midterm_solutions_(2010)

midterm_solutions_(2010) - EE261 Raj Bhatnagar Summer...

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EE261 Raj Bhatnagar Summer 2009-2010 EE 261 The Fourier Transform and its Applications Midterm Examination 19 July 2010 (a) This exam consists of 4 questions with 12 total subparts for a total of 50 points . (b) The questions di er in length and di culty. Do not spend too much time on any one part; be sure to attempt all parts. (c) The duration of this exam is 90 minutes . Open course reader, class handouts, class notes and homework. (d) Students taking the exam on-campus are required to write all answers in the blue books provided. (e) Do not forget to write your name on your exam book. 1
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Some short (independent) problems about sinc (a) (4 points) Find the Fourier transform of this sinc: (b) (4 points) Evaluate the following integral: Z -∞ sinc 3 ( t ) dt (c) (4 points) Find the repeated convolution of sinc: (sinc * sinc * ··· * sinc | {z } n convolutions )( t ) Solution: (a) First we note that the sinc has been scaled by 1 /b , shifted by c , and multiplied by a (in that order). So our sinc is c σ 1 /b sinc( t ) = a sinc ± t - c b . Calculating the Fourier transform, F c σ 1 /b sinc( s ) = ae - 2 πisc | b | σ b Π( s ) = a | b | e - 2 πisc Π( sb ) (b) Evaluating the Fourier transform of sinc 3 at zero gives us the desired integral: F{ sinc 3 } (0) = Z -∞ sinc 3 ( t ) e - 2 πi (0) t dt = Z -∞ sinc 3 ( t ) dt The Fourier transform of sinc 3 is Π * Π * Π which we know is Λ * Π . We just need to know the value of this convolution at zero, which corresponds to the area of overlap of the rectangle and triangle functions when both are centered at the origin. Drawing a sketch, we see this value is 3/4. (c) Repeated convolution becomes repeated multiplication in the frequency domain. So our problem corresponds to Π · Π ··· · Π( s ) which fortunately is just Π( s ) . We conclude the repeated convolution of sinc is sinc. 2
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