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Show that the system is linear?

  1. Feb 11, 2008 #1
    1. The problem statement, all variables and given/known data
    Is the following system linear:
    [tex] y(n) = x(n) + 0.8y(n-1) [/tex]

    3. The attempt at a solution
    I'm really stuck on this one.

    My idea was to assume that the system is casual, find the impulse response and then use the fact that: [tex] y(n)=x(n)*h(n) [/tex]. However, if I assume that the output is the convolution of the impulse response and the input, I have ALREADY ASSUMED (and not proven) that the system is linear.

    So I am lost.

    I can't wrap my brain around the fact that the output depends on the input and the previous output.

    Another idea I had is that if I show that I input a sinusoid and I get a scaled and phase shifted sinusoid, then the system is LTI.

    Any help?
  2. jcsd
  3. Feb 12, 2008 #2


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    Science Advisor
    Homework Helper

    What if, assuming y(0) = x(0), you wrote:

    y(n) = x(n) + 0.8y(n-1) = x(n) + 0.8x(n-1)... + (0.8)^n x(0),

    would that help?
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