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Math Help - Convolution integral

  1. #1
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    Convolution integral

    Hey guys. just wondering if anyone can help me with this piece of convolution integral.

    i am asked to convolute f(t) and g(t) where

    f(t) = 2r(t-1) - 2r(t-2) - u(t-2) - u(t-3) where r is a ramp function and
    u is a unit step function

    and

    g(t) = d(t-2) - d(t-3) where d is the unit impulse
    function or delta dirac function


    i use the graphical approach and make use of the sifting property of the impulse function. does that sound right? i have an answer but i'm not sure if rite. any help or input would be much appreciated.

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  2. #2
    Grand Panjandrum
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    By definition:

    (f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\;d\tau

    and so for scalars a and b :

    (f*(a\times g+b \times h)(t)=a(f*g)(t)+b(f*h)(t)

    Again by definition:

    (f*\delta_k)(t)=\int_{-\infty}^{\infty} f(\tau)\delta((t-\tau)-k)\;d\tau=f(t-k)

    Which should allow you to write down the result you require (there may be a problem if f is discontinuous at k).

    CB
    Last edited by CaptainBlack; December 3rd 2009 at 11:24 PM.
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  3. #3
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    thanks for the quick response CaptainBlack. this is my answer

    for t <3, y(t) = 0
    for 3<t<4, y(t) = 2(t-1)
    for 4<t<5, y(t) = 2(t-1) + 1
    for 5<t<6,y(t) = 1

    where y(t) is the convolution of f(t) and g(t).

    there is a discontinuity of f(t) at t=2 but i have taken that into consideration in the answer above. can you confirm if this is correct?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by squirby View Post
    thanks for the quick response CaptainBlack. this is my answer

    for t <3, y(t) = 0
    for 3<t<4, y(t) = 2(t-1)
    for 4<t<5, y(t) = 2(t-1) + 1
    for 5<t<6,y(t) = 1

    where y(t) is the convolution of f(t) and g(t).

    there is a discontinuity of f(t) at t=2 but i have taken that into consideration in the answer above. can you confirm if this is correct?
    Well other than any exceptional points, if my algebra is right I have:

    (f*g)(t)=2r(t-3)-4r(t-4)+2r(t-5)-u(t-4)+u(t-6)

    CB
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