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Writing in terms of step function

  1. Nov 23, 2009 #1

    rock.freak667

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    1. The problem statement, all variables and given/known data

    f(t)= 4t t<1
    f(t)= 4 t≥1

    I need to write f(t) in terms of H(t)



    2. Relevant equations

    H(t-t0)=0 for t<t0
    = 1 for t≥t0

    3. The attempt at a solution

    I am not too sure how it is done, as what I read online, people can just immediately write it down but I don't get the the thought process.
     
  2. jcsd
  3. Nov 23, 2009 #2

    LCKurtz

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    Here's the idea. Say you want your function to be f(t) for t < 0, g(t) between 0 and 2, h(t) between 2 and 4, and 0 for t > 4.

    So start building your function with f(t). At t=0 you want to get rid of f and start g:

    f(t) + (g(t) - f(t))u(t)

    At 2 you want to get rid of the g and start h

    f(t) + (g(t) - f(t))u(t) + (h(t) - g(t))u(t-2)

    At t = 4 you want to get rid of the h.

    f(t) + (g(t) - f(t))u(t) + (h(t) - g(t))u(t-2) - h(t)u(t-4)

    and you are done. Equivalently you could collect terms on the functions:

    f(t)(1 - u(t)) + g(t)(u(t)-u(t-2)) + h(t)(u(t-2)-u(t-4))

    Sometimes terms like u(t-b)-u(t-a) are called filter functions because they just pick up whatever they are multiplied by between a and b.
     
  4. Nov 24, 2009 #3

    rock.freak667

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    Thank you, I wish my lecturer would have explained it like this :biggrin:
     
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