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convol - Convolution Recall that the convolution of two...

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Page 1 - 9 Convolution Recall that the convolution of two functions f 1 ( t ) and f 2 ( t ) is written f 1 * f 2 (t) and is defined by the convolution integral: f 1 * f 2 ( t ) = f 1 ( ! ) "# + # $ f 2 ( t " ! ) d ! In Maple this integral can be found using: > f1*f2(t) = int( f1(tau)* f2(t-tau) , tau=-infinity .. infinity); However, even Maple often has trouble evaluating such integrals in closed form. It can however, always evaluate the integral numerically and create a graph of the resulting convolution. Thus you will always be able to create an accurate graph of the convolution of any two functions. Step 0 - Set up our usual definitions and aliases for the singularity functions u and r . (unit step and ramp) alias(u=Heaviside): r := t -> t*u(t): Exercise 1 - Define and plot the two piecewise-linear functions x ( t ) and h ( t ) using these pictures as guides. x ( t ) h ( t ) x := t -> 2*u(t-2) -2* u(t-4): h:= t -> u(t) -2*u(t-1) + u(t-2): Plot them both to be sure they have been entered correctly. plot( [x(t), h(t) ], t=-4..6, color = [red, green], thickness =[1,2], ytickmarks=4, font=[TIMES, 18,BOLD]); Now obtain the convolution c ( t ) using the convolution integral and then plot it. c := t -> int( x(tau)* h(t-tau), tau=-infinity .. infinity); c(t); plot(c(t), t=-4..8, thickness=2, color=blue, font=[TIMES, PLAIN,24], gridlines=true); You can specify the font for the graph. Specify whatever thickness you want for each graph.
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Page 2 - 9 Notice Maple did not evaluate the integral in a simple closed form. However, you can write out the answer by inspecting the above graph and using our expansion for any piecewise linear function. Express the convolution c ( t ) in terms of ramp and/or unit step functions. Note the slope jumps at t = 2, 3, 5 and 6. We will denote your visually determined answer as Y ( t ) to distinguish it from c ( t ). Y ( t ) = ………………………………………………………………………………………………….. Then plot your exact expression Y ( t ) simultaneously with c ( t ), (determined by numerical integration), and confirm they are identical. Y := t -> 2*r(t-2) - 4*r(t-3) + 4* r(t-5) - 2*r(t-6); plot( [Y(t), c(t)], t=-4..8, thickness=2, color=[blue,red], thickness=[4,1], font=[TIMES, PLAIN,24], title="Double Check"); Convolution Animation Create an animation to illustrate the convolution process. Just enter the following code. We will start from scratch. Step 1: Define the two functions and find their convolvution. restart; alias(u=Heaviside): r := t -> t*u(t): x := t -> 2*u(t-2) -2* u(t-4): h:= t -> u(t) -2*u(t-1) + u(t-2): y := t -> int( x(tau)* h(t-tau), tau=-infinity .. infinity); Step 2: Create the convolution animation. The first line gives the range for the tau and t variables.
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