# Convolution Problems

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07-Dec-2015

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Convolution Problems

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• Acknowledgment: Multiple slides are from internet and Prof. Mark Fowler in Binghamton University.

Convolution Examples

• 2

)dx(t)h(t=h(t)x(t)=f(t)

-

*

• Convolution Example

5

• ( ) ( ) ( ) ( ) ( )

( ) ( )

( )

Let d where

0 1 0 01 1 2 0 1

and 2 2 3 0 10 3

Find .

y t f t g t f g t

t tt t t

f t g tt t

ty t

= =

<

• 1

g( ) g(- )

0 0 -1

• Band 1: Sliding Function Method

0 -1

g(- )

• Band 2: Sliding Function Method

• Band 3: Sliding Function Method

• Band 4: Sliding Function Method

• Band 5: Sliding Function Method

• Band 6: Sliding Function Method

• Convolution Example Table view

h(-m)

h(1-m)

• Discrete-Time Convolution Example: Sliding Tape View

• D-T Convolution Examples( ) ]4[][][][21][ == nununhnunx n

-3 -2 -1 1 2 3 4 5 6 7 8 9

][ih

i

-3 -2 -1 1 2 3 4 5 6 7 8 9

][ix

i

-3 -2 -1 1 2 3 4 5 6 7 8 9

]0[ ih

i

Choose to flip and slide h[n] This shows h[n-i] for n = 0

lyLine

lyRectangle

lyRectangle

• For n < 0 h[n-i]x(i) = 0 i00][
• -3 -2 -1 1 2 3 4 5 6 7 8 9

][ix

i

n = 4 case

-3 -2 -1 1 2 3 4 5 6 7 8 9

]4[][ ihinh

=

i

Now for n = 4, n = 5,

13 == ni 4== nin = 5 case

-3 -2 -1 1 2 3 4 5 6 7 8 9

]5[][ ihinh =

i

23 == ni 5== ni

• Notice that: for n = 4, 5, 6,

( )( ) ( )

!2

112

12

1

,...6,5,421][

133

simplifythen

nforny

nn

in

ni

=

==+

=

Then we can write out the solution as:

( )[ ]( ) ( )[ ]

==