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To determine whether it's LTI system

  1. Apr 11, 2012 #1
    1. The problem statement, all variables and given/known data

    this system satisfies the condition of initial rest.

    Y[n] =(1/2)y[n-1] + x[n]

    I'm trying to prove that this is linear and time invariant.


    3. The attempt at a solution

    I'm trying it this way.
    To be LTI system,
    It has to satisfy below.
    aX1[n] -> aY1[n]
    bX2[n] -> bY2[n]
    aX1 + bX2 -> aY1 + bY2

    1. firstly, I don't see
    aX[n] -> aY[n]

    Since if I put aX[n] then (1/2)Y[n-1] + aX[n] =! {aY[n]= a(1/2)Y[n-1]+aX[n]}
    i.e., aX[n] -> aY[n] is not true in my view.

    2. Secondly,
    Just let me try this without multiplying constant

    X1[n] -> Y1[n]
    X2[n] -> Y2[n]
    X1 + X2 -> Y1 + Y2

    Y1[n] = (1/2)Y1[n-1] + X1[n]
    Y2[n] = (1/2)Y2[n-1] + X2[n]

    X3[n] = X1[n] + X2[n]

    Y3[n] = (1/2)Y3[n-1] + X3[n] = (1/2)Y3[n-1] + {X1[n] + X2[n]}

    After this I don't seem to find what to do with Y3[n-1] and since this, I can't prove
    Y3[n] = Y1[n] + Y2[n]

    Could somebody help me and tell me what I'm doing wrong?
    Appreciate for it.
     
  2. jcsd
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