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:

    ( )[ ]( ) ( )[ ]

    ==