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Math Help - Convolution

  1. #1
    Member
    Joined
    May 2010
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    Convolution

    Hi there. I must calculate the convolution between these functions
    f(t)= e^{-t} H(t)
    g(t)=e^t H(t) H(t) the unit step Heaviside function.
    So I have to find: f \star g

    This is what I did:
    f \star g=\displaystyle\int_{-\infty}^{\infty}e^{-\lambda}e^{t-\lambda}H(\lambda)H(t-\lambda)d\lambda=\displaystyle\int_{-\infty}^{\infty}e^{t-2\lambda} \left [ H(\lambda)-H(\lambda-t) \right ] d\lambda=\displaystyle\int_{-\infty}^{\infty}e^{t-2\lambda}H(\lambda)d\lambda-\displaystyle\int_{-\infty}^{\infty}e^{t-2\lambda} H(\lambda-t)d\lambda=\displaystyle\int_{0}^{\infty}e^{t-2\lambda}d\lambda-\displaystyle\int_{t}^{\infty}e^{t-2\lambda} d\lambda=\displaystyle\int_{0}^{t}e^{t-2\lambda} d\lambda=-2e^{-t}+2e^{t}

    Is this fine?
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  2. #2
    Senior Member
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    Re: Convolution

    In the last line, you were meant to divide by 2, not multiply. Everything else looks OK.
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