final_fall07 - Carleton University Department o f Systems...

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Carleton University Department of Systems & Computer Engineering SYSC-5602 Digital Signal Processing Fall Semester 2007 Final Exam, December 5th, 2007 This examination paper must be handed in with your answer booklet Answer the following Questions Question 1 Consider the system given by the following block diagram x(t) Ideal LTI Ideal y( t) NO D/A ~ ~ Lowpass --? System Lowpass Converter ~ Converter Pre-filter H(z) Post-filter I r Is = 48000 Hz Is = 48000 Hz where the ideal pre- and post- filters are appropriately chosen such as to avoid aliasing, for a sys- tem that uses a sampling rate of Is = 48000 Hz. The discrete time LTI system shown is causal with a transfer function 1 + Z-2 H(z) = 1 _ 0.5z-2 a) Draw the pole-zero diagram for the discrete-time filter H(z) b) List all frequencies that if present in x(t) would not be passed to yet). Consider the effects of each block. c) Determine the output yet) of the system given the input x(t) = 2 + 3cos( 16000nt +~) + 4 cos (24000nt +~) Use any information about the system that could simplify determining yet). 5+o.rd c;{oYl.e. ~sume that the input signal has the power spectrum shown below. fXcnl 2 -20000 20000 I Hz Give a reasonable estimate for the power spectrum (I Ycn1 2 ) of the output signal yet). Your answer may be given in the form of a reasonable sketch or in an analytical form.
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~uestion2 Consider the following diagram of an unknown LTI filter. With an input x(n) = 8(n), the filter zero-state response was found to be yen) = (]rCos(~ + ~)u(n). x(n) Unknown yen) LTI Filter -"' a) Is this filter causal? Is it stable? Justify your answers ./' b) Is this filter an FIR or IIR system? Write a difference equation for the output yen) in terms of the input x(n). _ {) c) Determine the filter's zero-state response if the input is a unit step sequence. YLo) .
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final_fall07 - Carleton University Department o f Systems...

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