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midterm1_sp08

midterm1_sp08 - N jω = 0 for | ω |> π T s For the...

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Dr. Raviv Raich Midterm I (May 07, 2008) The duration of the exam is fifty minutes (1:00pm-1:50pm). Only one sheet of formulae is allowed. No calculators are allowed. Return this copy of the exam form along with formulae sheet and your notes. 1. ( 100% + 5% bonus) In the following, we will consider the problem of reconstruction in the presence of interference. We consider the following system: × + × x ( t ) p ( t ) = s n = -∞ δ ( t - nT s ) x δ ( t ) y ( t ) n ( t ) n g ( t ) g ( t ) ω ω X ( ) π T s π T s - π T s - π T s N ( ) 1 1 In (a)-(b), we assume that n ( t ) = 0 . (Ignore N ( ) in the figure). (a) (20%) Express X δ ( ) (the Fourier Transform (FT) of x δ ( t ) ) in terms of X ( ) (the FT of x ( t ) ). Sketch X δ ( ω ) . (b) (20%) How would you reconstruct x ( t ) from x δ ( t ) ? Use the Fourier domain to explain your approach. From here on (i.e., (c)-(f)), we make the assumption that n ( t ) is a bandlimited signal such that its FT
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Unformatted text preview: N ( jω ) = 0 for | ω | > π T s . For the spectrum of N ( jω ) refer to the figure. (c) (20%) Let g ( t ) be a periodic rectangular wave train with period T s . We define g ( t ) over one period [0 ,T s ) as g ( t ) = 1 ≤ t < T s 2 T s 2 ≤ t < T s Derive and sketch G ( jω ) (the FT of g ( t ) ). (d) (20%) After multiplying the interference n ( t ) with the periodic signal g ( t ) , we obtain n g ( t ) = n ( t ) g ( t ) . Find N g ( jω ) (the FT of n g ( t ) ) in terms of N ( jω ) . Sketch N g ( jω ) . (e) (5%) Sketch Y ( jω ) (the FT of y ( t ) ). (f) (20%) Propose a system that reconstructs x ( t ) from y ( t ) in the presence of interference n ( t ) ....
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