) when 𝑛 ≤ 6 ?[𝑛] = ?[𝑛] ∗ ℎ[𝑛] ?[𝑛] = ∑ ?[𝑘]ℎ[𝑛 − 𝑘] 𝑘=−∞ = ∑ (− 1 2 ) 𝑘 4 𝑛−𝑘 𝑘=4 = ∑ (− 1 2 ) 𝑘 4 𝑛−𝑘 𝑘=4 = 4 𝑛 ∑ (− 1 8 ) 𝑘 𝑘=4 = 4 𝑛 (− 1 8 ) 4 − (− 1 8 ) 1 − (− 1 8 ) ?[𝑛] = 4 𝑛 8 9 (− 1 8 ) 4 = 4 𝑛 4608 when 𝑛 ≥ 7 ?[𝑛] = ?[𝑛] ∗ ℎ[𝑛] ?[𝑛] = ∑ ?[𝑘]ℎ[𝑛 − 𝑘] 𝑘=−∞ = (− 1 2 ) 𝑘 4 𝑛−𝑘 𝑘=𝑛−2 = (− 1 2 ) 𝑘 4 𝑛−𝑘 𝑘=𝑛−2 = 4 𝑛 (− 1 8 ) 𝑘 𝑘=𝑛−2
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= 4 𝑛 (− 1 8 ) 𝑛−2 − (− 1 8 ) 1 − (− 1 8 ) ?[𝑛] = 4 𝑛 8 9 (− 1 8 ) 𝑛−2 = 4 𝑛 4608 Problem 2
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