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Université d’Ottawa Faculté de Génie,

École d'Ingénierie et des Technologies de l'Information

University of Ottawa Faculty of Engineering, School of Information Technology and Engineering

ELG 4172 Digital signal processing

Professor Miodrag Bolic

Midterm

12/02/2008 12 pages This exam is 75 minutes long. • calculators are not allowed • notes and textbooks are not allowed (closed book exam) Last name: First name: Student #: Problem Maximum Score 1 5 2 6 3 4 4 5 5 5 Total 25

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Question 1 (5 points) The system function of a linear time-invariant system has the zero-pole plot shown in the figure bellow. Specify and explain whether each of the following statements is true, false or it cannot be determined from the information given:

a) The system is stable b) The system is causal c) If the system is causal then it must be stable d) If the system is stable then it must have two-sided impulse response e)

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. Question 2 (6 points)

a) (/1.5) A signal }6,5,1{)( =nx defined for 20 ≤≤ n has Fourier transform )( ωjeX . Express analytically

)( ωjeX .

b) (/1.5) Discrete Fourier transform (DFT) )(kX is obtained by sampling )( ωjeX at 2 points (N=2). Find numeric values of )(kX ( 10 ≤≤ k ). c) (/1.5) x(n) is a 3-point sequence and DFT )(kX is a 2-point sequence. Explain if we will obtain x(n) by doing IDFT{ )(kX }. d) (/1.5) Compute IDFT{ )(kX }

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Question 3 (4 points)Suppose that we are given 10s of speech that has been sampled at a rate of 8kHz and we would like to filter it with a filter h(n) of length K=64. We are using overlap and save method with 1063-points DFTs. How many DFTs and inverse DFTs are necessary to perform the convolution? Explain.

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Question 4 (5 points) Calculate the linear convolution of:

}5,1{)( =nx 10 ≤≤ n with

}2{)( =nh 0=n , by using the DFT and inverse DFT (other solutions will not be accepted).

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Question 5 (5 points) Calculate the DFT )(kX of size N=4 of the signal }4,6,5,1{)( =nx defined for 30 ≤≤ n , by using the approach of the Radix-2 FFT with decimation in time. Write the numerical values obtained at each stage.

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