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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 unitstep 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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 Spring '08
 SHIN
 Digital Signal Processing

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