FunctionsConvolutionsHO-2

# FunctionsConvolutionsHO-2 - Functions!Convolutions...

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8/26/2009 1 Functions & Convolutions 0 x x step b ! "# \$% &’ 2 1 x step ( ! Step function xx 1 x 0 0 x x ) 1 00 0 0, 0.5, 1, bb x x step b b b x x + , , ! , - - . , , / , 0

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8/26/2009 2 0 x x rect b ! "# \$% &’ 2 2 x rect ( 1 Rectangle (rect) function xx 1 x 0 1 area b - b 0 00 0 1 0, 2 1 0.5, 2 1 1, 2 b x x x x rect bb b ) ! / , , , !! , - - . , , ! + , , 0 0 x x ramp b ! 1 2 x ramp ! ! Ramp function 1 x 0 1 x 0 + b 1 slope b - 0 0 0, , x x ramp b x x b b b ) 2 , , ! , - . , ! , / , 0
8/26/2009 3 0 x x tri b ! "# \$% &’ 2 2 x tri ( 1 Triangle (tri) function xx 1 x 0 1 area b - b 3 0 0 00 0, 1 1 , 1 b tri b x x x x bb ) ! 4 , , ! , - . , !! , ! + , 0 1 sin x c b b 1 area b - 2 sinc x b sinc and sinc 2 functions x area - b 3 x b 3 b 2 b 3 b 4 b b ! 2 b ! b 2 b 3 b 4 b b ! 2 b ! 0 0 0 sin sinc x x b b b 5 ! ! - ! 2 2 sinc sinc x x x x 6 7 " # " # - \$ % \$ % 8 9 & & : ;

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8/26/2009 4 1 x Gaus b "# \$% &’ area b - Gaussian function x b b ! 0.043 0.5 0.47 b 2 00 exp x x x x Gaus bb 5 6 7 !! " # " # -! 8 9 \$ % \$ % & & 8 9 : ; <= 0 x x > ! < = 21 x ! Delta function (Dirac delta function, impulse function) xx 1 x 0 1 1 ( ) lim x G 0 0 0 ( ) 1, ( ) 0, b xG aus b x x b x x x x x x x x >> ? - ! ! - - ! - @
8/26/2009 5 < = x >> 0 x x b ! "# \$% &’ Impulse pair function xx 1 1 x 0 x 0 + b x 0 " b < = < = 0 00 b x x b x x b b > ! 6 7 - ! ( ( ! !

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FunctionsConvolutionsHO-2 - Functions!Convolutions...

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