Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Causality with time invariance

  1. Jun 30, 2012 #1
    Assume u:R[itex]\rightarrow[/itex] C^n and define shift operator S([itex]\tau[/itex]) with

    S([itex]\tau[/itex])u(t)=u(t-[itex]\tau[/itex])

    and truncation operator P([itex]\tau[/itex]) with

    P([itex]\tau[/itex])u(t)=u(t) for t[itex]\leq[/itex][itex]\tau[/itex] and 0 for t>[itex]\tau[/itex]

    Then P([itex]\tau[/itex])S([itex]\tau[/itex])=S([itex]\tau[/itex])P(0) for every [itex]\tau[/itex]>=0.

    Can someone please prove last statement..
     
    Last edited: Jun 30, 2012
  2. jcsd
  3. Jun 30, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Looks like pretty direct computation. If u(t) is any such function, then what is[itex]SD(\tau)u[/itex]? What is [itex]P(\tau)S(\tau)u[/itex]? Then turn around and find [itex]S(\tau)P(0)u[/itex].
     
  4. Jun 30, 2012 #3
    Yes, I tried that, and it just doesn't fit..

    P([itex]\tau[/itex])S([itex]\tau[/itex])u(t)=P([itex]\tau[/itex])u(t-[itex]\tau[/itex])=u(t-[itex]\tau[/itex]) if t-[itex]\tau[/itex]<=[itex]\tau[/itex] and 0 for t-[itex]\tau[/itex]>[itex]\tau[/itex]

    S([itex]\tau[/itex])P(0)u(t)=S([itex]\tau[/itex])u(t) for t<=0 and 0 otherwise=u(t-[itex]\tau[/itex]) if t<=0 and 0 otherwise..

    Well, something's got to be wrong here, but I can't see what..
     
  5. Jul 1, 2012 #4
    I think your last equation is wrong. As, if we have:

    $$P(0)u(t)=u(t) \mbox{ if } t\leq 0 \mbox{ and } 0 \mbox{ otherwise }$$

    than:

    $$S(\tau)P(0)u(t)=u(t-\tau) \mbox{ if } t-\tau\leq 0 \mbox{ and } 0 \mbox{ if } t-\tau>0$$

    Still, I'm not able to prove the statement as in the first case you have $$t-\tau\leq\tau$$ and in this case there is $$t-\tau\leq 0$$. :tongue: I'm sorry...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Causality with time invariance
  1. Invariant subspaces (Replies: 6)

  2. Invariant Subspace (Replies: 6)

  3. Invariant subspace (Replies: 3)

Loading...