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Signal strength of a wave packet

  1. Nov 22, 2016 #1
    1. The problem statement, all variables and given/known data
    Assume a wave packet is has contributions from various frequencies, give by g(ω)=C for |ω|<ω0, and g(ω) =0 for elsewhere.

    a)What is the signal strength as a function of time, i.e., V(t)=?

    b) Sketch g(ω) and V(t); You can use fooplots.com, for example, or python.

    c) Indicate Δω and Δt in the above plots; Does the products of these two satisfy ΔωΔt>1/2?

    2. Relevant equations
    V(t) = 1/√(2π) * integral from -∞ to +∞ of [ g(ω)*exp(iωt)] dω

    exp(iωt) = cos(ωt) + isin(ωt)
    3. The attempt at a solution

    a.) V(t) = 1/√(2π) * integral from -ω0 to +ω0 of [ C*exp(iωt)] dω

    using eulers formula and the properties of even and odd functions

    V(t) = 2/√(2π) * integral from 0 to +ω0 of [ C*cos(ωt)] dω

    V(t) = 2C/√(2π) * sin(ω0t)/t = √(2/π)*ω0C * sin(ω0t)/ω0t

    b.) the sketches for g(ω) and V(t) are in the attachments

    c.) im not really sure what Δω and Δt in these graphs
     

    Attached Files:

  2. jcsd
  3. Nov 22, 2016 #2

    blue_leaf77

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    Shouldn't the lower integral limit be ##-\omega_0##?
    Use the definition of standard deviation. For example for V(t), you will use
    $$
    \Delta t = \sqrt{E[t^2]-(E[t])^2}
    $$
    where ##E[\,\,]## means taking average over the intensity ##|V(t)|^2##. Similar arguments for ##g(\omega)##.
     
  4. Nov 22, 2016 #3
    I have a factor of 2 in the front to account for that. Is that not right?
     
  5. Nov 22, 2016 #4

    blue_leaf77

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    Ah sorry I missed that, you are right.
     
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