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finals00 - Sample Final Exam Covering Chapters 10-17...

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Sample Final Exam Covering Chapters 10-17 (finals00) 1 Sample Final Exam (finals00) Covering Chapters 10-17 of Fundamentals of Signals & Systems Problem 1 (10 marks) Consider the causal FIR filter described by its impulse response 11 [] [ 1 ] [ 2 ] 22 hn n n n δδ δ =− + − − . (a) [3 marks] Draw a block diagram of the direct form realization of the filter. Write the transfer function H z () of the filter and specify its ROC. (b) [5 marks] Find the frequency response He j ω of the filter and give its magnitude and its phase. Sketch its magnitude j for [, ] ωπ π ∈− . What type of filter is it? (low-pass, band-pass or high-pass?) (c) [2 marks] Compute the -3dB cutoff frequency c of the filter. Answer: (a) [3 marks] 2 12 2 110 . 50 . 5 , 0 zz Hz z z z z −− −+ =− + = > + yn z 1 x n z 1 1 05 . .

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Sample Final Exam Covering Chapters 10-17 (finals00) 2 (b) [5 marks] Frequency response: () 2 11 1 1 1 22 2 2 1c o s jj j j j j j He e e e e e e ωω ω −−  =− + =   =− ()1 c o s j j ∠= Highpass filter (c) [2 marks] -3dB cutoff frequency c of the filter: 1 cos cos(1 ) 1.2735 cc a −= = Problem 2 (20 marks) The system shown below processes the continuous-time signal 12 n yt yt yt n t =++ composed of the sum of: Signal 1 yt which is a lower single-sideband amplitude modulation, suppressed carrier (lower SSB-AM/SC) of the corresponding message signal 1 x t , Signal 2 which is an upper single-sideband amplitude modulation, suppressed carrier (upper SSB-AM/SC) of the corresponding message signal 2 x t , a noise signal nt . Both SSB-AM/SC signals share the same carrier signal cos( ) c t of frequency c . Also assume that the sidebands have the same magnitude as the spectra of the message signals. The antialiasing filter a Hj is a perfect unity-gain lowpass filter with cutoff frequency a . The antialiased signal wt is first converted to a discrete-time signal [] wn via the CT/DT operator. Signal is processed in parallel by two discrete-time filters: a lowpass filter j lp , and a highpass filter j hp . Then, the output of either filter, as selected by a switch, is the input to a discrete-time synchronous demodulator with frequency c which is used to demodulate either of the two discretized message signals. Finally, the DT/CT operator followed by a continuous-time lowpass filter reconstruct the desired continuous-time message signal. j H e ω 0.5π −π 1 −0.5π π
Sample Final Exam Covering Chapters 10-17 (finals00) 3 The modulating (or message signals) 12 () , () x txt have spectra X jX j ωω as shown below. The spectrum Nj ω of the noise signal is also shown below. (a) [5 marks] Find the maximum carrier frequency c and the corresponding minimum antialiasing filter's cutoff frequency a that will avoid any unrepairable distortion of the modulated signals due to the additive noise nt . Sketch the spectrum Wj of the signal wt for the frequencies c and a that you found. Indicate the important frequencies and magnitudes on your sketch.

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This note was uploaded on 09/16/2011 for the course ECSE 361 taught by Professor Franciscodgaliana during the Winter '09 term at McGill.

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finals00 - Sample Final Exam Covering Chapters 10-17...

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