finals01

# finals01 - Sample Final Exam Covering Chapters...

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Sample Final Exam Covering Chapters 10-17 (finals01) 1 Sample Final Exam (finals01) Covering Chapters 10-17 of Fundamentals of Signals & Systems Problem 1 (15 marks) Consider the system depicted below used for discrete-time processing of continuous-time signals. The sampling period is 1 microsecond ( 6 10 T = s). The discrete-time filter () d Hz is given by the following block diagram: (a) [5 marks] Write the transfer function d of the filter, specify its ROC, and sketch its pole-zero plot. Compute the DC gain of d . CT/DT Τ DT/CT Τ [] d x n xt c yt c d yn d H z + d z 1 d x n z 1 0 2 2

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Sample Final Exam Covering Chapters 10-17 (finals01) 2 Answer : 2 2 2 22 () 2 2 , 0 z Hz z z z + =+ = > . DC gain is (1) 4 H = . (b) [5 marks] Compute the frequency response () j He of the filter and sketch its magnitude and its phase over [, ] ππ Ω∈ − . Answer: 2 2 2 4 0 . 5 0 . 5 4cos jj j j j j e e e e e Ω− −Ω = + =Ω ()4 c o s , 2 , 2 , 2 j j π −Ω Ω ≤ ∠= + < −Ω − − < Ω < − 1 1 Re{ z } Im{ z ROC: all z except 0 -1
Sample Final Exam Covering Chapters 10-17 (finals01) 3 (c) [5 marks] Suppose that the continuous-time signal to be filtered is given by: ( ) cos(500000 ) 250000cos(500000 )sinc(250000 ) c x tt t t ππ =+ . Sketch the spectra of the continuous-time and discrete-time versions of the input signal () c X j ω and j d X e . Sketch the spectra j d Ye of the discrete-time output signal and c Yj of the continuous-time output signal c yt . Answer: Recall that sin sinc( ) x x x π = and the perfect unit-magnitude lowpass filter of bandwidth W has impulse response: sin sinc( ) WW W t t t = . j He j

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Sample Final Exam Covering Chapters 10-17 (finals01) 4 Problem 2 (15 marks) Design an envelope detector to demodulate the AM signal: 6 () [1 0 .3 () ]cos (2 10 ) yt xt t π =+ where () x t is the periodic modulating signal shown below. That is, draw a circuit diagram of the envelope detector and compute the values of the circuit components. Justify all of your approximations and assumptions. Provide rough sketches of the carrier signal, the modulated signal and the signal at the output of the detector. What is the modulation index m of the AM signal? 500000π −500000π −250000π −750000π −1000000π 250000π 750000π 1000000π .5π −.5π .25π −.75π −π .25π .75π π ω 500000π 1/2 −500000π −250000π −750000π −1000000π 250000π 750000π 1000000π c X j ω .5π 500000 −.5π .25π −.75π −π .25π .75π π j d X e j c Ye j d 1000000 2 2
Sample Final Exam Covering Chapters 10-17 (finals01) 5 Answer: An envelope detector can be implemented with the following simple RC circuit with a diode. The output voltage of the detector, when it goes from one peak at voltage v 1 to the next when it intersects the modulated carrier at voltage v 2 after approximately one period T s = 1 µ of the carrier, is given by: vv e TRC 21 / Since the time constant τ = RC of the detector should be large with respect to T s = 1 , we can use a first-order approximation of the exponential such that T R C 1 ≅− () .

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finals01 - Sample Final Exam Covering Chapters...

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