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Unformatted text preview: EE 261 The Fourier Transform and its Applications Fall 2009 Problem Set Three Due Wednesday, October 14, 2009 1. (25 points) Piecewise linear approximations and Fourier transforms. (a) The stretched triangle function is defined by a ( t ) = ( t/a ) = braceleftBigg a-| t | a , | t | a , | t | &gt; a Find F a ( s ). (b) Find the Fourier transform of the following signal. 1 2 2 2 . 5 4 6 t Hint: Think s. (c) Consider a signal f ( t ) defined on an interval from 0 to D with f (0) = 0 and f ( D ) = 0. We get a uniform, piecewise linear approximation to f ( t ) by dividing the inter- val into n equal subintervals of length T = D/n , and then joining the values 0 = f (0) , f ( T ) , f (2 T ) , . . . , f ( nT ) = f ( D ) = 0 by consecutive line segments. Let g ( t ) be the linear approximation of a signal f ( t ), obtained in this manner, as illustrated in the following figure where T = 1 and D = 6. 1 1 1 2 2 3 3 4 5 6 t f ( t ) g ( t ) Find F g ( s ) for the general problem ( not for the example given in the figure above) using any necessary information about the signal f ( t ) or its Fourier transform F f ( s ). Think s, again. 2. (15 points) Hubbard-Stratonovich Formula Show that integraldisplay - e- x 2 e- 2 sx dx = e s 2 ....
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