Convolution 4327

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    The Convolution Integral

    Convolution operation given symbol *

    dthxthtxty )(*)(

    y equals x convolved with h

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    The Convolution Integral

    The time domain output of an LTI system is

    equal to the convolution of the impulse

    response of the system with the input signal

    Much simpler relationship between

    frequency domain input and output

    First look at graphical interpretation of

    convolution integral

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    Graphical Interpretation of

    Convolution Integral

    To correctly understand convolution it is

    often easier to think graphically

    h(

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    Graphical Interpretation of

    Convolution Integral

    h(

    h(-

    Take impulse response and reverse it in time

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    Graphical Interpretation of

    Convolution Integral

    h(-

    Then shift it by time t

    h(t-

    t

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    Graphical Interpretation of

    Convolution Integral

    Overlay input functionx(t) and integrate over times

    where functions overlap - in this case between a and t

    h(t-

    ta

    x(

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    Graphical Interpretation of the

    Convolution Integral

    Convolving two functions involves

    flipping or reversing one function in time

    sliding this reversed or flipped function over

    the other and

    integrating between the times when BOTH

    functions overlap

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    Example

    Convolution of two gate pulses each of

    height 1

    0 1

    x1(

    0 2

    x2(

    dtxxxxy2121

    *

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    Example

    -2 0 2

    x2(x2(-

    Reverse function

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    Example

    -1 0 1

    x1(x2(-

    Reverse function, slidex2overx1 and evaluate integralt

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    Example

    0 1

    x1(x2(t-

    t

    tdtxxy

    tt

    0

    21 1*

    10for

    Area of overlap is increasing linearly

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    Example

    0 1

    x1(

    x2(t-

    t)(

    pulsesmallerofarea

    1*

    21for

    1

    21

    x

    xxy

    t

    Area of overlap constant

    t-2

    1

    0

    1

    0111 d

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    Example

    0 1

    x1(

    x2(t-

    t

    txxy

    t

    3*

    32for

    21

    Area declining linearly -

    width of shaded area = 1-(t-2)=3-t

    t-2 ttdt

    t

    32111

    1

    2

    2

    2

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    Example

    0 1

    x1(

    x2(t-

    t

    0*

    3for

    21

    xxy

    t

    After time t=3 the convolution integral is zero

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    Example

    0 1 2 3

    x1(t)*x2(t)

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    tint=0;

    tfinal=10;

    tstep=.01;

    t=tint:tstep:tfinal;x=5*((t>=0)&(t=0)&(t

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    Example 2

    Convolve the following functions

    0 1 t

    1.0

    x1(t)

    0 1 t

    x2(t)

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    Example 2

    0 1

    x2

    -1

    Reversal

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    Example 2

    0 t 1

    x2t

    -1

    Shift reversed function

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    Example 2

    0 t 1

    x2t

    -1

    Overlay shift reversed function ontoother function and integrate overlapping

    section

    x1

    tdtxx

    t

    t

    0

    21 1*

    10for

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    Example 2

    0 1 t

    x2t

    -1

    Overlay shift reversed function onto

    other function and integrate overlapping

    section

    x1

    tdtxx

    t

    t

    21*

    21for

    1

    1

    21

    t-1

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    Example 2

    x1(t)*x2(t)

    0 1 2

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    Example 3

    3.heightofpulsesecond4aishinput whicantodue

    systemthisofoutputthecompute5)(

    issystemLTIanofresponseimpulseGiven the

    2 u(t)etht

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    Example 3

    5 )(5)(2

    tuetht

    t0 4

    3

    )(tx

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    Example 3

    5

    )( h

    Reverse h(

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    Example 3

    5

    )( th

    Shift the reversed h(by t

    t 4

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    Example 3

    5

    )( th

    Performing integral for 0

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    Example 3

    t

    t

    t

    t

    t

    t

    t

    ety

    ee

    deedety

    20

    22

    0

    22

    0

    2

    15.7)(

    2

    115

    1515)(

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    Example 3

    5

    )( th

    Performing integral for t>4

    t 4

    4

    0

    215)( dety t

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    Example 3

    15.72

    115

    1515)(

    82

    4

    0

    22

    4

    0

    22

    4

    0

    2

    eeee

    deedety

    tt

    tt

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    Example 3

    415.7

    4015.7

    00

    )(

    82

    2

    tee

    te

    t

    ty

    t

    t

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    Commutativity of Convolution

    Operation

    The actions of flipping and shifting can be

    applied to EITHER function

    )(*)(

    )(*)(

    txthdtxh

    dthxthtx

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    Example 4

    Repeat example 3 by flipping and shifting

    x(t) rather than h(t)

    0 t

    tt

    dedety

    t

    0

    2

    0

    21553)(

    40for

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    Example 4

    0 t

    t

    tt

    ety

    edety

    t

    2

    02

    0

    2

    15.7)(

    5.715)(

    40for

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    Example 4

    0 t

    t

    t

    dety

    t

    4

    215)(

    4for

    t-4

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    Example 4

    ttt

    t

    t

    t

    t

    eeeety

    edety

    t

    28242

    4

    2

    4

    2

    15.75.7)(

    2

    11515)(

    4for

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    Example 4

    415.7

    4015.7

    00

    )(

    82

    2

    tee

    te

    t

    ty

    t

    t

    Same result as before