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Properties of Systems Memoryless, Causal, time invariant, linear

  1. Aug 20, 2011 #1
    Hi, I need a hand with my reasoning on the following question.
    I have answered all the questions, but not too sure weather they are correct.
    Please guide me or point me on the right direction if they are not correct.
    Regards

    1. A system defined by the following equation y(t)=[x(t-2)]^2
    Is the system Memoryless, Causal, Time Invariant, linear

    The attempt at a solution

    Memoryless ?

    y(t)=x(t-2) has memory, the output is equal to the input two seconds ago.
    y(t)=[x(t)]^2 = memoryless, the output at time t0 depends only on the input value at time t0.

    Now, y(t)=[x(t-2)]^2 can be thought as y(t)=[x(t-2)*x(t-2)] right?, if so, my best guess would be that the system y(t)=[x(t-2)]^2 has memory.

    Causal?
    by definition; A system is causal if the output at any time t0 is dependent on the input only for t<=t0.
    y(t)=[x(t-2)]^2 = causal

    Time Invariant?
    by definition if
    x(t)--->y(t), then
    x(t-t0)--->y(t-t0)
    so;
    x(t-t0)--->[x((t-t0)-2)]^2
    y(t-t0)=[x((t-t0)-2)]^2,
    Therefore, y(t)=[x(t-2)]^2 is time invariant.

    linear ?
    A system is linear if it is homogeneous and additive.
    Homogeneous Property
    if x(t)--->y(t), then
    ax(t)--->ay(t) a= alpha
    Additive Property
    if x1(t)--->y1(t), and x2(t)--->y2(t), then
    x1(t)+x2(t)--->y1(t)+y2(t),

    x(t)--->y(t)=[x(t-2)]^2
    multiply input by alpha a, then
    ax(t)--->[ax(t-2)]^2
    ay(t)--->a[(x(t-2))^2]
    system y(t)=[x(t-2)]^2 is not homogeneous, thus not linear.
     
  2. jcsd
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