Transcription of ECE 45 Homework 3 Solutions
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ECE 45 Homework 3 Solutions
UC San DiegoJ. ConnellyECE 45 Homework 3 SolutionsProblem the Fourier transform of the function (t) = 1 2|t| |t| 1 the statement of Problem to verify your answer. Note: the function (t) is sometimes called the unit triangle function, as it a triangular pulse with height 1, width 1, and is centered at : Recall the trig identity 1 cos(2x) = 2 sin2(x). Also, the time-reversal property can be :Letz(t)= 1 2t0 t 1/20otherwiseThen (t) =z(t) +z( t), so by the linearity and time reversal properties of the Fourier Transform,F( ) =Z( ) +Z( ).Z( ) =Z z(t)e j tdt=Z1/20(1 2t)e j tdt=j (2t 1) + 2(j )2e j t 1/2t=0=2 j 2e j /2 2 ThereforeD( ) =2 j 2e j /2 2+2+j 2ej /2 2=4 2 ej /2+e j /2 2=4 4cos( /2) 2=8sin2( /4) 2=sinc2( /4)2wheresinc (x) = corresponds to the function in Problem withA=W= 1andt0= report any typos/errors to , W,andt0be real numbers such thatA, W >0, and suppose thatg(t)is given byg(t)At0t0 W2t0+W2 Show the Fourier transform ofg(t)is equal toAW2sinc2(W /4)e j t0 Wusing the results of Problem and the properties of the Fourier : You do NOT have to
Problem 3.2 Let A,W, and t 0 be real numbers such that A,W > 0, and suppose that g(t) is given by g(t) A t 0 t 0 − W 2 t 0 + W 2 Show the Fourier transform of g(t) is equal to AW 2 sinc2(Wω/4) e−jωt0 W using the results of Problem3.1 and the propertiesof the Fourier transform.
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