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A simple delta function properties, sifting property

  1. Sep 9, 2015 #1
    • Member warned to use the formatting template
    upload_2015-9-9_18-27-14.png

    I don't know why this is possible

    To use delta function properties( sifting property)

    integral range have to (-inf ,inf)

    or at least variable s should be included in [t_0,t_0+T]

    but there is no conditions at all (i.e. t_0 < s < t_0+T)

    am I wrong?
     
  2. jcsd
  3. Sep 9, 2015 #2

    blue_leaf77

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    It is the sameness of the integration limits that eliminates the need for them to be +- infinity. Consider the integral over ##t## only
    $$
    \int_{t_0}^{t_0+T} \delta(t-s) dt
    $$.
    The integrand is a delta function centered at ##t=s##, but the range of ##s## is the same as the range of ##t##. This means the integrand ##\delta(t-s)## is guaranteed to always be located inside the integration limit of ##t##. So,
    $$
    \int_{t_0}^{t_0+T} \delta(t-s) dt = 1
    $$.
     
  4. Sep 9, 2015 #3
    u mean delta function property does not care about integral range?
     
  5. Sep 9, 2015 #4

    blue_leaf77

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    This is only true under the situtaion being considered, namely there is a second integral with respect to ##s## present with the same limits as the first integral.

    EDIT: It might have been more justifiable if I had written
    $$
    \int_{t_0}^{t_0+T} \delta(t-s) dt = 1 \hspace{1.2cm} \text{for} \hspace{1.22cm} s=[t_0,t_0+T]
    $$
     
    Last edited: Sep 9, 2015
  6. Sep 13, 2015 #5

    rude man

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    Taking one integral at a time: ∫ δ(t-s)dt from t0 to t0 + T = 0 unless s is within range of t0 to t0 + T, in which case it = 1. So you have to assume s is within range of t0 to t0 + T, and I agree that should have been specified.

    Then the second integral is obviously T so the whole thing is N0/2 times 1 times T = N0T/2.
     
  7. Sep 13, 2015 #6

    Ray Vickson

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    Do you mean "shifting" or "sifting"? These are both relevant aspects of transforms such as the delta function, but they refer to vastly different properties.
     
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