midterm1_w09

midterm1_w09 - . (b) (20%) How would you reconstruct x ( t...

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Dr. Raviv Raich Midterm I (Feb 11, 2009) The duration of the exam is fifty minutes (3:00pm-3:50pm). 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%+10% bonus) In the following, we will consider the problem of reconstruction from an incomplete set of samples. We are given a sampling system in which each third sample is set to zero due to a circuit malfunction. We use the following diagram to represent the system: × × x ( t ) X ( ) W - W x δ ( t ) p ( t ) q [ n ] x [ n ] y [ n ] C/D Where p ( t ) = n = -∞ δ ( t - nT s ) and q [ n ] = k = -∞ δ [ n - 3 k ] + δ [ n - 3 k - 1] . (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 )
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Unformatted text preview: . (b) (20%) How would you reconstruct x ( t ) from x [ n ] ? Explain the process in both the frequency domain and time domain (use diagrams and equations to make your explanation precise). (c) (20%) Find Q ( e j ) the DTFT of q [ n ] and sketch it. (d) (20%) Express Y ( e j ) in terms of X ( e j ) . Express Y ( e j ) in terms of X ( j ) . Sketch Y ( e j ) . (e) (20%) For W = 3 T s , how would you reconstruct x ( t ) from y [ n ] ? Explain the process in both the frequency domain and time domain (use diagrams and equations to make your precise). (f) Bonus (10%) For W = 2 3 T s , how would you reconstruct x ( t ) from y [ n ] ?...
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