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Discrete time system that is homogenous but not additive

  1. Jul 14, 2013 #1
    I have just started working with discrete time signals, more specifically various system properties. I am wondering if their is a discrete time system that is homogeneous but not additive? This is basically testing the linearity of a signal with the additive and homogeneous criteria.
     
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  3. Jul 14, 2013 #2

    jbunniii

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    What does homogeneous mean in this context? Simple examples of non-additive systems would be ##x \mapsto x^2##, ##x \mapsto \sin(x)## and so forth.
     
  4. Jul 14, 2013 #3
    T{x1[n] +x2[n]} = T{x1[n]} + T{x2[n]}

    It is using linear to algebra to see if the system is linear. It has to be both have the additivity and homogeneous/scaling property
     
  5. Jul 15, 2013 #4

    jbunniii

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    What if you define something like
    $$T(x[n]) = \begin{cases}x[n] & \text{ if }x[0] \neq x[1] \\
    0 & \text{ if }x[0] = x[1]\end{cases}$$
    In other words, ##T## maps the sequence ##\{x[n]\}## to itself if ##x[0] \neq x[1]##, and it maps ##\{x[n]\}## to the zero sequence if ##x[0] = x[1]##.

    Verification that ##T## is homogeneous is straightforward.

    Case 1: ##c = 0##. Then clearly ##cx[0] = cx[1]##, so ##T(cx[n]) = 0 = cT(x[n])##.

    Case 2: ##c \neq 0## and ##x[0] \neq x[1]##. Then ##cx[0] \neq cx[1]##, so ##T(cx[n]) = cx[n] = cT(x[n])##.

    Case 3: ##c \neq 0## and ##x[0] = x[1]##. Then ##cx[0] = cx[1]##, so ##T(cx[n]) = 0 = cT(x[n])##.

    On the other hand, ##T## is not additive. For example, let
    $$x_1[n] = \begin{cases}1 & \text{ if }n = 0 \\
    0 & \text{ if }n \neq 0\end{cases}$$
    $$x_2[n] = \begin{cases}1 & \text{ if }n = 1 \\
    0 & \text{ if }n \neq 1\end{cases}$$
    Then ##T## maps ##\{x_1[n]\}## to itself, and ##\{x_2[n]\}## to itself, but ##T## maps ##\{x_1[n] + x_2[n]\}## to the zero sequence.
     
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