hw09su09_soln

# hw09su09_soln - GEORGIA INSTITUTE OF TECHNOLOGY ECE 2025...

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GEORGIA INSTITUTE OF TECHNOLOGY ECE 2025 SOLUTIONS Problem Set # 9 Summer 2009 Problem 9.1 (b) From: sin(4 πt ) ←→ jπδ ( ω + 4 π ) - jπδ ( ω - 4 π ) sin(50 πt ) ←→ jπδ ( ω + 50 π ) - jπδ ( ω - 50 π ) sin(4 πt )sin(50 πt ) ←→ 1 2 π ( ) 2 [ δ ( ω + 4 π ) - δ ( ω - 4 π )] * [ δ ( ω + 50 π ) - δ ( ω - 50 π )] = - π 2 [ δ ( ω + 54 π ) - δ ( ω + 46 π ) - δ ( ω ( - 46 π ) + δ ( ω - 54 π )] . (c) From: sin(4 πt ) πt ←→ u ( ω + 4 π ) - u ( ω - 4 π )] sin(50 πt ) ←→ jπδ ( ω + 25 π ) - jπδ ( ω - 25 π ) sin(4 πt ) πt sin(50 πt ) ←→ 1 2 π ( )[ u ( ω + 4 π ) - u ( ω - 4 π )] * [ δ ( ω + 50 π ) - δ ( ω - 50 π )] = j 2 [ u ( ω + 54 π ) - δ ( ω + 46 π ) - δ ( ω ( - 24 π ) + δ ( ω - 54 π )] . Problem 9.2 (a) e - 2 t u ( t ), delayed by 3. Thus e - 2 t +6 u ( t - 3) . (b) e - 2 t u ( t ), diﬀerentiated. Thus - 2e - 2 t u ( t ) + e - 2 t δ ( t ) = - 2e - 2 t u ( t ) . 1

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(c) e - 2 t u ( t ), delayed and diﬀerentiated. Thus - 2e - 2 t +6 u ( t - 3) . (d) 2sin ω ω ←→ u ( t + 1) - u ( t - 1), and k = -∞ π 5 δ ( ω - 2 πk/ 10) ←→ k = -∞ δ ( t - 10 k ). Thus X ( ) ←→ X k = -∞ [ u ( t + 1) - u ( t - 1)] * X k = -∞ δ ( t - 10 k ) = X k = -∞ [ u ( t + 1 - 10 k ) - u ( t - 1 - 10 k )] . Problem 9.3
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## This note was uploaded on 01/24/2010 for the course ECE 2025 taught by Professor Juang during the Spring '08 term at Georgia Tech.

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hw09su09_soln - GEORGIA INSTITUTE OF TECHNOLOGY ECE 2025...

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