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Finding LTI (Linear Time-Invariant Systems)

  1. Sep 14, 2014 #1
    1. The problem statement, all variables and given/known data

    Tell whether following systems are i. linear and ii. time-invariant:

    $$y(t) = \int_{-\infty}^t x(\tau)d\tau$$
    $$y(t) = \int_{0}^t x(\tau)d\tau$$
    $$y(t) = \int_{t-1}^{t+1} x(\tau)d\tau$$​


    2. Relevant equations
    N/A


    3. The attempt at a solution

    I'm a little thrown off by the integrals but here's my best explanation, all 3 of them are linear, why? Because the constants can be pulled out of the integrals thus fulfilling the scaling property that y(t) = x(t) and Cy(x) = C(x); similarly the concept of an integral inherently proves the additive property necessary for linearity.

    As for time-invariance, I'm also inclined to say all of them are time-invarient, why? If we take a look at the systems ( once they've already been integrated we can see this - example of the first integral):

    $$y(t) = C(x_{integrated}(t) - x_{integrated}(-\infty))$$​

    It seems to me that this clearly shows that if you delay t by 1 your output will also be delayed by one.

    Let me know if I have any errors in my thought process or how I went about this, LTI systems are still a little difficult for me to "reason" through, although I am very comfortable with what they stand for the analysis is just...magic.
     
  2. jcsd
  3. Sep 15, 2014 #2
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