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lab6 - Lab 6 Fourier Transforms in Maple and Matlab Part A...

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Page 1-8 Lab 6: Fourier Transforms in Maple and Matlab Part A: Fourier Transforms in Maple The Fourier transform sometimes involves challenging integrals. For electrical engineers, the input functions are often piecewise defined signals. Although this presents a little extra computational difficulty, using Maple, you will be able to check all your problems. The problems below are slight modifications of problems from your text. I. Finite Window Signals As an exercise, we will find the Fourier Transform for each of the following signals. Notice that each signal occupies a finite window. Signals a, b and c are non-zero on the interval [0,5]. Signal d is non-zero on the interval [-1,1]. Since signals with finite windows are not periodic, we must use the Fourier transform - not Fourier series. a. Rectangular Pulse from 0 to 5. b. Exponential starts at 0 and stops at 5. c. Ramp starts at 0 and stops at 5. d. Two periods of a sine wave from -1 to +1 1. First establish our usual notations and functions. > alias(u=Heaviside); interface(imaginaryunit=j); 2. Next enter each of the four functions x a ( t ), x b ( t ), x c ( t )and x d ( t ). Try to do this without looking at the Maple code below. Recall that any function which is confined to a window such as the interval [ a , b ] can be written in the form: f ( t ) = f r ( t ) u ( t ! a ) ! u ( t ! b ) ( ) where f r ( t ) denotes its restriction to the window. The window is represented by the difference of two unit step functions. > xa := t -> u(t) - u(t-5): > xb := t -> exp(-2*t) * (u(t) - u(t-5)): > xc := t -> t * (u(t) - u(t-5)): > xd := t -> sin(2*Pi*t) * (u(t+1) - u(t-1)):
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Page 2-8 3. Plot each of the functions and verify they resemble the pictures shown above. Try to do this without looking at the code below. > plot( xa(t), t=-3..8, thickness=3); > plot( xb(t), t=-1..6, thickness=3); > plot( xc(t), t=-1..6, thickness=3); > plot( xd(t), t=-2..2, thickness=3); 4. Now we are ready to calculate the Fourier transforms. Recall the Fourier transform is a function defined by an infinite (or improper) integral.
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