ece3101_Homework3_SOLNS_F19 (1).docx - B Olson Basic Fourier Transform Analysis 1 Determine the Fourier transforms of the following functions and sketch

# ece3101_Homework3_SOLNS_F19 (1).docx - B Olson Basic...

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B. Olson Basic Fourier Transform Analysis 1) Determine the Fourier transforms of the following functions and sketch the Fourier Transform as a function of a) f ( t )= 3 δ ( t ) F ( ω )= 3 −∞ δ ( t ) e jωt dt = 3 e jωt | t = 0 = 3 e 0 = 3 b) f ( t )= 4 f ( t )= 4 ℑ{ 4 }= 4 −∞ e jω t dt = 4 2 πδ ( ω ) = 8 πδ ( ω )
c) f ( t )= cos 2 ( ω 0 t ) f ( t )= cos 2 ( ω 0 t )= 1 2 + 1 2 cos ( 2 ω 0 t ) ℑ{ f ( t )}=ℑ { 1 2 + 1 2 1 2 ( e + j 2 ω 0 t + e j 2 ω 0 t ) ) { e j 2 ω 0 t ¿ = 2 πδ ( ω + 2 ω 0 ) { e + j 2 ω 0 t ¿ = 2 πδ ( ω 2 ω 0 ) { 1 = e + j 0 t ¿ = 2 πδ ( ω ) ℑ{ f ( t )}= πδ ( ω )+ π 2 δ ( ω 2 ω 0 )+ π 2 δ ( ω + 2 ω 0 ) d) f ( t )= 1 + sin ( ω 0 t ) ℑ{ f ( t )}=ℑ { 1 + 1 2 j ( e + 0 t e 0 t ) } { e + 0 t } = 2 πδ ( ω ω 0 ) { e 0 t } = 2 πδ ( ω + ω 0 ) { 1 } = 2 πδ ( ω ) ℑ{ f ( t )}= 2 πδ ( ω ) + π j δ ( ω ω 0 )− π j δ ( ω + ω 0 ) -20 0 20 F() 2 2 0 0 F() 2 j - j -0
e) f ( t )= k =− 100 100 2 −| k | e j 100 k t F ( ω )= −∞ f ( t ) e jωt dt = −∞ k =− 100 100 2 −| k | e j 100 k t e jωt dt = k =− 100 100 2 −| k | −∞ e j 100 k t e jωt dt = k =− 100

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