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Homework2

Homework2 - x t = sin4 πt πt sin(50 πt ↔ X jω =(d x t...

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ELCT 321 Digital Signal Processing Homework Assignment # 2 (Chapter 11: Continuous Time Fourier Transfrom) by Dr. Yong-June Shin Assigned: September 1, 2011 Due: September 8, 2011 1. (20 Points) The delay property of Fourier transform states that if X ( ) is the Fourier transform of x ( t ), then x ( t - t d ) e - jωt d X ( ) Use this property to find the Fourier transform of the following signals: (a) x ( t ) = δ ( t + 1) + 2 δ ( t ) + δ ( t - 1) (b) x ( t ) = sin(100 π ( t - 2)) π ( t - 2) (c) x ( t ) = e - t u ( t ) - e - t u ( t - 4) Note: The “ ” symbol stands for “Fourier transform” 2. (20 Points) For each of the following cases, use the table of known Fourier transform pairs to find x ( t ) when (a) X ( ) = 0 . 1+ e - 0 . 2 (b) X ( ) = 2 + 2 cos( ω ) (c) X ( ) = 1 1+ - 1 2+ (d) X ( ) = ( ω - 100 π ) - ( ω + 100 π ) 3. (20 Points) In each of the following cases, use known Fourier trans- form pairs together with Fourier transform properties to complete the following Fourier transform pair relationships: (a) x ( t ) = u ( t + 3) u (3 - t ) X ( ) = (b) x ( t ) = sin(4 πt ) sin(50 πt ) X

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Unformatted text preview: x ( t ) = sin4 πt πt sin(50 πt ) ↔ X ( jω ) = (d) x ( t ) = ↔ X ( jω ) = ( sin(200 ω ) ω ) 2 (e) x ( t ) = ↔ X ( jω ) = cos 2 ( ω ) 1 4. (20 Points) Determine the inverse Fourier transform of X ( jω ) = 5 (3 + jω ) 2 Although there is no table entry for this case, the convolution property can be applied if X ( jω ) is written as a product. 5. (20 Points) Prove that the Fourier transform of the unit-step signal, u ( t ), is U ( jω ) = 1 jω + Kδ ( ω ) where K = π . At ﬁrst glance, it might seem that the impulse term Kδ ( ω ) should be zero, but show that K 6 = 0 because the following signal s ( t ) is an odd symmetric signal. s ( t ) = ( u ( t )-1 2 if t 6 = 0; if t = 0; Note: Even though s ( t ) and u ( t )-1 2 diﬀer at one isolated point, t = 0; they still have the same Fourier transform. 2...
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